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# Program to find minimum cost to pick up gold in given two locations in Python

Suppose we have a 2d matrix and some other values like row, col, erow0, ecol0, erow1, and ecol1. If our current position is matrix [row, col] and we want to pick up gold that is at matrix [erow0, ecol0] and matrix [erow1, ecol1]. We can go up, down, left, and right but when we are at a cell (r, c), we have to pay cost matrix [r, c], although if we land at a cell more than once, we do not need to pay cost for that cell again. We have to find the minimum cost to pick up gold at both locations.

So, if the input is like

1 | 1 | 1 | 1 | 1 |

1 | 10 | 10 | 10 | 10 |

1 | 1 | 1 | 10 | 10 |

row = 0, col = 0, erow0 = 0, ecol0 = 3, erow1 = 2, ecol1 = 2, then the output will be 8, as we are at (0, 0) and want to pick gold from location (0, 3) and (2, 2). So first move from (0, 0) to (0, 3) by three steps then come back to (0,0) then go to (2, 2) by following the 1 marked cells.

To solve this, we will follow these steps −

Define a function is_valid() . This will take x, y

return true when x and y are in range of matrix, otherwise false

Define a function min_cost() . This will take sx, sy

heap := a heap with item (matrix[sx, sy], sx, sy)

dists := a matrix of same size of given matrix and fill with inf

dists[sx, sy] := matrix[sx, sy]

while heap is not empty, do

(cost, x, y) := first element of heap and delete first element from heap

for each pair (nx, ny) in [(x, y - 1) ,(x + 1, y) ,(x - 1, y) ,(x, y + 1) ], do

if is_valid(nx, ny) and matrix[nx, ny] + cost < dists[nx, ny] is non-zero, then

edge := matrix[nx, ny]

dists[nx, ny] := edge + cost

insert (edge + cost, nx, ny) into heap

return dists

From the main method do the following −

res := inf

a := min_cost(row, col), b := min_cost(erow0, ecol0), c := min_cost(erow1, ecol1)

for i in range 0 to row count of matrix, do

for j in range 0 to column count of matrix, do

res := minimum of res and (a[i, j] + b[i, j] + c[i, j] - 2 * matrix[i, j])

return res

## Example

Let us see the following implementation to get better understanding −

import heapq import math class Solution: def solve(self, matrix, row, col, erow0, ecol0, erow1, ecol1): def is_valid(x, y): return x >= 0 and y >= 0 and x < len(matrix) and y < len(matrix[0]) def min_cost(sx, sy): heap = [(matrix[sx][sy], sx, sy)] dists = [[math.inf] * len(matrix[0]) for _ in range(len(matrix))] dists[sx][sy] = matrix[sx][sy] while heap: cost, x, y = heapq.heappop(heap) for nx, ny in [(x, y - 1), (x + 1, y), (x - 1, y), (x, y + 1)]: if is_valid(nx, ny) and matrix[nx][ny] + cost < dists[nx][ny]: edge = matrix[nx][ny] dists[nx][ny] = edge + cost heapq.heappush(heap, (edge + cost, nx, ny)) return dists res = math.inf a, b, c = min_cost(row, col), min_cost(erow0, ecol0), min_cost(erow1, ecol1) for i in range(len(matrix)): for j in range(len(matrix[0])): res = min(res, a[i][j] + b[i][j] + c[i][j] - 2 * matrix[i][j]) return res ob = Solution() matrix = [ [1, 1, 1, 1, 1], [1, 10, 10, 10, 10], [1, 1, 1, 10, 10] ] row = 0 col = 0 erow0 = 0 ecol0 = 3 erow1 = 2 ecol1 = 2 print(ob.solve(matrix, row, col, erow0, ecol0, erow1, ecol1))

## Input

[ [1, 1, 1, 1, 1], [1, 10, 10, 10, 10], [1, 1, 1, 10, 10] ], 0, 0, 0, 3, 2, 2

## Output

8

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