Problem:
Score : 1200 points

Problem Statement

We have a weighted directed graph with N vertices numbered 0 to N-1.

The graph initially has N-1 edges. The i-th edge (0 \leq i \leq N-2) is directed from Vertex i to Vertex i+1 and has a weight of 0.

Snuke will now add a new edge (i → j) for every pair i, j (0 \leq i,j \leq N-1,\ i \neq j).
The weight of the edge will be -1 if i < j, and 1 otherwise.

Ringo is a boy. A negative cycle (a cycle whose total weight is negative) in a graph makes him sad.
He will delete some of the edges added by Snuke so that the graph will no longer contain a negative cycle.
The cost of deleting the edge (i → j) is A_{i,j}. He cannot delete edges that have been present from the beginning.

Find the minimum total cost required to achieve Ringo's objective.

Constraints

3 \leq N \leq 500

1 \leq A_{i,j} \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_{0,1} A_{0,2} A_{0,3} \cdots A_{0,N-1}
A_{1,0} A_{1,2} A_{1,3} \cdots A_{1,N-1}
A_{2,0} A_{2,1} A_{2,3} \cdots A_{2,N-1}
\vdots
A_{N-1,0} A_{N-1,1} A_{N-1,2} \cdots A_{N-1,N-2}

Output

Print the minimum total cost required to achieve Ringo's objective.

Sample Input 1

3
2 1
1 4
3 3

Sample Output 1

2

If we delete the edge (0 → 1) added by Snuke, the graph will no longer contain a negative cycle.
The cost will be 2 in this case, which is the minimum possible.

Sample Input 2

4
1 1 1
1 1 1
1 1 1
1 1 1

Sample Output 2

2

If we delete the edges (1 → 2) and (3 → 0) added by Snuke, the graph will no longer contain a negative cycle.
The cost will be 1+1=2 in this case, which is the minimum possible.

Sample Input 3

10
190587 2038070 142162180 88207341 215145790 38 2 5 20
32047998 21426 4177178 52 734621629 2596 102224223 5 1864
41 481241221 1518272 51 772 146 8805349 3243297 449
918151 126080576 5186563 46354 6646 491776 5750138 2897 161
3656 7551068 2919714 43035419 495 3408 26 3317 2698
455357 3 12 1857 5459 7870 4123856 2402 258
3 25700 16191 102120 971821039 52375 40449 20548149 16186673
2 16 130300357 18 6574485 29175 179 1693 2681
99 833 131 2 414045824 57357 56 302669472 95
8408 7 1266941 60620177 129747 41382505 38966 187 5151064

Sample Output 3

2280211

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include<iostream>
#include<iomanip>
#include<cstring>
#include<cmath>
#include<cstdio>
#include<queue>
#include<algorithm>
using namespace std;
int read()
{
	char c=getchar();
	int res=0;
	while(c<'0'||c>'9') c=getchar();
	while(c>='0'&&c<='9') res=(res<<1)+(res<<3)+(c^48),c=getchar();
	return res;
}
int n;
int a[505][505];
long long dp[505][505],sum[505][505],sum2[505][505];
long long ask(int k,int j,int i)
{
	return sum[j+1][i]+sum2[n][j]-sum2[n][k]-sum2[i][j]+sum2[i][k];
}
int main()
{
	n=read();
	for(int i=1;i<=n;i++)
	for(int j=1;j<=n;j++) if(i!=j) a[i][j]=read();
	for(int j=1;j<=n;j++)
	for(int i=j;i>=1;i--) sum[i][j]=sum[i+1][j]+sum[i][j-1]-sum[i+1][j-1]+a[i][j];
	for(int i=1;i<=n;i++)
	for(int j=1;j<=n;j++) sum2[i][j]=sum2[i-1][j]+sum2[i][j-1]-sum2[i-1][j-1]+a[i][j];
	memset(dp,0x3f,sizeof(dp));
	for(int i=1;i<n;i++)
	{
		dp[i][0]=ask(0,0,i);
		for(int j=1;j<i;j++)
		for(int k=0;k<j;k++) dp[i][j]=min(dp[i][j],dp[j][k]+ask(k,j,i));
	}
	long long ans=1e18;
	for(int i=1;i<n;i++)
	for(int j=0;j<i;j++) ans=min(ans,dp[i][j]+ask(j,i,n));
	printf("%lld\n",ans);
}