Minimum Number of Operations to Make X and Y Equal - Solution & Explanation

Q: Is Minimum Number of Operations to Make X and Y Equal easy or hard?

Minimum Number of Operations to Make X and Y Equal is rated Medium difficulty on LeetCode. The main challenge is recognizing the problem as a shortest-path search on integers and efficiently exploring valid operations without revisiting states.

Q: Minimum Number of Operations to Make X and Y Equal Python or Java solution

Python and Java implementations typically use a queue for BFS and a set to track visited numbers. Each step generates neighbors using +1, -1, divide by 5, and divide by 11 when divisible. The algorithm stops once Y is reached, returning the number of BFS levels as the answer.

Q: How to solve Minimum Number of Operations to Make X and Y Equal efficiently?

Use BFS to treat each number as a graph node and apply transitions for +1, -1, divide by 5, and divide by 11 when valid. Push the starting value into a queue and explore neighbors while tracking visited numbers. The first time you reach Y gives the minimum operations because BFS processes nodes by increasing distance.

Q: What is the best approach for Minimum Number of Operations to Make X and Y Equal?

Breadth-First Search (BFS) is the most reliable approach because the problem can be modeled as a shortest-path search where each integer value is a node and each operation is an edge. BFS explores states level by level and guarantees the first time you reach Y uses the minimum number of operations. Time complexity is O(N) with O(N) space for visited states.

Q: What data structure is used in Minimum Number of Operations to Make X and Y Equal?

The primary data structure is a queue used for Breadth-First Search traversal. A hash set or boolean visited array tracks numbers that were already processed to prevent cycles. These structures ensure each state is explored once while maintaining optimal BFS traversal order.

Q: What is the time complexity of Minimum Number of Operations to Make X and Y Equal?

The BFS solution runs in O(N) time where N is the range of numbers explored between X and Y. Each state is visited at most once and generates up to four transitions. A greedy division-based strategy can reduce the complexity to about O(log max(x, y)) in practice because division operations shrink the value quickly.

Q: Is Minimum Number of Operations to Make X and Y Equal asked at Google or Amazon interviews?

Problems involving transforming numbers using minimum operations frequently appear in interviews at companies like Google, Amazon, and Meta. They test whether candidates recognize the hidden graph structure and apply BFS or dynamic programming techniques to compute the shortest transformation sequence.

MediumDynamic Programming Breadth-First Search Memoization17 min readAsked at: Microsoft, Groww

Practice this problem

Problem Statement

You are given two positive integers x and y.

In one operation, you can do one of the four following operations:

Divide x by 11 if x is a multiple of 11.
Divide x by 5 if x is a multiple of 5.
Decrement x by 1.
Increment x by 1.

Return the minimum number of operations required to make x and y equal.

Example 1:

Input: x = 26, y = 1
Output: 3
Explanation: We can make 26 equal to 1 by applying the following operations: 
1. Decrement x by 1
2. Divide x by 5
3. Divide x by 5
It can be shown that 3 is the minimum number of operations required to make 26 equal to 1.

Example 2:

Input: x = 54, y = 2
Output: 4
Explanation: We can make 54 equal to 2 by applying the following operations: 
1. Increment x by 1
2. Divide x by 11 
3. Divide x by 5
4. Increment x by 1
It can be shown that 4 is the minimum number of operations required to make 54 equal to 2.

Example 3:

Input: x = 25, y = 30
Output: 5
Explanation: We can make 25 equal to 30 by applying the following operations: 
1. Increment x by 1
2. Increment x by 1
3. Increment x by 1
4. Increment x by 1
5. Increment x by 1
It can be shown that 5 is the minimum number of operations required to make 25 equal to 30.

Constraints:

1 <= x, y <= 10⁴

Approach Overview

Problem Overview: You are given two integers x and y. The goal is to transform x into y using the minimum number of operations. Allowed operations include dividing by 5 or 11 when divisible, or adjusting the value by +1 or -1. The challenge is finding the shortest sequence of operations without exploring unnecessary states.

Approach 1: Greedy Adjustment + Division (Time: O(log max(x,y)), Space: O(1))

The greedy idea is to reduce large numbers quickly using division because division shrinks the value much faster than repeated decrements. When x > y, try to make x divisible by 11 or 5 using the fewest +1 or -1 adjustments, then perform the division. Each division significantly reduces the search space. If x <= y, the best strategy is simply applying +1 operations until the values match. This method avoids exploring every possible path and works efficiently for large values.

Approach 2: Breadth-First Search (BFS) (Time: O(N), Space: O(N))

Model the problem as a shortest path search where each integer value represents a node in a graph. From a node v, you can move to v+1, v-1, v/5 if divisible, and v/11 if divisible. Using Breadth-First Search guarantees the first time you reach y is through the minimum number of operations. A queue stores the next states to process while a visited set prevents revisiting numbers. BFS works well when the search range is bounded and guarantees correctness for all cases.

Some implementations also cache results with memoization or structure the transitions similarly to dynamic programming to avoid recomputation. However, BFS already ensures minimal operations due to level-by-level exploration.

Recommended for interviews: The BFS approach is the safest answer because it models the problem directly as a shortest-path search and guarantees correctness. Interviewers often expect you to recognize the implicit graph and apply BFS. The greedy strategy is more optimized and shows deeper insight into how division operations reduce the search space, but BFS demonstrates clear reasoning and correctness under all constraints.

Approach 1: Greedy Approach

The greedy strategy involves reducing the number in a way that it quickly converges towards the target. By greedily dividing by 11 or 5 whenever possible, we attempt to minimize the number rapidly before fine-tuning through increments or decrements.

