Enough of the tiling problems. Time for some 2D dynamic programming. You can work in any programming language of your choice - Python, C++, Java, or anything else you want. Let’s get started!

Problem Statement

You are given a grid with $n$ rows and $m$ columns. Some of the grid locations are blocked, and cannot be entered. Find the number of paths from the top-left corner of the grid to the bottom right corner.

Your solution should have run-time O($nm$).

Examples

Example 1:

....
....
....

Number of paths = 10

Example 2:

..#.
#.#.
....

Number of paths = 1

Example 3:

...#
.#..
....

Number of paths = 3

Submitting your solution

Once you are confidant of your solution, take the quiz! It will provide you with some grids to run your code on.

Progressive ‘hint-by-hint’ solution

Only look at the hints if you can’t solve the problem for at-least 20-minutes. Read only one hint that you could not figure out on your own. Then spend at-least 20-minutes trying to solve the problem again.

Hint 1 (dp state):

count(i, j) = number of paths from (0, 0) to (i, j)

Hint 2 (recursion):

You can either come from top, or from the left. Hence, if square (i, j) is not blocked, then count(i, j) = count(i-1, j) + count(i, j-1).

Full solution

The solution for this assignment is included at the end of the quiz.