You are given a table, $a$, with $n$ rows and $m$ columns. The top-left corner of the table has coordinates $(0,0)$, and the bottom-right corner has coordinates $(n-1,m-1)$. The $i^{\mbox{th}}$ cell contains integer $a_{i,j}$.

A path in the table is a sequence of cells $(r_1,c_1),(r_2,c_2),\ldots,(r_k,c_k)$ such that for each $i\in\{1,\ldots,k-1\}$, cell $(r_{i},c_{i})$ and cell $(r_{i+1},c_{i+1})$ share a side. 

The weight of the path $(r_1,c_1),(r_2,c_2),\ldots,(r_k,c_k)$ is defined by $\sum_{i=1}^{k}a_{r_{i},c_{i}}$ where $a_{r_i,c_i}$ is the weight of the cell $(r_{i},c_{i})$.

You must answer $q$ queries. In each query, you are given the coordinates of two cells, $(r_1,c_1)$ and $(r_2,c_2)$. You must find and print the minimum possible weight of a path connecting them.

Note: A cell can share sides with at most $4$ other cells. A cell with coordinates $(r,c)$ shares sides with $(r-1,c)$, $(r+1,c)$, $(r,c-1)$ and $(r,c+1)$.

Input Format

The first line contains $2$ space-separated integers, $n$ (the number of rows in $a$) and $m$ (the number of columns in $a$), respectively. 

Each of $n$ subsequent lines contains $m$ space-separated integers. The $j^{th}$ integer in the $i^{\mbox{th}}$ line denotes the value of $a_{i,j}$. 

The next line contains a single integer, $\textit{q}$, denoting the number of queries. 

Each of the $\textit{q}$ subsequent lines describes a query in the form of $A$ space-separated integers: $r_1$, $c_{1}$, $r_2$, and $c_2$, respectively. 

Constraints

$1\leq n\leq7$
$1\leq m\leq5\times10^3$
$0\leq a_{i,j}\leq3\times10^3$
$1\leq q\leq3\times10^4$

For each query:

$0\leq r_{1},r_{2}<n$
$0\leq c_1,c_2\lt m$

Output Format

On a new line for each query, print a single integer denoting the minimum possible weight of a path between $(r_1,c_1)$ and $(r_2,c_2)$.

Sample Input
3 5
0 0 0 0 0
1 9 9 9 1
0 0 0 0 0
3
0 0 2 4
0 3 2 3
1 1 1 3

Sample Output
1
1
18

Explanation

The input table looks like this:

The first two queries are explained below:

In the first query, we have to find the minimum possible weight of a path connecting $(0,0)$ and $(2,4)$. Here is one possible path:

The total weight of the path is $0+1+0+0+0+0+0=1$.

In the second query, we have to find the minimum possible weight of a path connecting $(0,3)$ and $(2,3)$. Here is one possible path:

The total weight of the path is $0+0+1+0+0=1$.

Source: hackerrank | Difficulty: MEDIUM_HARD
Tags: Graph algorithms


Solutions:
def heuristic(board, r1, c1, r2, c2):
	return 0

def astar(board, r1, c1, r2, c2):
	open_nodes = {(r1, c1): (board[r1][c1], board[r1][c1] + heuristic(board, r1, c1, r2, c2))}
	closed_nodes = set()
	while open_nodes:
		(cur_est, cur_pos, cur_cost) = min(((est, pos, cost) for (pos, (cost, est)) in open_nodes.items()))
		del open_nodes[cur_pos]
		if cur_pos == (r2, c2):
			return cur_cost
		(r, c) = cur_pos
		neighbours = []
		if r > 0:
			neighbours.append((r - 1, c))
		if c > 0:
			neighbours.append((r, c - 1))
		if r < len(board) - 1:
			neighbours.append((r + 1, c))
		if c < len(board[0]) - 1:
			neighbours.append((r, c + 1))
		for nb_pos in neighbours:
			(r, c) = nb_pos
			cost = cur_cost + board[r][c]
			est = cost + heuristic(board, r, c, r2, c2)
			if nb_pos in closed_nodes:
				continue
			if nb_pos in open_nodes and open_nodes[nb_pos][0] <= cost:
				continue
			open_nodes[nb_pos] = (cost, est)
		closed_nodes.add(cur_pos)
(n, m) = map(int, input().strip().split())
board = []
for _ in range(n):
	board.append(list(map(int, input().strip().split())))
q = int(input())
for _ in range(q):
	(r1, c1, r2, c2) = map(int, input().strip().split())
	print(astar(board, r1, c1, r2, c2))


import sys
from collections import deque
from math import inf
from heapq import heappush, heappop

