ctci/08. Recursion/8.2. Robot in grid.md

8.2. Robot in a Grid

Imagine a robot sitting on the upper left corner of grid with r rows and c columns. The robot can only move in two directions, right and down, but certain cells are "off limits" such that the robot cannot step on them. Design an algorithm to find a path for the robot from the top left to the bottom right.

Start going down, if you find some cell that can't be stepped on, go one right and down again. And so on.

def go_bottom_down(r: int, c: int):
    pos = [0, 0]
    while pos[0] < r and pos[1] < c:
        pos[0] += 1
        step = can_step(pos)
        while not step:
            pos[1] += 1
            step = can_step(pos)

Hints

For the robot to reach the last cell, it must find a path to the second-to-last cells. For it to find a path to the second-to-last cells, it must find a path to the tird-to-last cells.

I'm missing something.

Simplify this problem by first figuring out if there's a path. Then, modify your algorithm to track the path.

But if there's no path to going down, I can go one right and try again. Why check if there's a path and then track it?

Solution

To reach the point (r, c) we have to reach either (r-1, c) or (r, c-1). To reach (r-1, c), we can either reach (r-2, c) or (r-1, c-1). This memoization could be improved by storing the grid in memory, and whether it can be accessed or not.

You create a maze with True or False for the positions that can be stepped on.

def get_path(maze):
    if not maze or len(maze) == 0:
        return None

    path = []
    if is_path(maze, len(maze)-1, len(maze[0])-1, path):
        return path
    return None

def is_path(maze, row, col, path):
    # if out of bounds or not available, return
    if col < 0 or row < 0 or not maze[row][col]:
        return False
    
    isAtOrigin = (row == 0) and (col == 0)

    #if there's a path from the start to here, add my location
    if isAtOrigin or isPath(maze, row, col-1, path) or isPath(maze, row-1, col,path):
        path.append((row, col))
        return True
    return False

print(getPath([[True, True],[True,True]]))

This works, but many many locations are visited twice. Using memoization:

def get_path_memoized(maze):
    if maze == None or len(maze) == 0:
        return None

    path = []
    visited_points = []
    if is_path_memoized(maze, len(maze) - 1, len(maze[0]) - 1, path, visited_points):
        return path
    return None

def is_path_memoized(maze, row, col, path, visited_points):
    if row < 0 or col < 0 or not maze[row][col] or (row, col) in visited_points:
        return False

    is_at_origin = row == 0 && col == 0

    if is_at_origin or is_path_memoized(maze, row-1, col, path) or is_path_memoized(maze, row, col-1, path):
        path.append((row, col))
        return True
    visited_points.append((row, col))
    return False
  
print(get_path_memoized([[True,True], [False,True]]))