Simplifier variants

README.md

These files are variants of simplifier.cpp from https://github.com/zeux/meshoptimizer, created for the article https://zeuxcg.org/2019/01/17/is-c-fast/

simplifier.c

// This file is part of meshoptimizer library; see meshoptimizer.h for version/license details
#include "meshoptimizer.h"

#include <assert.h>
#include <float.h>
#include <math.h>
#include <string.h>
#include <stdlib.h>
#include <stdbool.h>

#ifndef TRACE
#define TRACE 0
#endif

#if TRACE
#include <stdio.h>
#endif

typedef struct meshopt_Allocator
{
	size_t count;
	void* blocks[16];
} meshopt_Allocator;

static void* meshopt_allocate(meshopt_Allocator* alloc, size_t size)
{
	void* result = malloc(size);
	alloc->blocks[alloc->count++] = result;
	return result;
}

static void meshopt_deallocate(meshopt_Allocator* alloc)
{
	for (size_t i = 0; i < alloc->count; ++i)
		free(alloc->blocks[i]);
}

// This work is based on:
// Michael Garland and Paul S. Heckbert. Surface simplification using quadric error metrics. 1997
// Michael Garland. Quadric-based polygonal surface simplification. 1999
typedef struct EdgeAdjacency
{
	unsigned int* counts;
	unsigned int* offsets;
	unsigned int* data;
} EdgeAdjacency;

static void buildEdgeAdjacency(EdgeAdjacency* adjacency, const unsigned int* indices, size_t index_count, size_t vertex_count, meshopt_Allocator* allocator)
{
	size_t face_count = index_count / 3;

	// allocate arrays
	adjacency->counts = (unsigned int*)meshopt_allocate(allocator, vertex_count * sizeof(unsigned int));
	adjacency->offsets = (unsigned int*)meshopt_allocate(allocator, vertex_count * sizeof(unsigned int));
	adjacency->data = (unsigned int*)meshopt_allocate(allocator, index_count * sizeof(unsigned int));

	// fill edge counts
	memset(adjacency->counts, 0, vertex_count * sizeof(unsigned int));

	for (size_t i = 0; i < index_count; ++i)
	{
		assert(indices[i] < vertex_count);

		adjacency->counts[indices[i]]++;
	}

	// fill offset table
	unsigned int offset = 0;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		adjacency->offsets[i] = offset;
		offset += adjacency->counts[i];
	}

	assert(offset == index_count);

	// fill edge data
	for (size_t i = 0; i < face_count; ++i)
	{
		unsigned int a = indices[i * 3 + 0], b = indices[i * 3 + 1], c = indices[i * 3 + 2];

		adjacency->data[adjacency->offsets[a]++] = b;
		adjacency->data[adjacency->offsets[b]++] = c;
		adjacency->data[adjacency->offsets[c]++] = a;
	}

	// fix offsets that have been disturbed by the previous pass
	for (size_t i = 0; i < vertex_count; ++i)
	{
		assert(adjacency->offsets[i] >= adjacency->counts[i]);

		adjacency->offsets[i] -= adjacency->counts[i];
	}
}

typedef struct PositionHasher
{
	const float* vertex_positions;
	size_t vertex_stride_float;
} PositionHasher;

static size_t hasherHash(const PositionHasher* hasher, unsigned int index)
{
	// MurmurHash2
	const unsigned int m = 0x5bd1e995;
	const int r = 24;

	unsigned int h = 0;
	const unsigned int* key = (const unsigned int*)(hasher->vertex_positions + index * hasher->vertex_stride_float);

	for (size_t i = 0; i < 3; ++i)
	{
		unsigned int k = key[i];

		k *= m;
		k ^= k >> r;
		k *= m;

		h *= m;
		h ^= k;
	}

	return h;
}

static bool hasherEqual(const PositionHasher* hasher, unsigned int lhs, unsigned int rhs)
{
	return memcmp(hasher->vertex_positions + lhs * hasher->vertex_stride_float, hasher->vertex_positions + rhs * hasher->vertex_stride_float, sizeof(float) * 3) == 0;
}

static size_t hashBuckets2(size_t count)
{
	size_t buckets = 1;
	while (buckets < count)
		buckets *= 2;

	return buckets;
}

static unsigned int* hashLookup2(unsigned int* table, size_t buckets, const PositionHasher* hasher, unsigned int key, unsigned int empty)
{
	assert(buckets > 0);
	assert((buckets & (buckets - 1)) == 0);

	size_t hashmod = buckets - 1;
	size_t bucket = hasherHash(hasher, key) & hashmod;

	for (size_t probe = 0; probe <= hashmod; ++probe)
	{
		unsigned int* item = &table[bucket];

		if (*item == empty)
			return item;

		if (hasherEqual(hasher, *item, key))
			return item;

		// hash collision, quadratic probing
		bucket = (bucket + probe + 1) & hashmod;
	}

	assert(false && "Hash table is full");
	return 0;
}

static void buildPositionRemap(unsigned int* remap, unsigned int* wedge, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride, meshopt_Allocator* allocator)
{
	PositionHasher hasher = {vertex_positions_data, vertex_positions_stride / sizeof(float)};

	size_t table_size = hashBuckets2(vertex_count);
	unsigned int* table = (unsigned int*)meshopt_allocate(allocator, table_size * sizeof(unsigned int));
	memset(table, -1, table_size * sizeof(unsigned int));

	// build forward remap: for each vertex, which other (canonical) vertex does it map to?
	// we use position equivalence for this, and remap vertices to other existing vertices
	for (size_t i = 0; i < vertex_count; ++i)
	{
		unsigned int index = (unsigned)(i);
		unsigned int* entry = hashLookup2(table, table_size, &hasher, index, ~0u);

		if (*entry == ~0u)
			*entry = index;

		remap[index] = *entry;
	}

	// build wedge table: for each vertex, which other vertex is the next wedge that also maps to the same vertex?
	// entries in table form a (cyclic) wedge loop per vertex; for manifold vertices, wedge[i] == remap[i] == i
	for (size_t i = 0; i < vertex_count; ++i)
		wedge[i] = (unsigned)(i);

	for (size_t i = 0; i < vertex_count; ++i)
		if (remap[i] != i)
		{
			unsigned int r = remap[i];

			wedge[i] = wedge[r];
			wedge[r] = (unsigned)(i);
		}
}

enum VertexKind
{
	Kind_Manifold, // not on an attribute seam, not on any boundary
	Kind_Border,   // not on an attribute seam, has exactly two open edges
	Kind_Seam,     // on an attribute seam with exactly two attribute seam edges
	Kind_Locked,   // none of the above; these vertices can't move

	Kind_Count
};

// manifold vertices can collapse on anything except locked
// border/seam vertices can only be collapsed onto border/seam respectively
const unsigned char kCanCollapse[Kind_Count][Kind_Count] = {
    {1, 1, 1, 1},
    {0, 1, 0, 0},
    {0, 0, 1, 0},
    {0, 0, 0, 0},
};

// if a vertex is manifold or seam, adjoining edges are guaranteed to have an opposite edge
// note that for seam edges, the opposite edge isn't present in the attribute-based topology
// but is present if you consider a position-only mesh variant
const unsigned char kHasOpposite[Kind_Count][Kind_Count] = {
    {1, 1, 1, 1},
    {1, 0, 1, 0},
    {1, 1, 1, 1},
    {1, 0, 1, 0},
};

static bool hasEdge(const EdgeAdjacency* adjacency, unsigned int a, unsigned int b)
{
	unsigned int count = adjacency->counts[a];
	const unsigned int* data = adjacency->data + adjacency->offsets[a];

	for (size_t i = 0; i < count; ++i)
		if (data[i] == b)
			return true;

	return false;
}

static unsigned int findWedgeEdge(const EdgeAdjacency* adjacency, const unsigned int* wedge, unsigned int a, unsigned int b)
{
	unsigned int v = a;

	do
	{
		if (hasEdge(adjacency, v, b))
			return v;

		v = wedge[v];
	} while (v != a);

	return ~0u;
}

static size_t countOpenEdges(const EdgeAdjacency* adjacency, unsigned int vertex, unsigned int* last)
{
	size_t result = 0;

	unsigned int count = adjacency->counts[vertex];
	const unsigned int* data = adjacency->data + adjacency->offsets[vertex];

	for (size_t i = 0; i < count; ++i)
		if (!hasEdge(adjacency, data[i], vertex))
		{
			result++;

			if (last)
				*last = data[i];
		}

	return result;
}

static void classifyVertices(unsigned char* result, unsigned int* loop, size_t vertex_count, const EdgeAdjacency* adjacency, const unsigned int* remap, const unsigned int* wedge)
{
	for (size_t i = 0; i < vertex_count; ++i)
		loop[i] = ~0u;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		if (remap[i] == i)
		{
			if (wedge[i] == i)
			{
				// no attribute seam, need to check if it's manifold
				unsigned int v = 0;
				size_t edges = countOpenEdges(adjacency, (unsigned)(i), &v);

				// note: we classify any vertices with no open edges as manifold
				// this is technically incorrect - if 4 triangles share an edge, we'll classify vertices as manifold
				// it's unclear if this is a problem in practice
				// also note that we classify vertices as border if they have *one* open edge, not two
				// this is because we only have half-edges - so a border vertex would have one incoming and one outgoing edge
				if (edges == 0)
				{
					result[i] = Kind_Manifold;
				}
				else if (edges == 1)
				{
					result[i] = Kind_Border;
					loop[i] = v;
				}
				else
				{
					result[i] = Kind_Locked;
				}
			}
			else if (wedge[wedge[i]] == i)
			{
				// attribute seam; need to distinguish between Seam and Locked
				unsigned int a = 0;
				size_t a_count = countOpenEdges(adjacency, (unsigned)(i), &a);
				unsigned int b = 0;
				size_t b_count = countOpenEdges(adjacency, wedge[i], &b);

				// seam should have one open half-edge for each vertex, and the edges need to "connect" - point to the same vertex post-remap
				if (a_count == 1 && b_count == 1)
				{
					unsigned int ao = findWedgeEdge(adjacency, wedge, a, wedge[i]);
					unsigned int bo = findWedgeEdge(adjacency, wedge, b, (unsigned)(i));

					if (ao != ~0u && bo != ~0u)
					{
						result[i] = Kind_Seam;

						loop[i] = a;
						loop[wedge[i]] = b;
					}
					else
					{
						result[i] = Kind_Locked;
					}
				}
				else
				{
					result[i] = Kind_Locked;
				}
			}
			else
			{
				// more than one vertex maps to this one; we don't have classification available
				result[i] = Kind_Locked;
			}
		}
		else
		{
			assert(remap[i] < i);

			result[i] = result[remap[i]];
		}
	}
}

typedef struct Vector3
{
	float x, y, z;
} Vector3;

static void rescalePositions(Vector3* result, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride)
{
	size_t vertex_stride_float = vertex_positions_stride / sizeof(float);

	float minv[3] = {FLT_MAX, FLT_MAX, FLT_MAX};
	float maxv[3] = {-FLT_MAX, -FLT_MAX, -FLT_MAX};

	for (size_t i = 0; i < vertex_count; ++i)
	{
		const float* v = vertex_positions_data + i * vertex_stride_float;

		result[i].x = v[0];
		result[i].y = v[1];
		result[i].z = v[2];

		for (int j = 0; j < 3; ++j)
		{
			float vj = v[j];

			minv[j] = minv[j] > vj ? vj : minv[j];
			maxv[j] = maxv[j] < vj ? vj : maxv[j];
		}
	}

	float extent = 0.f;

	extent = (maxv[0] - minv[0]) < extent ? extent : (maxv[0] - minv[0]);
	extent = (maxv[1] - minv[1]) < extent ? extent : (maxv[1] - minv[1]);
	extent = (maxv[2] - minv[2]) < extent ? extent : (maxv[2] - minv[2]);

	float scale = extent == 0 ? 0.f : 1.f / extent;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		result[i].x = (result[i].x - minv[0]) * scale;
		result[i].y = (result[i].y - minv[1]) * scale;
		result[i].z = (result[i].z - minv[2]) * scale;
	}
}

typedef struct Quadric
{
	float a00;
	float a10, a11;
	float a20, a21, a22;
	float b0, b1, b2, c;
} Quadric;

typedef struct Collapse
{
	unsigned int v0;
	unsigned int v1;

	union {
		unsigned int bidi;
		float error;
		unsigned int errorui;
	};
} Collapse;

static float normalize(Vector3* v)
{
	float length = sqrtf(v->x * v->x + v->y * v->y + v->z * v->z);

	if (length > 0)
	{
		v->x /= length;
		v->y /= length;
		v->z /= length;
	}

	return length;
}

static void quadricAdd(Quadric* Q, const Quadric* R)
{
	Q->a00 += R->a00;
	Q->a10 += R->a10;
	Q->a11 += R->a11;
	Q->a20 += R->a20;
	Q->a21 += R->a21;
	Q->a22 += R->a22;
	Q->b0 += R->b0;
	Q->b1 += R->b1;
	Q->b2 += R->b2;
	Q->c += R->c;
}

static void quadricMul(Quadric* Q, float s)
{
	Q->a00 *= s;
	Q->a10 *= s;
	Q->a11 *= s;
	Q->a20 *= s;
	Q->a21 *= s;
	Q->a22 *= s;
	Q->b0 *= s;
	Q->b1 *= s;
	Q->b2 *= s;
	Q->c *= s;
}

static float quadricError(const Quadric* Q, const Vector3* v)
{
	float rx = Q->b0;
	float ry = Q->b1;
	float rz = Q->b2;

	rx += Q->a10 * v->y;
	ry += Q->a21 * v->z;
	rz += Q->a20 * v->x;

	rx *= 2;
	ry *= 2;
	rz *= 2;

	rx += Q->a00 * v->x;
	ry += Q->a11 * v->y;
	rz += Q->a22 * v->z;

	float r = Q->c;
	r += rx * v->x;
	r += ry * v->y;
	r += rz * v->z;

	return fabsf(r);
}

static void quadricFromPlane(Quadric* Q, float a, float b, float c, float d)
{
	Q->a00 = a * a;
	Q->a10 = b * a;
	Q->a11 = b * b;
	Q->a20 = c * a;
	Q->a21 = c * b;
	Q->a22 = c * c;
	Q->b0 = d * a;
	Q->b1 = d * b;
	Q->b2 = d * c;
	Q->c = d * d;
}

static void quadricFromTriangle(Quadric* Q, const Vector3* p0, const Vector3* p1, const Vector3* p2)
{
	Vector3 p10 = {p1->x - p0->x, p1->y - p0->y, p1->z - p0->z};
	Vector3 p20 = {p2->x - p0->x, p2->y - p0->y, p2->z - p0->z};

