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 LuaJIT (2.0.4) (judge verdict: Runtime Error)

local function uf_initialize(n)
  local parent = {}
  for i = 1, n do parent[i] = i end
  return parent
end

local function uf_findroot(idx, parent)
  local idx_update = idx
  while parent[idx] ~= idx do
    idx = parent[idx]
  end
  while parent[idx_update] ~= idx do
    parent[idx_update], idx_update = idx, parent[idx_update]
  end
  return idx
end

local function kruskal(n, line)
  -- line.c, line.i1, line.i2
  table.sort(line, function(a, b) return a.c < b.c end)
  local parent = uf_initialize(n)
  local totalcost = 0
  local used = 0
  local linenum = #line
  for i = 1, linenum do
    local c, i1, i2 = line[i].c, line[i].i1, line[i].i2
    local r1, r2 = uf_findroot(i1, parent), uf_findroot(i2, parent)
    parent[i1], parent[i2] = r1, r2
    if r1 ~= r2 then
      parent[r2] = r1
      parent[i2] = r1
      totalcost = totalcost + c
      used = used + 1
      if used == n - 1 then break end
    end
  end
  return totalcost
end

local n, m = io.read("*n", "*n")
local edge = {}
for i = 1, n do
  local c = io.read("*n")
  edge[i] = {}
  edge[i].c, edge[i].i1, edge[i].i2 = c, i, n + 1
end
for i = 1, m do
  local a, b, r = io.read("*n", "*n", "*n")
  edge[n + i] = {}
  edge[n + i].c, edge[n + i].i1, edge[n + i].i2 = r, a, b
end
local ret = kruskal(n + 1, edge)
if 99999999999999 < ret then
  print("100000000000000")
else
  print(ret)
end