"Hdoj" "2829" Lawrence

dp/Quadrilateral Inequalities

After the POJ 1739 Post office that question, it is easy to write out the rules of the equation:

DP[I][J]=MIN{DP[I-1][K]+W[K+1][J]} (indicates the minimum cost of dividing the first J points into blocks i)

$w (l,r) =\sum_{i=l}^{r}\sum_{j=i+1}^{r}a[i]*a[j]$

Then there is $w (l,r+1) =w (l,r) +a[j]*\sum\limits_{i=l}^{r}a[i]$

So: W[i][j] clearly satisfies the monotonicity of the interval inclusions

Then we boldly guessed that the small (BU) Heart (Yong) proves that w[i][j] satisfies the quadrilateral inequality, so the question is the same as the post office.

All right, well, as a moral integrity, I try to prove it (the result is that it takes longer to prove than I write the code ...). ）

First move the definition of W (i,j) Down: \[W (l,r) =\sum\limits_{i=l}^{r}\sum\limits_{j=i+1}^{r}a[i]*a[j] \]

The image is:

For $ i\leq i ' < J \leq J ' $

The middle is counted two times, the remainder:

(left) represents W (i,i '-1), [left] represents $\sum_{k=i}^{i ' -1}a[k] $

(medium) means w (i ', j), [medium] means $\sum_{k=i '}^j A[k] $

(right) means W (j+1,j '), [right] means $\sum_{k=j+1}^{j '} A[k] $

\[W (i,j) +w (i ', j ') = (left) +[left]*[]+ (right) +[right]*[] \ \ w (i,j ') +w (i ', j) = (left + right) +[left + right]*[] \]

　　　　

where \[[left + right]*[in]=[left]*[]+[right]*[] \]

But \[(left + right) = (left) + (right) +[left]*[right] \]

So \[(left + right) > (left) + (right) \]

So \[w (i,j) +w (i ', J ') \leq W (i,j ') +w (i ', j) \]

1 //hdoj 28292#include <cmath>3#include <vector>4#include <cstdio>5#include <cstring>6#include <cstdlib>7#include <iostream>8#include <algorithm>9 #defineRep (i,n) for (int i=0;i<n;++i)Ten #defineF (i,j,n) for (int i=j;i<=n;++i) One #defineD (i,j,n) for (int i=j;i>=n;--i) A #definePB Push_back - #defineCC (A, B) memset (A,b,sizeof (a)) - using namespacestd; the intGetint () { -     intv=0, sign=1;CharCh=GetChar (); -      while(!isdigit (CH)) {if(ch=='-') sign=-1; Ch=GetChar ();} -      while(IsDigit (CH)) {v=v*Ten+ch-'0'; Ch=GetChar ();} +     returnv*Sign ; - } + Const intn=1010, inf=~0u>>2; A Const Doubleeps=1e-8; at #defineDebug - /*******************template********************/ - intdp[n][n],s[n][n],w[n][n],b[n],a[n],n,m; -  - intMain () { -      while(SCANF ("%d%d", &n,&m)!=eof &&N) { inm++; -F (I,1, N) a[i]=getint (); toF (I,1, N) { +b[i]=A[i]; -w[i][i]=0; theF (j,i+1, N) { *w[i][j]=w[i][j-1]+a[j]*B[i]; $b[i]+=A[j];Panax Notoginseng             } -         } theF (I,1, N) F (J,1, m) dp[j][i]=INF; +F (I,1, N) { Adp[1][i]=w[1][i]; thes[1][i]=0; +         } -F (I,2, M) { $s[i][n+1]=N; $ D (j,n,i) -F (k,s[i-1][j],s[i][j+1]) -                     if(dp[i-1][k]+w[k+1][j]<Dp[i][j]) { thes[i][j]=K; -dp[i][j]=dp[i-1][k]+w[k+1][j];Wuyi                     } the         } -printf"%d\n", Dp[m][n]); Wu     } -     return 0; About}

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"Hdoj" "2829" Lawrence