	Vector3 normal = {p10.y * p20.z - p10.z * p20.y, p10.z * p20.x - p10.x * p20.z, p10.x * p20.y - p10.y * p20.x};
	float area = normalize(&normal);

	float distance = normal.x * p0->x + normal.y * p0->y + normal.z * p0->z;

	quadricFromPlane(Q, normal.x, normal.y, normal.z, -distance);

	quadricMul(Q, area);
}

static void quadricFromTriangleEdge(Quadric* Q, const Vector3* p0, const Vector3* p1, const Vector3* p2, float weight)
{
	Vector3 p10 = {p1->x - p0->x, p1->y - p0->y, p1->z - p0->z};
	float length = normalize(&p10);

	Vector3 p20 = {p2->x - p0->x, p2->y - p0->y, p2->z - p0->z};
	float p20p = p20.x * p10.x + p20.y * p10.y + p20.z * p10.z;

	Vector3 normal = {p20.x - p10.x * p20p, p20.y - p10.y * p20p, p20.z - p10.z * p20p};
	normalize(&normal);

	float distance = normal.x * p0->x + normal.y * p0->y + normal.z * p0->z;

	quadricFromPlane(Q, normal.x, normal.y, normal.z, -distance);

	quadricMul(Q, length * length * weight);
}

static void fillFaceQuadrics(Quadric* vertex_quadrics, const unsigned int* indices, size_t index_count, const Vector3* vertex_positions, const unsigned int* remap)
{
	for (size_t i = 0; i < index_count; i += 3)
	{
		unsigned int i0 = indices[i + 0];
		unsigned int i1 = indices[i + 1];
		unsigned int i2 = indices[i + 2];

		Quadric Q;
		quadricFromTriangle(&Q, &vertex_positions[i0], &vertex_positions[i1], &vertex_positions[i2]);

		quadricAdd(&vertex_quadrics[remap[i0]], &Q);
		quadricAdd(&vertex_quadrics[remap[i1]], &Q);
		quadricAdd(&vertex_quadrics[remap[i2]], &Q);
	}
}

static void fillEdgeQuadrics(Quadric* vertex_quadrics, const unsigned int* indices, size_t index_count, const Vector3* vertex_positions, const unsigned int* remap, const unsigned char* vertex_kind, const unsigned int* loop)
{
	for (size_t i = 0; i < index_count; i += 3)
	{
		static const int next[3] = {1, 2, 0};

		for (int e = 0; e < 3; ++e)
		{
			unsigned int i0 = indices[i + e];
			unsigned int i1 = indices[i + next[e]];

			unsigned char k0 = vertex_kind[i0];
			unsigned char k1 = vertex_kind[i1];

			// check that i0 and i1 are border/seam and are on the same edge loop
			// loop[] tracks half edges so we only need to check i0->i1
			if (k0 != k1 || (k0 != Kind_Border && k0 != Kind_Seam) || loop[i0] != i1)
				continue;

			unsigned int i2 = indices[i + next[next[e]]];

			// we try hard to maintain border edge geometry; seam edges can move more freely
			// due to topological restrictions on collapses, seam quadrics slightly improves collapse structure but aren't critical
			const float kEdgeWeightSeam = 1.f;
			const float kEdgeWeightBorder = 10.f;

			float edgeWeight = (k0 == Kind_Seam) ? kEdgeWeightSeam : kEdgeWeightBorder;

			Quadric Q;
			quadricFromTriangleEdge(&Q, &vertex_positions[i0], &vertex_positions[i1], &vertex_positions[i2], edgeWeight);

			quadricAdd(&vertex_quadrics[remap[i0]], &Q);
			quadricAdd(&vertex_quadrics[remap[i1]], &Q);
		}
	}
}

static size_t pickEdgeCollapses(Collapse* collapses, const unsigned int* indices, size_t index_count, const unsigned int* remap, const unsigned char* vertex_kind, const unsigned int* loop)
{
	size_t collapse_count = 0;

	for (size_t i = 0; i < index_count; i += 3)
	{
		static const int next[3] = {1, 2, 0};

		for (int e = 0; e < 3; ++e)
		{
			unsigned int i0 = indices[i + e];
			unsigned int i1 = indices[i + next[e]];

			// this can happen either when input has a zero-length edge, or when we perform collapses for complex
			// topology w/seams and collapse a manifold vertex that connects to both wedges onto one of them
			// we leave edges like this alone since they may be important for preserving mesh integrity
			if (remap[i0] == remap[i1])
				continue;

			unsigned char k0 = vertex_kind[i0];
			unsigned char k1 = vertex_kind[i1];

			// the edge has to be collapsible in at least one direction
			if (!(kCanCollapse[k0][k1] | kCanCollapse[k1][k0]))
				continue;

			// manifold and seam edges should occur twice (i0->i1 and i1->i0) - skip redundant edges
			if (kHasOpposite[k0][k1] && remap[i1] > remap[i0])
				continue;

			// two vertices are on a border or a seam, but there's no direct edge between them
			// this indicates that they belong to two different edge loops and we should not collapse this edge
			// loop[] tracks half edges so we only need to check i0->i1
			if (k0 == k1 && (k0 == Kind_Border || k0 == Kind_Seam) && loop[i0] != i1)
				continue;

			// edge can be collapsed in either direction - we will pick the one with minimum error
			// note: we evaluate error later during collapse ranking, here we just tag the edge as bidirectional
			if (kCanCollapse[k0][k1] & kCanCollapse[k1][k0])
			{
				Collapse c = {i0, i1, {/* bidi= */ 1}};
				collapses[collapse_count++] = c;
			}
			else
			{
				// edge can only be collapsed in one direction
				unsigned int e0 = kCanCollapse[k0][k1] ? i0 : i1;
				unsigned int e1 = kCanCollapse[k0][k1] ? i1 : i0;

				Collapse c = {e0, e1, {/* bidi= */ 0}};
				collapses[collapse_count++] = c;
			}
		}
	}

	return collapse_count;
}

static void rankEdgeCollapses(Collapse* collapses, size_t collapse_count, const Vector3* vertex_positions, const Quadric* vertex_quadrics, const unsigned int* remap)
{
	for (size_t i = 0; i < collapse_count; ++i)
	{
		Collapse* c = &collapses[i];

		unsigned int i0 = c->v0;
		unsigned int i1 = c->v1;

		// most edges are bidirectional which means we need to evaluate errors for two collapses
		// to keep this code branchless we just use the same edge for unidirectional edges
		unsigned int j0 = c->bidi ? i1 : i0;
		unsigned int j1 = c->bidi ? i0 : i1;

		float ei = quadricError(&vertex_quadrics[remap[i0]], &vertex_positions[i1]);
		float ej = quadricError(&vertex_quadrics[remap[j0]], &vertex_positions[j1]);

		// pick edge direction with minimal error
		c->v0 = ei <= ej ? i0 : j0;
		c->v1 = ei <= ej ? i1 : j1;
		c->error = ei <= ej ? ei : ej;
	}
}

#if TRACE > 1
static void dumpEdgeCollapses(const Collapse* collapses, size_t collapse_count, const unsigned char* vertex_kind)
{
	size_t ckinds[Kind_Count][Kind_Count] = {};
	float cerrors[Kind_Count][Kind_Count] = {};

	for (int k0 = 0; k0 < Kind_Count; ++k0)
		for (int k1 = 0; k1 < Kind_Count; ++k1)
			cerrors[k0][k1] = FLT_MAX;

	for (size_t i = 0; i < collapse_count; ++i)
	{
		unsigned int i0 = collapses[i].v0;
		unsigned int i1 = collapses[i].v1;

		unsigned char k0 = vertex_kind[i0];
		unsigned char k1 = vertex_kind[i1];

		ckinds[k0][k1]++;
		cerrors[k0][k1] = (collapses[i].error < cerrors[k0][k1]) ? collapses[i].error : cerrors[k0][k1];
	}

	for (int k0 = 0; k0 < Kind_Count; ++k0)
		for (int k1 = 0; k1 < Kind_Count; ++k1)
			if (ckinds[k0][k1])
				printf("collapses %d -> %d: %d, min error %e\n", k0, k1, int(ckinds[k0][k1]), cerrors[k0][k1]);
}

static void dumpLockedCollapses(const unsigned int* indices, size_t index_count, const unsigned char* vertex_kind)
{
	size_t locked_collapses[Kind_Count][Kind_Count] = {};

	for (size_t i = 0; i < index_count; i += 3)
	{
		static const int next[3] = {1, 2, 0};

		for (int e = 0; e < 3; ++e)
		{
			unsigned int i0 = indices[i + e];
			unsigned int i1 = indices[i + next[e]];

			unsigned char k0 = vertex_kind[i0];
			unsigned char k1 = vertex_kind[i1];

			locked_collapses[k0][k1] += !kCanCollapse[k0][k1] && !kCanCollapse[k1][k0];
		}
	}

	for (int k0 = 0; k0 < Kind_Count; ++k0)
		for (int k1 = 0; k1 < Kind_Count; ++k1)
			if (locked_collapses[k0][k1])
				printf("locked collapses %d -> %d: %d\n", k0, k1, int(locked_collapses[k0][k1]));
}
#endif

static void sortEdgeCollapses(unsigned int* sort_order, const Collapse* collapses, size_t collapse_count)
{
	const int sort_bits = 11;

	// fill histogram for counting sort
	unsigned int histogram[1 << sort_bits];
	memset(histogram, 0, sizeof(histogram));

	for (size_t i = 0; i < collapse_count; ++i)
	{
		// skip sign bit since error is non-negative
		unsigned int key = (collapses[i].errorui << 1) >> (32 - sort_bits);

		histogram[key]++;
	}

	// compute offsets based on histogram data
	size_t histogram_sum = 0;

	for (size_t i = 0; i < (size_t)1 << sort_bits; ++i)
	{
		size_t count = histogram[i];
		histogram[i] = (unsigned)(histogram_sum);
		histogram_sum += count;
	}

	assert(histogram_sum == collapse_count);

	// compute sort order based on offsets
	for (size_t i = 0; i < collapse_count; ++i)
	{
		// skip sign bit since error is non-negative
		unsigned int key = (collapses[i].errorui << 1) >> (32 - sort_bits);

		sort_order[histogram[key]++] = (unsigned)(i);
	}
}

static size_t performEdgeCollapses(unsigned int* collapse_remap, unsigned char* collapse_locked, Quadric* vertex_quadrics, const Collapse* collapses, size_t collapse_count, const unsigned int* collapse_order, const unsigned int* remap, const unsigned int* wedge, const unsigned char* vertex_kind, size_t triangle_collapse_goal, float error_limit)
{
	size_t edge_collapses = 0;
	size_t triangle_collapses = 0;

	for (size_t i = 0; i < collapse_count; ++i)
	{
		const Collapse* c = &collapses[collapse_order[i]];

		if (c->error > error_limit)
			break;

		if (triangle_collapses >= triangle_collapse_goal)
			break;

		unsigned int r0 = remap[c->v0];
		unsigned int r1 = remap[c->v1];

		// we don't collapse vertices that had source or target vertex involved in a collapse
		// it's important to not move the vertices twice since it complicates the tracking/remapping logic
		// it's important to not move other vertices towards a moved vertex to preserve error since we don't re-rank collapses mid-pass
		if (collapse_locked[r0] | collapse_locked[r1])
			continue;

		assert(collapse_remap[r0] == r0);
		assert(collapse_remap[r1] == r1);

		quadricAdd(&vertex_quadrics[r1], &vertex_quadrics[r0]);

		if (vertex_kind[c->v0] == Kind_Seam)
		{
			// remap v0 to v1 and seam pair of v0 to seam pair of v1
			unsigned int s0 = wedge[c->v0];
			unsigned int s1 = wedge[c->v1];

			assert(s0 != c->v0 && s1 != c->v1);
			assert(wedge[s0] == c->v0 && wedge[s1] == c->v1);

			collapse_remap[c->v0] = c->v1;
			collapse_remap[s0] = s1;
		}
		else
		{
			assert(wedge[c->v0] == c->v0);

			collapse_remap[c->v0] = c->v1;
		}

		collapse_locked[r0] = 1;
		collapse_locked[r1] = 1;

		// border edges collapse 1 triangle, other edges collapse 2 or more
		triangle_collapses += (vertex_kind[c->v0] == Kind_Border) ? 1 : 2;
		edge_collapses++;
	}

	return edge_collapses;
}

static size_t remapIndexBuffer(unsigned int* indices, size_t index_count, const unsigned int* collapse_remap)
{
	size_t write = 0;

	for (size_t i = 0; i < index_count; i += 3)
	{
		unsigned int v0 = collapse_remap[indices[i + 0]];
		unsigned int v1 = collapse_remap[indices[i + 1]];
		unsigned int v2 = collapse_remap[indices[i + 2]];

		// we never move the vertex twice during a single pass
		assert(collapse_remap[v0] == v0);
		assert(collapse_remap[v1] == v1);
		assert(collapse_remap[v2] == v2);

		if (v0 != v1 && v0 != v2 && v1 != v2)
		{
			indices[write + 0] = v0;
			indices[write + 1] = v1;
			indices[write + 2] = v2;
			write += 3;
		}
	}

	return write;
}

static void remapEdgeLoops(unsigned int* loop, size_t vertex_count, const unsigned int* collapse_remap)
{
	for (size_t i = 0; i < vertex_count; ++i)
	{
		if (loop[i] != ~0u)
		{
			unsigned int l = loop[i];
			unsigned int r = collapse_remap[l];

			// i == r is a special case when the seam edge is collapsed in a direction opposite to where loop goes
			loop[i] = (i == r) ? loop[l] : r;
		}
	}
}

#if TRACE
unsigned char* meshopt_simplifyDebugKind = 0;
unsigned int* meshopt_simplifyDebugLoop = 0;
#endif

size_t meshopt_simplify(unsigned int* destination, const unsigned int* indices, size_t index_count, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride, size_t target_index_count, float target_error)
{
	assert(index_count % 3 == 0);
	assert(vertex_positions_stride > 0 && vertex_positions_stride <= 256);
	assert(vertex_positions_stride % sizeof(float) == 0);
	assert(target_index_count <= index_count);

	meshopt_Allocator allocator = {0};

	unsigned int* result = destination;