This method prioritizes division operations first because they have a more significant effect on the number's value, bringing it closer to the desired value faster than increment or decrement steps.

The code checks conditions in a loop to differentiate between division and increment/decrement operations.

If x is larger than y and can be evenly divided by 11, it performs that division since it's the largest divisor. Otherwise, it checks divisibility by 5, performs subtractions if still larger than y, or adds if smaller.

Code

C C++Java Python C#JavaScript

C++

Java

Python

JavaScript

Complexity

Time Complexity: O(log n), where n is the maximum of x or y. Each division operation halves the problem size.

Space Complexity: O(1), as only a fixed amount of additional storage is used.

Try this approach in the editor →

Approach 2: Breadth-First Search (BFS) Approach

This approach treats each possible value of x as a node in a graph, with edges representing the operations that can be performed. Using BFS allows exploring shorter paths first, ensuring the fewest operations are discovered to convert x to y. The BFS traverses the graph until we locate the node corresponding to y, returning the level it appears on as the minimal operations count.

The BFS approach uses a queue to manage states of x, exploring each one by applying all possible operations. It marks states as visited to avoid re-processing, efficiently identifying the shortest path through levels in the graph

Code

C C++Java Python C#JavaScript

C++

Java

Python

JavaScript

Complexity

Time Complexity: O(V+E), where V is the vertex count (possible values of x) and E is edge operations.

Space Complexity: O(V) for tracking visited states and queue storage.

Try this approach in the editor →

Approach 3: Default Approach

Code

Python Java C++Go TypeScript

Python

Java

C++

TypeScript

Try this approach in the editor →

Complexity Comparison

Approach	Complexity
Greedy Approach	Time Complexity: O(log n), where n is the maximum of x or y. Each division operation halves the problem size. Space Complexity: O(1), as only a fixed amount of additional storage is used.
Breadth-First Search (BFS) Approach	Time Complexity: O(V+E), where V is the vertex count (possible values of x) and E is edge operations. Space Complexity: O(V) for tracking visited states and queue storage.
Default Approach	—

Detailed Complexity Analysis

Approach	Time	Space	When to Use
Greedy Adjustment + Division	O(log max(x,y))	O(1)	Best when large numbers allow frequent division by 5 or 11
Breadth-First Search (BFS)	O(N)	O(N)	General case when you need guaranteed minimum operations

Video Solution

Minimum Number of Operations to Make X and Y Equal | BFS | Graph | Similar to Race Car • Aryan Mittal • 3,781 views views

Watch 8 more video solutions →

Frequently Asked Questions

Is Minimum Number of Operations to Make X and Y Equal easy or hard?

Minimum Number of Operations to Make X and Y Equal is rated Medium difficulty on LeetCode. The main challenge is recognizing the problem as a shortest-path search on integers and efficiently exploring valid operations without revisiting states.

Minimum Number of Operations to Make X and Y Equal Python or Java solution

Python and Java implementations typically use a queue for BFS and a set to track visited numbers. Each step generates neighbors using +1, -1, divide by 5, and divide by 11 when divisible. The algorithm stops once Y is reached, returning the number of BFS levels as the answer.

How to solve Minimum Number of Operations to Make X and Y Equal efficiently?

Use BFS to treat each number as a graph node and apply transitions for +1, -1, divide by 5, and divide by 11 when valid. Push the starting value into a queue and explore neighbors while tracking visited numbers. The first time you reach Y gives the minimum operations because BFS processes nodes by increasing distance.

What is the best approach for Minimum Number of Operations to Make X and Y Equal?

Breadth-First Search (BFS) is the most reliable approach because the problem can be modeled as a shortest-path search where each integer value is a node and each operation is an edge. BFS explores states level by level and guarantees the first time you reach Y uses the minimum number of operations. Time complexity is O(N) with O(N) space for visited states.

What data structure is used in Minimum Number of Operations to Make X and Y Equal?

The primary data structure is a queue used for Breadth-First Search traversal. A hash set or boolean visited array tracks numbers that were already processed to prevent cycles. These structures ensure each state is explored once while maintaining optimal BFS traversal order.

What is the time complexity of Minimum Number of Operations to Make X and Y Equal?

The BFS solution runs in O(N) time where N is the range of numbers explored between X and Y. Each state is visited at most once and generates up to four transitions. A greedy division-based strategy can reduce the complexity to about O(log max(x, y)) in practice because division operations shrink the value quickly.

Is Minimum Number of Operations to Make X and Y Equal asked at Google or Amazon interviews?

Problems involving transforming numbers using minimum operations frequently appear in interviews at companies like Google, Amazon, and Meta. They test whether candidates recognize the hidden graph structure and apply BFS or dynamic programming techniques to compute the shortest transformation sequence.

Ready to solve this problem?

Practice Minimum Number of Operations to Make X and Y Equal with our built-in code editor and test cases.

Practice on FleetCode

Minimum Moves to Spread Stones Over Grid

Shortest Bridge

Problem Info

DifficultyMedium

Acceptance48.4%

Approaches3

Reading time17 min

Asked at

Microsoft Groww

Practice this problem

Open in Editor

Problem Statement

Approach Overview

Approach 1: Greedy Approach

Code

Complexity

Approach 2: Breadth-First Search (BFS) Approach

Code

Complexity

Approach 3: Default Approach

Code

Complexity Comparison

Detailed Complexity Analysis

Video Solution

Frequently Asked Questions

Related Problems

Ready to solve this problem?

Problem Info

Table of Contents