POJ 1651 Multiplication Puzzle （区间dp）

题目大意：对n个数组成的序列取数，规定最两边不能取，每次取一个a[i]，得到 a[l] * a[i] * a[r] 的分数(a[l]是a[i]左边的数,a[r]是a[i]右边的数)，并把这个数从序列中移走，求n-2次取数后的得分和的最小值

分析：正着确定状态不好做，不如反着来，设dp[l][r]为向区间[l, r]中填满数所得到分数和的最小值，考虑最近一次填数的位置，不难得出：

dp[l][r] = fmin(dp[l][m] + dp[m][r] + a[l] * a[m] * a[r]) (l<m<r);

实现方式可以选择记忆化递归，也可直接以区间长度为阶段来递推，各有优缺点，代码如下：

  1 /**********************************************

  2 ***    Problem:

  3 ***    Author:        JKL

  4 ***    University:    CSUST

  5 ***    Team:          __Dream

  6 ***    Email:          [email protected]

  7 ***    My Blog:        http://www.cnblogs.com/jklongint/

  8 ***********************************************/

  9 //===================================================

 10 #include <iostream>

 11 #include <fstream>

 12 #include <sstream>

 13 #include <iomanip>

 14 #include <cstdio>

 15 #include <cstdlib>

 16 #include <cmath>

 17 #include <cassert>

 18 #include <numeric>

 19 #include <ctime>

 20 #include <algorithm>

 21 #include <cstring>

 22 #include <string>

 23 #include <vector>

 24 #include <queue>

 25 #include <map>

 26 #include <stack>

 27 #include <list>

 28 #include <set>

 29 #include <bitset>

 30 #include <deque>

 31 using namespace std;

 32 //---------------------------------------------------

 33 #define mem(a,b) memset(a,b,sizeof(a))

 34 #define GO cout<<"HelloWorld!"<<endl

 35 #define Case(x) cout<<"Case "<<x<<":"

 36 #define foru(i,n) for(int i=1; i <= n; i++)

 37 #define ford(i,n) for(int i = n; i >= 1; i--)

 38  #define fin freopen("input.txt","r",stdin);

 39  #define fout freopen("output.txt","w",stdout)

 40 #define lson  l, m, rt << 1

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

 42 

 43 #define sqr(a)  ((a)*(a))

 44 #define abs(a) ((a>0)?(a):-(a))

 45 #define pii pair<int,int>

 46 

 47 #define fmax(a,b) max(a,b)

 48 #define fmin(a,b) min(a,b)

 49 #define fmax3(a,b,c)  (fmax(a,fmax(a,b)))

 50 #define fmin3(a,b,c)  (fmin(a,fmin(a,b)))

 51 

 52 #define sfi(x) scanf("%d",&x)

 53 #define sfL(x) scanf("%I64d",&x)

 54 #define sfc(x) scanf("%c",&x)

 55 #define sfd(x) scanf("%lf",&x)

 56 #define sfs(x) scanf("%s",x)

 57 #define sfii(a,b) scanf("%d%d",&a,&b)

 58 #define sfLL(a,b) scanf("%I64d%I64d",&a,&b)

 59 #define sfcc(a,b) scanf("%c%c",&a,&b)

 60 #define sfdd(a,b) scanf("%lf%lf",&a,&b)

 61 #define sfss(a,b) scanf("%s%s",a,b)

 62 

 63 #define pfi(x) printf("%d",x)

 64 #define pfL(x) printf("%I64d",x)

 65 #define pfs(x) printf("%s",x)

 66 #define pfd(x) printf("%lf",x)

 67 #define pfc(x) print("%c",x)

 68 #define newLine pfs("\n")

 69 #define space pfs(" ")

 70 

 71 //--------------------------------------------------------

 72 typedef long long LL;

 73 typedef unsigned long long ULL;

 74 typedef __int64 __LL;

 75 typedef unsigned __int64 __ULL;

 76 

 77 typedef vector<int> vi;

 78 typedef vector<LL> vL;

 79 typedef vector<string> vs;

 80 typedef set<int> si;

 81 typedef map<int,int> mii;

 82 typedef map<LL,LL> mLL;

 83 typedef map<string,int> msi;

 84 typedef map<char,int> mci;

 85 //--------------------------------------------------------

 86 const int dx[4]={1,-1,0,0};

 87 const int dy[4]={0,0,1,-1};

 88  const int day[13]={0,31,28,31,30,31,30,31,31,30,31,30,31};

 89  const int N6=1000006;

 90  const int N5=100006;

 91  const int N4=10006;

 92  const int N3=1006;

 93  const int N2=106;

 94  const int N=1009;

 95  const int MOD=1000000007;

 96  const LL LMAX=0x7fffffffffffffff;

 97  const LL IMAX=0x3fffffff;

 98  const double PI=3.14159265359;

 99 //--------------------------------------------------------

100 template< class T > T gcd(T a, T b) { return (b != 0 ? gcd<T>(b, a%b) : a); }

101 template< class T > T lcm(T a, T b) { return (a / gcd<T>(a, b) * b); }

102 

103 //------------------------------------------------------------

104 struct TreeNode{

105     LL  sum, add;

106 };

107 //=================================================================

108 TreeNode tree[N << 3];

109 int flag[N][N], a[N], dp[N][N];

110 int dpSearch(int l, int r)

111 {

112     if(l + 1 == r)return 0;

113     if(flag[l][r])return dp[l][r];

114     dp[l][r] = IMAX;

115     for(int j = l + 1; j < r; j++){

116         int s = dpSearch(l, j), t = dpSearch(j, r);

117         //if(l == 1 && r == 4)cout<< s<< t<< endl;

118         dp[l][r] = fmin(dp[l][r], s + t + a[l] * a[j] * a[r]);

119     }

120     flag[l][r] = 1;

121     //cout<<l<<r<<dp[l][r]<<endl;

122     return dp[l][r];

123 }

124 int main()

125 {

126     //fin;//fout;//freopen("input.txt","r",stdin);

127     int n;

128     cin>>n;

129     foru(i, n)sfi(a[i]);

130     cout << dpSearch(1, n) << endl;

131     return 0;

132 }

View Code