	// build adjacency information
	EdgeAdjacency adjacency = {0};
	buildEdgeAdjacency(&adjacency, indices, index_count, vertex_count, &allocator);

	// build position remap that maps each vertex to the one with identical position
	unsigned int* remap = (unsigned int*)meshopt_allocate(&allocator, vertex_count * sizeof(unsigned int));
	unsigned int* wedge = (unsigned int*)meshopt_allocate(&allocator, vertex_count * sizeof(unsigned int));
	buildPositionRemap(remap, wedge, vertex_positions_data, vertex_count, vertex_positions_stride, &allocator);

	// classify vertices; vertex kind determines collapse rules, see kCanCollapse
	unsigned char* vertex_kind = (unsigned char*)meshopt_allocate(&allocator, vertex_count);
	unsigned int* loop = (unsigned int*)meshopt_allocate(&allocator, vertex_count * sizeof(unsigned int));
	classifyVertices(vertex_kind, loop, vertex_count, &adjacency, remap, wedge);

#if TRACE
	size_t unique_positions = 0;
	for (size_t i = 0; i < vertex_count; ++i)
		unique_positions += remap[i] == i;

	printf("position remap: %d vertices => %d positions\n", int(vertex_count), int(unique_positions));

	size_t kinds[Kind_Count] = {};
	for (size_t i = 0; i < vertex_count; ++i)
		kinds[vertex_kind[i]] += remap[i] == i;

	printf("kinds: manifold %d, border %d, seam %d, locked %d\n",
	       int(kinds[Kind_Manifold]), int(kinds[Kind_Border]), int(kinds[Kind_Seam]), int(kinds[Kind_Locked]));
#endif

	Vector3* vertex_positions = (Vector3*)meshopt_allocate(&allocator, vertex_count * sizeof(Vector3));
	rescalePositions(vertex_positions, vertex_positions_data, vertex_count, vertex_positions_stride);

	Quadric* vertex_quadrics = (Quadric*)meshopt_allocate(&allocator, vertex_count * sizeof(Quadric));
	memset(vertex_quadrics, 0, vertex_count * sizeof(Quadric));

	fillFaceQuadrics(vertex_quadrics, indices, index_count, vertex_positions, remap);
	fillEdgeQuadrics(vertex_quadrics, indices, index_count, vertex_positions, remap, vertex_kind, loop);

	if (result != indices)
		memcpy(result, indices, index_count * sizeof(unsigned int));

#if TRACE
	size_t pass_count = 0;
	float worst_error = 0;
#endif

	Collapse* edge_collapses = (Collapse*)meshopt_allocate(&allocator, index_count * sizeof(Collapse));
	unsigned int* collapse_order = (unsigned int*)meshopt_allocate(&allocator, index_count * sizeof(unsigned int));
	unsigned int* collapse_remap = (unsigned int*)meshopt_allocate(&allocator, vertex_count * sizeof(unsigned int));
	unsigned char* collapse_locked = (unsigned char*)meshopt_allocate(&allocator, vertex_count);

	size_t result_count = index_count;

	while (result_count > target_index_count)
	{
		size_t edge_collapse_count = pickEdgeCollapses(edge_collapses, result, result_count, remap, vertex_kind, loop);

		// no edges can be collapsed any more due to topology restrictions
		if (edge_collapse_count == 0)
			break;

		rankEdgeCollapses(edge_collapses, edge_collapse_count, vertex_positions, vertex_quadrics, remap);

#if TRACE > 1
		dumpEdgeCollapses(edge_collapses, edge_collapse_count, vertex_kind);
#endif

		sortEdgeCollapses(collapse_order, edge_collapses, edge_collapse_count);

		// most collapses remove 2 triangles; use this to establish a bound on the pass in terms of error limit
		// note that edge_collapse_goal is an estimate; triangle_collapse_goal will be used to actually limit collapses
		size_t triangle_collapse_goal = (result_count - target_index_count) / 3;
		size_t edge_collapse_goal = triangle_collapse_goal / 2;

		// we limit the error in each pass based on the error of optimal last collapse; since many collapses will be locked
		// as they will share vertices with other successfull collapses, we need to increase the acceptable error by this factor
		const float kPassErrorBound = 1.5f;

		float error_goal = edge_collapse_goal < edge_collapse_count ? edge_collapses[collapse_order[edge_collapse_goal]].error * kPassErrorBound : FLT_MAX;
		float error_limit = error_goal > target_error ? target_error : error_goal;

		for (size_t i = 0; i < vertex_count; ++i)
			collapse_remap[i] = (unsigned)(i);

		memset(collapse_locked, 0, vertex_count);

		size_t collapses = performEdgeCollapses(collapse_remap, collapse_locked, vertex_quadrics, edge_collapses, edge_collapse_count, collapse_order, remap, wedge, vertex_kind, triangle_collapse_goal, error_limit);

		// no edges can be collapsed any more due to hitting the error limit or triangle collapse limit
		if (collapses == 0)
			break;

		remapEdgeLoops(loop, vertex_count, collapse_remap);

		size_t new_count = remapIndexBuffer(result, result_count, collapse_remap);
		assert(new_count < result_count);

#if TRACE
		float pass_error = 0.f;
		for (size_t i = 0; i < edge_collapse_count; ++i)
		{
			Collapse& c = edge_collapses[collapse_order[i]];

			if (collapse_remap[c.v0] == c.v1)
				pass_error = c.error;
		}

		pass_count++;
		worst_error = (worst_error < pass_error) ? pass_error : worst_error;

		printf("pass %d: triangles: %d -> %d, collapses: %d/%d (goal: %d), error: %e (limit %e goal %e)\n", int(pass_count), int(result_count / 3), int(new_count / 3), int(collapses), int(edge_collapse_count), int(edge_collapse_goal), pass_error, error_limit, error_goal);
#endif

		result_count = new_count;
	}

#if TRACE
	printf("passes: %d, worst error: %e\n", int(pass_count), worst_error);
#endif

#if TRACE > 1
	dumpLockedCollapses(result, result_count, vertex_kind);
#endif

#if TRACE
	if (meshopt_simplifyDebugKind)
		memcpy(meshopt_simplifyDebugKind, vertex_kind, vertex_count);

	if (meshopt_simplifyDebugLoop)
		memcpy(meshopt_simplifyDebugLoop, loop, vertex_count * sizeof(unsigned int));
#endif

	meshopt_deallocate(&allocator);

	return result_count;
}

simplifier.cpp

// This file is part of meshoptimizer library; see meshoptimizer.h for version/license details
#include "meshoptimizer.h"

#include <assert.h>
#include <float.h>
#include <math.h>
#include <string.h>

#ifndef TRACE
#define TRACE 0
#endif

#if TRACE
#include <stdio.h>
#endif

// This work is based on:
// Michael Garland and Paul S. Heckbert. Surface simplification using quadric error metrics. 1997
// Michael Garland. Quadric-based polygonal surface simplification. 1999
namespace meshopt
{

struct EdgeAdjacency
{
	unsigned int* counts;
	unsigned int* offsets;
	unsigned int* data;
};

static void buildEdgeAdjacency(EdgeAdjacency& adjacency, const unsigned int* indices, size_t index_count, size_t vertex_count, meshopt_Allocator& allocator)
{
	size_t face_count = index_count / 3;

	// allocate arrays
	adjacency.counts = allocator.allocate<unsigned int>(vertex_count);
	adjacency.offsets = allocator.allocate<unsigned int>(vertex_count);
	adjacency.data = allocator.allocate<unsigned int>(index_count);

	// fill edge counts
	memset(adjacency.counts, 0, vertex_count * sizeof(unsigned int));

	for (size_t i = 0; i < index_count; ++i)
	{
		assert(indices[i] < vertex_count);

		adjacency.counts[indices[i]]++;
	}

	// fill offset table
	unsigned int offset = 0;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		adjacency.offsets[i] = offset;
		offset += adjacency.counts[i];
	}

	assert(offset == index_count);

	// fill edge data
	for (size_t i = 0; i < face_count; ++i)
	{
		unsigned int a = indices[i * 3 + 0], b = indices[i * 3 + 1], c = indices[i * 3 + 2];

		adjacency.data[adjacency.offsets[a]++] = b;
		adjacency.data[adjacency.offsets[b]++] = c;
		adjacency.data[adjacency.offsets[c]++] = a;
	}

	// fix offsets that have been disturbed by the previous pass
	for (size_t i = 0; i < vertex_count; ++i)
	{
		assert(adjacency.offsets[i] >= adjacency.counts[i]);

		adjacency.offsets[i] -= adjacency.counts[i];
	}
}

struct PositionHasher
{
	const float* vertex_positions;
	size_t vertex_stride_float;

	size_t hash(unsigned int index) const
	{
		// MurmurHash2
		const unsigned int m = 0x5bd1e995;
		const int r = 24;

		unsigned int h = 0;
		const unsigned int* key = reinterpret_cast<const unsigned int*>(vertex_positions + index * vertex_stride_float);

		for (size_t i = 0; i < 3; ++i)
		{
			unsigned int k = key[i];

			k *= m;
			k ^= k >> r;
			k *= m;

			h *= m;
			h ^= k;
		}

		return h;
	}

	bool equal(unsigned int lhs, unsigned int rhs) const
	{
		return memcmp(vertex_positions + lhs * vertex_stride_float, vertex_positions + rhs * vertex_stride_float, sizeof(float) * 3) == 0;
	}
};

static size_t hashBuckets2(size_t count)
{
	size_t buckets = 1;
	while (buckets < count)
		buckets *= 2;

	return buckets;
}

template <typename T, typename Hash>
static T* hashLookup2(T* table, size_t buckets, const Hash& hash, const T& key, const T& empty)
{
	assert(buckets > 0);
	assert((buckets & (buckets - 1)) == 0);

	size_t hashmod = buckets - 1;
	size_t bucket = hash.hash(key) & hashmod;

	for (size_t probe = 0; probe <= hashmod; ++probe)
	{
		T& item = table[bucket];

		if (item == empty)
			return &item;

		if (hash.equal(item, key))
			return &item;

		// hash collision, quadratic probing
		bucket = (bucket + probe + 1) & hashmod;
	}

	assert(false && "Hash table is full");
	return 0;
}

static void buildPositionRemap(unsigned int* remap, unsigned int* wedge, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride, meshopt_Allocator& allocator)
{
	PositionHasher hasher = {vertex_positions_data, vertex_positions_stride / sizeof(float)};

	size_t table_size = hashBuckets2(vertex_count);
	unsigned int* table = allocator.allocate<unsigned int>(table_size);
	memset(table, -1, table_size * sizeof(unsigned int));

	// build forward remap: for each vertex, which other (canonical) vertex does it map to?
	// we use position equivalence for this, and remap vertices to other existing vertices
	for (size_t i = 0; i < vertex_count; ++i)
	{
		unsigned int index = unsigned(i);
		unsigned int* entry = hashLookup2(table, table_size, hasher, index, ~0u);

		if (*entry == ~0u)
			*entry = index;

		remap[index] = *entry;
	}

	// build wedge table: for each vertex, which other vertex is the next wedge that also maps to the same vertex?
	// entries in table form a (cyclic) wedge loop per vertex; for manifold vertices, wedge[i] == remap[i] == i
	for (size_t i = 0; i < vertex_count; ++i)
		wedge[i] = unsigned(i);

	for (size_t i = 0; i < vertex_count; ++i)
		if (remap[i] != i)
		{
			unsigned int r = remap[i];

			wedge[i] = wedge[r];
			wedge[r] = unsigned(i);
		}
}

enum VertexKind
{
	Kind_Manifold, // not on an attribute seam, not on any boundary
	Kind_Border,   // not on an attribute seam, has exactly two open edges
	Kind_Seam,     // on an attribute seam with exactly two attribute seam edges
	Kind_Locked,   // none of the above; these vertices can't move

	Kind_Count
};

// manifold vertices can collapse on anything except locked
// border/seam vertices can only be collapsed onto border/seam respectively
const unsigned char kCanCollapse[Kind_Count][Kind_Count] = {
    {1, 1, 1, 1},
    {0, 1, 0, 0},
    {0, 0, 1, 0},
    {0, 0, 0, 0},
};

// if a vertex is manifold or seam, adjoining edges are guaranteed to have an opposite edge
// note that for seam edges, the opposite edge isn't present in the attribute-based topology
// but is present if you consider a position-only mesh variant
const unsigned char kHasOpposite[Kind_Count][Kind_Count] = {
    {1, 1, 1, 1},
    {1, 0, 1, 0},
    {1, 1, 1, 1},
    {1, 0, 1, 0},
};

static bool hasEdge(const EdgeAdjacency& adjacency, unsigned int a, unsigned int b)
{
	unsigned int count = adjacency.counts[a];
	const unsigned int* data = adjacency.data + adjacency.offsets[a];

	for (size_t i = 0; i < count; ++i)
		if (data[i] == b)
			return true;

	return false;
}

static unsigned int findWedgeEdge(const EdgeAdjacency& adjacency, const unsigned int* wedge, unsigned int a, unsigned int b)
{
	unsigned int v = a;

	do
	{
		if (hasEdge(adjacency, v, b))
			return v;

		v = wedge[v];
	} while (v != a);

	return ~0u;
}

static size_t countOpenEdges(const EdgeAdjacency& adjacency, unsigned int vertex, unsigned int* last = 0)
{
	size_t result = 0;

	unsigned int count = adjacency.counts[vertex];
	const unsigned int* data = adjacency.data + adjacency.offsets[vertex];

	for (size_t i = 0; i < count; ++i)
		if (!hasEdge(adjacency, data[i], vertex))
		{
			result++;

			if (last)
				*last = data[i];
		}

	return result;
}

static void classifyVertices(unsigned char* result, unsigned int* loop, size_t vertex_count, const EdgeAdjacency& adjacency, const unsigned int* remap, const unsigned int* wedge)
{
	for (size_t i = 0; i < vertex_count; ++i)
		loop[i] = ~0u;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		if (remap[i] == i)
		{
			if (wedge[i] == i)
			{
				// no attribute seam, need to check if it's manifold
				unsigned int v = 0;
				size_t edges = countOpenEdges(adjacency, unsigned(i), &v);