def shortestPath(matrix, query, numRows, numCols):
	start = (query[0], query[1])
	goal = (query[2], query[3])
	pathWeight = matrix[start]
	pathQueue = []
	heappush(pathQueue, [pathWeight, start[0], start[1]])
	seen = {start: pathWeight}
	minWeight = inf
	while pathQueue:
		pathCell = heappop(pathQueue)
		(row, col) = (pathCell[1], pathCell[2])
		pathWeight = pathCell[0]
		if (row, col) == goal:
			if pathWeight < minWeight:
				minWeight = pathWeight
			continue
		if pathWeight >= minWeight:
			continue
		for (newRow, newCol) in ((row + 1, col), (row - 1, col), (row, col + 1), (row, col - 1)):
			if 0 <= newRow < numRows and 0 <= newCol < numCols:
				newPathWeight = pathWeight + matrix[newRow, newCol]
				if (newRow, newCol) in seen:
					if seen[newRow, newCol] <= newPathWeight:
						continue
				heappush(pathQueue, [newPathWeight, newRow, newCol])
				seen[newRow, newCol] = newPathWeight
	return minWeight
(m, n) = input().strip().split(' ')
(m, n) = [int(m), int(n)]
matrix = {}
for matrixRow in range(m):
	matrixRowList = [int(weight) for weight in input().strip().split(' ')]
	for matrixCol in range(n):
		matrix[matrixRow, matrixCol] = matrixRowList[matrixCol]
q = int(input().strip())
for queries in range(q):
	query = [int(query_temp) for query_temp in input().strip().split(' ')]
	print(shortestPath(matrix, query, m, n))


def heuristic(board, r1, c1, r2, c2):
	return 0

def astar(board, r1, c1, r2, c2):
	open_nodes = {(r1, c1): (board[r1][c1], board[r1][c1] + heuristic(board, r1, c1, r2, c2))}
	closed_nodes = set()
	while open_nodes:
		(cur_est, cur_pos, cur_cost) = min(((est, pos, cost) for (pos, (cost, est)) in open_nodes.items()))
		del open_nodes[cur_pos]
		if cur_pos == (r2, c2):
			return cur_cost
		(r, c) = cur_pos
		neighbours = []
		if r > 0:
			neighbours.append((r - 1, c))
		if c > 0:
			neighbours.append((r, c - 1))
		if r < len(board) - 1:
			neighbours.append((r + 1, c))
		if c < len(board[0]) - 1:
			neighbours.append((r, c + 1))
		for nb_pos in neighbours:
			(r, c) = nb_pos
			cost = cur_cost + board[r][c]
			est = cost + heuristic(board, r, c, r2, c2)
			if nb_pos in closed_nodes:
				continue
			if nb_pos in open_nodes and open_nodes[nb_pos][0] <= cost:
				continue
			open_nodes[nb_pos] = (cost, est)
		closed_nodes.add(cur_pos)
(n, m) = map(int, input().strip().split())
board = []
for _ in range(n):
	board.append(list(map(int, input().strip().split())))
q = int(input())
for _ in range(q):
	(r1, c1, r2, c2) = map(int, input().strip().split())
	print(astar(board, r1, c1, r2, c2))


def heuristic(board, r1, c1, r2, c2):
	return 0

def astar(board, r1, c1, r2, c2):
	open_nodes = {(r1, c1): (board[r1][c1], board[r1][c1] + heuristic(board, r1, c1, r2, c2))}
	closed_nodes = set()
	while open_nodes:
		(cur_est, cur_pos, cur_cost) = min(((est, pos, cost) for (pos, (cost, est)) in open_nodes.items()))
		del open_nodes[cur_pos]
		if cur_pos == (r2, c2):
			return cur_cost
		(r, c) = cur_pos
		neighbours = []
		if r > 0:
			neighbours.append((r - 1, c))
		if c > 0:
			neighbours.append((r, c - 1))
		if r < len(board) - 1:
			neighbours.append((r + 1, c))
		if c < len(board[0]) - 1:
			neighbours.append((r, c + 1))
		for nb_pos in neighbours:
			(r, c) = nb_pos
			cost = cur_cost + board[r][c]
			est = cost + heuristic(board, r, c, r2, c2)
			if nb_pos in closed_nodes:
				continue
			if nb_pos in open_nodes and open_nodes[nb_pos][0] <= cost:
				continue
			open_nodes[nb_pos] = (cost, est)
		closed_nodes.add(cur_pos)
(n, m) = map(int, input().strip().split())
board = []
for _ in range(n):
	board.append(list(map(int, input().strip().split())))
q = int(input())
for _ in range(q):
	(r1, c1, r2, c2) = map(int, input().strip().split())
	print(astar(board, r1, c1, r2, c2))


class Node:

	def __init__(self, r, c, w):
		self.row = r
		self.col = c
		self.weight = w

	def weight(self):
		return self.weight

	def position(self):
		return (self.row, self.col)

	def __lt__(self, other):
		if self.weight < other.weight:
			return True
		return False

	def __gt__(self, other):
		if self.weight > other.weight:
			return True
		return False

	def __eq__(self, other):
		if self.weight == other.weight:
			return True
		return False

