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#696593#6402. MEXimum Spanning TreeNTTRE 45ms3796kbC++237.1kb2024-10-31 23:42:342024-10-31 23:42:36

Judging History

你现在查看的是最新测评结果

  • [2024-10-31 23:42:36]
  • 评测
  • 测评结果:RE
  • 用时:45ms
  • 内存:3796kb
  • [2024-10-31 23:42:34]
  • 提交

answer

#pragma GCC optimize("Ofast")
//verification of Huang, Chien-Chung; Kakimura, Naonori; Kamiyama, Naoyuki (2019-09-01). "Exact and approximation algorithms for weighted matroid intersection". Mathematical Programming. 177 (1): 85–112. doi:10.1007/s10107-018-1260-x. hdl:2324/1474903. ISSN 1436-4646. S2CID 254138118.
//based on https://qoj.ac/submission/548815
#include <bits/stdc++.h>
// #include"C:/code/deb_20.cpp"
using ll = long long;
using i128=__int128;
using ld = long double;
#define int ll
template<typename T>using V=std::vector<T>;
using vi=V<int>;
using vvi=V<vi>;
using namespace std;
constexpr ll M=ll(1e18)+9;

#define len(a) int((a).size())
#define all(a) begin(a), end(a)
#define rep(i, n) for (int i = 0; i < (n); i++)

template<typename T, typename A, typename B>
vector<T> matroid_intersection(const vector<T>& ground_set, const A& matroid1, const B& matroid2) {
    int n = ground_set.size();
    vector<char> in_set(n), in_matroid1(n), in_matroid2(n);
    vector<bool> used(n);
    vector<int> par(n), left, right;
    left.reserve(n);
    right.reserve(n);

    while (true) {
        A m1 = matroid1;
        B m2 = matroid2;
        left.clear();
        right.clear();
        for (int i = 0; i < n; i++) {
            if (in_set[i]) {
                m1.add(ground_set[i]);
                m2.add(ground_set[i]);
                left.push_back(i);
            } else {
                right.push_back(i);
            }
        }

        fill(all(in_matroid1), 0);
        fill(all(in_matroid2), 0);
        bool found = false;
        for (int i : right) {
            in_matroid1[i] = m1.independed_with(ground_set[i]);
            in_matroid2[i] = m2.independed_with(ground_set[i]);
            if (in_matroid1[i] && in_matroid2[i]) {
                in_set[i] = 1;
                found = true;
                break;
            }
        }

        if (found) {
            continue;
        }

        fill(all(used), false);
        fill(all(par), -1);
        queue<int> que;
        for (int i : right) {
            if (in_matroid1[i]) {
                used[i] = true;
                que.push(i);
            }
        }

        while (!que.empty() && !found) {
            int v = que.front();
            que.pop();

            if (in_set[v]) {
                A m = matroid1;
                for (auto i : left) {
                    if (i != v) {
                        m.add(ground_set[i]);
                    }
                }

                for (auto u : right) {
                    if (!used[u] && m.independed_with(ground_set[u])) {
                        par[u] = v;
                        used[u] = true;
                        que.push(u);

                        if (in_matroid2[u]) {
                            found = true;
                            for (; u != -1; u = par[u]) {
                                in_set[u] ^= 1;
                            }
                            break;
                        }
                    }
                }
            } else {
                B m = m2;
                m.add_extra(ground_set[v]);
                for (auto u : left) {
                    if (!used[u] && m.independed_without(ground_set[u])) {
                        par[u] = v;
                        used[u] = true;
                        que.push(u);
                    }
                }
            }
        }

        if (!found) {
            break;
        }
    }

    vector<T> res;
    for (int i = 0; i < n; i++) {
        if (in_set[i]) {
            res.push_back(ground_set[i]);
        }
    }
    return res;
}

struct item {
    int v, u, w;
};

struct colorful_matroid {
    vector<int> cnt;
    int cnt_bad = 0;

    colorful_matroid(int n) : cnt(n + 1) {}

    void add(const item& item) {
        auto x = item.w;
        assert(cnt[x] == 0);
        cnt[x]++;
    }

    bool independed_with(const item& item) const {
        auto x = item.w;
        return cnt[x] == 0;
    }

    void add_extra(const item& item) {
        auto x = item.w;
        cnt_bad += cnt[x] == 1;
        cnt[x]++;
    }

    bool independed_without(const item& item) const {
        auto x = item.w;
        return cnt_bad == 0 || cnt[x] == 2;
    }
};

struct graph_matroid {
    vector<int> par;

    graph_matroid(int n) : par(n) {
        iota(all(par), 0);
    }

    int root(int v) {
        return par[v] == v ? v : par[v] = root(par[v]);
    }

    bool independed_with(const item& item) {
        int v = item.v, u = item.u;
        return root(v) != root(u);
    }

    void add(const item& item) {
        int v = item.v, u = item.u;
        assert(root(v) != root(u));
        par[root(v)] = root(u);
    }
};