				// note: we classify any vertices with no open edges as manifold
				// this is technically incorrect - if 4 triangles share an edge, we'll classify vertices as manifold
				// it's unclear if this is a problem in practice
				// also note that we classify vertices as border if they have *one* open edge, not two
				// this is because we only have half-edges - so a border vertex would have one incoming and one outgoing edge
				if (edges == 0)
				{
					result[i] = Kind_Manifold;
				}
				else if (edges == 1)
				{
					result[i] = Kind_Border;
					loop[i] = v;
				}
				else
				{
					result[i] = Kind_Locked;
				}
			}
			else if (wedge[wedge[i]] == i)
			{
				// attribute seam; need to distinguish between Seam and Locked
				unsigned int a = 0;
				size_t a_count = countOpenEdges(adjacency, unsigned(i), &a);
				unsigned int b = 0;
				size_t b_count = countOpenEdges(adjacency, wedge[i], &b);

				// seam should have one open half-edge for each vertex, and the edges need to "connect" - point to the same vertex post-remap
				if (a_count == 1 && b_count == 1)
				{
					unsigned int ao = findWedgeEdge(adjacency, wedge, a, wedge[i]);
					unsigned int bo = findWedgeEdge(adjacency, wedge, b, unsigned(i));

					if (ao != ~0u && bo != ~0u)
					{
						result[i] = Kind_Seam;

						loop[i] = a;
						loop[wedge[i]] = b;
					}
					else
					{
						result[i] = Kind_Locked;
					}
				}
				else
				{
					result[i] = Kind_Locked;
				}
			}
			else
			{
				// more than one vertex maps to this one; we don't have classification available
				result[i] = Kind_Locked;
			}
		}
		else
		{
			assert(remap[i] < i);

			result[i] = result[remap[i]];
		}
	}
}

struct Vector3
{
	float x, y, z;
};

static void rescalePositions(Vector3* result, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride)
{
	size_t vertex_stride_float = vertex_positions_stride / sizeof(float);

	float minv[3] = {FLT_MAX, FLT_MAX, FLT_MAX};
	float maxv[3] = {-FLT_MAX, -FLT_MAX, -FLT_MAX};

	for (size_t i = 0; i < vertex_count; ++i)
	{
		const float* v = vertex_positions_data + i * vertex_stride_float;

		result[i].x = v[0];
		result[i].y = v[1];
		result[i].z = v[2];

		for (int j = 0; j < 3; ++j)
		{
			float vj = v[j];

			minv[j] = minv[j] > vj ? vj : minv[j];
			maxv[j] = maxv[j] < vj ? vj : maxv[j];
		}
	}

	float extent = 0.f;

	extent = (maxv[0] - minv[0]) < extent ? extent : (maxv[0] - minv[0]);
	extent = (maxv[1] - minv[1]) < extent ? extent : (maxv[1] - minv[1]);
	extent = (maxv[2] - minv[2]) < extent ? extent : (maxv[2] - minv[2]);

	float scale = extent == 0 ? 0.f : 1.f / extent;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		result[i].x = (result[i].x - minv[0]) * scale;
		result[i].y = (result[i].y - minv[1]) * scale;
		result[i].z = (result[i].z - minv[2]) * scale;
	}
}

struct Quadric
{
	float a00;
	float a10, a11;
	float a20, a21, a22;
	float b0, b1, b2, c;
};

struct Collapse
{
	unsigned int v0;
	unsigned int v1;

	union {
		unsigned int bidi;
		float error;
		unsigned int errorui;
	};
};

static float normalize(Vector3& v)
{
	float length = sqrtf(v.x * v.x + v.y * v.y + v.z * v.z);

	if (length > 0)
	{
		v.x /= length;
		v.y /= length;
		v.z /= length;
	}

	return length;
}

static void quadricAdd(Quadric& Q, const Quadric& R)
{
	Q.a00 += R.a00;
	Q.a10 += R.a10;
	Q.a11 += R.a11;
	Q.a20 += R.a20;
	Q.a21 += R.a21;
	Q.a22 += R.a22;
	Q.b0 += R.b0;
	Q.b1 += R.b1;
	Q.b2 += R.b2;
	Q.c += R.c;
}

static void quadricMul(Quadric& Q, float s)
{
	Q.a00 *= s;
	Q.a10 *= s;
	Q.a11 *= s;
	Q.a20 *= s;
	Q.a21 *= s;
	Q.a22 *= s;
	Q.b0 *= s;
	Q.b1 *= s;
	Q.b2 *= s;
	Q.c *= s;
}

static float quadricError(const Quadric& Q, const Vector3& v)
{
	float rx = Q.b0;
	float ry = Q.b1;
	float rz = Q.b2;

	rx += Q.a10 * v.y;
	ry += Q.a21 * v.z;
	rz += Q.a20 * v.x;

	rx *= 2;
	ry *= 2;
	rz *= 2;

	rx += Q.a00 * v.x;
	ry += Q.a11 * v.y;
	rz += Q.a22 * v.z;

	float r = Q.c;
	r += rx * v.x;
	r += ry * v.y;
	r += rz * v.z;

	return fabsf(r);
}

static void quadricFromPlane(Quadric& Q, float a, float b, float c, float d)
{
	Q.a00 = a * a;
	Q.a10 = b * a;
	Q.a11 = b * b;
	Q.a20 = c * a;
	Q.a21 = c * b;
	Q.a22 = c * c;
	Q.b0 = d * a;
	Q.b1 = d * b;
	Q.b2 = d * c;
	Q.c = d * d;
}

static void quadricFromTriangle(Quadric& Q, const Vector3& p0, const Vector3& p1, const Vector3& p2)
{
	Vector3 p10 = {p1.x - p0.x, p1.y - p0.y, p1.z - p0.z};
	Vector3 p20 = {p2.x - p0.x, p2.y - p0.y, p2.z - p0.z};

	Vector3 normal = {p10.y * p20.z - p10.z * p20.y, p10.z * p20.x - p10.x * p20.z, p10.x * p20.y - p10.y * p20.x};
	float area = normalize(normal);

	float distance = normal.x * p0.x + normal.y * p0.y + normal.z * p0.z;

	quadricFromPlane(Q, normal.x, normal.y, normal.z, -distance);

	quadricMul(Q, area);
}

static void quadricFromTriangleEdge(Quadric& Q, const Vector3& p0, const Vector3& p1, const Vector3& p2, float weight)
{
	Vector3 p10 = {p1.x - p0.x, p1.y - p0.y, p1.z - p0.z};
	float length = normalize(p10);

	Vector3 p20 = {p2.x - p0.x, p2.y - p0.y, p2.z - p0.z};
	float p20p = p20.x * p10.x + p20.y * p10.y + p20.z * p10.z;

	Vector3 normal = {p20.x - p10.x * p20p, p20.y - p10.y * p20p, p20.z - p10.z * p20p};
	normalize(normal);

	float distance = normal.x * p0.x + normal.y * p0.y + normal.z * p0.z;

	quadricFromPlane(Q, normal.x, normal.y, normal.z, -distance);

	quadricMul(Q, length * length * weight);
}

static void fillFaceQuadrics(Quadric* vertex_quadrics, const unsigned int* indices, size_t index_count, const Vector3* vertex_positions, const unsigned int* remap)
{
	for (size_t i = 0; i < index_count; i += 3)
	{
		unsigned int i0 = indices[i + 0];
		unsigned int i1 = indices[i + 1];
		unsigned int i2 = indices[i + 2];

		Quadric Q;
		quadricFromTriangle(Q, vertex_positions[i0], vertex_positions[i1], vertex_positions[i2]);

		quadricAdd(vertex_quadrics[remap[i0]], Q);
		quadricAdd(vertex_quadrics[remap[i1]], Q);
		quadricAdd(vertex_quadrics[remap[i2]], Q);
	}
}

static void fillEdgeQuadrics(Quadric* vertex_quadrics, const unsigned int* indices, size_t index_count, const Vector3* vertex_positions, const unsigned int* remap, const unsigned char* vertex_kind, const unsigned int* loop)
{
	for (size_t i = 0; i < index_count; i += 3)
	{
		static const int next[3] = {1, 2, 0};

		for (int e = 0; e < 3; ++e)
		{
			unsigned int i0 = indices[i + e];
			unsigned int i1 = indices[i + next[e]];

			unsigned char k0 = vertex_kind[i0];
			unsigned char k1 = vertex_kind[i1];

			// check that i0 and i1 are border/seam and are on the same edge loop
			// loop[] tracks half edges so we only need to check i0->i1
			if (k0 != k1 || (k0 != Kind_Border && k0 != Kind_Seam) || loop[i0] != i1)
				continue;

			unsigned int i2 = indices[i + next[next[e]]];

			// we try hard to maintain border edge geometry; seam edges can move more freely
			// due to topological restrictions on collapses, seam quadrics slightly improves collapse structure but aren't critical
			const float kEdgeWeightSeam = 1.f;
			const float kEdgeWeightBorder = 10.f;

			float edgeWeight = (k0 == Kind_Seam) ? kEdgeWeightSeam : kEdgeWeightBorder;

			Quadric Q;
			quadricFromTriangleEdge(Q, vertex_positions[i0], vertex_positions[i1], vertex_positions[i2], edgeWeight);

			quadricAdd(vertex_quadrics[remap[i0]], Q);
			quadricAdd(vertex_quadrics[remap[i1]], Q);
		}
	}
}

static size_t pickEdgeCollapses(Collapse* collapses, const unsigned int* indices, size_t index_count, const unsigned int* remap, const unsigned char* vertex_kind, const unsigned int* loop)
{
	size_t collapse_count = 0;

	for (size_t i = 0; i < index_count; i += 3)
	{
		static const int next[3] = {1, 2, 0};

		for (int e = 0; e < 3; ++e)
		{
			unsigned int i0 = indices[i + e];
			unsigned int i1 = indices[i + next[e]];

			// this can happen either when input has a zero-length edge, or when we perform collapses for complex
			// topology w/seams and collapse a manifold vertex that connects to both wedges onto one of them
			// we leave edges like this alone since they may be important for preserving mesh integrity
			if (remap[i0] == remap[i1])
				continue;

			unsigned char k0 = vertex_kind[i0];
			unsigned char k1 = vertex_kind[i1];

			// the edge has to be collapsible in at least one direction
			if (!(kCanCollapse[k0][k1] | kCanCollapse[k1][k0]))
				continue;

			// manifold and seam edges should occur twice (i0->i1 and i1->i0) - skip redundant edges
			if (kHasOpposite[k0][k1] && remap[i1] > remap[i0])
				continue;

			// two vertices are on a border or a seam, but there's no direct edge between them
			// this indicates that they belong to two different edge loops and we should not collapse this edge
			// loop[] tracks half edges so we only need to check i0->i1
			if (k0 == k1 && (k0 == Kind_Border || k0 == Kind_Seam) && loop[i0] != i1)
				continue;

			// edge can be collapsed in either direction - we will pick the one with minimum error
			// note: we evaluate error later during collapse ranking, here we just tag the edge as bidirectional
			if (kCanCollapse[k0][k1] & kCanCollapse[k1][k0])
			{
				Collapse c = {i0, i1, {/* bidi= */ 1}};
				collapses[collapse_count++] = c;
			}
			else
			{
				// edge can only be collapsed in one direction
				unsigned int e0 = kCanCollapse[k0][k1] ? i0 : i1;
				unsigned int e1 = kCanCollapse[k0][k1] ? i1 : i0;

				Collapse c = {e0, e1, {/* bidi= */ 0}};
				collapses[collapse_count++] = c;
			}
		}
	}

	return collapse_count;
}

static void rankEdgeCollapses(Collapse* collapses, size_t collapse_count, const Vector3* vertex_positions, const Quadric* vertex_quadrics, const unsigned int* remap)
{
	for (size_t i = 0; i < collapse_count; ++i)
	{
		Collapse& c = collapses[i];

		unsigned int i0 = c.v0;
		unsigned int i1 = c.v1;

		// most edges are bidirectional which means we need to evaluate errors for two collapses
		// to keep this code branchless we just use the same edge for unidirectional edges
		unsigned int j0 = c.bidi ? i1 : i0;
		unsigned int j1 = c.bidi ? i0 : i1;

		float ei = quadricError(vertex_quadrics[remap[i0]], vertex_positions[i1]);
		float ej = quadricError(vertex_quadrics[remap[j0]], vertex_positions[j1]);

		// pick edge direction with minimal error
		c.v0 = ei <= ej ? i0 : j0;
		c.v1 = ei <= ej ? i1 : j1;
		c.error = ei <= ej ? ei : ej;
	}
}

#if TRACE > 1
static void dumpEdgeCollapses(const Collapse* collapses, size_t collapse_count, const unsigned char* vertex_kind)
{
	size_t ckinds[Kind_Count][Kind_Count] = {};
	float cerrors[Kind_Count][Kind_Count] = {};

	for (int k0 = 0; k0 < Kind_Count; ++k0)
		for (int k1 = 0; k1 < Kind_Count; ++k1)
			cerrors[k0][k1] = FLT_MAX;

	for (size_t i = 0; i < collapse_count; ++i)
	{
		unsigned int i0 = collapses[i].v0;
		unsigned int i1 = collapses[i].v1;

		unsigned char k0 = vertex_kind[i0];
		unsigned char k1 = vertex_kind[i1];

		ckinds[k0][k1]++;
		cerrors[k0][k1] = (collapses[i].error < cerrors[k0][k1]) ? collapses[i].error : cerrors[k0][k1];
	}

	for (int k0 = 0; k0 < Kind_Count; ++k0)
		for (int k1 = 0; k1 < Kind_Count; ++k1)
			if (ckinds[k0][k1])
				printf("collapses %d -> %d: %d, min error %e\n", k0, k1, int(ckinds[k0][k1]), cerrors[k0][k1]);
}

static void dumpLockedCollapses(const unsigned int* indices, size_t index_count, const unsigned char* vertex_kind)
{
	size_t locked_collapses[Kind_Count][Kind_Count] = {};

	for (size_t i = 0; i < index_count; i += 3)
	{
		static const int next[3] = {1, 2, 0};

		for (int e = 0; e < 3; ++e)
		{
			unsigned int i0 = indices[i + e];
			unsigned int i1 = indices[i + next[e]];

			unsigned char k0 = vertex_kind[i0];
			unsigned char k1 = vertex_kind[i1];

			locked_collapses[k0][k1] += !kCanCollapse[k0][k1] && !kCanCollapse[k1][k0];
		}
	}

	for (int k0 = 0; k0 < Kind_Count; ++k0)
		for (int k1 = 0; k1 < Kind_Count; ++k1)
			if (locked_collapses[k0][k1])
				printf("locked collapses %d -> %d: %d\n", k0, k1, int(locked_collapses[k0][k1]));
}
#endif

static void sortEdgeCollapses(unsigned int* sort_order, const Collapse* collapses, size_t collapse_count)
{
	const int sort_bits = 11;

