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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;

typedef long long ll;

const int MAXN = 505;
const ll INF = 1e18;

int N;
ll a[MAXN][MAXN];
ll dp[MAXN][MAXN];

void load() {
	scanf("%d", &N);
	for (int i = 1; i <= N; i++)
		for (int j = 1; j <= N; j++)
			if (j != i)
				scanf("%lld", a[i] + j);
}

ll rectangle(int x1, int x2, int y1, int y2) {
	return a[x2][y2] + a[x1 - 1][y1 - 1] - a[x1 - 1][y2] - a[x2][y1 - 1];
}

ll rec(int l, int r) {
	ll &ref = dp[l][r];
	if (ref != INF)
		return ref;
	if (l == 1)
		return ref = rectangle(l, r, l, r);
	for (int i = 1; i < l; i++)
		ref = min(ref, rec(i, l - 1) + rectangle(l, r, i, r));
	return ref;
}

ll solve() {
	for (int i = 0; i <= N; i++)
		for (int j = 0; j <= N; j++)
			dp[i][j] = INF;
			
	ll sum = 0;
	for (int i = 1; i <= N; i++)
		for (int j = 1; j < i; j++) {
			sum += a[i][j];
			a[i][j] = -a[i][j];
		}
		
	for (int i = 1; i <= N; i++)
		for (int j = 1; j <= N; j++)
			a[i][j] += a[i][j - 1] + a[i - 1][j] - a[i - 1][j - 1];
	
	ll best = INF;
	for (int i = 1; i <= N; i++)
		best = min(best, rec(i, N));
	
	return sum + best;
}

int main() {
	load();
	printf("%lld\n", solve());
	return 0;
}