Count all paths from top left to bottom right of MxN matrix

If you want to practice data structure and algorithm programs, you can go through Java coding interview questions.

In this post, we will see about how to count all paths from top left to bottom right of MxN matrix.


Problem

We need to count all paths from top left to bottom right of MxN matrix. You can either move down or right.

Matrix paths


Solution

You can solve this problem using recursion.

Recursion

Recursion will work fine but time complexity of this solution will be exponential as there are lots of overlapping subproblems here.

We can use dynamic programming to solve the problem. We won’t recompute any subproblem more than once.

Dynamic programming

Here is simple algorithm for dynamic programming solution.

  • Initialize an array dp with matrix’s dimensions. This array will give us final count, once we reach to bottom right.
  • Fill the first row with 1, as you can reach here from one direction(by going right)
  • Fill the first column with 1 as well, as you can reach here from one direction(by going down).
  • Iterate over all other rows and columns and use formula dp=dp[i-1][j] + dp[i][j-1], as you can reach here from two directions (By going right or By going down)
  • return dp[matrix.length-1][matrix[0].length-1]

Here is a diagramtic illustration of the algorithm.

Matrix Paths DP

Here is complete program to Count all paths from top left to bottom right of MxN matrix.

Here is output of the above program.

Total paths to reach top left to bottom right using recursion: 6
Total paths to reach top left to bottom right using DP: 6

That’s all about counting all paths from top left to bottom right of MxN matrix.

Related Posts

  • Maximum number of vowels in a Substring of given length
    18 June

    Maximum Number of Vowels in a Substring of Given Length

    Table of ContentsApproach – 1 Generate All Substrings Using substring() MethodApproach – 2 Using Sliding Window Method (Linear Time Solution) In this article, we will look at an interesting problem related to the Strings and [Sliding-Window Algorithm](https://java2blog.com/sliding-window-maximum-java/ “Sliding-Window Algorithm”). The problem is : "Given a String we have to Find the Maximum Number of Vowel […]

  • Find first and last position of element in sorted array
    04 June

    Search for a range Leetcode – Find first and last position of element in sorted array

    Table of ContentsApproach 1 (Using Linear Search)Approach 2 (Using Modified Binary Search-Optimal) In this article, we will look into an interesting problem asked in Coding Interviews related to Searching Algorithms. The problem is: Given a Sorted Array, we need to find the first and last position of an element in Sorted array. This problem is […]

  • 30 April

    Convert Postfix to Infix in Java

    Learn about how to convert Postfix to Infix in java.

  • Prefix to postfix in java
    30 April

    Convert Prefix to Postfix in Java

    Learn about how to convert Prefix to Postfix in java.

  • Data structures in java
    16 April

    Data Structures in java

    Table of ContentsArray Declare and initialize array in javaAdvantages of arrayDisadvantages of array ExampleArray practice programsStackStack implementation using ArrayStack implementation using LinkedListImplementationPractice ProgramsQueueQueue implementation using arrayQueue implementation using LinkedListImplementationLinkedListImplementationLinkedList Practice ProgramsBinary treeImplementationBinary tree practice programsBinary Search treeImplementationBinary search tree Practice programsTrieImplementationHeapImplementationGraphImplementation Inbuild data structures in javaStringHashMapLinkedHashMapArrayListLinkedListHashSet In this post, we will see about various data […]

  • 29 November

    Top 100+ Java coding interview questions

    Table of ContentsStringQuestion 1 : How to reverse a String in java? Can you write a program without using any java inbuilt methods?Question 2 : Write a java program to check if two Strings are anagram in java?Question 3 : Write a program to check if String has all unique characters in java?Question 4 : […]

Leave a Reply

Your email address will not be published. Required fields are marked *

Subscribe to our newletter

Get quality tutorials to your inbox. Subscribe now.