	// fill histogram for counting sort
	unsigned int histogram[1 << sort_bits];
	memset(histogram, 0, sizeof(histogram));

	for (size_t i = 0; i < collapse_count; ++i)
	{
		// skip sign bit since error is non-negative
		unsigned int key = (collapses[i].errorui << 1) >> (32 - sort_bits);

		histogram[key]++;
	}

	// compute offsets based on histogram data
	size_t histogram_sum = 0;

	for (size_t i = 0; i < 1 << sort_bits; ++i)
	{
		size_t count = histogram[i];
		histogram[i] = unsigned(histogram_sum);
		histogram_sum += count;
	}

	assert(histogram_sum == collapse_count);

	// compute sort order based on offsets
	for (size_t i = 0; i < collapse_count; ++i)
	{
		// skip sign bit since error is non-negative
		unsigned int key = (collapses[i].errorui << 1) >> (32 - sort_bits);

		sort_order[histogram[key]++] = unsigned(i);
	}
}

static size_t performEdgeCollapses(unsigned int* collapse_remap, unsigned char* collapse_locked, Quadric* vertex_quadrics, const Collapse* collapses, size_t collapse_count, const unsigned int* collapse_order, const unsigned int* remap, const unsigned int* wedge, const unsigned char* vertex_kind, size_t triangle_collapse_goal, float error_limit)
{
	size_t edge_collapses = 0;
	size_t triangle_collapses = 0;

	for (size_t i = 0; i < collapse_count; ++i)
	{
		const Collapse& c = collapses[collapse_order[i]];

		if (c.error > error_limit)
			break;

		if (triangle_collapses >= triangle_collapse_goal)
			break;

		unsigned int r0 = remap[c.v0];
		unsigned int r1 = remap[c.v1];

		// we don't collapse vertices that had source or target vertex involved in a collapse
		// it's important to not move the vertices twice since it complicates the tracking/remapping logic
		// it's important to not move other vertices towards a moved vertex to preserve error since we don't re-rank collapses mid-pass
		if (collapse_locked[r0] | collapse_locked[r1])
			continue;

		assert(collapse_remap[r0] == r0);
		assert(collapse_remap[r1] == r1);

		quadricAdd(vertex_quadrics[r1], vertex_quadrics[r0]);

		if (vertex_kind[c.v0] == Kind_Seam)
		{
			// remap v0 to v1 and seam pair of v0 to seam pair of v1
			unsigned int s0 = wedge[c.v0];
			unsigned int s1 = wedge[c.v1];

			assert(s0 != c.v0 && s1 != c.v1);
			assert(wedge[s0] == c.v0 && wedge[s1] == c.v1);

			collapse_remap[c.v0] = c.v1;
			collapse_remap[s0] = s1;
		}
		else
		{
			assert(wedge[c.v0] == c.v0);

			collapse_remap[c.v0] = c.v1;
		}

		collapse_locked[r0] = 1;
		collapse_locked[r1] = 1;

		// border edges collapse 1 triangle, other edges collapse 2 or more
		triangle_collapses += (vertex_kind[c.v0] == Kind_Border) ? 1 : 2;
		edge_collapses++;
	}

	return edge_collapses;
}

static size_t remapIndexBuffer(unsigned int* indices, size_t index_count, const unsigned int* collapse_remap)
{
	size_t write = 0;

	for (size_t i = 0; i < index_count; i += 3)
	{
		unsigned int v0 = collapse_remap[indices[i + 0]];
		unsigned int v1 = collapse_remap[indices[i + 1]];
		unsigned int v2 = collapse_remap[indices[i + 2]];

		// we never move the vertex twice during a single pass
		assert(collapse_remap[v0] == v0);
		assert(collapse_remap[v1] == v1);
		assert(collapse_remap[v2] == v2);

		if (v0 != v1 && v0 != v2 && v1 != v2)
		{
			indices[write + 0] = v0;
			indices[write + 1] = v1;
			indices[write + 2] = v2;
			write += 3;
		}
	}

	return write;
}

static void remapEdgeLoops(unsigned int* loop, size_t vertex_count, const unsigned int* collapse_remap)
{
	for (size_t i = 0; i < vertex_count; ++i)
	{
		if (loop[i] != ~0u)
		{
			unsigned int l = loop[i];
			unsigned int r = collapse_remap[l];

			// i == r is a special case when the seam edge is collapsed in a direction opposite to where loop goes
			loop[i] = (i == r) ? loop[l] : r;
		}
	}
}

} // namespace meshopt

#if TRACE
unsigned char* meshopt_simplifyDebugKind = 0;
unsigned int* meshopt_simplifyDebugLoop = 0;
#endif

size_t meshopt_simplify(unsigned int* destination, const unsigned int* indices, size_t index_count, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride, size_t target_index_count, float target_error)
{
	using namespace meshopt;

	assert(index_count % 3 == 0);
	assert(vertex_positions_stride > 0 && vertex_positions_stride <= 256);
	assert(vertex_positions_stride % sizeof(float) == 0);
	assert(target_index_count <= index_count);

	meshopt_Allocator allocator;

	unsigned int* result = destination;

	// build adjacency information
	EdgeAdjacency adjacency = {};
	buildEdgeAdjacency(adjacency, indices, index_count, vertex_count, allocator);

	// build position remap that maps each vertex to the one with identical position
	unsigned int* remap = allocator.allocate<unsigned int>(vertex_count);
	unsigned int* wedge = allocator.allocate<unsigned int>(vertex_count);
	buildPositionRemap(remap, wedge, vertex_positions_data, vertex_count, vertex_positions_stride, allocator);

	// classify vertices; vertex kind determines collapse rules, see kCanCollapse
	unsigned char* vertex_kind = allocator.allocate<unsigned char>(vertex_count);
	unsigned int* loop = allocator.allocate<unsigned int>(vertex_count);
	classifyVertices(vertex_kind, loop, vertex_count, adjacency, remap, wedge);

#if TRACE
	size_t unique_positions = 0;
	for (size_t i = 0; i < vertex_count; ++i)
		unique_positions += remap[i] == i;

	printf("position remap: %d vertices => %d positions\n", int(vertex_count), int(unique_positions));

	size_t kinds[Kind_Count] = {};
	for (size_t i = 0; i < vertex_count; ++i)
		kinds[vertex_kind[i]] += remap[i] == i;

	printf("kinds: manifold %d, border %d, seam %d, locked %d\n",
	       int(kinds[Kind_Manifold]), int(kinds[Kind_Border]), int(kinds[Kind_Seam]), int(kinds[Kind_Locked]));
#endif

	Vector3* vertex_positions = allocator.allocate<Vector3>(vertex_count);
	rescalePositions(vertex_positions, vertex_positions_data, vertex_count, vertex_positions_stride);

	Quadric* vertex_quadrics = allocator.allocate<Quadric>(vertex_count);
	memset(vertex_quadrics, 0, vertex_count * sizeof(Quadric));

	fillFaceQuadrics(vertex_quadrics, indices, index_count, vertex_positions, remap);
	fillEdgeQuadrics(vertex_quadrics, indices, index_count, vertex_positions, remap, vertex_kind, loop);

	if (result != indices)
		memcpy(result, indices, index_count * sizeof(unsigned int));

#if TRACE
	size_t pass_count = 0;
	float worst_error = 0;
#endif

	Collapse* edge_collapses = allocator.allocate<Collapse>(index_count);
	unsigned int* collapse_order = allocator.allocate<unsigned int>(index_count);
	unsigned int* collapse_remap = allocator.allocate<unsigned int>(vertex_count);
	unsigned char* collapse_locked = allocator.allocate<unsigned char>(vertex_count);

	size_t result_count = index_count;

	while (result_count > target_index_count)
	{
		size_t edge_collapse_count = pickEdgeCollapses(edge_collapses, result, result_count, remap, vertex_kind, loop);

		// no edges can be collapsed any more due to topology restrictions
		if (edge_collapse_count == 0)
			break;

		rankEdgeCollapses(edge_collapses, edge_collapse_count, vertex_positions, vertex_quadrics, remap);

#if TRACE > 1
		dumpEdgeCollapses(edge_collapses, edge_collapse_count, vertex_kind);
#endif

		sortEdgeCollapses(collapse_order, edge_collapses, edge_collapse_count);

		// most collapses remove 2 triangles; use this to establish a bound on the pass in terms of error limit
		// note that edge_collapse_goal is an estimate; triangle_collapse_goal will be used to actually limit collapses
		size_t triangle_collapse_goal = (result_count - target_index_count) / 3;
		size_t edge_collapse_goal = triangle_collapse_goal / 2;

		// we limit the error in each pass based on the error of optimal last collapse; since many collapses will be locked
		// as they will share vertices with other successfull collapses, we need to increase the acceptable error by this factor
		const float kPassErrorBound = 1.5f;

		float error_goal = edge_collapse_goal < edge_collapse_count ? edge_collapses[collapse_order[edge_collapse_goal]].error * kPassErrorBound : FLT_MAX;
		float error_limit = error_goal > target_error ? target_error : error_goal;

		for (size_t i = 0; i < vertex_count; ++i)
			collapse_remap[i] = unsigned(i);

		memset(collapse_locked, 0, vertex_count);

		size_t collapses = performEdgeCollapses(collapse_remap, collapse_locked, vertex_quadrics, edge_collapses, edge_collapse_count, collapse_order, remap, wedge, vertex_kind, triangle_collapse_goal, error_limit);

		// no edges can be collapsed any more due to hitting the error limit or triangle collapse limit
		if (collapses == 0)
			break;

		remapEdgeLoops(loop, vertex_count, collapse_remap);

		size_t new_count = remapIndexBuffer(result, result_count, collapse_remap);
		assert(new_count < result_count);

#if TRACE
		float pass_error = 0.f;
		for (size_t i = 0; i < edge_collapse_count; ++i)
		{
			Collapse& c = edge_collapses[collapse_order[i]];

			if (collapse_remap[c.v0] == c.v1)
				pass_error = c.error;
		}

		pass_count++;
		worst_error = (worst_error < pass_error) ? pass_error : worst_error;

		printf("pass %d: triangles: %d -> %d, collapses: %d/%d (goal: %d), error: %e (limit %e goal %e)\n", int(pass_count), int(result_count / 3), int(new_count / 3), int(collapses), int(edge_collapse_count), int(edge_collapse_goal), pass_error, error_limit, error_goal);
#endif

		result_count = new_count;
	}

#if TRACE
	printf("passes: %d, worst error: %e\n", int(pass_count), worst_error);
#endif

#if TRACE > 1
	dumpLockedCollapses(result, result_count, vertex_kind);
#endif

#if TRACE
	if (meshopt_simplifyDebugKind)
		memcpy(meshopt_simplifyDebugKind, vertex_kind, vertex_count);

	if (meshopt_simplifyDebugLoop)
		memcpy(meshopt_simplifyDebugLoop, loop, vertex_count * sizeof(unsigned int));
#endif

	return result_count;
}

simplifierb.cpp

// This file is part of meshoptimizer library; see meshoptimizer.h for version/license details
#include "meshoptimizer.h"

#include <cassert>
#include <cfloat>
#include <cmath>
#include <cstring>

#ifndef TRACE
#define TRACE 0
#endif

#if TRACE
#include <stdio.h>
#endif

// This work is based on:
// Michael Garland and Paul S. Heckbert. Surface simplification using quadric error metrics. 1997
// Michael Garland. Quadric-based polygonal surface simplification. 1999
namespace meshopt
{

template <typename T> struct meshopt_Buffer
{
	T* data;
	size_t size;

	meshopt_Buffer()
		: data(0)
		, size(0)
	{
	}

	meshopt_Buffer(size_t size)
		: data(new T[size])
		, size(size)
	{
	}

	~meshopt_Buffer()
	{
		delete[] data;
	}

	void resize(size_t new_size)
	{
		assert(!data);
		data = new T[new_size];
		size = new_size;
	}

	const T& operator[](size_t index) const
	{
		assert(index < size);
		return data[index];
	}

	T& operator[](size_t index)
	{
		assert(index < size);
		return data[index];
	}
};

struct EdgeAdjacency
{
	meshopt_Buffer<unsigned int> counts;
	meshopt_Buffer<unsigned int> offsets;
	meshopt_Buffer<unsigned int> data;
};

static void buildEdgeAdjacency(EdgeAdjacency& adjacency, const unsigned int* indices, size_t index_count, size_t vertex_count)
{
	size_t face_count = index_count / 3;

	// allocate arrays
	adjacency.counts.resize(vertex_count);
	adjacency.offsets.resize(vertex_count);
	adjacency.data.resize(index_count);

	// fill edge counts
	memset(&adjacency.counts[0], 0, vertex_count * sizeof(unsigned int));

	for (size_t i = 0; i < index_count; ++i)
	{
		assert(indices[i] < vertex_count);

		adjacency.counts[indices[i]]++;
	}

	// fill offset table
	unsigned int offset = 0;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		adjacency.offsets[i] = offset;
		offset += adjacency.counts[i];
	}

	assert(offset == index_count);

	// fill edge data
	for (size_t i = 0; i < face_count; ++i)
	{
		unsigned int a = indices[i * 3 + 0], b = indices[i * 3 + 1], c = indices[i * 3 + 2];

		adjacency.data[adjacency.offsets[a]++] = b;
		adjacency.data[adjacency.offsets[b]++] = c;
		adjacency.data[adjacency.offsets[c]++] = a;
	}

