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# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using System.Diagnostics;
using Debug = System.Diagnostics.Trace;
using SB = System.Text.StringBuilder;
using static System.Math;

using Number = System.Int64;
using static System.Numerics.BigInteger;
namespace Program {
	public class Solver {
		Random rnd = new Random(0);
		public void Solve() {
			var n = ri;
			var a = Enumerate(n, x => new long[n]);
			for (int i = 0; i < n; i++)
				for (int j = 0; j < n; j++)
					if (i != j) a[i][j] = rl;
			for (int i = 0; i < n; i++)
				Debug.WriteLine(a[i].AsJoinedString("\t"));
			var b = Enumerate(n, x => new long[n]);
			for (int i = 0; i < n; i++)
				for (int j = 0; j < i; j++) b[i][j] = a[i][j];
			var A = new RectangleSum(a);
			var B = new RectangleSum(b);
			var dp = Enumerate(n, n, (x, y) => -INF);
			for (int r = 0; r < n; r++) {
				dp[0][r] = B.Sum(0, 0, r + 1, r + 1);
			}
			for (int nl = 1; nl < n; nl++) {
				var r = nl - 1;
				for (int l = 0; l <= r; l++)
					for (int nr = nl; nr < n; nr++) {
						var u = dp[l][r] + A.Sum(0, nl, r + 1, nr + 1) + A.Sum(nl, l, nr + 1, r + 1) + B.Sum(nl, nl, nr + 1, nr + 1);
						dp[nl][nr] = Max(dp[nl][nr], u);
					}
			}
			var sum = 0L;
			for (int i = 0; i < n; i++)
				for (int j = 0; j < n; j++) sum += a[i][j];
			var ans = INF;
			for (int l = 0; l < n; l++) ans = Min(ans, sum - dp[l][n - 1]);
			Console.WriteLine(ans);
		}
		const long INF = 1L << 60;
		static int[] dx = { -1, 0, 1, 0 };
		static int[] dy = { 0, 1, 0, -1 };
		int ri { get { return sc.Integer(); } }
		long rl { get { return sc.Long(); } }
		double rd { get { return sc.Double(); } }
		string rs { get { return sc.Scan(); } }
		public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());
		static T[] Enumerate<T>(int n, Func<int, T> f) {
			var a = new T[n]; for (int i = 0; i < a.Length; ++i) a[i] = f(i); return a;
		}
		static T[][] Enumerate<T>(int n, int m, Func<int, int, T> f) {
			return Enumerate(n, x => Enumerate(m, y => f(x, y)));
		}
		static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
	}
}

#region main
static class Ex {
	static public string AsString(this IEnumerable<char> ie) { return new string(ie.ToArray()); }
	static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") {
		return string.Join(st, ie);
	}
	static public void Main() {
		Console.SetOut(new Program.IO.Printer(Console.OpenStandardOutput()) { AutoFlush = false });
		var solver = new Program.Solver();
		//* 
		var t = new System.Threading.Thread(solver.Solve, 200000000);
		t.Start();
		t.Join();
		//*/
		// solver.Solve();
		Console.Out.Flush();
	}
}
#endregion
#region Ex
namespace Program.IO {
	using System.IO;
	using System.Text;
	using System.Globalization;

	public class Printer : StreamWriter {
		public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
		public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
	}

	public class StreamScanner {
		public StreamScanner(Stream stream) { str = stream; }

		public readonly Stream str;
		private readonly byte[] buf = new byte[1024];
		private int len, ptr;
		public bool isEof = false;
		public bool IsEndOfStream { get { return isEof; } }

		private byte read() {
			if (isEof) return 0;
			if (ptr >= len) {
				ptr = 0;
				if ((len = str.Read(buf, 0, 1024)) <= 0) {
					isEof = true;
					return 0;
				}
			}
			return buf[ptr++];
		}

		public char Char() {
			byte b = 0;
			do b = read(); while ((b < 33 || 126 < b) && !isEof);
			return (char)b;
		}
		public string Scan() {
			var sb = new StringBuilder();
			for (var b = Char(); b >= 33 && b <= 126; b = (char)read()) sb.Append(b);
			return sb.ToString();
		}
		public string ScanLine() {
			var sb = new StringBuilder();
			for (var b = Char(); b != '\n' && b != 0; b = (char)read()) if (b != '\r') sb.Append(b);
			return sb.ToString();
		}
		public long Long() { return isEof ? long.MinValue : long.Parse(Scan()); }
		public int Integer() { return isEof ? int.MinValue : int.Parse(Scan()); }
		public double Double() { return isEof ? double.NaN : double.Parse(Scan(), CultureInfo.InvariantCulture); }
	}
}

#endregion

public class RectangleSum {
	int n, m;
	long[][] s;
	public RectangleSum(long[][] a) {
		n = a.Length;
		m = a.Length;
		s = new long[n + 1][];
		for (int i = 0; i < n + 1; i++)
			s[i] = new long[m + 1];
		for (int i = 0; i < n; i++)
			for (int j = 0; j < m; j++)
				s[i + 1][j + 1] = s[i + 1][j] + s[i][j + 1] - s[i][j] + a[i][j];
	}
	public RectangleSum(int[][] a) {
		var n = a.Length;
		var m = a.Length;
		s = new long[n + 1][];
		for (int i = 0; i < n + 1; i++)
			s[i] = new long[m + 1];
		for (int i = 0; i < n; i++)
			for (int j = 0; j < m; j++)
				s[i + 1][j + 1] = s[i + 1][j] + s[i][j + 1] - s[i][j] + a[i][j];
	}
	/// <summary>
	/// [t,b),[l,r) で作られる矩形領域の和を求める
	/// </summary>
	public long Sum(int top, int left, int bottom, int right) {
		if (top < 0 || left < 0 || bottom > n || right > m || top >= bottom || left >= right) return 0;
		return s[bottom][right] - s[top][right] - s[bottom][left] + s[top][left];
	}
}