Maximum Value of Concatenated Binary Segments - Solution & Explanation

Problem Overview: You are given multiple binary segments and must concatenate them in some order to produce the largest possible binary value. The task is essentially choosing the best ordering of segments so the final concatenated binary string represents the maximum decimal value.

Approach 1: Brute Force Permutations (O(n! * k) time, O(n) space)

Generate every possible ordering of the binary segments using permutations. For each permutation, concatenate the segments and compute the resulting value (or compare the binary strings directly). Track the maximum result seen so far. This approach works for very small n but quickly becomes infeasible because the number of permutations grows factorially. It is mainly useful for understanding the problem and validating test cases.

Approach 2: Greedy Sorting by Concatenation Order (O(n log n * k) time, O(n) space)

The key insight: for two segments a and b, the better order is whichever produces a larger binary string between a + b and b + a. If a + b is larger, place a before b; otherwise place b first. Sorting all segments using this custom comparator produces the optimal global order. The comparison works because maximizing the most significant bits of the final concatenation dominates the value. Implement the comparator with string concatenation and lexicographic comparison. This pattern is similar to the classic “largest number by concatenation” problem and is naturally implemented with sorting and greedy reasoning.

Approach 3: Bit-Length Aware Comparison (O(n log n) comparisons, O(n) space)

Instead of building large intermediate strings repeatedly, treat each segment as a binary number with a known bit length. When comparing a before b versus b before a, compute the shifted values: (a << len(b)) | b and (b << len(a)) | a. Compare these two numbers to determine order. This reduces temporary string allocations and highlights the bit-level behavior of the concatenation. The technique relies on concepts from bit manipulation and works well when segment lengths are manageable.

Recommended for interviews: The greedy sorting comparator is what interviewers typically expect. Starting with the brute force permutation shows you understand the search space, but recognizing the a+b vs b+a ordering rule demonstrates the key greedy insight and reduces the complexity to O(n log n), which is scalable for large inputs.