Problem:
Score : 1000 points

Problem Statement

There are N barbecue restaurants along a street.
The restaurants are numbered 1 through N from west to east, and the distance between restaurant i and restaurant i+1 is A_i.

Joisino has M tickets, numbered 1 through M.
Every barbecue restaurant offers barbecue meals in exchange for these tickets.
Restaurant i offers a meal of deliciousness B_{i,j} in exchange for ticket j.
Each ticket can only be used once, but any number of tickets can be used at a restaurant.

Joisino wants to have M barbecue meals by starting from a restaurant of her choice, then repeatedly traveling to another barbecue restaurant and using unused tickets at the restaurant at her current location.
Her eventual happiness is calculated by the following formula: "(The total deliciousness of the meals eaten) - (The total distance traveled)".
Find her maximum possible eventual happiness.

Constraints

All input values are integers.

2≤N≤5×10^3

1≤M≤200

1≤A_i≤10^9

1≤B_{i,j}≤10^9

Input

The input is given from Standard Input in the following format:

N M
A_1 A_2 ... A_{N-1}
B_{1,1} B_{1,2} ... B_{1,M}
B_{2,1} B_{2,2} ... B_{2,M}
:
B_{N,1} B_{N,2} ... B_{N,M}

Output

Print Joisino's maximum possible eventual happiness.

Sample Input 1

3 4
1 4
2 2 5 1
1 3 3 2
2 2 5 1

Sample Output 1

11

The eventual happiness can be maximized by the following strategy: start from restaurant 1 and use tickets 1 and 3, then move to restaurant 2 and use tickets 2 and 4.

Sample Input 2

5 3
1 2 3 4
10 1 1
1 1 1
1 10 1
1 1 1
1 1 10

Sample Output 2

20

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Memory Limit Exceeded)

import java.io.IOException;
import java.util.*;
import java.io.ByteArrayInputStream;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.InputMismatchException;

public class Main {
	public static void main(String[] args) throws Exception {
		new Main().run();
	}

	void solve() {
		int N = ni();
		int M = ni();
		long[] A = new long[N];// 8*N<=4*10^4=40KB
		for (int i = 1; i < N; ++i) {
			A[i] = ni();
		}
		int[][] B = new int[N][M];// (4*N)*(4*M)<=40KB*800B=32MB
		for (int i = 0; i < N; ++i) {
			for (int j = 0; j < M; ++j) {
				B[i][j] = ni();
			}
		}
		int[][] a = new int[N][2];// 4*2*N<=80KB
		long[][] sum = new long[N + 1][N + 1];// 1600MB
		int[] mate = new int[N];// 4*N<=40KB
		boolean[] vis = new boolean[N];// 4*N<=40KB
		for (int i = 0; i < M; ++i) {
			if (Runtime.getRuntime().freeMemory() / 1000 < 10) {
				System.gc();
			}
			for (int j = 0; j < N; ++j) {
				a[j][0] = j;
				a[j][1] = B[j][i];
			}
			Arrays.sort(a, new Comparator<int[]>() {
				@Override
				public int compare(int[] o1, int[] o2) {
					return Integer.compare(o1[1], o2[1]);
				}
			});
			Arrays.fill(vis, false);
			for (int j = 0; j < N; ++j) {
				mate[j] = j;
			}
			for (int j = 0; j < N; ++j) {
				int idx = a[j][0];
				int l = idx, r = idx;
				if (idx > 0 && vis[idx - 1]) {
					l = mate[idx - 1];
				}
				if (idx + 1 < N && vis[idx + 1]) {
					r = mate[idx + 1];
				}
				sum[l][idx] += a[j][1];
				sum[idx + 1][r + 1] += a[j][1];
				sum[l][r + 1] -= a[j][1];
				sum[idx + 1][idx] -= a[j][1];
				vis[idx] = true;
				if (idx > 0 && idx + 1 < N && vis[idx - 1] && vis[idx + 1]) {
					mate[l] = r;
					mate[r] = l;
				} else if (idx > 0 && vis[idx - 1]) {
					mate[l] = idx;
					mate[idx] = l;
				} else if (idx + 1 < N && vis[idx + 1]) {
					mate[r] = idx;
					mate[idx] = r;
				}
			}
		}
		for (int i = 0; i < N; ++i) {
			for (int j = 0; j < N; ++j) {
				if (j > 0)
					sum[i][j] += sum[i][j - 1];
			}
		}
		for (int i = 0; i < N; ++i) {
			for (int j = 0; j < N; ++j) {
				if (j > 0)
					sum[j][i] += sum[j - 1][i];
			}
		}
		for (int i = 0; i < N; ++i) {
			if (i > 0)
				A[i] += A[i - 1];
		}
		long best = 0;
		for (int i = 0; i < N; ++i) {
			for (int j = i; j < N; ++j) {
				best = Math.max(best, sum[i][j] - (A[j] - A[i]));
			}
		}
		out.println(best);
	}

