Get least sum of a combination of matrix elements

Question

Yesterday one of my friend came with a problem, asking me to find the solution.

The problem

I have a matrix(n x m). I need to

Answer 1

An update to MBo's answer. Given a n*m (n=3, m=4 in your post) The space consumed can be reduce to O(N) by only remembering the result for previous line (column).

Cost[0] = matrix[0, 0]
for x = 1 to m - 1
    Cost[x] = matrix[0, x] + Cost[x-1]
for y = 1 to n - 1
    Cost[0] += matrix[y, 0]
    for x = 1 to m - 1
        Cost[x] = matrix[y, x] + Min(Cost[x-1], Cost[x])
output(Cost[n-1])

Don't know how to write in PHP... Here's python sample code

matrix = [
    [3, 44, 75],
    [21, 98, 60],
  ]

n = len(matrix)
m = len(matrix[0])

cost = [0] * m

cost[0] = matrix[0][0]
for x in xrange(1, m):
    cost[x] = matrix[0][x] + cost[x-1]
for y in xrange(1, n):
    cost[0] += matrix[y][0]
    for x in xrange(1, m):
        cost[x] = matrix[y][x] + min(cost[x-1], cost[x])

print cost[-1]