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graph/find_path_through_specified.hpp
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- Last update: 2024-04-19 02:20:22+09:00
- Include:
#include "graph/find_path_through_specified.hpp"
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#include "graph/base.hpp"
#include "random/base.hpp"
#include "nt/GF2.hpp"
// (s,t) path であって need をすべて通るものの最小長さを求める.
// s=t でもよい. 最初の 1 歩目も求める.
// {length, first v}. なければ {-1,-1}.
// O(2^K(N+M)L), L = shortest path length
template <typename GT>
pair<int, int> find_path_through_specified(GT& G, int s, int t, vc<int> need) {
static_assert(!GT::is_directed);
int N = G.N;
assert(0 <= s && s < N && 0 <= t && t < N);
using F = GF2<64>;
// frm, to, wt
vc<tuple<int, int, F>> edges;
vc<int> S, T;
for (auto& e: G.edges) {
int a = e.frm, b = e.to;
if ((a == s && b == t) || (a == t && b == s)) {
if (need.empty()) return {1, t};
continue;
}
if (a == s || b == s) { S.eb(a ^ b ^ s); }
if (a == t || b == t) { T.eb(a ^ b ^ t); }
if (a != s && a != t && b != s && b != t) {
F x = RNG_64();
edges.eb(a, b, x), edges.eb(b, a, x);
}
}
int K = len(need);
vc<int> get(N);
FOR(k, K) get[need[k]] = 1 << k;
int M = len(edges);
vv(F, dp_e, 1 << K, M);
vv(F, dp_v, 1 << K, N);
for (auto& v: T) dp_v[get[v]][v] = RNG_64();
FOR(L, 1, N) {
for (auto& v: S) {
if (dp_v.back()[v] != F(0)) return {1 + L, v};
}
vv(F, newdp_e, 1 << K, M);
vv(F, newdp_v, 1 << K, N);
FOR(s, 1 << K) {
FOR(m, M) {
auto [a, b, c] = edges[m];
int t = s | get[b];
if (get[b] && s == t) continue;
F x = (dp_v[s][a] + dp_e[s][m ^ 1]) * c;
newdp_e[t][m] += x, newdp_v[t][b] += x;
}
}
swap(dp_v, newdp_v), swap(dp_e, newdp_e);
}
return {-1, -1};
}#line 2 "graph/base.hpp"
template <typename T>
struct Edge {
int frm, to;
T cost;
int id;
};
template <typename T = int, bool directed = false>
struct Graph {
static constexpr bool is_directed = directed;
int N, M;
using cost_type = T;
using edge_type = Edge<T>;
vector<edge_type> edges;
vector<int> indptr;
vector<edge_type> csr_edges;
vc<int> vc_deg, vc_indeg, vc_outdeg;
bool prepared;
class OutgoingEdges {
public:
OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}
const edge_type* begin() const {
if (l == r) { return 0; }
return &G->csr_edges[l];
}
const edge_type* end() const {
if (l == r) { return 0; }
return &G->csr_edges[r];
}
private:
const Graph* G;
int l, r;
};
bool is_prepared() { return prepared; }
Graph() : N(0), M(0), prepared(0) {}
Graph(int N) : N(N), M(0), prepared(0) {}
void build(int n) {
N = n, M = 0;
prepared = 0;
edges.clear();
indptr.clear();
csr_edges.clear();
vc_deg.clear();
vc_indeg.clear();
vc_outdeg.clear();
}
void add(int frm, int to, T cost = 1, int i = -1) {
assert(!prepared);
assert(0 <= frm && 0 <= to && to < N);
if (i == -1) i = M;
auto e = edge_type({frm, to, cost, i});
edges.eb(e);
++M;
}
#ifdef FASTIO
// wt, off
void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }
void read_graph(int M, bool wt = false, int off = 1) {