vi&operator+=(vi&a,const vi&b){for(int i=0;i<ssize(a);++i)(a[i]+=b[i])%=M;return a;}
vi&operator*=(vi&a,const ll&b){for(auto&&x:a)x=x*i128(b)%M;return a;}
#define GEN_OP(op) auto operator op(auto a,const auto&b){return a op##= b;}
GEN_OP(+)
GEN_OP(*)
// struct Matrix:vvi{
// 	
// };
ll qpow(i128 a,auto b){ll res=1;for(;b;a=a*a%M,b>>=1)if(b&1)res=res*a%M;return res;}
using Matrix=vvi;
int rnk(Matrix a){
	int n=ssize(a),m=ssize(a[0]),i=0,j=0;
	for(;i<n&&j<m;++j){
		int r=i;
		for(;r<n;++r)if(a[r][j])break;
		if(r<n&&a[r][j]){
			if(r!=i)std::swap(a[r],a[i]);
			for(int l=i+1;l<n;++l){
				i128 fct=i128(M-a[l][j])*qpow(a[i][j],M-2)%M;
				if(fct)a[l]+=a[i]*fct;
			}
			++i;
		}
	}
	return i;
}
mt19937_64 rng(19260817);
auto rnd(auto a,auto b){return std::uniform_int_distribution(a,b)(rng);}
signed main() {
    cin.tie(nullptr)->sync_with_stdio(false);

    int n, m;
    cin >> n >> m;
    vi rdw(m);
    vector<item> st(m);
    for (int i = 0; i < m; i++) {
    	rdw[i]=rnd(1ll,M-1);
        cin >> st[i].v >> st[i].u >> st[i].w;
        st[i].v--, st[i].u--;
        assert(st[i].v!=st[i].u);
    }
    std::ranges::sort(st,{},&item::w);

	int jcnt=0;
    auto possible = [&](int mex) {
    	int m=0;
    	// if constexpr(false){
	        vector<item> cur_st;
	        for (auto item : st) {
	            if (item.w < mex) {
	            	++m;
	                cur_st.push_back(item);
	            }
	        }
	        colorful_matroid cm(n);
	        graph_matroid g(n);
	        auto resoid=len(matroid_intersection(cur_st, g, cm));
	        if(++jcnt>=0)return resoid==mex;
    	// }else{
    		vvi ma(n+m,vi(m));
    		vi vis(mex,-1);
    		int resix=0;
    		for(int i=0;i<m;++i)if(st[i].w<mex){
    			resix--;
    			if(!~vis[st[i].w]){
    				vis[st[i].w]=i;
    				resix++;
    			}else{
    				ma[n+i][vis[st[i].w]]=-rdw[vis[st[i].w]];
    				ma[n+i][i]=rdw[i];
    			}
    			auto x=rnd(1ll,M-1);
    			ma[st[i].v][i]=+x;
    			ma[st[i].u][i]=M-x;
    		}
			resix+=rnk(ma);
    	// }
    	// debug(mex,resoid,resix);
    	// cerr<<mex<<' '<<resoid<<' '<<resix<<'\n';
    	assert(resix==resoid);
    	return resix==mex;
    };

    int lb = 0, rb = n + 1;
    while (rb - lb > 1) {
        int mid = (lb + rb) / 2;
        (possible(mid) ? lb : rb) = mid;
    }
    cout << lb << '\n';
}

詳細信息

Test #1:

score: 100
Accepted
time: 1ms
memory: 3568kb

input:

4 4
1 2 0
2 3 1
1 3 1
3 4 2

output:

3

result:

ok 1 number(s): "3"

Test #2:

score: 0
Accepted
time: 19ms
memory: 3780kb

input:

1000 1000
647 790 6
91 461 435
90 72 74
403 81 240
893 925 395
817 345 136
88 71 821
831 962 53
164 270 298
14 550 317
99 580 81
26 477 488
977 474 861
413 483 167
872 675 17
819 327 449
594 242 68
381 983 319
867 582 358
869 225 669
274 352 392
40 388 998
246 477 44
508 979 286
483 776 71
580 438 6...

output:

502

result:

ok 1 number(s): "502"

Test #3:

score: 0
Accepted
time: 45ms
memory: 3796kb

input:

900 1000
232 890 107
425 399 19
5 74 753
105 333 163
779 42 582
359 647 524
767 409 48
239 780 443
484 489 546
97 634 562
627 866 714
500 357 590
60 728 591
407 686 210
547 32 370
76 772 500
407 584 772
73 699 69
332 847 516
829 754 727
562 756 678
819 303 128
781 667 263
535 672 767
89 762 216
878 ...

output:

801

result:

ok 1 number(s): "801"

Test #4:

score: -100
Runtime Error

input:

500 1000
381 118 230
258 331 21
403 71 207
170 2 125
467 99 6
369 100 492
70 187 352
99 163 123
135 51 352
461 175 486
275 194 236
299 14 19
16 1 68
7 229 316
235 433 320
411 179 463
112 329 326
464 169 52
377 93 51
84 336 335
42 240 379
182 496 344
377 481 195
88 286 491
199 425 165
37 292 44
197 2...

output:


result: