HDU 2829 Lawrence (斜率优化DP或四边形不等式优化DP)

题意：给定 n 个数，要你将其分成m + 1组，要求每组数必须是连续的而且要求得到的价值最小。一组数的价值定义为该组内任意两个数乘积之和，如果某组中仅有一个数，那么该组数的价值为0。

析：DP状态方程很容易想出来，dp[i][j] 表示前 j 个数分成 i 组。但是复杂度是三次方的，肯定会超时，就要对其进行优化。

有两种方式，一种是斜率对其进行优化，是一个很简单的斜率优化

dp[i][j] = min{dp[i-1][k] - w[k] + sum[k]*sum[k] - sum[k]*sum[j]} + w[j] (i-1<=k<j)。sum[i]表示前i个数之和，w[i]表示前i个数分成一组的价值。

第二种方式就是四边形不等式进行优化，

dp[i][j] = min{dp[i-1][k] + w[k+1][j] } w[i][j] 表示第 i 个数到第 j 个数的价值和。

代码如下：

斜率优化：

#pragma comment(linker, "/STACK:1024000000,1024000000")

#include <cstdio>

#include <string>

#include <cstdlib>

#include <cmath>

#include <iostream>

#include <cstring>

#include <set>

#include <queue>

#include <algorithm>

#include <vector>

#include <map>

#include <cctype>

#include <cmath>

#include <stack>

#include <sstream>

#define debug() puts("++++");

#define gcd(a, b) __gcd(a, b)

#define lson l,m,rt<<1

#define rson m+1,r,rt<<1|1

#define freopenr freopen("in.txt", "r", stdin)

#define freopenw freopen("out.txt", "w", stdout)

using namespace std;

typedef long long LL;

typedef unsigned long long ULL;

typedef pair<int, int> P;

const int INF = 0x3f3f3f3f;

const LL LNF = 1e16;

const double inf = 0x3f3f3f3f3f3f;

const double PI = acos(-1.0);

const double eps = 1e-8;

const int maxn = 1e3 + 10;

const int mod = 1e9 + 7;

const int dr[] = {-1, 0, 1, 0};

const int dc[] = {0, 1, 0, -1};

const char *de[] = {"0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001", "1010", "1011", "1100", "1101", "1110", "1111"};

int n, m;

const int mon[] = {0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

const int monn[] = {0, 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

inline bool is_in(int r, int c){

  return r >= 0 && r < n && c >= 0 && c < m;

}

int dp[maxn][maxn], w[maxn];

int a[maxn], sum[maxn];

int q[maxn];

int getUP(int i, int j, int k){

  return dp[i-1][j] - w[j] + sum[j] * sum[j] - (dp[i-1][k] - w[k] + sum[k] * sum[k]);

}

int getDOWN(int i, int j){

  return sum[i] - sum[j];

}

int getDP(int i, int j){

  return dp[i-1][j] - w[j] + sum[j] * sum[j];

}

int main(){

  while(scanf("%d %d", &n, &m) == 2 && n+m){

    for(int i = 1; i <= n; ++i){

      scanf("%d", a+i);

      sum[i] = sum[i-1] + a[i];

      w[i] = w[i-1] + a[i] * sum[i-1];

      dp[0][i] = w[i];

    }

    for(int i = 1; i <= m; ++i){

      int fro = 0, rear = 0;

      q[++rear] = 0;

      for(int j = 1; j <= n; ++j){

        while(fro + 1 < rear && getUP(i, q[fro+2], q[fro+1]) <= sum[j]*getDOWN(q[fro+2], q[fro+1]))  ++fro;

        dp[i][j] = getDP(i, q[fro+1]) + w[j] - sum[j] * sum[q[fro+1]];

        while(fro + 1 < rear && getUP(i, j, q[rear])*getDOWN(q[rear], q[rear-1]) <= getUP(i, q[rear], q[rear-1])*getDOWN(j, q[rear]))  --rear;

        q[++rear] = j;

      }

    }

    printf("%d\n", dp[m][n]);

  }

  return 0;

}

　　

四边形不等式优化：

#pragma comment(linker, "/STACK:1024000000,1024000000")

#include <cstdio>

#include <string>

#include <cstdlib>

#include <cmath>

#include <iostream>

#include <cstring>

#include <set>

#include <queue>

#include <algorithm>

#include <vector>

#include <map>

#include <cctype>

#include <cmath>

#include <stack>

#include <sstream>

#define debug() puts("++++");

#define gcd(a, b) __gcd(a, b)

#define lson l,m,rt<<1

#define rson m+1,r,rt<<1|1

#define freopenr freopen("in.txt", "r", stdin)

#define freopenw freopen("out.txt", "w", stdout)

using namespace std;

typedef long long LL;

typedef unsigned long long ULL;

typedef pair<int, int> P;

const int INF = 0x3f3f3f3f;

const LL LNF = 1e16;

const double inf = 0x3f3f3f3f3f3f;

const double PI = acos(-1.0);

const double eps = 1e-8;

const int maxn = 1e3 + 10;

const int mod = 1e9 + 7;

const int dr[] = {-1, 0, 1, 0};

const int dc[] = {0, 1, 0, -1};

const char *de[] = {"0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001", "1010", "1011", "1100", "1101", "1110", "1111"};

int n, m;

const int mon[] = {0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

const int monn[] = {0, 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};

inline bool is_in(int r, int c){

  return r >= 0 && r < n && c >= 0 && c < m;

}

LL dp[maxn][maxn], w[maxn][maxn];

int a[maxn], sum[maxn], s[maxn][maxn];

int main(){

  while(scanf("%d %d", &n, &m) == 2 && n+m){

    for(int i = 1; i <= n; ++i){

      scanf("%d", a+i);

      sum[i] = sum[i-1] + a[i];

    }

    for(int i = 1; i <= n; ++i){

      w[i][i] = 0;

      for(int j = i+1; j <= n; ++j)

        w[i][j] = w[i][j-1] + a[j] * (sum[j-1]-sum[i-1]);

    }

    memset(s, 0, sizeof s);

    for(int i = 0; i <= m; ++i)

      fill(dp[i], dp[i]+n+1, LNF);

    for(int i = 1; i <= n; ++i)  dp[0][i] = w[1][i];

    for(int i = 1; i <= m; ++i){

      dp[0][i] = 0;

      s[i][n+1] = n;

      for(int j = n; j >= i; --j)

        for(int k = s[i-1][j]; k <= s[i][j+1]; ++k)

          if(dp[i][j] > dp[i-1][k] + w[k+1][j]){

            dp[i][j] = dp[i-1][k] + w[k+1][j];

            s[i][j] = k;

          }

    }

    printf("%I64d\n", dp[m][n]);

  }

  return 0;

}

　　

HDU 2829 Lawrence (斜率优化DP或四边形不等式优化DP)的相关教程结束。