def children(point, r, c):
	ans = []
	if point[0] > 0:
		ans.append((point[0] - 1, point[1]))
	if point[1] > 0:
		ans.append((point[0], point[1] - 1))
	if point[0] < r - 1:
		ans.append((point[0] + 1, point[1]))
	if point[1] < c - 1:
		ans.append((point[0], point[1] + 1))
	return ans

def bfs(a, start, finish, r, c):
	childs = [Node(start[0], start[1], a[start[0]][start[1]])]
	explored = {(start[0], start[1]): True}
	import heapq
	heapq.heapify(childs)
	while True:
		curnode = heapq.heappop(childs)
		pos = curnode.position()
		if curnode.position() == finish:
			return curnode.weight
		toexplore = children(curnode.position(), r, c)
		for ele in toexplore:
			if ele not in explored:
				heapq.heappush(childs, Node(ele[0], ele[1], curnode.weight + a[ele[0]][ele[1]]))
				explored[ele] = True

def shortestPath(a, queries):
	r = len(a)
	c = len(a[0])
	for i in range(len(queries)):
		print(bfs(a, (queries[i][0], queries[i][1]), (queries[i][2], queries[i][3]), r, c))
(r, c) = list(map(int, input().split(' ')))
a = []
for i in range(r):
	a.append(list(map(int, input().split(' '))))
q = int(input())
queries = []
for i in range(q):
	queries.append(list(map(int, input().split(' '))))
shortestPath(a, queries)


import sys
from collections import deque
from math import inf
from heapq import heappush, heappop

def shortestPath(matrix, query, numRows, numCols):
	start = (query[0], query[1])
	goal = (query[2], query[3])
	pathWeight = matrix[start]
	pathQueue = []
	heappush(pathQueue, [pathWeight, start[0], start[1]])
	seen = {start: pathWeight}
	minWeight = inf
	while pathQueue:
		pathCell = heappop(pathQueue)
		(row, col) = (pathCell[1], pathCell[2])
		pathWeight = pathCell[0]
		if (row, col) == goal:
			if pathWeight < minWeight:
				minWeight = pathWeight
			continue
		if pathWeight >= minWeight:
			continue
		for (newRow, newCol) in ((row + 1, col), (row - 1, col), (row, col + 1), (row, col - 1)):
			if 0 <= newRow < numRows and 0 <= newCol < numCols:
				newPathWeight = pathWeight + matrix[newRow, newCol]
				if (newRow, newCol) in seen:
					if seen[newRow, newCol] <= newPathWeight:
						continue
				heappush(pathQueue, [newPathWeight, newRow, newCol])
				seen[newRow, newCol] = newPathWeight
	return minWeight
(m, n) = input().strip().split(' ')
(m, n) = [int(m), int(n)]
matrix = {}
for matrixRow in range(m):
	matrixRowList = [int(weight) for weight in input().strip().split(' ')]
	for matrixCol in range(n):
		matrix[matrixRow, matrixCol] = matrixRowList[matrixCol]
q = int(input().strip())
for queries in range(q):
	query = [int(query_temp) for query_temp in input().strip().split(' ')]
	print(shortestPath(matrix, query, m, n))


import math
import os
import random
import re
import sys
from collections import deque
from math import inf
from heapq import heappush, heappop

def shortestPath(matrix, query, numRows, numCols):
	start = (query[0], query[1])
	goal = (query[2], query[3])
	pathWeight = matrix[start]
	pathQueue = []
	heappush(pathQueue, [pathWeight, start[0], start[1]])
	seen = {start: pathWeight}
	minWeight = inf
	while pathQueue:
		pathCell = heappop(pathQueue)
		(row, col) = (pathCell[1], pathCell[2])
		pathWeight = pathCell[0]
		if (row, col) == goal:
			if pathWeight < minWeight:
				minWeight = pathWeight
			continue
		if pathWeight >= minWeight:
			continue
		for (newRow, newCol) in ((row + 1, col), (row - 1, col), (row, col + 1), (row, col - 1)):
			if 0 <= newRow < numRows and 0 <= newCol < numCols:
				newPathWeight = pathWeight + matrix[newRow, newCol]
				if (newRow, newCol) in seen:
					if seen[newRow, newCol] <= newPathWeight:
						continue
				heappush(pathQueue, [newPathWeight, newRow, newCol])
				seen[newRow, newCol] = newPathWeight
	return minWeight
(m, n) = input().strip().split(' ')
(m, n) = [int(m), int(n)]
matrix = {}
for matrixRow in range(m):
	matrixRowList = [int(weight) for weight in input().strip().split(' ')]
	for matrixCol in range(n):
		matrix[matrixRow, matrixCol] = matrixRowList[matrixCol]
q = int(input().strip())
for queries in range(q):
	query = [int(query_temp) for query_temp in input().strip().split(' ')]
	print(shortestPath(matrix, query, m, n))