Given a distance ‘dist, count total number of ways to cover the distance with 1, 2 and 3 steps.

Examples :

Input: n = 3 Output: 4 Below are the four ways 1 step + 1 step + 1 step 1 step + 2 step 2 step + 1 step 3 step Input: n = 4 Output: 7

## C++

// A naive recursive C++ program to count number of ways to cover // a distance with 1, 2 and 3 steps #include<iostream> using namespace std; // Returns count of ways to cover 'dist' int printCountRec(int dist) { // Base cases if (dist<0) return 0; if (dist==0) return 1; // Recur for all previous 3 and add the results return printCountRec(dist-1) + printCountRec(dist-2) + printCountRec(dist-3); } // driver program int main() { int dist = 4; cout << printCountRec(dist); return 0; }

## Java

// A naive recursive Java program to count number // of ways to cover a distance with 1, 2 and 3 steps import java.io.*; class GFG { // Function returns count of ways to cover 'dist' static int printCountRec(int dist) { // Base cases if (dist<0) return 0; if (dist==0) return 1; // Recur for all previous 3 and add the results return printCountRec(dist-1) + printCountRec(dist-2) + printCountRec(dist-3); } // driver program public static void main (String[] args) { int dist = 4; System.out.println(printCountRec(dist)); } } // This code is contributed by Pramod Kumar

## Python3

# A naive recursive Python3 program # to count number of ways to cover # a distance with 1, 2 and 3 steps # Returns count of ways to # cover 'dist' def printCountRec(dist): # Base cases if dist < 0: return 0 if dist == 0: return 1 # Recur for all previous 3 and # add the results return (printCountRec(dist-1) + printCountRec(dist-2) + printCountRec(dist-3)) # Driver code dist = 4 print(printCountRec(dist)) # This code is contributed by Anant Agarwal.

## C#

// A naive recursive C# program to // count number of ways to cover a // distance with 1, 2 and 3 steps using System; class GFG { // Function returns count of // ways to cover 'dist' static int printCountRec(int dist) { // Base cases if (dist < 0) return 0; if (dist == 0) return 1; // Recur for all previous 3 // and add the results return printCountRec(dist - 1) + printCountRec(dist - 2) + printCountRec(dist - 3); } // Driver Code public static void Main () { int dist = 4; Console.WriteLine(printCountRec(dist)); } } // This code is contributed by Sam007.

## PHP

<?php // A naive recursive PHP program to // count number of ways to cover // a distance with 1, 2 and 3 steps // Returns count of ways to cover 'dist' function printCountRec( $dist) { // Base cases if ($dist<0) return 0; if ($dist==0) return 1; // Recur for all previous 3 // and add the results return printCountRec($dist - 1) + printCountRec($dist - 2) + printCountRec($dist - 3); } // Driver Code $dist = 4; echo printCountRec($dist); // This code is contributed by anuj_67. ?>

Output:

7

The time complexity of above solution is exponential, a close upper bound is O(3^{n}). If we draw the complete recursion tree, we can observer that many subproblems are solved again and again. For example, when we start from n = 6, we can reach 4 by subtracting one 2 times and by subtracting 2 one times. So the subproblem for 4 is called twice.

Since same suproblems are called again, this problem has Overlapping Subprolems property. So min square sum problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by constructing a temporary array count[] in bottom up manner.

Below is the implementation of above idea :

## C++

// A Dynamic Programming based C++ program to count number of ways // to cover a distance with 1, 2 and 3 steps #include<iostream> using namespace std; int printCountDP(int dist) { int count[dist+1]; // Initialize base values. There is one way to cover 0 and 1 // distances and two ways to cover 2 distance count[0] = 1, count[1] = 1, count[2] = 2; // Fill the count array in bottom up manner for (int i=3; i<=dist; i++) count[i] = count[i-1] + count[i-2] + count[i-3]; return count[dist]; } // driver program int main() { int dist = 4; cout << printCountDP(dist); return 0; }

## Java

// A Dynamic Programming based Java program // to count number of ways to cover a distance // with 1, 2 and 3 steps import java.io.*; class GFG { // Function returns count of ways to cover 'dist' static int printCountDP(int dist) { int[] count = new int[dist+1]; // Initialize base values. There is one way to // cover 0 and 1 distances and two ways to // cover 2 distance count[0] = 1; count[1] = 1; count[2] = 2; // Fill the count array in bottom up manner for (int i=3; i<=dist; i++) count[i] = count[i-1] + count[i-2] + count[i-3]; return count[dist]; } // driver program public static void main (String[] args) { int dist = 4; System.out.println(printCountDP(dist)); } } // This code is contributed by Pramod Kumar

## Python3

# A Dynamic Programming based on Python3 # program to count number of ways to # cover a distance with 1, 2 and 3 steps def printCountDP(dist): count = [0] * (dist + 1) # Initialize base values. There is # one way to cover 0 and 1 distances # and two ways to cover 2 distance count[0] = 1 count[1] = 1 count[2] = 2 # Fill the count array in bottom # up manner for i in range(3, dist + 1): count[i] = (count[i-1] + count[i-2] + count[i-3]) return count[dist]; # driver program dist = 4; print( printCountDP(dist)) # This code is contributed by Sam007.

## C#

// A Dynamic Programming based C# program // to count number of ways to cover a distance // with 1, 2 and 3 steps using System; class GFG { // Function returns count of ways // to cover 'dist' static int printCountDP(int dist) { int[] count = new int[dist + 1]; // Initialize base values. There is one // way to cover 0 and 1 distances // and two ways to cover 2 distance count[0] = 1; count[1] = 1; count[2] = 2; // Fill the count array // in bottom up manner for (int i = 3; i <= dist; i++) count[i] = count[i - 1] + count[i - 2] + count[i - 3]; return count[dist]; } // Driver Code public static void Main () { int dist = 4; Console.WriteLine(printCountDP(dist)); } } // This code is contributed by Sam007.

## PHP

<?php // A Dynamic Programming based PHP program // to count number of ways to cover a // distance with 1, 2 and 3 steps function printCountDP( $dist) { $count = array(); // Initialize base values. There is // one way to cover 0 and 1 distances // and two ways to cover 2 distance $count[0] = 1; $count[1] = 1; $count[2] = 2; // Fill the count array // in bottom up manner for ( $i = 3; $i <= $dist; $i++) $count[$i] = $count[$i - 1] + $count[$i - 2] + $count[$i - 3]; return $count[$dist]; } // Driver Code $dist = 4; echo printCountDP($dist); // This code is contributed by anuj_67. ?>

**Output :**

7

This article is contributed by Vignesh Venkatesan. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above