	static void tr(Object... objects) {
		System.out.println(Arrays.deepToString(objects));
	}

	InputStream is;
	PrintWriter out;
	String INPUT = "";

	void run() throws Exception {
		is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
		out = new PrintWriter(System.out);

		long s = System.currentTimeMillis();
		solve();
		out.flush();
		if (!INPUT.isEmpty())
			tr(System.currentTimeMillis() - s + "ms");
		// Thread t = new Thread(null, null, "~",
		// Runtime.getRuntime().maxMemory()){
		// @Override
		// public void run() {
		// long s = System.currentTimeMillis();
		// solve();
		// out.flush();
		// if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
		// }
		// };
		// t.start();
		// t.join();
	}

	private byte[] inbuf = new byte[1024];
	public int lenbuf = 0, ptrbuf = 0;

	private int readByte() {
		if (lenbuf == -1)
			throw new InputMismatchException();
		if (ptrbuf >= lenbuf) {
			ptrbuf = 0;
			try {
				lenbuf = is.read(inbuf);
			} catch (IOException e) {
				throw new InputMismatchException();
			}
			if (lenbuf <= 0)
				return -1;
		}
		return inbuf[ptrbuf++];
	}

	private boolean isSpaceChar(int c) {
		return !(c >= 33 && c <= 126);
	}

	private int skip() {
		int b;
		while ((b = readByte()) != -1 && isSpaceChar(b))
			;
		return b;
	}

	private double nd() {
		return Double.parseDouble(ns());
	}

	private char nc() {
		return (char) skip();
	}

	private String ns() {
		int b = skip();
		StringBuilder sb = new StringBuilder();
		while (!(isSpaceChar(b))) { // when nextLine, (isSpaceChar(b) && b != '
									// ')
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}

	private char[] ns(int n) {
		char[] buf = new char[n];
		int b = skip(), p = 0;
		while (p < n && !(isSpaceChar(b))) {
			buf[p++] = (char) b;
			b = readByte();
		}
		return n == p ? buf : Arrays.copyOf(buf, p);
	}

	private int[] na(int n) {
		int[] a = new int[n];
		for (int i = 0; i < n; i++)
			a[i] = ni();
		return a;
	}

	private long[] nal(int n) {
		long[] a = new long[n];
		for (int i = 0; i < n; i++)
			a[i] = nl();
		return a;
	}

	private char[][] nm(int n, int m) {
		char[][] map = new char[n][];
		for (int i = 0; i < n; i++)
			map[i] = ns(m);
		return map;
	}

	private int[][] nmi(int n, int m) {
		int[][] map = new int[n][];
		for (int i = 0; i < n; i++)
			map[i] = na(m);
		return map;
	}

	private int ni() {
		return (int) nl();
	}

	private long nl() {
		long num = 0;
		int b;
		boolean minus = false;
		while ((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'))
			;
		if (b == '-') {
			minus = true;
			b = readByte();
		}

		while (true) {
			if (b >= '0' && b <= '9') {
				num = num * 10 + (b - '0');
			} else {
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
}