	// fix offsets that have been disturbed by the previous pass
	for (size_t i = 0; i < vertex_count; ++i)
	{
		assert(adjacency.offsets[i] >= adjacency.counts[i]);

		adjacency.offsets[i] -= adjacency.counts[i];
	}
}

struct PositionHasher
{
	const float* vertex_positions;
	size_t vertex_stride_float;

	size_t hash(unsigned int index) const
	{
		// MurmurHash2
		const unsigned int m = 0x5bd1e995;
		const int r = 24;

		unsigned int h = 0;
		const unsigned int* key = reinterpret_cast<const unsigned int*>(vertex_positions + index * vertex_stride_float);

		for (size_t i = 0; i < 3; ++i)
		{
			unsigned int k = key[i];

			k *= m;
			k ^= k >> r;
			k *= m;

			h *= m;
			h ^= k;
		}

		return h;
	}

	bool equal(unsigned int lhs, unsigned int rhs) const
	{
		return memcmp(vertex_positions + lhs * vertex_stride_float, vertex_positions + rhs * vertex_stride_float, sizeof(float) * 3) == 0;
	}
};

static size_t hashBuckets2(size_t count)
{
	size_t buckets = 1;
	while (buckets < count)
		buckets *= 2;

	return buckets;
}

template <typename T, typename Hash>
static T* hashLookup2(meshopt_Buffer<T>& table, size_t buckets, const Hash& hash, const T& key, const T& empty)
{
	assert(buckets > 0);
	assert((buckets & (buckets - 1)) == 0);

	size_t hashmod = buckets - 1;
	size_t bucket = hash.hash(key) & hashmod;

	for (size_t probe = 0; probe <= hashmod; ++probe)
	{
		T& item = table[bucket];

		if (item == empty)
			return &item;

		if (hash.equal(item, key))
			return &item;

		// hash collision, quadratic probing
		bucket = (bucket + probe + 1) & hashmod;
	}

	assert(false && "Hash table is full");
	return 0;
}

static void buildPositionRemap(meshopt_Buffer<unsigned int>& remap, meshopt_Buffer<unsigned int>& wedge, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride)
{
	PositionHasher hasher = {vertex_positions_data, vertex_positions_stride / sizeof(float)};

	size_t table_size = hashBuckets2(vertex_count);
	meshopt_Buffer<unsigned int> table(table_size);
	memset(&table[0], -1, table_size * sizeof(unsigned int));

	// build forward remap: for each vertex, which other (canonical) vertex does it map to?
	// we use position equivalence for this, and remap vertices to other existing vertices
	for (size_t i = 0; i < vertex_count; ++i)
	{
		unsigned int index = unsigned(i);
		unsigned int* entry = hashLookup2(table, table_size, hasher, index, ~0u);

		if (*entry == ~0u)
			*entry = index;

		remap[index] = *entry;
	}

	// build wedge table: for each vertex, which other vertex is the next wedge that also maps to the same vertex?
	// entries in table form a (cyclic) wedge loop per vertex; for manifold vertices, wedge[i] == remap[i] == i
	for (size_t i = 0; i < vertex_count; ++i)
		wedge[i] = unsigned(i);

	for (size_t i = 0; i < vertex_count; ++i)
		if (remap[i] != i)
		{
			unsigned int r = remap[i];

			wedge[i] = wedge[r];
			wedge[r] = unsigned(i);
		}
}

enum VertexKind
{
	Kind_Manifold, // not on an attribute seam, not on any boundary
	Kind_Border,   // not on an attribute seam, has exactly two open edges
	Kind_Seam,     // on an attribute seam with exactly two attribute seam edges
	Kind_Locked,   // none of the above; these vertices can't move

	Kind_Count
};

// manifold vertices can collapse on anything except locked
// border/seam vertices can only be collapsed onto border/seam respectively
const unsigned char kCanCollapse[Kind_Count][Kind_Count] = {
    {1, 1, 1, 1},
    {0, 1, 0, 0},
    {0, 0, 1, 0},
    {0, 0, 0, 0},
};

// if a vertex is manifold or seam, adjoining edges are guaranteed to have an opposite edge
// note that for seam edges, the opposite edge isn't present in the attribute-based topology
// but is present if you consider a position-only mesh variant
const unsigned char kHasOpposite[Kind_Count][Kind_Count] = {
    {1, 1, 1, 1},
    {1, 0, 1, 0},
    {1, 1, 1, 1},
    {1, 0, 1, 0},
};

static bool hasEdge(const EdgeAdjacency& adjacency, unsigned int a, unsigned int b)
{
	unsigned int count = adjacency.counts[a];
	const unsigned int* data = &adjacency.data[adjacency.offsets[a]];

	for (size_t i = 0; i < count; ++i)
		if (data[i] == b)
			return true;

	return false;
}

static unsigned int findWedgeEdge(const EdgeAdjacency& adjacency, const meshopt_Buffer<unsigned int>& wedge, unsigned int a, unsigned int b)
{
	unsigned int v = a;

	do
	{
		if (hasEdge(adjacency, v, b))
			return v;

		v = wedge[v];
	} while (v != a);

	return ~0u;
}

static size_t countOpenEdges(const EdgeAdjacency& adjacency, unsigned int vertex, unsigned int* last = 0)
{
	size_t result = 0;

	unsigned int count = adjacency.counts[vertex];
	const unsigned int* data = &adjacency.data[adjacency.offsets[vertex]];

	for (size_t i = 0; i < count; ++i)
		if (!hasEdge(adjacency, data[i], vertex))
		{
			result++;

			if (last)
				*last = data[i];
		}

	return result;
}

static void classifyVertices(meshopt_Buffer<unsigned char>& result, meshopt_Buffer<unsigned int>& loop, size_t vertex_count, const EdgeAdjacency& adjacency, const meshopt_Buffer<unsigned int>& remap, const meshopt_Buffer<unsigned int>& wedge)
{
	for (size_t i = 0; i < vertex_count; ++i)
		loop[i] = ~0u;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		if (remap[i] == i)
		{
			if (wedge[i] == i)
			{
				// no attribute seam, need to check if it's manifold
				unsigned int v = 0;
				size_t edges = countOpenEdges(adjacency, unsigned(i), &v);

				// note: we classify any vertices with no open edges as manifold
				// this is technically incorrect - if 4 triangles share an edge, we'll classify vertices as manifold
				// it's unclear if this is a problem in practice
				// also note that we classify vertices as border if they have *one* open edge, not two
				// this is because we only have half-edges - so a border vertex would have one incoming and one outgoing edge
				if (edges == 0)
				{
					result[i] = Kind_Manifold;
				}
				else if (edges == 1)
				{
					result[i] = Kind_Border;
					loop[i] = v;
				}
				else
				{
					result[i] = Kind_Locked;
				}
			}
			else if (wedge[wedge[i]] == i)
			{
				// attribute seam; need to distinguish between Seam and Locked
				unsigned int a = 0;
				size_t a_count = countOpenEdges(adjacency, unsigned(i), &a);
				unsigned int b = 0;
				size_t b_count = countOpenEdges(adjacency, wedge[i], &b);

				// seam should have one open half-edge for each vertex, and the edges need to "connect" - point to the same vertex post-remap
				if (a_count == 1 && b_count == 1)
				{
					unsigned int ao = findWedgeEdge(adjacency, wedge, a, wedge[i]);
					unsigned int bo = findWedgeEdge(adjacency, wedge, b, unsigned(i));

					if (ao != ~0u && bo != ~0u)
					{
						result[i] = Kind_Seam;

						loop[i] = a;
						loop[wedge[i]] = b;
					}
					else
					{
						result[i] = Kind_Locked;
					}
				}
				else
				{
					result[i] = Kind_Locked;
				}
			}
			else
			{
				// more than one vertex maps to this one; we don't have classification available
				result[i] = Kind_Locked;
			}
		}
		else
		{
			assert(remap[i] < i);

			result[i] = result[remap[i]];
		}
	}
}

struct Vector3
{
	float x, y, z;
};

static void rescalePositions(meshopt_Buffer<Vector3>& result, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride)
{
	size_t vertex_stride_float = vertex_positions_stride / sizeof(float);

	float minv[3] = {FLT_MAX, FLT_MAX, FLT_MAX};
	float maxv[3] = {-FLT_MAX, -FLT_MAX, -FLT_MAX};

	for (size_t i = 0; i < vertex_count; ++i)
	{
		const float* v = vertex_positions_data + i * vertex_stride_float;

		result[i].x = v[0];
		result[i].y = v[1];
		result[i].z = v[2];

		for (int j = 0; j < 3; ++j)
		{
			float vj = v[j];

			minv[j] = minv[j] > vj ? vj : minv[j];
			maxv[j] = maxv[j] < vj ? vj : maxv[j];
		}
	}

	float extent = 0.f;

	extent = (maxv[0] - minv[0]) < extent ? extent : (maxv[0] - minv[0]);
	extent = (maxv[1] - minv[1]) < extent ? extent : (maxv[1] - minv[1]);
	extent = (maxv[2] - minv[2]) < extent ? extent : (maxv[2] - minv[2]);

	float scale = extent == 0 ? 0.f : 1.f / extent;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		result[i].x = (result[i].x - minv[0]) * scale;
		result[i].y = (result[i].y - minv[1]) * scale;
		result[i].z = (result[i].z - minv[2]) * scale;
	}
}

struct Quadric
{
	float a00;
	float a10, a11;
	float a20, a21, a22;
	float b0, b1, b2, c;
};

struct Collapse
{
	unsigned int v0;
	unsigned int v1;

	union {
		unsigned int bidi;
		float error;
		unsigned int errorui;
	};
};

static float normalize(Vector3& v)
{
	float length = sqrtf(v.x * v.x + v.y * v.y + v.z * v.z);

	if (length > 0)
	{
		v.x /= length;
		v.y /= length;
		v.z /= length;
	}

	return length;
}

static void quadricAdd(Quadric& Q, const Quadric& R)
{
	Q.a00 += R.a00;
	Q.a10 += R.a10;
	Q.a11 += R.a11;
	Q.a20 += R.a20;
	Q.a21 += R.a21;
	Q.a22 += R.a22;
	Q.b0 += R.b0;
	Q.b1 += R.b1;
	Q.b2 += R.b2;
	Q.c += R.c;
}

static void quadricMul(Quadric& Q, float s)
{
	Q.a00 *= s;
	Q.a10 *= s;
	Q.a11 *= s;
	Q.a20 *= s;
	Q.a21 *= s;
	Q.a22 *= s;
	Q.b0 *= s;
	Q.b1 *= s;
	Q.b2 *= s;
	Q.c *= s;
}

static float quadricError(const Quadric& Q, const Vector3& v)
{
	float rx = Q.b0;
	float ry = Q.b1;
	float rz = Q.b2;

	rx += Q.a10 * v.y;
	ry += Q.a21 * v.z;
	rz += Q.a20 * v.x;

	rx *= 2;
	ry *= 2;
	rz *= 2;

	rx += Q.a00 * v.x;
	ry += Q.a11 * v.y;
	rz += Q.a22 * v.z;

	float r = Q.c;
	r += rx * v.x;
	r += ry * v.y;
	r += rz * v.z;

	return fabsf(r);
}

static void quadricFromPlane(Quadric& Q, float a, float b, float c, float d)
{
	Q.a00 = a * a;
	Q.a10 = b * a;
	Q.a11 = b * b;
	Q.a20 = c * a;
	Q.a21 = c * b;
	Q.a22 = c * c;
	Q.b0 = d * a;
	Q.b1 = d * b;
	Q.b2 = d * c;
	Q.c = d * d;
}

static void quadricFromTriangle(Quadric& Q, const Vector3& p0, const Vector3& p1, const Vector3& p2)
{
	Vector3 p10 = {p1.x - p0.x, p1.y - p0.y, p1.z - p0.z};
	Vector3 p20 = {p2.x - p0.x, p2.y - p0.y, p2.z - p0.z};

	Vector3 normal = {p10.y * p20.z - p10.z * p20.y, p10.z * p20.x - p10.x * p20.z, p10.x * p20.y - p10.y * p20.x};
	float area = normalize(normal);

	float distance = normal.x * p0.x + normal.y * p0.y + normal.z * p0.z;

	quadricFromPlane(Q, normal.x, normal.y, normal.z, -distance);

	quadricMul(Q, area);
}

static void quadricFromTriangleEdge(Quadric& Q, const Vector3& p0, const Vector3& p1, const Vector3& p2, float weight)
{
	Vector3 p10 = {p1.x - p0.x, p1.y - p0.y, p1.z - p0.z};
	float length = normalize(p10);

	Vector3 p20 = {p2.x - p0.x, p2.y - p0.y, p2.z - p0.z};
	float p20p = p20.x * p10.x + p20.y * p10.y + p20.z * p10.z;

	Vector3 normal = {p20.x - p10.x * p20p, p20.y - p10.y * p20p, p20.z - p10.z * p20p};
	normalize(normal);

	float distance = normal.x * p0.x + normal.y * p0.y + normal.z * p0.z;

	quadricFromPlane(Q, normal.x, normal.y, normal.z, -distance);

	quadricMul(Q, length * length * weight);
}

static void fillFaceQuadrics(meshopt_Buffer<Quadric>& vertex_quadrics, const unsigned int* indices, size_t index_count, const meshopt_Buffer<Vector3>& vertex_positions, const meshopt_Buffer<unsigned int>& remap)
{
	for (size_t i = 0; i < index_count; i += 3)
	{
		unsigned int i0 = indices[i + 0];
		unsigned int i1 = indices[i + 1];
		unsigned int i2 = indices[i + 2];

		Quadric Q;
		quadricFromTriangle(Q, vertex_positions[i0], vertex_positions[i1], vertex_positions[i2]);

		quadricAdd(vertex_quadrics[remap[i0]], Q);
		quadricAdd(vertex_quadrics[remap[i1]], Q);
		quadricAdd(vertex_quadrics[remap[i2]], Q);
	}
}

static void fillEdgeQuadrics(meshopt_Buffer<Quadric>& vertex_quadrics, const unsigned int* indices, size_t index_count, const meshopt_Buffer<Vector3>& vertex_positions, const meshopt_Buffer<unsigned int>& remap, const meshopt_Buffer<unsigned char>& vertex_kind, const meshopt_Buffer<unsigned int>& loop)
{
	for (size_t i = 0; i < index_count; i += 3)
	{
		static const int next[3] = {1, 2, 0};

		for (int e = 0; e < 3; ++e)
		{
			unsigned int i0 = indices[i + e];
			unsigned int i1 = indices[i + next[e]];

			unsigned char k0 = vertex_kind[i0];
			unsigned char k1 = vertex_kind[i1];

			// check that i0 and i1 are border/seam and are on the same edge loop
			// loop[] tracks half edges so we only need to check i0->i1
			if (k0 != k1 || (k0 != Kind_Border && k0 != Kind_Seam) || loop[i0] != i1)
				continue;

			unsigned int i2 = indices[i + next[next[e]]];

			// we try hard to maintain border edge geometry; seam edges can move more freely
			// due to topological restrictions on collapses, seam quadrics slightly improves collapse structure but aren't critical
			const float kEdgeWeightSeam = 1.f;
			const float kEdgeWeightBorder = 10.f;

			float edgeWeight = (k0 == Kind_Seam) ? kEdgeWeightSeam : kEdgeWeightBorder;

