Construct the Lexicographically Largest Valid Sequence - Video Solutions

Problem Overview: You need to build an array of length 2n - 1 using numbers from 1 to n. Each number k > 1 must appear exactly twice and the distance between the two occurrences must be k. Number 1 appears once. Among all valid sequences, return the lexicographically largest one.

Approach 1: Backtracking with Greedy Placement (Time: O(n!), Space: O(n))

This is the most common solution and the one typically expected in interviews. Use backtracking to fill the sequence from left to right. At each empty index, try placing numbers from n down to 1. For values greater than 1, you must check whether index i + k is within bounds and also empty before placing both occurrences. Mark the number as used, continue recursion, and revert the placement if it leads to a dead end. Trying larger numbers first guarantees the first valid sequence you find is already the lexicographically largest. The search space is exponential, but pruning invalid placements early keeps it manageable.

Approach 2: Iterative Construction (Time: O(n^2), Space: O(n))

An iterative strategy builds the sequence while scanning the array and attempting placements based on available numbers. Maintain a result array and a structure that tracks which numbers are still unused. When encountering an empty position, try inserting the largest remaining number that satisfies the distance constraint. This approach relies on repeated checks and adjustments as the sequence grows. It works well for moderate n values and avoids recursion overhead, though it still performs many placement validations.

Approach 3: Divide and Conquer Placement (Time: O(n!), Space: O(n))

You can model the problem as recursively solving subsegments of the array. Pick the largest unused number and attempt to place it at valid positions within the current segment. Once placed, the array splits into smaller independent segments that must also satisfy the same constraints. The recursive structure resembles divide-and-conquer: each placement reduces the problem size and creates smaller subproblems. In practice this behaves similarly to backtracking but conceptually emphasizes segment decomposition within the array.

Approach 4: Heap-based Ordering (Time: O(n log n), Space: O(n))

A heap can help prioritize which numbers to attempt first. Store unused numbers in a max-heap so the largest value is always considered before smaller ones. While scanning the sequence, extract the largest candidate and check if its required distance placement is valid. If placement fails, temporarily skip it and reinsert it later. The heap guarantees lexicographic preference toward larger numbers while maintaining efficient retrieval.

Recommended for interviews: The backtracking solution is the expected answer. Interviewers want to see how you model constraints, prune invalid states, and enforce lexicographic ordering by trying larger numbers first. Starting with brute-force placement shows understanding, but implementing efficient backtracking with early validation demonstrates strong problem-solving skills.