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

-----Sample Output-----
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.

Difficulty: competition

Starter code:

Solutions:
def main():
    import sys
    from array import array
    input = sys.stdin.readline

    class Bit:
        def __init__(self, n):
            self.size = n
            self.size_bit_length = n.bit_length()
            self.tree = array('h', [0] * (n+1))

        def reset(self):
            self.tree = array('h', [0] * (self.size+1))

        def sum(self, i):
            s = 0
            while i > 0:
                s += self.tree[i]
                i -= i & -i
            return s

        def add(self, i, x):
            while i <= self.size:
                self.tree[i] += x
                i += i & -i

        def lower_bound(self, w):
            if w <= 0:
                return 0
            x = 0
            k = 1 << (self.size_bit_length - 1)
            while k:
                if x + k <= self.size and self.tree[x + k] < w:
                    w -= self.tree[x + k]
                    x += k
                k >>= 1
            return x + 1

    N, M = list(map(int, input().split()))
    dist = [0] + list(map(int, input().split()))
    for i in range(N-1):
        dist[i+1] += dist[i]
    B = [0] * (M * N)
    for i in range(N):
        BB = list(map(int, input().split()))
        for j in range(M):
            B[j * N + i] = BB[j] * (N+1) + i+1

    imos = []
    for i in range(N+1):
        imos.append([0] * (N+1 - i))
    bit = Bit(N)
    for m in range(M):
        bit.reset()
        for bi in sorted(B[m*N: (m+1) * N], reverse=True):
            b, i = divmod(bi, N+1)
            k = bit.sum(i)
            l = bit.lower_bound(k)
            r = bit.lower_bound(k+1)
            imos[l+1][i - (l+1)] += b
            if i != N:
                imos[i+1][0] -= b
            if r != N+1:
                imos[l+1][r - (l+1)] -= b
            if i != N and r != N+1:
                imos[i+1][r - (i+1)] += b
            bit.add(i, 1)

    for i in range(1, N+1):
        for j in range(i+1, N+1):
            imos[i][j - i] += imos[i][j-1-i]
    for i in range(2, N + 1):
        for j in range(i, N + 1):
            imos[i][j-i] += imos[i - 1][j - (i-1)]
    ans = 0
    for i in range(1, N + 1):
        for j in range(i, N + 1):
            ans = max(ans, imos[i][j-i] - (dist[j - 1] - dist[i - 1]))
    print(ans)


def __starting_point():
    main()

__starting_point()

import sys
readline = sys.stdin.readline

N, M = map(int, readline().split())
A = [0] + list(map(int, readline().split()))
for i in range(1, N):
    A[i] += A[i-1]

INF = 10**9+7
B = [list(map(int, readline().split())) for _ in range(N)] + [[INF]*M] + [[0]*M]
right = [[0]*M for _ in range(N)]

for m in range(M):
    L = list(range(N))
    L.sort(key = lambda x: B[x][m])
    RR = list(range(1, N+3))
    LL = list(range(-1, N+1))
    for vn in L:
        vn += 1
        right[vn-1][m] = RR[vn]-1
        RR[LL[vn]] = RR[vn]
        LL[RR[vn]] = LL[vn]

diff = [0]*(N+1)
for m in range(M):
    diff[0] += B[0][m]
    pre = B[0][m]
    for i in range(1, N):
        if B[i][m] > pre:
            diff[i] += B[i][m]-pre
            pre = B[i][m]

offset = 0
ans = 0
for l in range(N):
    offset += diff[l]
    for m in range(M):
        if B[l][m] < B[l-1][m]:
            oldr = right[l-1][m]
            diff[oldr] -= B[oldr][m] - B[l-1][m]
            offset += B[l][m] - B[l-1][m]
            cnt = l
            while cnt < N and right[cnt][m] <= oldr:
                rcm = right[cnt][m]
                diff[rcm] += B[rcm][m] - B[cnt][m]
                cnt = rcm
    ans = max(ans, offset)
    roff = offset
    for r in range(l+1, N):
        roff += diff[r]
        ans = max(ans, roff - (A[r]-A[l]))
        #print(l, r, roff - (A[r] - A[l]))

print(ans)