Largest Integer With Given Digit Sum - Solution & Explanation

Problem Overview: You need to construct the numerically largest integer whose digits add up to a given target sum. The key observation is that larger leading digits always produce a larger overall number, so the solution depends on how you distribute the digit sum efficiently.

Approach 1: Brute Force Enumeration (Exponential Time)

This approach generates every possible number whose digit sum matches the target and keeps track of the maximum value found. You can use recursion or backtracking to try all digit combinations from 0 to 9. While this proves correctness for small inputs, the search space grows exponentially because each position branches into multiple digit choices. Time complexity is O(10^n) in the worst case, and space complexity is O(n) for recursion depth.

Approach 2: Greedy Construction (O(n))

The optimal solution uses a greedy strategy. To maximize the integer, you place the largest possible digit at every position. Repeatedly append 9 while the remaining sum is at least 9, then append the leftover value if it is greater than 0. This works because any smaller leading digit would reduce the final number regardless of later digits. Time complexity is O(n), where n is the number of digits in the result, and space complexity is O(n) for storing the output string.

The greedy solution is a classic optimization pattern from Greedy problems. Instead of exploring all combinations, you make the locally optimal choice at each step and still guarantee the globally largest number. Since each iteration reduces the remaining sum, the implementation stays simple and efficient.

Approach 3: String Builder Optimization (O(n))

For languages where repeated string concatenation is expensive, use a mutable structure such as StringBuilder in Java or a character list in Python. The logic stays identical to the greedy method, but append operations become more efficient for large digit sums. This version is preferred in production-quality implementations because it avoids unnecessary string copies. Time complexity remains O(n) and space complexity remains O(n).

Problems like this often appear alongside Math and Implementation patterns because the challenge is mostly about numeric reasoning and constructing the output correctly.

Recommended for interviews: Interviewers expect the greedy solution because it demonstrates that you can recognize optimal digit placement without brute force search. Mentioning the exhaustive recursive approach first shows problem-solving depth, but implementing the greedy construction shows stronger optimization skills.