Least Operators to Express Number

Least Operators to Express Number

What is this problem about?

The "Least Operators to Express Number interview question" is a challenging math-oriented problem. You are given a base x and a target number. You need to represent the target using the base x and the operators '+', '-', '', and '/'. The goal is to find the minimum number of operators required. You can use any number of x's. For example, if x=3 and target=19, you might use 333 - 33 + 3/3. This "Least Operators to Express Number coding problem" is a test of your ability to break down a number using its base representation and dynamic programming.

Why is this asked in interviews?

This problem is typically reserved for senior roles or highly competitive companies like Snap. It assesses a candidate's mastery of "Dynamic Programming interview pattern" and "Memoization interview pattern". It requires deep mathematical insight into how numbers can be represented in different bases and how to handle the "carry" or "remainder" when moving between powers of the base.

Algorithmic pattern used

This problem is solved using recursion with memoization. The idea is to express the target in terms of powers of x. At each step, you can either reach the target by adding some multiples of the current power of x or by going past the target and then subtracting. This creates a decision tree where memoization is essential to avoid redundant calculations. The cost of operators (multiplication and division) varies depending on the power of x being used.

Example explanation

Suppose x = 3 and target = 5. We look for the power of 3 closest to 5, which is 3^1 = 3 and 3^2 = 9.

• Option 1: Use 3^1. We need 5 - 3 = 2 more. The cost for 3^1 is 1 operator (multiplication).
• Option 2: Use 3^2. We need 9 - 5 = 4 less. The cost for 3^2 is 2 operators (3*3). We recursively find the cost to represent the remainders (2 and 4) and choose the path that yields the minimum total operators.

Common mistakes candidates make

• Ignoring the subtraction case: Many only think about adding up to the target, but sometimes overshooting and subtracting is cheaper.
• Incorrect operator counting: Forgetting that x/x counts as one operator ('/') and provides the value 1.
• Not using memoization: The state space grows quickly, and without memoization, the solution will time out.

Interview preparation tip

When you see problems involving "least" or "minimum" ways to reach a target using a set of operations, think of BFS or DP. For large targets, DP with memoization is usually the way to go. Brush up on base conversions as they are often the foundation for these types of problems.