Submission #1301432

#TimeUsernameProblemLanguageResultExecution timeMemory
1301432BLOBVISGODKnapsack (NOI18_knapsack)C++20
73 / 100
189 ms327680 KiB
#include "bits/stdc++.h"
using namespace std;
#define rep(i,a,b) for(int i=(a); i<(b); ++i)
#define all(x) x.begin(),x.end()
#define sz(x) int(x.size())
typedef long long ll;
typedef unsigned long long ull;
typedef vector<int> vi;
typedef vector<vi> vvi;

typedef long double T; // long double, Rational, double + mod<P>...
typedef vector<T> vd;
typedef vector<vd> vvd;

const T eps = 1e-8, inf = 1/.0;
#define MP make_pair
#define ltj(X) if(s == -1 || MP(X[j],N[j]) < MP(X[s],N[s])) s=j

struct LPSolver {
	int m, n;
	vi N, B;
	vvd D;

	LPSolver(const vvd& A, const vd& b, const vd& c) :
		m(sz(b)), n(sz(c)), N(n+1), B(m), D(m+2, vd(n+2)) {
			rep(i,0,m) rep(j,0,n) D[i][j] = A[i][j];
			rep(i,0,m) { B[i] = n+i; D[i][n] = -1; D[i][n+1] = b[i];}
			rep(j,0,n) { N[j] = j; D[m][j] = -c[j]; }
			N[n] = -1; D[m+1][n] = 1;
		}

	void pivot(int r, int s) {
		T *a = D[r].data(), inv = 1 / a[s];
		rep(i,0,m+2) if (i != r && abs(D[i][s]) > eps) {
			T *b = D[i].data(), inv2 = b[s] * inv;
			rep(j,0,n+2) b[j] -= a[j] * inv2;
			b[s] = a[s] * inv2;
		}
		rep(j,0,n+2) if (j != s) D[r][j] *= inv;
		rep(i,0,m+2) if (i != r) D[i][s] *= -inv;
		D[r][s] = inv;
		swap(B[r], N[s]);
	}

	bool simplex(int phase) {
		int x = m + phase - 1;
		for (;;) {
			int s = -1;
			rep(j,0,n+1) if (N[j] != -phase) ltj(D[x]);
			if (D[x][s] >= -eps) return true;
			int r = -1;
			rep(i,0,m) {
				if (D[i][s] <= eps) continue;
				if (r == -1 || MP(D[i][n+1] / D[i][s], B[i])
				             < MP(D[r][n+1] / D[r][s], B[r])) r = i;
			}
			if (r == -1) return false;
			pivot(r, s);
		}
	}

	T solve(vd &x) {
		int r = 0;
		rep(i,1,m) if (D[i][n+1] < D[r][n+1]) r = i;
		if (D[r][n+1] < -eps) {
			pivot(r, n);
			if (!simplex(2) || D[m+1][n+1] < -eps) return -inf;
			rep(i,0,m) if (B[i] == -1) {
				int s = 0;
				rep(j,1,n+1) ltj(D[i]);
				pivot(i, s);
			}
		}
		bool ok = simplex(1); x = vd(n);
		rep(i,0,m) if (B[i] < n) x[B[i]] = D[i][n+1];
		return ok ? D[m][n+1] : inf;
	}
};

struct ILP{
    int n, m;
    vvd A; vd b, c, lo, hi; T ans = -inf;

    ILP(vvd A, vd b, vd c) : n(sz(b)), m(sz(c)), 
        A(A), b(b), c(c), lo(m), hi(m,inf) { }

    void solve(){
        T bal = 0;
        rep(i,0,m) {
            bal += c[i]*lo[i];
            rep(j,0,n) b[j] -= A[j][i]*lo[i];
            if (hi[i] < inf){
                vd e(m); e[i] = 1;
                A.push_back(e);
                b.push_back(hi[i]-lo[i]);
            }
        }
        LPSolver lp(A, b, c);
        vd x(m), sm(n); T res = floor(bal + lp.solve(x) + .1), me = bal;
        A.resize(n), b.resize(n);
        pair<T, pair<ll,ll>> w = {};
        rep(i,0,m){
            T r = floor(x[i] + eps), d = x[i] - r;
            w = max(w,{min(d,1-d),{i,r}}), me += r * c[i];
            rep(j,0,n) d = A[j][i] * lo[i], 
                b[j] += d, sm[j] += r * A[j][i] + d;
        }
        rep(i,0,n) if (sm[i] > b[i]) me = -inf;
        ans = max(ans, me);
        if (res > ans and w.first > eps) {
            auto [i,v] = w.second; T mx = hi[i];
            hi[i] = lo[i] + v, solve();
            hi[i] = mx, lo[i] += v + 1, solve();
            lo[i] -= v + 1;
        } 
    }
};

int main(){
    cin.tie(NULL),ios::sync_with_stdio(false);
    
    int s; cin >> s;
    int n; cin >> n;

    vvd A(1,vd(n));
    vd b(1,s);
    vd c(n);
    vi cnts(n);

    rep(i,0,n){
        int v,w,k; cin >> v >> w >> k;
        c[i] = v;
        A[0][i] = w;
        cnts[i] = k;
    }

    ILP ilp(A,b,c);
    rep(i,0,n) ilp.hi[i] = cnts[i];

    ilp.solve();

    cout << llround(ilp.ans) << '\n';
}
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