			Quadric Q;
			quadricFromTriangleEdge(Q, vertex_positions[i0], vertex_positions[i1], vertex_positions[i2], edgeWeight);

			quadricAdd(vertex_quadrics[remap[i0]], Q);
			quadricAdd(vertex_quadrics[remap[i1]], Q);
		}
	}
}

static size_t pickEdgeCollapses(meshopt_Buffer<Collapse>& collapses, const unsigned int* indices, size_t index_count, const meshopt_Buffer<unsigned int>& remap, const meshopt_Buffer<unsigned char>& vertex_kind, const meshopt_Buffer<unsigned int>& loop)
{
	size_t collapse_count = 0;

	for (size_t i = 0; i < index_count; i += 3)
	{
		static const int next[3] = {1, 2, 0};

		for (int e = 0; e < 3; ++e)
		{
			unsigned int i0 = indices[i + e];
			unsigned int i1 = indices[i + next[e]];

			// this can happen either when input has a zero-length edge, or when we perform collapses for complex
			// topology w/seams and collapse a manifold vertex that connects to both wedges onto one of them
			// we leave edges like this alone since they may be important for preserving mesh integrity
			if (remap[i0] == remap[i1])
				continue;

			unsigned char k0 = vertex_kind[i0];
			unsigned char k1 = vertex_kind[i1];

			// the edge has to be collapsible in at least one direction
			if (!(kCanCollapse[k0][k1] | kCanCollapse[k1][k0]))
				continue;

			// manifold and seam edges should occur twice (i0->i1 and i1->i0) - skip redundant edges
			if (kHasOpposite[k0][k1] && remap[i1] > remap[i0])
				continue;

			// two vertices are on a border or a seam, but there's no direct edge between them
			// this indicates that they belong to two different edge loops and we should not collapse this edge
			// loop[] tracks half edges so we only need to check i0->i1
			if (k0 == k1 && (k0 == Kind_Border || k0 == Kind_Seam) && loop[i0] != i1)
				continue;

			// edge can be collapsed in either direction - we will pick the one with minimum error
			// note: we evaluate error later during collapse ranking, here we just tag the edge as bidirectional
			if (kCanCollapse[k0][k1] & kCanCollapse[k1][k0])
			{
				Collapse c = {i0, i1, {/* bidi= */ 1}};
				collapses[collapse_count++] = c;
			}
			else
			{
				// edge can only be collapsed in one direction
				unsigned int e0 = kCanCollapse[k0][k1] ? i0 : i1;
				unsigned int e1 = kCanCollapse[k0][k1] ? i1 : i0;

				Collapse c = {e0, e1, {/* bidi= */ 0}};
				collapses[collapse_count++] = c;
			}
		}
	}

	return collapse_count;
}

static void rankEdgeCollapses(meshopt_Buffer<Collapse>& collapses, size_t collapse_count, const meshopt_Buffer<Vector3>& vertex_positions, const meshopt_Buffer<Quadric>& vertex_quadrics, const meshopt_Buffer<unsigned int>& remap)
{
	for (size_t i = 0; i < collapse_count; ++i)
	{
		Collapse& c = collapses[i];

		unsigned int i0 = c.v0;
		unsigned int i1 = c.v1;

		// most edges are bidirectional which means we need to evaluate errors for two collapses
		// to keep this code branchless we just use the same edge for unidirectional edges
		unsigned int j0 = c.bidi ? i1 : i0;
		unsigned int j1 = c.bidi ? i0 : i1;

		float ei = quadricError(vertex_quadrics[remap[i0]], vertex_positions[i1]);
		float ej = quadricError(vertex_quadrics[remap[j0]], vertex_positions[j1]);

		// pick edge direction with minimal error
		c.v0 = ei <= ej ? i0 : j0;
		c.v1 = ei <= ej ? i1 : j1;
		c.error = ei <= ej ? ei : ej;
	}
}

#if TRACE > 1
static void dumpEdgeCollapses(const Collapse* collapses, size_t collapse_count, const unsigned char* vertex_kind)
{
	size_t ckinds[Kind_Count][Kind_Count] = {};
	float cerrors[Kind_Count][Kind_Count] = {};

	for (int k0 = 0; k0 < Kind_Count; ++k0)
		for (int k1 = 0; k1 < Kind_Count; ++k1)
			cerrors[k0][k1] = FLT_MAX;

	for (size_t i = 0; i < collapse_count; ++i)
	{
		unsigned int i0 = collapses[i].v0;
		unsigned int i1 = collapses[i].v1;

		unsigned char k0 = vertex_kind[i0];
		unsigned char k1 = vertex_kind[i1];

		ckinds[k0][k1]++;
		cerrors[k0][k1] = (collapses[i].error < cerrors[k0][k1]) ? collapses[i].error : cerrors[k0][k1];
	}

	for (int k0 = 0; k0 < Kind_Count; ++k0)
		for (int k1 = 0; k1 < Kind_Count; ++k1)
			if (ckinds[k0][k1])
				printf("collapses %d -> %d: %d, min error %e\n", k0, k1, int(ckinds[k0][k1]), cerrors[k0][k1]);
}

static void dumpLockedCollapses(const unsigned int* indices, size_t index_count, const unsigned char* vertex_kind)
{
	size_t locked_collapses[Kind_Count][Kind_Count] = {};

	for (size_t i = 0; i < index_count; i += 3)
	{
		static const int next[3] = {1, 2, 0};

		for (int e = 0; e < 3; ++e)
		{
			unsigned int i0 = indices[i + e];
			unsigned int i1 = indices[i + next[e]];

			unsigned char k0 = vertex_kind[i0];
			unsigned char k1 = vertex_kind[i1];

			locked_collapses[k0][k1] += !kCanCollapse[k0][k1] && !kCanCollapse[k1][k0];
		}
	}

	for (int k0 = 0; k0 < Kind_Count; ++k0)
		for (int k1 = 0; k1 < Kind_Count; ++k1)
			if (locked_collapses[k0][k1])
				printf("locked collapses %d -> %d: %d\n", k0, k1, int(locked_collapses[k0][k1]));
}
#endif

static void sortEdgeCollapses(meshopt_Buffer<unsigned int>& sort_order, const meshopt_Buffer<Collapse>& collapses, size_t collapse_count)
{
	const int sort_bits = 11;

	// fill histogram for counting sort
	unsigned int histogram[1 << sort_bits];
	memset(histogram, 0, sizeof(histogram));

	for (size_t i = 0; i < collapse_count; ++i)
	{
		// skip sign bit since error is non-negative
		unsigned int key = (collapses[i].errorui << 1) >> (32 - sort_bits);

		histogram[key]++;
	}

	// compute offsets based on histogram data
	size_t histogram_sum = 0;

	for (size_t i = 0; i < 1 << sort_bits; ++i)
	{
		size_t count = histogram[i];
		histogram[i] = unsigned(histogram_sum);
		histogram_sum += count;
	}

	assert(histogram_sum == collapse_count);

	// compute sort order based on offsets
	for (size_t i = 0; i < collapse_count; ++i)
	{
		// skip sign bit since error is non-negative
		unsigned int key = (collapses[i].errorui << 1) >> (32 - sort_bits);

		sort_order[histogram[key]++] = unsigned(i);
	}
}

static size_t performEdgeCollapses(meshopt_Buffer<unsigned int>& collapse_remap, meshopt_Buffer<unsigned char>& collapse_locked, meshopt_Buffer<Quadric>& vertex_quadrics, const meshopt_Buffer<Collapse>& collapses, size_t collapse_count, const meshopt_Buffer<unsigned int>& collapse_order, const meshopt_Buffer<unsigned int>& remap, const meshopt_Buffer<unsigned int>& wedge, const meshopt_Buffer<unsigned char>& vertex_kind, size_t triangle_collapse_goal, float error_limit)
{
	size_t edge_collapses = 0;
	size_t triangle_collapses = 0;

	for (size_t i = 0; i < collapse_count; ++i)
	{
		const Collapse& c = collapses[collapse_order[i]];

		if (c.error > error_limit)
			break;

		if (triangle_collapses >= triangle_collapse_goal)
			break;

		unsigned int r0 = remap[c.v0];
		unsigned int r1 = remap[c.v1];

		// we don't collapse vertices that had source or target vertex involved in a collapse
		// it's important to not move the vertices twice since it complicates the tracking/remapping logic
		// it's important to not move other vertices towards a moved vertex to preserve error since we don't re-rank collapses mid-pass
		if (collapse_locked[r0] | collapse_locked[r1])
			continue;

		assert(collapse_remap[r0] == r0);
		assert(collapse_remap[r1] == r1);

		quadricAdd(vertex_quadrics[r1], vertex_quadrics[r0]);

		if (vertex_kind[c.v0] == Kind_Seam)
		{
			// remap v0 to v1 and seam pair of v0 to seam pair of v1
			unsigned int s0 = wedge[c.v0];
			unsigned int s1 = wedge[c.v1];

			assert(s0 != c.v0 && s1 != c.v1);
			assert(wedge[s0] == c.v0 && wedge[s1] == c.v1);

			collapse_remap[c.v0] = c.v1;
			collapse_remap[s0] = s1;
		}
		else
		{
			assert(wedge[c.v0] == c.v0);

			collapse_remap[c.v0] = c.v1;
		}

		collapse_locked[r0] = 1;
		collapse_locked[r1] = 1;

		// border edges collapse 1 triangle, other edges collapse 2 or more
		triangle_collapses += (vertex_kind[c.v0] == Kind_Border) ? 1 : 2;
		edge_collapses++;
	}

	return edge_collapses;
}

static size_t remapIndexBuffer(unsigned int* indices, size_t index_count, const meshopt_Buffer<unsigned int>& collapse_remap)
{
	size_t write = 0;

	for (size_t i = 0; i < index_count; i += 3)
	{
		unsigned int v0 = collapse_remap[indices[i + 0]];
		unsigned int v1 = collapse_remap[indices[i + 1]];
		unsigned int v2 = collapse_remap[indices[i + 2]];

		// we never move the vertex twice during a single pass
		assert(collapse_remap[v0] == v0);
		assert(collapse_remap[v1] == v1);
		assert(collapse_remap[v2] == v2);

		if (v0 != v1 && v0 != v2 && v1 != v2)
		{
			indices[write + 0] = v0;
			indices[write + 1] = v1;
			indices[write + 2] = v2;
			write += 3;
		}
	}

	return write;
}

static void remapEdgeLoops(meshopt_Buffer<unsigned int>& loop, size_t vertex_count, const meshopt_Buffer<unsigned int>& collapse_remap)
{
	for (size_t i = 0; i < vertex_count; ++i)
	{
		if (loop[i] != ~0u)
		{
			unsigned int l = loop[i];
			unsigned int r = collapse_remap[l];

			// i == r is a special case when the seam edge is collapsed in a direction opposite to where loop goes
			loop[i] = (i == r) ? loop[l] : r;
		}
	}
}

} // namespace meshopt

#if TRACE
unsigned char* meshopt_simplifyDebugKind = 0;
unsigned int* meshopt_simplifyDebugLoop = 0;
#endif

size_t meshopt_simplify(unsigned int* destination, const unsigned int* indices, size_t index_count, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride, size_t target_index_count, float target_error)
{
	using namespace meshopt;

	assert(index_count % 3 == 0);
	assert(vertex_positions_stride > 0 && vertex_positions_stride <= 256);
	assert(vertex_positions_stride % sizeof(float) == 0);
	assert(target_index_count <= index_count);

	meshopt_Allocator allocator;

	unsigned int* result = destination;

	// build adjacency information
	EdgeAdjacency adjacency = {};
	buildEdgeAdjacency(adjacency, indices, index_count, vertex_count);

	// build position remap that maps each vertex to the one with identical position
	meshopt_Buffer<unsigned int> remap(vertex_count);
	meshopt_Buffer<unsigned int> wedge(vertex_count);
	buildPositionRemap(remap, wedge, vertex_positions_data, vertex_count, vertex_positions_stride);

	// classify vertices; vertex kind determines collapse rules, see kCanCollapse
	meshopt_Buffer<unsigned char> vertex_kind(vertex_count);
	meshopt_Buffer<unsigned int> loop(vertex_count);
	classifyVertices(vertex_kind, loop, vertex_count, adjacency, remap, wedge);

#if TRACE
	size_t unique_positions = 0;
	for (size_t i = 0; i < vertex_count; ++i)
		unique_positions += remap[i] == i;

	printf("position remap: %d vertices => %d positions\n", int(vertex_count), int(unique_positions));

	size_t kinds[Kind_Count] = {};
	for (size_t i = 0; i < vertex_count; ++i)
		kinds[vertex_kind[i]] += remap[i] == i;

	printf("kinds: manifold %d, border %d, seam %d, locked %d\n",
	       int(kinds[Kind_Manifold]), int(kinds[Kind_Border]), int(kinds[Kind_Seam]), int(kinds[Kind_Locked]));
#endif

	meshopt_Buffer<Vector3> vertex_positions(vertex_count);
	rescalePositions(vertex_positions, vertex_positions_data, vertex_count, vertex_positions_stride);

	meshopt_Buffer<Quadric> vertex_quadrics(vertex_count);
	memset(&vertex_quadrics[0], 0, vertex_count * sizeof(Quadric));

	fillFaceQuadrics(vertex_quadrics, indices, index_count, vertex_positions, remap);
	fillEdgeQuadrics(vertex_quadrics, indices, index_count, vertex_positions, remap, vertex_kind, loop);

	if (result != indices)
		memcpy(result, indices, index_count * sizeof(unsigned int));

#if TRACE
	size_t pass_count = 0;
	float worst_error = 0;
#endif

	meshopt_Buffer<Collapse> edge_collapses(index_count);
	meshopt_Buffer<unsigned int> collapse_order(index_count);
	meshopt_Buffer<unsigned int> collapse_remap(vertex_count);
	meshopt_Buffer<unsigned char> collapse_locked(vertex_count);

	size_t result_count = index_count;

	while (result_count > target_index_count)
	{
		size_t edge_collapse_count = pickEdgeCollapses(edge_collapses, result, result_count, remap, vertex_kind, loop);

		// no edges can be collapsed any more due to topology restrictions
		if (edge_collapse_count == 0)
			break;

		rankEdgeCollapses(edge_collapses, edge_collapse_count, vertex_positions, vertex_quadrics, remap);

