Minimum moves to make M and N equal by repeated addition of divisors except 1 using Dynamic Programming
Given two integers N and M, the task is to calculate the minimum number of moves to change N to M, where In one move it is allowed to add any divisor of the current value of N to N itself except 1. Print “-1” if it is not possible.
Example:
Input: N = 4, M = 24
Output: 5
Explanation: The given value of N can be converted into M using the following steps: (4)+2 -> (6)+2 -> (8)+4 -> (12)+6 -> (18)+6 -> 24. Hence, the count of moves is 5 which is the minimum possible.
Input: N = 4, M = 576
Output: 14
BFS Approach: The given problem has already been discussed in Set 1 of this article which using the Breadth First Traversal to solve the given problem.
Approach: This article focused on a different approach based on Dynamic Programming. Below are the steps to follow:
- Create a 1-D array dp[], where dp[i] stores the minimum number of operations to reach i from N. Initially, dp[N+1… M] = {INT_MAX} and dp[N] = 0.
- Iterate through the range [N, M] using a variable i and for each i, iterate through all the factors of the given number i. For a factor X, the DP relation can be define as follows:
dp[i + X] = min( dp[i], dp[i + X])
- The value stored at dp[M] is the required answer.
Below is the implementation of the above approach:
C++
#include <bits/stdc++.h>
using namespace std;
int minOperationCnt( int N, int M)
{
int dp[M + 1];
for ( int i = N + 1; i <= M; i++) {
dp[i] = INT_MAX;
}
dp[N] = 0;
for ( int i = N; i <= M; i++) {
if (dp[i] == INT_MAX) {
continue ;
}
for ( int j = 2; j * j <= i; j++) {
if (i % j == 0) {
if (i + j <= M) {
dp[i + j] = min(dp[i + j], dp[i] + 1);
}
if (i + i / j <= M) {
dp[i + i / j]
= min(dp[i + i / j], dp[i] + 1);
}
}
}
}
return (dp[M] == INT_MAX) ? -1 : dp[M];
}
int main()
{
int N = 4;
int M = 576;
cout << minOperationCnt(N, M);
return 0;
}
|
Java
class GFG {
public static int minOperationCnt( int N, int M) {
int [] dp = new int [M + 1 ];
for ( int i = N + 1 ; i <= M; i++) {
dp[i] = Integer.MAX_VALUE;
}
dp[N] = 0 ;
for ( int i = N; i <= M; i++) {
if (dp[i] == Integer.MAX_VALUE) {
continue ;
}
for ( int j = 2 ; j * j <= i; j++) {
if (i % j == 0 ) {
if (i + j <= M) {
dp[i + j] = Math.min(dp[i + j], dp[i] + 1 );
}
if (i + i / j <= M) {
dp[i + i / j] = Math.min(dp[i + i / j], dp[i] + 1 );
}
}
}
}
return (dp[M] == Integer.MAX_VALUE) ? - 1 : dp[M];
}
public static void main(String args[]) {
int N = 4 ;
int M = 576 ;
System.out.println(minOperationCnt(N, M));
}
}
|
Python3
import math
INT_MAX = 2147483647
def minOperationCnt(N, M):
dp = [ 0 for _ in range (M + 1 )]
for i in range (N + 1 , M + 1 ):
dp[i] = INT_MAX
dp[N] = 0
for i in range (N, M + 1 ):
if (dp[i] = = INT_MAX):
continue
for j in range ( 2 , int (math.sqrt(i)) + 1 ):
if (i % j = = 0 ):
if (i + j < = M):
dp[i + j] = min (dp[i + j], dp[i] + 1 )
if (i + i / / j < = M):
dp[i + i / / j] = min (dp[i + i / / j], dp[i] + 1 )
if dp[M] = = INT_MAX:
return - 1
else :
return dp[M]
if __name__ = = "__main__" :
N = 4
M = 576
print (minOperationCnt(N, M))
|
C#
using System;
class GFG
{
public static int minOperationCnt( int N, int M)
{
int [] dp = new int [M + 1];
for ( int i = N + 1; i <= M; i++)
{
dp[i] = int .MaxValue;
}
dp[N] = 0;
for ( int i = N; i <= M; i++)
{
if (dp[i] == int .MaxValue)
{
continue ;
}
for ( int j = 2; j * j <= i; j++)
{
if (i % j == 0)
{
if (i + j <= M)
{
dp[i + j] = Math.Min(dp[i + j], dp[i] + 1);
}
if (i + i / j <= M)
{
dp[i + i / j] = Math.Min(dp[i + i / j], dp[i] + 1);
}
}
}
}
return (dp[M] == int .MaxValue) ? -1 : dp[M];
}
public static void Main()
{
int N = 4;
int M = 576;
Console.Write(minOperationCnt(N, M));
}
}
|
Javascript
<script>
function minOperationCnt(N, M) {
let dp = new Array(M + 1)
for (let i = N + 1; i <= M; i++) {
dp[i] = Number.MAX_VALUE;
}
dp[N] = 0;
for (let i = N; i <= M; i++) {
if (dp[i] == Number.MAX_VALUE) {
continue ;
}
for (let j = 2; j * j <= i; j++) {
if (i % j == 0) {
if (i + j <= M) {
dp[i + j] = Math.min(dp[i + j], dp[i] + 1);
}
if (i + i / j <= M) {
dp[i + i / j]
= Math.min(dp[i + i / j], dp[i] + 1);
}
}
}
}
return (dp[M] == Number.MAX_VALUE) ? -1 : dp[M];
}
let N = 4;
let M = 576;
document.write(minOperationCnt(N, M));
</script>
|
Time Complexity: O((M – N)*√(M – N))
Auxiliary Space: O(M)
Last Updated :
28 Nov, 2022
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