Problem:
Score : 1000 points

Problem Statement

Problem F and F2 are the same problem, but with different constraints and time limits.

We have a board divided into N horizontal rows and N vertical columns of square cells.
The cell at the i-th row from the top and the j-th column from the left is called Cell (i,j).
Each cell is either empty or occupied by an obstacle.
Also, each empty cell has a digit written on it.
If A_{i,j}= 1, 2, ..., or 9, Cell (i,j) is empty and the digit A_{i,j} is written on it.
If A_{i,j}= #, Cell (i,j) is occupied by an obstacle.

Cell Y is reachable from cell X when the following conditions are all met:

Cells X and Y are different.

Cells X and Y are both empty.

One can reach from Cell X to Cell Y by repeatedly moving right or down to an adjacent empty cell.

Consider all pairs of cells (X,Y) such that cell Y is reachable from cell X.
Find the sum of the products of the digits written on cell X and cell Y for all of those pairs.

Constraints

1 \leq N \leq 500

A_{i,j} is one of the following characters: 1, 2, ... 9 and #.

Input

Input is given from Standard Input in the following format:

N
A_{1,1}A_{1,2}...A_{1,N}
A_{2,1}A_{2,2}...A_{2,N}
:
A_{N,1}A_{N,2}...A_{N,N}

Output

Print the sum of the products of the digits written on cell X and cell Y for all pairs (X,Y) such that cell Y is reachable from cell X.

Sample Input 1

2
11
11

Sample Output 1

5

There are five pairs of cells (X,Y) such that cell Y is reachable from cell X, as follows:

X=(1,1), Y=(1,2)

X=(1,1), Y=(2,1)

X=(1,1), Y=(2,2)

X=(1,2), Y=(2,2)

X=(2,1), Y=(2,2)

The product of the digits written on cell X and cell Y is 1 for all of those pairs, so the answer is 5.

Sample Input 2

4
1111
11#1
1#11
1111

Sample Output 2

47

Sample Input 3

10
76##63##3#
8445669721
75#9542133
3#285##445
749632##89
2458##9515
5952578#77
1#3#44196#
4355#99#1#
#298#63587

Sample Output 3

36065

Sample Input 4

10
4177143673
7#########
5#1716155#
6#4#####5#
2#3#597#6#
6#9#8#3#5#
5#2#899#9#
1#6#####6#
6#5359657#
5#########

Sample Output 4

6525

Solution in Go (1.6) (judge verdict: Time Limit Exceeded)

package main
import "fmt"
var n int
var a,g [][]int
var u []bool
func main() {
  var s string
  fmt.Scan(&n)
  a = make([][]int,n+1)
  a[n] = make([]int,n+1)
  for j:=0;j<n+1;j++ { a[n][j] = -13 }
  for i:=0;i<n;i++ {
    a[i] = make([]int,n+1)
    fmt.Scan(&s)
    s += "#"
    for j,r := range s { a[i][j] = int(r)-48 }
  }
  g = make([][]int,n*n)
  for i:=0;i<n;i++ {
    for j:=0;j<n;j++ {
      if a[i][j] < 0 { continue }
      x := en(i,j)
      if a[i][j+1] > 0 { g[x] = append(g[x],en(i,j+1)) }
      if a[i+1][j] > 0 { g[x] = append(g[x],en(i+1,j)) }
    }
  }
  ans := 0
  u = make([]bool,n*n)
  for k:=0;k<2*n-1;k++ {
    for i:=0;i<=k;i++ {
      if i >= n { break }
      j := k-i
      if j >= n { continue }
      if a[i][j] < 0 { continue }
      for t:=0;t<n*n;t++ { u[t] = false }
      ans += (dfs(en(i,j))-a[i][j])*a[i][j]
    }
  }
  fmt.Println(ans)
}
func dfs(x int) int {
  u[x] = true
  i,j := de(x)
  s := 0
  for _,y := range g[x] {
    if u[y] { continue }
    s += dfs(y)
  }
  return s+a[i][j]
}
func en(i,j int) int { return i*n+j }
func de(x int) (int,int) { return x/n,x%n }