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IDProblemSubmitterResultTimeMemoryLanguageFile sizeSubmit timeJudge time
#110512#6542. Optimal Quadratic Functionhjroh0315WA 19ms4388kbC++203.9kb2023-06-02 18:06:432023-06-02 18:06:45

Judging History

你现在查看的是最新测评结果

  • [2023-08-10 23:21:45]
  • System Update: QOJ starts to keep a history of the judgings of all the submissions.
  • [2023-06-02 18:06:45]
  • 评测
  • 测评结果:WA
  • 用时:19ms
  • 内存:4388kb
  • [2023-06-02 18:06:43]
  • 提交

answer

#include<bits/stdc++.h>
using namespace std;
using lf=long double;

// Two-phase simplex algorithm for solving linear programs of the form
//
//     maximize     c^T x
//     subject to   Ax <= b
//                  x >= 0
// OUTPUT: value of the optimal solution (inf if unbounded
//         above, -inf if infeasible)
//
// To use this code, create an LPSolver object with A, b, and c as
// arguments.  Then, call Solve(x).

template<typename DOUBLE>
struct Simplex_Steep {
  using VD = vector<DOUBLE>;
  using VVD = vector<VD>;
  using VI = vector<int>;
  const DOUBLE EPS = 1e-12L;
  int m, n;
  VI B, N;
  VVD D;

  int iteration_cnt = 0;

  Simplex_Steep(const VVD &A, const VD &b, const VD &c) :
    m(b.size()), n(c.size()), B(m), N(n+1), D(m+2, VD(n+2)) {
    for (int i = 0; i < m; i++) for (int j = 0; j < n; j++) D[i][j] = A[i][j];
    for (int i = 0; i < m; i++) { B[i] = n+i; D[i][n] = -1; D[i][n+1] = b[i]; }
    for (int j = 0; j < n; j++) { N[j] = j; D[m][j] = -c[j]; }
    N[n] = -1; D[m+1][n] = 1;
  }

  void Pivot(int r, int s) {
    for (int i = 0; i < m+2; i++) if (i != r)
      for (int j = 0; j < n+2; j++) if (j != s)
        D[i][j] -= D[r][j] * D[i][s] / D[r][s];
    for (int j = 0; j < n+2; j++) if (j != s) D[r][j] /= D[r][s];
    for (int i = 0; i < m+2; i++) if (i != r) D[i][s] /= -D[r][s];
    D[r][s] = 1.0 / D[r][s];
    swap(B[r], N[s]);
  }

  bool Simplex(int phase) {
    int x = phase == 1 ? m+1 : m;
    while (true) {
      ++iteration_cnt;
      int s = -1;
      DOUBLE c_val = -1;
      for (int j = 0; j <= n; j++) {
        if (phase == 2 && N[j] == -1) continue;
        DOUBLE norm_sq = 0;
        for(int k=0; k<=m;++k){ norm_sq+= D[k][j]*D[k][j];}
        if(norm_sq < EPS) norm_sq = EPS; // stop division by 0
        DOUBLE c_val_j = D[x][j] / sqrtl(norm_sq);
        if (s == -1 || c_val_j < c_val  || (c_val == c_val_j && N[j] < N[s])){
            s = j; c_val = c_val_j;
        }
      }
      if (D[x][s] >= -EPS) return true;
      int r = -1;
      for (int i = 0; i < m; i++) {
        if (D[i][s] <= EPS) continue;
        if (r == -1 || D[i][n+1] / D[i][s] < D[r][n+1] / D[r][s] ||
            (D[i][n+1] / D[i][s] == D[r][n+1] / D[r][s] && B[i] < B[r])) r = i;
      }
      if (r == -1) return false;
      Pivot(r, s);
    }
  }

  DOUBLE solve(VD &x) {
    int r = 0;
    for (int i = 1; i < m; i++) if (D[i][n+1] < D[r][n+1]) r = i;
    if (D[r][n+1] <= -EPS) {
      Pivot(r, n);
      if (!Simplex(1) || D[m+1][n+1] < -EPS) return -numeric_limits<DOUBLE>::infinity();
      for (int i = 0; i < m; i++) if (B[i] == -1) {
        int s = -1;
        for (int j = 0; j <= n; j++)
          if (s == -1 || D[i][j] < D[i][s] || (D[i][j] == D[i][s] && N[j] < N[s])) s = j;
        Pivot(i, s);
      }
    }
    if (!Simplex(2)) return numeric_limits<DOUBLE>::infinity();
    x = VD(n);
    for (int i = 0; i < m; i++) if (B[i] < n) x[B[i]] = D[i][n+1];
    //cerr << "Steepest edge iterations: " << iteration_cnt << "\n";
    return D[m][n+1];
  }
};

int main()
{
    cin.tie(0)->sync_with_stdio(0);
	// simplex with 7 dimensions, each being r, a1, a2, b1, b2, c1, c2 (1 and 2 exist due to bypassing nonnegativity constraints)
    // max -r
    // s.t. -r - a1*x^2 + a2*x^2 - b1*x + b2*x - c1 + c2 <= -y
    //      -r + a1*x^2 - a2*x^2 + b1*x - b2*x + c1 - c2 <= y
    vector<lf>c{-1,0,0,0,0,0,0};
    int q;cin>>q;
    while(q--)
    {
        vector<vector<lf>>A;
        vector<lf>B;
        vector<lf>x;
        int n;cin>>n;
        while(n--)
        {
            lf x,y;
            cin>>x>>y;
            B.push_back(-y);
            A.push_back({-1,-x*x,x*x,-x,x,-1,1});
            B.push_back(y);
            A.push_back({-1,x*x,-x*x,x,-x,1,-1});
        }
        lf ans=Simplex_Steep<lf>(A,B,c).solve(x);
        cout<<setprecision(10)<<fixed<<ans*ans<<"\n";
    }
}

Details

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Test #1:

score: 100
Accepted
time: 2ms
memory: 3596kb

input:

1
4
0 0
1 3
2 9
3 0

output:

5.0625000000

result:

ok found '5.0625000', expected '5.0625000', error '0.0000000'

Test #2:

score: -100
Wrong Answer
time: 19ms
memory: 4388kb

input:

60
1
1000 -990
2
171 -638
949 -99
2
633 227
-257 -602
3
634 -994
633 999
-374 995
3
445 -110
586 -121
462 29
9
-995 -224
-458 -833
691 -670
456 -259
-376 55
-563 -12
834 827
-826 -220
299 744
17
997 991
997 976
997 988
998 -986
999 -982
999 -980
999 -996
998 -988
998 -991
997 987
1000 996
999 -1000
...

output:

0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
543160.1259963981
121.0000000000
0.8320061812
412780.6071794284
12.2500000000
15750.2500000000
118751.3800860984
880245.5054735195
1.0000000000
15.3633733603
854986.7131649867
20.2500000000
24567.6673810630
881031.5630210808
306.250000...

result:

wrong answer 30th numbers differ - expected: '32672008516.0000000', found: '664725471785.0579834', error = '19.3454119'