dodge-the-lasers.cpp

#define _CRT_SECURE_NO_WARNINGS

#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <limits>
#include <ctime>
#include <cassert>
#include <map>
#include <set>
#include <iostream>
#include <memory>
#include <string>
#include <vector>
#include <algorithm>
#include <functional>
#include <sstream>
#include <stack>
#include <queue>
#include <numeric>
#include <iterator>
#include <bitset>
#include <unordered_map>
#include <unordered_set>

using namespace std;
#define FOR(i,n) for(int i = 0; i < (n); i++)
#define sz(c) ((int)c.size())
#define ten(n) ((int)1e##n)
using ll = long long;
using ull = unsigned long long;
using Pii = pair<int, int>;
using Pll = pair<ll, ll>;

template<typename ...> static inline int getchar_unlocked(void) { return getchar(); }
template<typename ...> static inline void putchar_unlocked(int c) { putchar(c); }
#define mygc(c) (c)=getchar_unlocked()
#define mypc(c) putchar_unlocked(c)
void reader(int& x) { int k, m = 0; x = 0; for (;;) { mygc(k); if (k == '-') { m = 1; break; }if ('0' <= k && k <= '9') { x = k - '0'; break; } }for (;;) { mygc(k); if (k < '0' || k>'9')break; x = x * 10 + k - '0'; }if (m) x = -x; }
void reader(ll& x) { int k, m = 0; x = 0; for (;;) { mygc(k); if (k == '-') { m = 1; break; }if ('0' <= k && k <= '9') { x = k - '0'; break; } }for (;;) { mygc(k); if (k < '0' || k>'9')break; x = x * 10 + k - '0'; }if (m) x = -x; }
int reader(char c[]) { int i, s = 0; for (;;) { mygc(i); if (i != ' ' && i != '\n' && i != '\r' && i != '\t' && i != EOF) break; }c[s++] = i; for (;;) { mygc(i); if (i == ' ' || i == '\n' || i == '\r' || i == '\t' || i == EOF) break; c[s++] = i; }c[s] = '\0'; return s; }
int reader(string& c) { c.clear(); int i; for (;;) { mygc(i); if (i != ' ' && i != '\n' && i != '\r' && i != '\t' && i != EOF) break; }c.push_back(i); for (;;) { mygc(i); if (i == ' ' || i == '\n' || i == '\r' || i == '\t' || i == EOF) break; c.push_back(i); }; return sz(c); }
template <class T, class S> void reader(T& x, S& y) { reader(x); reader(y); }
template <class T, class S, class U> void reader(T& x, S& y, U& z) { reader(x); reader(y); reader(z); }
template <class T, class S, class U, class V> void reader(T& x, S& y, U& z, V& w) { reader(x); reader(y); reader(z); reader(w); }
void writer(int x, char c) { int s = 0, m = 0; char f[10]; if (x < 0)m = 1, x = -x; while (x)f[s++] = x % 10, x /= 10; if (!s)f[s++] = 0; if (m)mypc('-'); while (s--)mypc(f[s] + '0'); mypc(c); }
void writer(ll x, char c) { int s = 0, m = 0; char f[20]; if (x < 0)m = 1, x = -x; while (x)f[s++] = x % 10, x /= 10; if (!s)f[s++] = 0; if (m)mypc('-'); while (s--)mypc(f[s] + '0'); mypc(c); }
void writer(const char c[]) { int i; for (i = 0; c[i] != '\0'; i++)mypc(c[i]); }
void writer(const string& x, char c) { int i; for (i = 0; x[i] != '\0'; i++)mypc(x[i]); mypc(c); }
void writer(const char x[], char c) { int i; for (i = 0; x[i] != '\0'; i++)mypc(x[i]); mypc(c); }
template<class T> void writerLn(T x) { writer(x, '\n'); }
template<class T, class S> void writerLn(T x, S y) { writer(x, ' '); writer(y, '\n'); }
template<class T, class S, class U> void writerLn(T x, S y, U z) { writer(x, ' '); writer(y, ' '); writer(z, '\n'); }
template<class T, class S, class U, class V> void writerLn(T x, S y, U z, V v) { writer(x, ' '); writer(y, ' '); writer(z, ' '); writer(v, '\n'); }
template<class T> void writerArr(T x[], int n) { if (!n) { mypc('\n'); return; }FOR(i, n - 1)writer(x[i], ' '); writer(x[n - 1], '\n'); }
template<class T> void writerArr(vector<T>& x) { writerArr(x.data(), (int)x.size()); }
void reader(ull& x) { int k, m = 0; x = 0; for (;;) { mygc(k); if (k == '-') { m = 1; break; }if ('0' <= k && k <= '9') { x = ull(k - '0'); break; } }for (;;) { mygc(k); if (k < '0' || k>'9')break; x = x * 10 + k - '0'; } }
void writer(ull x, char c) { int s = 0, m = 0; char f[20]; while (x)f[s++] = x % 10, x /= 10; if (!s)f[s++] = 0; if (m)mypc('-'); while (s--)mypc(f[s] + '0'); mypc(c); }
template<class T> vector<T> readerArray(int n) { vector<T> ret(n); for (int i = 0; i < n; i++) reader(ret[i]); return ret; }

