Least Operators to Express Number - Video Solutions

Problem Overview: You are given a number x and a target value. The task is to build the target using only the number x and operators +, -, *, and /. The goal is to minimize the number of operators used while respecting normal arithmetic precedence.

Approach 1: Recursive Approach with Memoization (Time: O(logx(target)), Space: O(logx(target)))

This method treats the problem as repeatedly decomposing the target using powers of x. For each target value, compute the closest power x^k. You then explore two choices: represent the target as x^k + remainder or overshoot using x^(k+1) and subtract the difference. A recursive function calculates the minimum operator count for each remainder while storing results in a memo table to avoid recomputation. The recursion depth stays proportional to log_x(target), making it efficient for large targets.

Approach 2: Dynamic Programming Using Base-x Representation (Time: O(logx(target)), Space: O(logx(target)))

This approach converts the target into base x. Each digit represents how many times a power of x contributes to the expression. You iterate through digits from least significant to most significant while maintaining two states: the cost of building the value directly and the cost if you carry to the next digit. Each step updates operator counts based on whether it is cheaper to add multiples of x^k or subtract from the next power. The DP transition captures the tradeoff between constructing the exact digit or adjusting via carry operations. This approach is deterministic and avoids recursion.

Approach 3: Greedy Backtracking with Power Selection (Time: O(logx(target)), Space: O(logx(target)))

The greedy idea selects the largest power of x that does not exceed the remaining target. After choosing that power, you recursively compute the cost of expressing the remainder. A second branch considers overshooting with the next power and subtracting the difference. Backtracking compares both paths and picks the minimum operator count. Although it looks greedy, the branching ensures correctness by evaluating both under- and over-approximation cases.

Recommended for interviews: The memoized recursion or the base-x dynamic programming solution is typically expected. They clearly demonstrate understanding of number decomposition and state reuse. The brute reasoning (choosing powers of x) shows intuition, while the optimized solution using memoization or dynamic programming highlights strong algorithmic design. The problem also relies heavily on number manipulation concepts from math.