Shortest Path in a Binary Matrix

Posted: 9 Sep, 2020
Difficulty: Moderate

PROBLEM STATEMENT

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You have been given a binary matrix of size 'N' * 'M' where each element is either 0 or 1. You are also given a source and a destination cell, both of them lie within the matrix.

Your task is to find the length of the shortest path from the source cell to the destination cell only consisting of 1s. If there is no path from source to destination cell, return -1.

Note:
1. Coordinates of the cells are given in 0-based indexing.
2. You can move in 4 directions (Up, Down, Left, Right) from a cell.
3. The length of the path is the number of 1s lying in the path.
4. The source cell is always filled with 1.
For example -
1 0 1
1 1 1
1 1 1
For the given binary matrix and source cell(0,0) and destination cell(0,2). Few valid paths consisting of only 1s are

X 0 X     X 0 X 
X X X     X 1 X 
1 1 1     X X X 
The length of the shortest path is 5.
Input Format:
The first line of input contains two integers 'N' and 'M' separated by a single space representing the number of rows and columns in the binary matrix respectively.

The next 'N' lines of the input contain 'M' single space-separated integers each representing the 'i'-th row of the Binary Matrix.

The next line of input contains two single space-separated integers 'SOURCEX' and 'SOURCEY' representing the coordinates of the source cell.

The next line of input contains two single space-separated integers 'DESTX' and 'DESTY' representing the coordinates of the destination cell.
Constraints :
1 <= N <= 500
1 <= M <= 500
MAT[i] = {0, 1}

0 <= SOURCEX <= N - 1
0 <= SOURCEY <= M - 1
0 <= DESTX <= N - 1
0 <= DESTY <= M - 1
MAT[SOURCEX][SOURCEY] = 1

Time Limit: 1 sec
Output Format :
For each test case, print a single line that contains a single integer i.e. length of the shortest path from the source cell to the destination cell.
Note:
You do not need to print anything; it has already been taken care of. Just implement the given function.
Approach 1

To find the shortest path in the Binary Matrix, we search for all possible paths in the Binary Matrix from the source cell to the destination cell until all possibilities are exhausted. We can easily achieve this with the help of backtracking.

 

We start from the given source cell in the matrix and explore all four paths possible and recursively check if they will lead to the destination or not. Out of all those possible paths, we will pick the minimum length path from source to destination. If a path doesn't reach the destination cell or we have explored all possible routes from the current cell, we backtrack. 

 

To make sure that the path is simple and doesn't contain any cycles, we keep track of cells involved in the current path in a boolean matrix ‘VISITED‘, and before exploring any cell, we ignore the cell if it is already covered in the current path.

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