template<class T> void chmin(T& a, const T& b) { if (a > b) a = b; }
template<class T> void chmax(T& a, const T& b) { if (a < b) a = b; }

template<class T> T gcd(T a, T b) { return (b != 0) ? gcd(b, a % b) : a; }
template<class T> T lcm(T a, T b) { return a / gcd(a, b) * b; }
ll mod_pow(ll a, ll n, ll mod) {
    ll ret = 1;
    ll p = a % mod;
    while (n) {
        if (n & 1) ret = ret * p % mod;
        p = p * p % mod;
        n >>= 1;
    }
    return ret;
}
template<class T> T extgcd(T a, T b, T& x, T& y) { for (T u = y = 1, v = x = 0; a;) { T q = b / a; swap(x -= q * u, u); swap(y -= q * v, v); swap(b -= q * a, a); } return b; }
template<class T> T mod_inv(T a, T m) { T x, y; extgcd<T>(a, m, x, y); return (m + x % m) % m; }

#ifdef _MSC_VER
// #ifdef _DEBUG
template <typename ... Args>
void debugPrintf(const char* format, Args const& ... args) {
    fprintf(stderr, format, args ...);
    fflush(stderr);
}
#else
#define debugPrintf(...)
#endif

ll nSqrt(ll x) {
    ll p = (ll)sqrt(x);
    while ((p + 1) * (p + 1) <= x) p++;
    while (p * p > x) p--;
    return p;
}

Pll sqrt2DiophantineApproximation(int cnt) {
    ll q = 0, p = 1;
    FOR(i, cnt) {
        ll np = 2 * p + q;
        ll nq = p;
        q = nq;
        p = np;
    }
    return Pll(q + p, p);
}

ll naive(ll n) {
    ll ans = 0;
    for (ll i = 1; i <= n; i++) {
        ll val = nSqrt(ll(i) * i * 2);
        ans += val;
    }
    return ans;
}

bool isLargerThanSqrt2(Pll qp) {
    return 2 * qp.second * qp.second < qp.first * qp.first;
}

ll sumDiophantineApproximatedLatticePoints(Pll qp) {
    ll s2 = qp.first * qp.second;
    ll b = qp.first + qp.second + 1;
    ll i = (s2 - b) / 2 + 1;
    bool larger = isLargerThanSqrt2(qp);
    if (larger) {
        debugPrintf("[*] %lld/%lld is slightly larger than sqrt(2)\n", qp.first, qp.second);
    }
    return i + qp.first - larger;
}

ll solve(ll n) {
    vector<Pll> points;
    for (int i = 2;; i++) {
        Pll cur = sqrt2DiophantineApproximation(i);
        points.push_back(cur);
        if (cur.second > n) break;
    }
    ll ans = 0;
    ll cur = 0;
    bool isSq2SlightlySmallerThanInteger = false;
    while (n - cur > 20) {
        int id = 0;
        while (points[id + 1].second < n - cur) id++;

        ll baseLine = nSqrt(2 * cur * cur);
        if (isSq2SlightlySmallerThanInteger) {
                baseLine++;
        }
        const ll dalAns = sumDiophantineApproximatedLatticePoints(points[id]);
        const ll add = dalAns + baseLine * points[id].second;

        printf("  cumulated range (%lld, %lld]. baseLine = %lld, dalAns = %lld, add = %lld, isSq2SlightlySmallerThanInteger = %d\n",
            cur, cur + points[id].second, baseLine, dalAns, add, isSq2SlightlySmallerThanInteger);
        if (isSq2SlightlySmallerThanInteger) {
            printf("  baseline was incremented since the baseLine is slightly smaller than integer.\n");
        }

        isSq2SlightlySmallerThanInteger = isLargerThanSqrt2(points[id]);
        ans += add;
        cur += points[id].second;
    }
    if (cur != 0) {
        cur++; // x = cur was calculated above
    }
    while (cur <= n) {
        ll val = nSqrt(2 * cur * cur);
        ans += val;
        cur++;
    }

    return ans;
}

void compare(int i) {
    puts("--------------------");
    printf("answer for %d\n", i);
    ll ans = naive(i);
    ll a2 = solve(i);
    printf("naive = %lld, solve = %lld\n", ans, a2);
    if (ans != a2) {
        cout << "?";
    }
}

int main() {
    //compare(77);

    //for(int i = 10; i < 500; i++) {
    //    compare(i);
    //}

    for (int i = 1000; i < ten(7); i *= 2) {
        compare(i);
    }

    return 0;
}