Suppose we have a matrix M where M[r][c] represents the height of that cell. If we are currently at top left corner and want to go to the bottom right corner. We can move to adjacent cells (up, down, left, right) only if that the height of that adjacent cell is less than or equal to the current cell's height. We can increase the height of any number of cells before we move, so we have to find the minimum total height that needs to be increased so that we can go to the bottom right cell.
So, if the input is like
| 2 | 4 | 5 |
| 8 | 6 | 1 |
then the output will be 4, as we can take the following path [2, 4, 5, 1] and change the heights to this configuration −
| 5 | 5 | 5 |
| 8 | 6 | 1 |
To solve this, we will follow these steps −
INF := infinity
R, C := row number of matrix, column number of matrix
pq := make a priority queue using heap, and insert [0, R-1, C-1, M[-1, -1]] into it
dist := a map
dist[R-1, C-1, A[-1, -1]] := 0
while pq is not empty, do
delete one element from pq and store them into d, r, c, h
if dist[r, c, h] < d, then
go for the next iteration
if r and c are both 0, then
return d
for each pair (nr, nc) in [[r+1, c], [r, c+1], [r-1, c], [r, c-1]], do
if 0 <= nr < R and 0 <= nc < C, then
if d2 < dist[nr, nc, h2], then
dist[nr, nc, h2] := d2
insert [d2, nr, nc, h2] into pq
Let us see the following implementation to get better understanding −
Example
import collections
import heapq
class Solution:
def solve(self, A):
INF = float('inf')
R, C = len(A), len(A[0])
pq = [[0, R-1, C-1, A[-1][-1]]]
dist = collections.defaultdict(lambda: INF)
dist[R-1, C-1, A[-1][-1]] = 0
while pq:
d, r, c, h = heapq.heappop(pq)
if dist[r, c, h] < d:
continue
if r == c == 0:
return d
for nr, nc in [[r+1, c], [r, c+1], [r-1, c], [r, c-1]]:
if 0 <= nr < R and 0 <= nc < C:
h2 = max(A[nr][nc], h)
d2 = d + max(h2 - A[nr][nc], 0)
if d2 < dist[nr, nc, h2]:
dist[nr, nc, h2] = d2
heapq.heappush(pq, [d2, nr, nc, h2])
ob = Solution()
matrix = [
[2, 4, 5],
[8, 6, 1]
]
print(ob.solve(matrix))Input
[[2, 4, 5],[8, 6, 1]]
Output
4