#1718 - Construct the Lexicographically Largest Valid Sequence

Problem Description

Given an integer n, find a sequence that satisfies all of the following:

The integer 1 occurs once in the sequence.
Each integer between 2 and n occurs twice in the sequence.
For every integer i between 2 and n, the distance between the two occurrences of i is exactly i.

The distance between two numbers on the sequence, a[i] and a[j], is the absolute difference of their indices, |j - i|.

Return the lexicographically largest sequence. It is guaranteed that under the given constraints, there is always a solution.

A sequence a is lexicographically larger than a sequence b (of the same length) if in the first position where a and b differ, sequence a has a number greater than the corresponding number in b. For example, [0,1,9,0] is lexicographically larger than [0,1,5,6] because the first position they differ is at the third number, and 9 is greater than 5.

Solution

/**
 * @param {number} n
 * @return {number[]}
 */
var constructDistancedSequence = function(n) {
  const result = new Array(2 * n - 1).fill(0);
  const group = new Array(n + 1).fill(false);

  backtrack(0);

  function backtrack(index) {
    if (index === 2 * n - 1) {
      return true;
    } else if (result[index] !== 0) {
      return backtrack(index + 1);
    }
    for (let num = n; num >= 1; num--) {
      if (group[num]) {
        continue;
      }
      group[num] = true;
      result[index] = num;
      if (num === 1 || (index + num < 2 * n - 1 && result[index + num] === 0)) {
        if (num > 1) {
          result[index + num] = num;
        }
        if (backtrack(index + 1)) {
          return true;
        }
        if (num > 1) {
          result[index + num] = 0;
        }
      }
      result[index] = 0;
      group[num] = false;
    }
    return false;
  }

  return result;
};