#if TRACE > 1
		dumpEdgeCollapses(edge_collapses, edge_collapse_count, vertex_kind);
#endif

		sortEdgeCollapses(collapse_order, edge_collapses, edge_collapse_count);

		// most collapses remove 2 triangles; use this to establish a bound on the pass in terms of error limit
		// note that edge_collapse_goal is an estimate; triangle_collapse_goal will be used to actually limit collapses
		size_t triangle_collapse_goal = (result_count - target_index_count) / 3;
		size_t edge_collapse_goal = triangle_collapse_goal / 2;

		// we limit the error in each pass based on the error of optimal last collapse; since many collapses will be locked
		// as they will share vertices with other successfull collapses, we need to increase the acceptable error by this factor
		const float kPassErrorBound = 1.5f;

		float error_goal = edge_collapse_goal < edge_collapse_count ? edge_collapses[collapse_order[edge_collapse_goal]].error * kPassErrorBound : FLT_MAX;
		float error_limit = error_goal > target_error ? target_error : error_goal;

		for (size_t i = 0; i < vertex_count; ++i)
			collapse_remap[i] = unsigned(i);

		memset(&collapse_locked[0], 0, vertex_count);

		size_t collapses = performEdgeCollapses(collapse_remap, collapse_locked, vertex_quadrics, edge_collapses, edge_collapse_count, collapse_order, remap, wedge, vertex_kind, triangle_collapse_goal, error_limit);

		// no edges can be collapsed any more due to hitting the error limit or triangle collapse limit
		if (collapses == 0)
			break;

		remapEdgeLoops(loop, vertex_count, collapse_remap);

		size_t new_count = remapIndexBuffer(result, result_count, collapse_remap);
		assert(new_count < result_count);

#if TRACE
		float pass_error = 0.f;
		for (size_t i = 0; i < edge_collapse_count; ++i)
		{
			Collapse& c = edge_collapses[collapse_order[i]];

			if (collapse_remap[c.v0] == c.v1)
				pass_error = c.error;
		}

		pass_count++;
		worst_error = (worst_error < pass_error) ? pass_error : worst_error;

		printf("pass %d: triangles: %d -> %d, collapses: %d/%d (goal: %d), error: %e (limit %e goal %e)\n", int(pass_count), int(result_count / 3), int(new_count / 3), int(collapses), int(edge_collapse_count), int(edge_collapse_goal), pass_error, error_limit, error_goal);
#endif

		result_count = new_count;
	}

#if TRACE
	printf("passes: %d, worst error: %e\n", int(pass_count), worst_error);
#endif

#if TRACE > 1
	dumpLockedCollapses(result, result_count, vertex_kind);
#endif

#if TRACE
	if (meshopt_simplifyDebugKind)
		memcpy(meshopt_simplifyDebugKind, vertex_kind, vertex_count);

	if (meshopt_simplifyDebugLoop)
		memcpy(meshopt_simplifyDebugLoop, loop, vertex_count * sizeof(unsigned int));
#endif

	return result_count;
}

simplifierv.cpp

// This file is part of meshoptimizer library; see meshoptimizer.h for version/license details
#include "meshoptimizer.h"

#include <cassert>
#include <cfloat>
#include <cmath>
#include <cstring>

#include <vector>

#ifndef TRACE
#define TRACE 0
#endif

#if TRACE
#include <stdio.h>
#endif

// This work is based on:
// Michael Garland and Paul S. Heckbert. Surface simplification using quadric error metrics. 1997
// Michael Garland. Quadric-based polygonal surface simplification. 1999
namespace meshopt
{

struct EdgeAdjacency
{
	std::vector<unsigned int> counts;
	std::vector<unsigned int> offsets;
	std::vector<unsigned int> data;
};

static void buildEdgeAdjacency(EdgeAdjacency& adjacency, const unsigned int* indices, size_t index_count, size_t vertex_count)
{
	size_t face_count = index_count / 3;

	// allocate arrays
	adjacency.counts.resize(vertex_count);
	adjacency.offsets.resize(vertex_count);
	adjacency.data.resize(index_count);

	// fill edge counts
	for (size_t i = 0; i < index_count; ++i)
	{
		assert(indices[i] < vertex_count);

		adjacency.counts[indices[i]]++;
	}

	// fill offset table
	unsigned int offset = 0;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		adjacency.offsets[i] = offset;
		offset += adjacency.counts[i];
	}

	assert(offset == index_count);

	// fill edge data
	for (size_t i = 0; i < face_count; ++i)
	{
		unsigned int a = indices[i * 3 + 0], b = indices[i * 3 + 1], c = indices[i * 3 + 2];

		adjacency.data[adjacency.offsets[a]++] = b;
		adjacency.data[adjacency.offsets[b]++] = c;
		adjacency.data[adjacency.offsets[c]++] = a;
	}

	// fix offsets that have been disturbed by the previous pass
	for (size_t i = 0; i < vertex_count; ++i)
	{
		assert(adjacency.offsets[i] >= adjacency.counts[i]);

		adjacency.offsets[i] -= adjacency.counts[i];
	}
}

struct PositionHasher
{
	const float* vertex_positions;
	size_t vertex_stride_float;

	size_t hash(unsigned int index) const
	{
		// MurmurHash2
		const unsigned int m = 0x5bd1e995;
		const int r = 24;

		unsigned int h = 0;
		const unsigned int* key = reinterpret_cast<const unsigned int*>(vertex_positions + index * vertex_stride_float);

		for (size_t i = 0; i < 3; ++i)
		{
			unsigned int k = key[i];

			k *= m;
			k ^= k >> r;
			k *= m;

			h *= m;
			h ^= k;
		}

		return h;
	}

	bool equal(unsigned int lhs, unsigned int rhs) const
	{
		return memcmp(vertex_positions + lhs * vertex_stride_float, vertex_positions + rhs * vertex_stride_float, sizeof(float) * 3) == 0;
	}
};

static size_t hashBuckets2(size_t count)
{
	size_t buckets = 1;
	while (buckets < count)
		buckets *= 2;

	return buckets;
}

template <typename T, typename Hash>
static T* hashLookup2(std::vector<T>& table, size_t buckets, const Hash& hash, const T& key, const T& empty)
{
	assert(buckets > 0);
	assert((buckets & (buckets - 1)) == 0);

	size_t hashmod = buckets - 1;
	size_t bucket = hash.hash(key) & hashmod;

	for (size_t probe = 0; probe <= hashmod; ++probe)
	{
		T& item = table[bucket];

		if (item == empty)
			return &item;

		if (hash.equal(item, key))
			return &item;

		// hash collision, quadratic probing
		bucket = (bucket + probe + 1) & hashmod;
	}

	assert(false && "Hash table is full");
	return 0;
}

static void buildPositionRemap(std::vector<unsigned int>& remap, std::vector<unsigned int>& wedge, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride)
{
	PositionHasher hasher = {vertex_positions_data, vertex_positions_stride / sizeof(float)};

	size_t table_size = hashBuckets2(vertex_count);
	std::vector<unsigned int> table(table_size);
	memset(&table[0], -1, table_size * sizeof(unsigned int));

	// build forward remap: for each vertex, which other (canonical) vertex does it map to?
	// we use position equivalence for this, and remap vertices to other existing vertices
	for (size_t i = 0; i < vertex_count; ++i)
	{
		unsigned int index = unsigned(i);
		unsigned int* entry = hashLookup2(table, table_size, hasher, index, ~0u);

		if (*entry == ~0u)
			*entry = index;

		remap[index] = *entry;
	}

	// build wedge table: for each vertex, which other vertex is the next wedge that also maps to the same vertex?
	// entries in table form a (cyclic) wedge loop per vertex; for manifold vertices, wedge[i] == remap[i] == i
	for (size_t i = 0; i < vertex_count; ++i)
		wedge[i] = unsigned(i);

	for (size_t i = 0; i < vertex_count; ++i)
		if (remap[i] != i)
		{
			unsigned int r = remap[i];

			wedge[i] = wedge[r];
			wedge[r] = unsigned(i);
		}
}

enum VertexKind
{
	Kind_Manifold, // not on an attribute seam, not on any boundary
	Kind_Border,   // not on an attribute seam, has exactly two open edges
	Kind_Seam,     // on an attribute seam with exactly two attribute seam edges
	Kind_Locked,   // none of the above; these vertices can't move

	Kind_Count
};

// manifold vertices can collapse on anything except locked
// border/seam vertices can only be collapsed onto border/seam respectively
const unsigned char kCanCollapse[Kind_Count][Kind_Count] = {
    {1, 1, 1, 1},
    {0, 1, 0, 0},
    {0, 0, 1, 0},
    {0, 0, 0, 0},
};

// if a vertex is manifold or seam, adjoining edges are guaranteed to have an opposite edge
// note that for seam edges, the opposite edge isn't present in the attribute-based topology
// but is present if you consider a position-only mesh variant
const unsigned char kHasOpposite[Kind_Count][Kind_Count] = {
    {1, 1, 1, 1},
    {1, 0, 1, 0},
    {1, 1, 1, 1},
    {1, 0, 1, 0},
};

static bool hasEdge(const EdgeAdjacency& adjacency, unsigned int a, unsigned int b)
{
	unsigned int count = adjacency.counts[a];
	const unsigned int* data = &adjacency.data[adjacency.offsets[a]];

	for (size_t i = 0; i < count; ++i)
		if (data[i] == b)
			return true;

	return false;
}

static unsigned int findWedgeEdge(const EdgeAdjacency& adjacency, const std::vector<unsigned int>& wedge, unsigned int a, unsigned int b)
{
	unsigned int v = a;

	do
	{
		if (hasEdge(adjacency, v, b))
			return v;

		v = wedge[v];
	} while (v != a);

	return ~0u;
}

static size_t countOpenEdges(const EdgeAdjacency& adjacency, unsigned int vertex, unsigned int* last = 0)
{
	size_t result = 0;

	unsigned int count = adjacency.counts[vertex];
	const unsigned int* data = &adjacency.data[adjacency.offsets[vertex]];

	for (size_t i = 0; i < count; ++i)
		if (!hasEdge(adjacency, data[i], vertex))
		{
			result++;

			if (last)
				*last = data[i];
		}

	return result;
}

static void classifyVertices(std::vector<unsigned char>& result, std::vector<unsigned int>& loop, size_t vertex_count, const EdgeAdjacency& adjacency, const std::vector<unsigned int>& remap, const std::vector<unsigned int>& wedge)
{
	for (size_t i = 0; i < vertex_count; ++i)
		loop[i] = ~0u;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		if (remap[i] == i)
		{
			if (wedge[i] == i)
			{
				// no attribute seam, need to check if it's manifold
				unsigned int v = 0;
				size_t edges = countOpenEdges(adjacency, unsigned(i), &v);

				// note: we classify any vertices with no open edges as manifold
				// this is technically incorrect - if 4 triangles share an edge, we'll classify vertices as manifold
				// it's unclear if this is a problem in practice
				// also note that we classify vertices as border if they have *one* open edge, not two
				// this is because we only have half-edges - so a border vertex would have one incoming and one outgoing edge
				if (edges == 0)
				{
					result[i] = Kind_Manifold;
				}
				else if (edges == 1)
				{
					result[i] = Kind_Border;
					loop[i] = v;
				}
				else
				{
					result[i] = Kind_Locked;
				}
			}
			else if (wedge[wedge[i]] == i)
			{
				// attribute seam; need to distinguish between Seam and Locked
				unsigned int a = 0;
				size_t a_count = countOpenEdges(adjacency, unsigned(i), &a);
				unsigned int b = 0;
				size_t b_count = countOpenEdges(adjacency, wedge[i], &b);

				// seam should have one open half-edge for each vertex, and the edges need to "connect" - point to the same vertex post-remap
				if (a_count == 1 && b_count == 1)
				{
					unsigned int ao = findWedgeEdge(adjacency, wedge, a, wedge[i]);
					unsigned int bo = findWedgeEdge(adjacency, wedge, b, unsigned(i));

					if (ao != ~0u && bo != ~0u)
					{
						result[i] = Kind_Seam;

						loop[i] = a;
						loop[wedge[i]] = b;
					}
					else
					{
						result[i] = Kind_Locked;
					}
				}
				else
				{
					result[i] = Kind_Locked;
				}
			}
			else
			{
				// more than one vertex maps to this one; we don't have classification available
				result[i] = Kind_Locked;
			}
		}
		else
		{
			assert(remap[i] < i);

			result[i] = result[remap[i]];
		}
	}
}

struct Vector3
{
	float x, y, z;
};

static void rescalePositions(std::vector<Vector3>& result, const float* vertex_positions_data, size_t vertex_count, size_t vertex_positions_stride)
{
	size_t vertex_stride_float = vertex_positions_stride / sizeof(float);

	float minv[3] = {FLT_MAX, FLT_MAX, FLT_MAX};
	float maxv[3] = {-FLT_MAX, -FLT_MAX, -FLT_MAX};

	for (size_t i = 0; i < vertex_count; ++i)
	{
		const float* v = vertex_positions_data + i * vertex_stride_float;

		result[i].x = v[0];
		result[i].y = v[1];
		result[i].z = v[2];

		for (int j = 0; j < 3; ++j)
		{
			float vj = v[j];

			minv[j] = minv[j] > vj ? vj : minv[j];
			maxv[j] = maxv[j] < vj ? vj : maxv[j];
		}
	}

	float extent = 0.f;

	extent = (maxv[0] - minv[0]) < extent ? extent : (maxv[0] - minv[0]);
	extent = (maxv[1] - minv[1]) < extent ? extent : (maxv[1] - minv[1]);
	extent = (maxv[2] - minv[2]) < extent ? extent : (maxv[2] - minv[2]);

	float scale = extent == 0 ? 0.f : 1.f / extent;

	for (size_t i = 0; i < vertex_count; ++i)
	{
		result[i].x = (result[i].x - minv[0]) * scale;
		result[i].y = (result[i].y - minv[1]) * scale;
		result[i].z = (result[i].z - minv[2]) * scale;
	}
}

struct Quadric
{
	float a00;
	float a10, a11;
	float a20, a21, a22;
	float b0, b1, b2, c;
};

struct Collapse
{
	unsigned int v0;
	unsigned int v1;

	union {
		unsigned int bidi;
		float error;
		unsigned int errorui;
	};
};

static float normalize(Vector3& v)
{
	float length = sqrtf(v.x * v.x + v.y * v.y + v.z * v.z);

	if (length > 0)
	{
		v.x /= length;
		v.y /= length;
		v.z /= length;
	}