for (int m = 0; m < M; ++m) {
INT(a, b);
a -= off, b -= off;
if (!wt) {
add(a, b);
} else {
T c;
read(c);
add(a, b, c);
}
}
build();
}
#endif
void build() {
assert(!prepared);
prepared = true;
indptr.assign(N + 1, 0);
for (auto&& e: edges) {
indptr[e.frm + 1]++;
if (!directed) indptr[e.to + 1]++;
}
for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
auto counter = indptr;
csr_edges.resize(indptr.back() + 1);
for (auto&& e: edges) {
csr_edges[counter[e.frm]++] = e;
if (!directed)
csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
}
}
OutgoingEdges operator[](int v) const {
assert(prepared);
return {this, indptr[v], indptr[v + 1]};
}
vc<int> deg_array() {
if (vc_deg.empty()) calc_deg();
return vc_deg;
}
pair<vc<int>, vc<int>> deg_array_inout() {
if (vc_indeg.empty()) calc_deg_inout();
return {vc_indeg, vc_outdeg};
}
int deg(int v) {
if (vc_deg.empty()) calc_deg();
return vc_deg[v];
}
int in_deg(int v) {
if (vc_indeg.empty()) calc_deg_inout();
return vc_indeg[v];
}
int out_deg(int v) {
if (vc_outdeg.empty()) calc_deg_inout();
return vc_outdeg[v];
}
#ifdef FASTIO
void debug() {
print("Graph");
if (!prepared) {
print("frm to cost id");
for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
} else {
print("indptr", indptr);
print("frm to cost id");
FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
}
}
#endif
vc<int> new_idx;
vc<bool> used_e;
// G における頂点 V[i] が、新しいグラフで i になるようにする
// {G, es}
Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
if (len(new_idx) != N) new_idx.assign(N, -1);
int n = len(V);
FOR(i, n) new_idx[V[i]] = i;
Graph<T, directed> G(n);
vc<int> history;
FOR(i, n) {
for (auto&& e: (*this)[V[i]]) {
if (len(used_e) <= e.id) used_e.resize(e.id + 1);
if (used_e[e.id]) continue;
int a = e.frm, b = e.to;
if (new_idx[a] != -1 && new_idx[b] != -1) {
history.eb(e.id);
used_e[e.id] = 1;
int eid = (keep_eid ? e.id : -1);
G.add(new_idx[a], new_idx[b], e.cost, eid);
}
}
}
FOR(i, n) new_idx[V[i]] = -1;
for (auto&& eid: history) used_e[eid] = 0;
G.build();
return G;
}
private:
void calc_deg() {
assert(vc_deg.empty());
vc_deg.resize(N);
for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
}
void calc_deg_inout() {
assert(vc_indeg.empty());
vc_indeg.resize(N);
vc_outdeg.resize(N);
for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
}
};
#line 2 "random/base.hpp"
u64 RNG_64() {
static uint64_t x_
= uint64_t(chrono::duration_cast<chrono::nanoseconds>(
chrono::high_resolution_clock::now().time_since_epoch())
.count())
* 10150724397891781847ULL;
x_ ^= x_ << 7;
return x_ ^= x_ >> 9;
}
u64 RNG(u64 lim) { return RNG_64() % lim; }
ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 1 "nt/GF2.hpp"
#include <emmintrin.h>
#include <smmintrin.h>
#include <wmmintrin.h>
__attribute__((target("pclmul"))) inline __m128i myclmul(const __m128i &a,
const __m128i &b) {
return _mm_clmulepi64_si128(a, b, 0);
}
// 2^n 元体
template <int K>
struct GF2 {
// https://oeis.org/A344141
// irreducible poly x^K + ...
