Greg has a weighed directed graph, consisting of n vertices. In this graph any pair of distinct vertices has an edge between them in both directions. Greg loves playing with the graph and now he has invented a new game:  The game consists of n steps.  On the i-th step Greg removes vertex number x_{i} from the graph. As Greg removes a vertex, he also removes all the edges that go in and out of this vertex.  Before executing each step, Greg wants to know the sum of lengths of the shortest paths between all pairs of the remaining vertices. The shortest path can go through any remaining vertex. In other words, if we assume that d(i, v, u) is the shortest path between vertices v and u in the graph that formed before deleting vertex x_{i}, then Greg wants to know the value of the following sum: $\sum_{v, u, v \neq u} d(i, v, u)$. 

Help Greg, print the value of the required sum before each step.


-----Input-----

The first line contains integer n (1 ≤ n ≤ 500) — the number of vertices in the graph.

Next n lines contain n integers each — the graph adjacency matrix: the j-th number in the i-th line a_{ij} (1 ≤ a_{ij} ≤ 10^5, a_{ii} = 0) represents the weight of the edge that goes from vertex i to vertex j.

The next line contains n distinct integers: x_1, x_2, ..., x_{n} (1 ≤ x_{i} ≤ n) — the vertices that Greg deletes.


-----Output-----

Print n integers — the i-th number equals the required sum before the i-th step.

Please, do not use the %lld specifier to read or write 64-bit integers in C++. It is preferred to use the cin, cout streams of the %I64d specifier.


-----Examples-----
Input
1
0
1

Output
0 
Input
2
0 5
4 0
1 2

Output
9 0 
Input
4
0 3 1 1
6 0 400 1
2 4 0 1
1 1 1 0
4 1 2 3

Output
17 23 404 0

Difficulty: MEDIUM_HARD
Tags: Dynamic programming

Graph algorithms

Shortest paths
Skill types: Dynamic programming

Starter code:

Solutions:
import sys
from array import array

def input():
	return sys.stdin.buffer.readline().decode('utf-8')
n = int(input())
matrix = [array('i', list(map(int, input().split()))) for _ in range(n)]
aa = tuple(map(lambda x: int(x) - 1, input().split()))
ans = [''] * n
for i in range(n - 1, -1, -1):
	x = aa[i]
	for a in range(n):
		for b in range(n):
			if matrix[a][b] > matrix[a][x] + matrix[x][b]:
				matrix[a][b] = matrix[a][x] + matrix[x][b]
	(lower, higher) = (0, 0)
	for a in aa[i:]:
		for b in aa[i:]:
			lower += matrix[a][b]
		if lower > 10 ** 9:
			higher += 1
			lower -= 10 ** 9
	ans[i] = str(10 ** 9 * higher + lower)
print(' '.join(ans))


import sys
from array import array

def input():
	return sys.stdin.buffer.readline().decode('utf-8')

def solve():
	n = int(input())
	matrix = [array('i', list(map(int, input().split()))) for _ in range(n)]
	a = tuple(map(lambda x: int(x) - 1, input().split()))
	ans = [''] * n
	for i in range(n - 1, -1, -1):
		x = a[i]
		for u in range(n):
			for v in range(n):
				if matrix[u][v] > matrix[u][x] + matrix[x][v]:
					matrix[u][v] = matrix[u][x] + matrix[x][v]
		(upper, lower) = (0, 0)
		for u in a[i:]:
			for v in a[i:]:
				lower += matrix[u][v]
			if lower >= 10 ** 9:
				upper += 1
				lower -= 10 ** 9
		ans[i] = str(upper * 10 ** 9 + lower)
	sys.stdout.buffer.write(' '.join(ans).encode('utf-8'))
solve()


import sys
from array import array
n = int(input())
matrix = [array('i', list(map(int, input().split()))) for _ in range(n)]
aa = tuple(map(lambda x: int(x) - 1, input().split()))
ans = [''] * n
for i in range(n - 1, -1, -1):
	x = aa[i]
	for a in range(n):
		for b in range(n):
			if matrix[a][b] > matrix[a][x] + matrix[x][b]:
				matrix[a][b] = matrix[a][x] + matrix[x][b]
	(val, overflow) = (0, 0)
	for a in aa[i:]:
		for b in aa[i:]:
			val += matrix[a][b]
		if val > 10 ** 9:
			overflow += 1
			val -= 10 ** 9
	ans[i] = str(10 ** 9 * overflow + val)
print(' '.join(ans))


import sys
from functools import lru_cache, cmp_to_key
from heapq import merge, heapify, heappop, heappush
from collections import defaultdict as dd, deque, Counter as C
from itertools import combinations as comb, permutations as perm
from bisect import bisect_left as bl, bisect_right as br, bisect
from time import perf_counter
from fractions import Fraction
import copy
import time
starttime = time.time()
mod = int(pow(10, 9) + 7)
mod2 = 998244353
from sys import stdin
input = stdin.readline

def data():
	return sys.stdin.readline().strip()

def out(*var, end='\n'):
	sys.stdout.write(' '.join(map(str, var)) + end)

def L():
	return list(sp())

def sl():
	return list(ssp())

def sp():
	return map(int, data().split())

def ssp():
	return map(str, data().split())

def l1d(n, val=0):
	return [val for i in range(n)]

def l2d(n, m, val=0):
	return [l1d(n, val) for j in range(m)]
try:
	sys.stdin = open('input.txt', 'r')
except:
	pass
from array import array

def input():
	return sys.stdin.buffer.readline().decode('utf-8')

def solve():
	n = int(input())
	matrix = [array('i', list(map(int, input().split()))) for _ in range(n)]
	a = tuple(map(lambda x: int(x) - 1, input().split()))
	ans = [''] * n
	for i in range(n - 1, -1, -1):
		x = a[i]
		for u in range(n):
			for v in range(n):
				if matrix[u][v] > matrix[u][x] + matrix[x][v]:
					matrix[u][v] = matrix[u][x] + matrix[x][v]
		(upper, lower) = (0, 0)
		for u in a[i:]:
			for v in a[i:]:
				lower += matrix[u][v]
			if lower >= 10 ** 9:
				upper += 1
				lower -= 10 ** 9
		ans[i] = str(upper * 10 ** 9 + lower)
	sys.stdout.buffer.write(' '.join(ans).encode('utf-8'))
solve()
endtime = time.time()


import sys
import io
import math
import collections
import decimal
import itertools
import bisect
import heapq
from array import array

def input():
	return sys.stdin.readline()[:-1]
INF = 10 ** 10
N = int(input())
A = [array('i', list(map(int, input().split()))) for _ in range(N)]
X = list(map(lambda x: int(x) - 1, input().split()))
result = []
for k in reversed(range(N)):
	x = X[k]
	for i in range(N):
		for j in range(N):
			d = A[i][x] + A[x][j]
			if d < A[i][j]:
				A[i][j] = d
	v = 0
	overflow = 0
	for i in X[k:]:
		for j in X[k:]:
			v += A[i][j]
			if v > 10 ** 9:
				v -= 10 ** 9
				overflow += 1
	result.append(str(v + 10 ** 9 * overflow))
print(' '.join(reversed(result)))