Minimum Operations to Reach a Number N

Introduction

In various problem-solving scenarios, it is often necessary to determine the minimum number of operations required to reach a specific number starting from a given value. In this article, we will explore an approach to solve the problem of finding the minimum number of operations required to reach a number N starting from 0. We will discuss an efficient algorithm to solve this problem, along with its time and space complexity.

Problem Statement

Given a number N, our goal is to find the minimum number of operations required to reach N starting from 0. We have two operations available: doubling the number and adding one to the number.

Approach

To solve this problem efficiently, we can use a dynamic programming approach. We will maintain an array dp[], where dp[i] represents the minimum number of operations required to reach the number i starting from 0. We will start populating the array from the base case dp[0] = 0, and then iterate from 1 to N.

For each number i, we have two options:

1. If i is even, we can reach i by doubling the previous number (i/2). In this case, the number of operations required would be dp[i/2] + 1.