Cracking the Coding Interview, 4 Edition - 150 Programming Interview Questions and Solutions

Solutions to Chapter 8 | Recursion
8.2 Imagine a robot sitting on the upper left hand corner of an NxN grid. The robot can
only move in two directions: right and down. How many possible paths are there for
the robot?
FOLLOW UP
Imagine certain squares are “off limits”, such that the robot can not step on them.
Design an algorithm to get all possible paths for the robot.
SOLUTION
Part 1: (For clarity, we will solve this part assuming an X by Y grid)
Each path has (X-1)+(Y-1) steps. Imagine the following paths:
X X Y Y X (move right -> right -> down -> down -> right)
X Y X Y X (move right -> down -> right -> down -> right)
...
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Each path can be fully represented by the moves at which we move right. That is, if I were to
ask you which path you took, you could simply say “I moved right on step 3 and 4.”
Since you must always move right X-1 times, and you have X-1 + Y-1 total steps, you have
to pick X-1 times to move right out of X-1+Y-1 choices. Thus, there are C(X-1, X-1+Y-1) paths
(e.g., X-1+Y-1 choose X-1):
Part 2: Code
(X-1 + Y-1)! / ((X-1)! * (Y-1)!)
We can implement a simple recursive algorithm with backtracking:
1 ArrayList current_path = new ArrayList();
2 public static boolean getPaths(int x, int y) {
3 Point p = new Point(x, y);
4 current_path.add(p);
5 if (0 == x && 0 == y) return true; // current_path
6 boolean success = false;
7 if (x >= 1 && is_free(x - 1, y)) { // Try right
8 success = getPaths(x - 1, y); // Free! Go right
9 }
10 if (!success && y >= 1 && is_free(x, y - 1)) { // Try down
11 success = getPaths(x, y - 1); // Free! Go down
12 }
13 if (!success) {
14 current_path.remove(p); // Wrong way!
15 }
16 return success;
17 }
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