Digit Operations to Make Two Integers Equal

Digit Operations to Make Two Integers Equal

What is this problem about?

In the Digit Operations to Make Two Integers Equal coding problem, you are given two integers, n and m. You can transform n into m by performing a series of steps. In each step, you can change a single digit of your current number to any other digit, provided that the new number created is not a prime number. Each step has a "cost" equal to the value of the new number created. You need to find the minimum cost to reach m from n, or return -1 if it's impossible.

Why is this asked in interviews?

Microsoft uses this problem to test a candidate's ability to combine multiple algorithmic concepts: Graph Traversal, Shortest Path, and Number Theory. It evaluations whether you can model an abstract state-transition problem as a graph problem, where numbers are nodes and valid digit changes are weighted edges.

Algorithmic pattern used

This is a Shortest Path on a Graph problem, typically solved using Dijkstra's Algorithm.

1. Nodes: All non-prime integers with the same number of digits as n and m.
2. Edges: A directed edge exists from $U$ to $V$ if they differ by exactly one digit and $V$ is not prime.
3. Weight: The weight of the edge from $U$ to $V$ is $V$.
4. Sieve of Eratosthenes: Precompute all prime numbers up to the maximum possible value (usually $10,000$ or $100,000$) to allow $O(1)$ primality checks.

Example explanation

Suppose n = 10, m = 12.

• Initial cost = 10.
• Step 1: Change 10 to 12. 12 is not prime. Cost = $10 + 12 = 22$.
• Alternatively: Change 10 to 14. 14 is not prime. Cost = $10 + 14 = 24$. The algorithm explores these paths using a Priority Queue to find the minimum total cost.

Common mistakes candidates make

• Incorrect Cost Calculation: Forgetting to include the starting number's value in the total cost if required, or adding the cost of the "from" node instead of the "to" node.
• Primality Check: Using an inefficient $O(\sqrt{N})$ primality check inside the Dijkstra loop instead of precomputing with a Sieve.
• State Space: Not correctly identifying that only numbers with the same number of digits are typically considered (unless the problem states otherwise).

Interview preparation tip

Whenever a problem asks for the "minimum cost" to transform one state into another through a series of discrete steps, your first thought should be Dijkstra's Algorithm. If all step costs were 1, you would use BFS. Precomputing constraints (like primality) is a common optimization that interviewers look for.