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April 22, 2021 16:05
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DSU on Tree
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# DSU on Tree | |
# Ref: https://codeforces.com/blog/entry/44351 | |
import random | |
import time | |
import sys | |
random.seed() | |
sys.setrecursionlimit(10**6) | |
N = 10000 | |
C = 10 | |
cols = [random.randint(0, C-1) for i in range(N)] | |
G = [[] for i in range(N)] | |
*rg, = range(N) | |
random.shuffle(rg) | |
for i in range(1, N): | |
j = random.randint(0, i-1) | |
G[rg[i]].append(rg[j]) | |
G[rg[j]].append(rg[i]) | |
# calc_sz: O(N) | |
def calc_sz(): | |
sz = [0]*N | |
def dfs_sz(v, p): | |
c = 0 | |
for w in G[v]: | |
if w == p: | |
continue | |
c += dfs_sz(w, v) | |
sz[v] = c | |
return c | |
dfs_sz(0, -1) | |
return sz | |
sz = calc_sz() | |
# calc_ett: O(N) | |
def calc_ett(): | |
st = [0]*N; ft = [0]*N | |
ver = [0]*N | |
idx = 0 | |
def dfs_ett(v, p): | |
nonlocal idx | |
st[v] = idx | |
ver[idx] = v | |
idx += 1 | |
for w in G[v]: | |
if w == p: | |
continue | |
dfs_ett(w, v) | |
ft[v] = idx | |
dfs_ett(0, -1) | |
return st, ft, ver | |
st, ft, ver = calc_ett() | |
# naive: O(N^2 + NC) | |
def solver_naive(N, G, C, cols, sz): | |
R = [[0]*C for i in range(N)] | |
cnts = [0]*C | |
def dfs_add(v, p, x): | |
cnts[cols[v]] += x | |
for w in G[v]: | |
if w != p: | |
dfs_add(w, v, x) | |
def dfs(v, p): | |
dfs_add(v, p, 1) | |
Rv = R[v] | |
for i in range(C): | |
Rv[i] = cnts[i] | |
dfs_add(v, p, -1) | |
for w in G[v]: | |
if w != p: | |
dfs(w, v) | |
dfs(0, -1) | |
return R | |
# impl1: O(N log^2 N + NC) | |
def solver_impl1(N, G, C, cols, sz): | |
R = [[0]*C for i in range(N)] | |
cnts = [None for i in range(N)] | |
sz = [0]*N | |
def dfs(v, p): | |
mx = heavy = -1 | |
for w in G[v]: | |
if w == p: | |
continue | |
dfs(w, v) | |
if mx < sz[w]: | |
mx = sz[w]; heavy = w | |
if heavy != -1: | |
cnts[v] = cnts[heavy] | |
else: | |
cnts[v] = {} | |
cv = cnts[v] | |
cv[cols[v]] = cv.get(cols[v], 0) + 1 | |
for w in G[v]: | |
if w == p or w == heavy: | |
continue | |
for c, val in cnts[w].items(): | |
cv[c] = cv.get(c, 0) + val | |
Rv = R[v] | |
for i in range(C): | |
Rv[i] = cv.get(i, 0) | |
dfs(0, -1) | |
return R | |
# impl2: O(N log N + NC) | |
def solver_impl2(N, G, C, cols, sz): | |
R = [[0]*C for i in range(N)] | |
vec = [None]*N | |
cnts = [0]*C | |
def dfs(v, p, keep): | |
mx = heavy = -1 | |
for w in G[v]: | |
if w == p: | |
continue | |
if mx < sz[w]: | |
mx = sz[w] | |
heavy = w | |
for w in G[v]: | |
if w == p or w == heavy: | |
continue | |
dfs(w, v, 0) | |
if heavy != -1: | |
dfs(heavy, v, 1) | |
vec[v] = vec[heavy] | |
else: | |
vec[v] = [] | |
vec[v].append(v) | |
cnts[cols[v]] += 1 | |
for w in G[v]: | |
if w == p or w == heavy: | |
continue | |
for x in vec[w]: | |
cnts[cols[x]] += 1 | |
vec[v].append(x) | |
Rv = R[v] | |
for i in range(C): | |
Rv[i] = cnts[i] | |
if keep == 0: | |
for x in vec[v]: | |
cnts[cols[x]] -= 1 | |
dfs(0, -1, 1) | |
return R | |
# impl3: O(N log N + NC) | |
def solver_impl3(N, G, C, cols, sz): | |
R = [[0]*C for i in range(N)] | |
cnts = [0]*C | |
hs = [0]*N | |
def dfs_add(v, p, x): | |
cnts[cols[v]] += x | |
for w in G[v]: | |
if w == p or hs[w]: | |
continue | |
dfs_add(w, v, x) | |
def dfs(v, p, keep): | |
mx = heavy = -1 | |
for w in G[v]: | |
if w == p: | |
continue | |
if mx < sz[w]: | |
mx = sz[w] | |
heavy = w | |
for w in G[v]: | |
if w != p and w != heavy: | |
dfs(w, v, 0) | |
if heavy != -1: | |
dfs(heavy, v, 1) | |
hs[heavy] = 1 | |
dfs_add(v, p, 1) | |
Rv = R[v] | |
for i in range(C): | |
Rv[i] = cnts[i] | |
if heavy != -1: | |
hs[heavy] = 0 | |
if keep == 0: | |
dfs_add(v, p, -1) | |
dfs(0, -1, 1) | |
return R | |
# impl4: O(N log N + NC) | |
def solver_impl4(N, G, C, cols, sz, st, ft, ver): | |
R = [[0]*C for i in range(N)] | |
cnts = [0]*C | |
def dfs(v, p, keep): | |
mx = heavy = -1 | |
for w in G[v]: | |
if w == p: | |
continue | |
if mx < sz[w]: | |
mx = sz[w] | |
heavy = w | |
for w in G[v]: | |
if w == p or w == heavy: | |
continue | |
dfs(w, v, 0) | |
if heavy != -1: | |
dfs(heavy, v, 1) | |
#for w in G[v]: | |
# if w == p or w == heavy: | |
# continue | |
# for p in range(st[w], ft[w]): | |
# cnts[cols[ver[p]]] += 1 | |
if heavy != -1: | |
for p in range(st[v]+1, st[heavy]): | |
cnts[cols[ver[p]]] += 1 | |
for p in range(ft[heavy], ft[v]): | |
cnts[cols[ver[p]]] += 1 | |
cnts[cols[v]] += 1 | |
Rv = R[v] | |
for i in range(C): | |
Rv[i] = cnts[i] | |
if keep == 0: | |
for p in range(st[v], ft[v]): | |
cnts[cols[ver[p]]] -= 1 | |
dfs(0, -1, 0) | |
return R | |
t0 = time.time() | |
R0 = solver_naive(N, G, C, cols, sz) | |
t1 = time.time() | |
R1 = solver_impl1(N, G, C, cols, sz) | |
t2 = time.time() | |
R2 = solver_impl2(N, G, C, cols, sz) | |
t3 = time.time() | |
R3 = solver_impl3(N, G, C, cols, sz) | |
t4 = time.time() | |
R4 = solver_impl4(N, G, C, cols, sz, st, ft, ver) | |
t5 = time.time() | |
print(R0 == R1 == R2 == R3 == R4) | |
print("solver_naive:", t1 - t0) | |
print("solver_impl1:", t2 - t1) | |
print("solver_impl2:", t3 - t2) | |
print("solver_impl3:", t4 - t3) | |
print("solver_impl4:", t5 - t4) |
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