Minimum Number of Operations to Make X and Y Equal

Minimum Number of Operations to Make X and Y Equal

1. What is this problem about?

The "Minimum Number of Operations to Make X and Y Equal" interview question is a classic optimization challenge where you are given two integers, X and Y. Your goal is to transform X into Y using a specific set of allowed operations in the fewest steps possible. Typically, these operations include incrementing X, decrementing X, or dividing X by certain constants (like 5 or 11) if X is divisible by them. This problem tests your ability to navigate a state space efficiently to find the shortest path between two numerical values.

2. Why is this asked in interviews?

Companies like Microsoft and Groww frequently include this coding problem to evaluate a candidate's grasp of state-space search and optimization. It requires a developer to think beyond simple greedy approaches, which often fail in such scenarios. Interviewers look for your ability to recognize that this is essentially a shortest-path problem in an implicit graph, where each number represents a node and each operation represents an edge.

3. Algorithmic pattern used

The most effective algorithmic pattern for this problem is Breadth-First Search (BFS) or Breadth-First Search with Memoization/Dynamic Programming. Since every operation can be viewed as having a weight of one, BFS is perfectly suited to find the minimum number of steps. Alternatively, a recursive approach with memoization can be used to explore different paths while avoiding redundant calculations for the same value of X.

4. Example explanation

Suppose you want to transform X = 26 to Y = 1. Your operations might include dividing by 5, dividing by 11, or adding/subtracting 1.

• You could subtract 1 repeatedly (25 steps), but that's inefficient.
• Instead, you could subtract 1 to get 25, then divide by 5 to get 5, then divide by 5 again to get 1. Total operations: 3.
• Or, you could add 7 to get 33, divide by 11 to get 3, then subtract 1 twice. Total: 4 operations. The BFS approach systematically explores these layers to confirm that 3 is the minimum.

5. Common mistakes candidates make

A common pitfall is attempting a purely greedy strategy, such as always choosing the division that brings you closest to Y. However, sometimes adding or subtracting a few units to make the number divisible by a larger factor (like 11) results in a much faster path. Another mistake is failing to use a "visited" set or memoization table, which leads to infinite loops or exponential time complexity as the algorithm revisits the same numbers.

6. Interview preparation tip

When dealing with "minimum operations" problems, always consider if the state space can be visualized as a graph. If all operations have the same "cost," BFS should be your go-to strategy. Practice identifying the boundaries of your search; for instance, in this problem, you rarely need to increase X significantly beyond its original value or Y's value, which helps in pruning the search space.