static constexpr int POLY[65]
= {0, 0, 3, 3, 3, 5, 3, 3, 27, 3, 9, 5, 9, 27, 33, 3, 43,
9, 9, 39, 9, 5, 3, 33, 27, 9, 27, 39, 3, 5, 3, 9, 141, 75,
27, 5, 53, 63, 99, 17, 57, 9, 39, 89, 33, 27, 3, 33, 45, 113, 29,
75, 9, 71, 125, 71, 149, 17, 99, 123, 3, 39, 105, 3, 27};
static constexpr u64 mask() { return u64(-1) >> (64 - K); }
__attribute__((target("sse4.2"))) static u64 mul(u64 a, u64 b) {
static bool prepared = 0;
static u64 MEMO[8][65536];
if (!prepared) {
prepared = 1;
vc<u64> tmp(128);
tmp[0] = 1;
FOR(i, 127) {
tmp[i + 1] = tmp[i] << 1;
if (tmp[i] >> (K - 1) & 1) {
tmp[i + 1] ^= POLY[K];
tmp[i + 1] &= mask();
}
}
FOR(k, 8) {
MEMO[k][0] = 0;
FOR(i, 16) {
FOR(s, 1 << i) { MEMO[k][s | 1 << i] = MEMO[k][s] ^ tmp[16 * k + i]; }
}
}
}
const __m128i a_ = _mm_set_epi64x(0, a);
const __m128i b_ = _mm_set_epi64x(0, b);
const __m128i c_ = myclmul(a_, b_);
u64 lo = _mm_extract_epi64(c_, 0);
u64 hi = _mm_extract_epi64(c_, 1);
u64 x = 0;
x ^= MEMO[0][lo & 65535];
x ^= MEMO[1][(lo >> 16) & 65535];
x ^= MEMO[2][(lo >> 32) & 65535];
x ^= MEMO[3][(lo >> 48) & 65535];
x ^= MEMO[4][hi & 65535];
x ^= MEMO[5][(hi >> 16) & 65535];
x ^= MEMO[6][(hi >> 32) & 65535];
x ^= MEMO[7][(hi >> 48) & 65535];
return x;
}
u64 val;
constexpr GF2(const u64 val = 0) noexcept : val(val & mask()) {}
bool operator<(const GF2 &other) const {
return val < other.val;
} // To use std::map
GF2 &operator+=(const GF2 &p) {
val ^= p.val;
return *this;
}
GF2 &operator-=(const GF2 &p) {
val ^= p.val;
return *this;
}
GF2 &operator*=(const GF2 &p) {
val = mul(val, p.val);
return *this;
}
GF2 &operator/=(const GF2 &p) {
*this *= p.inverse();
return *this;
}
GF2 operator-() const { return GF2(-val); }
GF2 operator+(const GF2 &p) const { return GF2(*this) += p; }
GF2 operator-(const GF2 &p) const { return GF2(*this) -= p; }
GF2 operator*(const GF2 &p) const { return GF2(*this) *= p; }
GF2 operator/(const GF2 &p) const { return GF2(*this) /= p; }
bool operator==(const GF2 &p) const { return val == p.val; }
bool operator!=(const GF2 &p) const { return val != p.val; }
GF2 inverse() const { return pow((u64(1) << K) - 2); }
GF2 pow(u64 n) const {
GF2 ret(1), mul(val);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
};
#ifdef FASTIO
template <int K>
void rd(GF2<K> &x) {
fastio::rd(x.val);
x &= GF2<K>::mask;
}
template <int K>
void wt(GF2<K> x) {
fastio::wt(x.val);
}
#endif
#line 4 "graph/find_path_through_specified.hpp"
// (s,t) path であって need をすべて通るものの最小長さを求める.
// s=t でもよい. 最初の 1 歩目も求める.
// {length, first v}. なければ {-1,-1}.
// O(2^K(N+M)L), L = shortest path length
template <typename GT>
pair<int, int> find_path_through_specified(GT& G, int s, int t, vc<int> need) {
static_assert(!GT::is_directed);
int N = G.N;
assert(0 <= s && s < N && 0 <= t && t < N);
using F = GF2<64>;
// frm, to, wt
vc<tuple<int, int, F>> edges;
vc<int> S, T;
for (auto& e: G.edges) {
int a = e.frm, b = e.to;
if ((a == s && b == t) || (a == t && b == s)) {
if (need.empty()) return {1, t};
continue;
}
if (a == s || b == s) { S.eb(a ^ b ^ s); }
if (a == t || b == t) { T.eb(a ^ b ^ t); }
if (a != s && a != t && b != s && b != t) {
F x = RNG_64();
edges.eb(a, b, x), edges.eb(b, a, x);
}
}
int K = len(need);
vc<int> get(N);
FOR(k, K) get[need[k]] = 1 << k;
int M = len(edges);
vv(F, dp_e, 1 << K, M);
vv(F, dp_v, 1 << K, N);
for (auto& v: T) dp_v[get[v]][v] = RNG_64();
FOR(L, 1, N) {
for (auto& v: S) {
if (dp_v.back()[v] != F(0)) return {1 + L, v};
}
vv(F, newdp_e, 1 << K, M);
vv(F, newdp_v, 1 << K, N);
FOR(s, 1 << K) {
FOR(m, M) {
auto [a, b, c] = edges[m];
int t = s | get[b];
if (get[b] && s == t) continue;
F x = (dp_v[s][a] + dp_e[s][m ^ 1]) * c;
newdp_e[t][m] += x, newdp_v[t][b] += x;
}
}
swap(dp_v, newdp_v), swap(dp_e, newdp_e);
}
return {-1, -1};
}