#1718

Medium

auto_awesomeBacktracking search with pruning

LeetCode Problem Workspace

Construct the Lexicographically Largest Valid Sequence

Construct the Lexicographically Largest Valid Sequence problem involves finding the largest sequence with backtracking and pruning.

array backtracking

Problem Statement

You are given an integer n. Your task is to construct a sequence consisting of elements in the range [1, n] that satisfies the following condition: the absolute difference between two numbers at positions i and j, |j - i|, equals the distance between those numbers. The sequence must be valid under this condition.

Your goal is to return the lexicographically largest sequence that satisfies this constraint. It is guaranteed that a solution always exists for the given constraints, where 1 <= n <= 20.

Examples

Example 1

Input: n = 3

Output: [3,1,2,3,2]

[2,3,2,1,3] is also a valid sequence, but [3,1,2,3,2] is the lexicographically largest valid sequence.

Example 2

Input: n = 5

Output: [5,3,1,4,3,5,2,4,2]

Example details omitted.

Constraints

1 <= n <= 20

Solution Approach

Backtracking with pruning

The solution uses backtracking to explore all possible sequences and checks for validity based on the distance condition. Pruning is applied to eliminate invalid sequences early, improving efficiency.

Lexicographical order optimization

During the backtracking process, sequences are constructed in a way that prioritizes larger values first, ensuring the lexicographically largest sequence is found.

Recursive exploration of sequence positions

The recursive function explores each position in the sequence, trying different values from 1 to n and backtracking when a valid sequence is not achievable, ensuring all possibilities are explored.

Complexity Analysis

Metric	Value
Time	O(n!)
Space	O(n)

The time complexity of this solution is O(n!) due to the backtracking nature, where each step explores n possibilities. The space complexity is O(n), as the recursion stack stores the current state of the sequence being built.

What Interviewers Usually Probe

Candidate demonstrates understanding of backtracking algorithms and pruning techniques.
Candidate is able to optimize the search space by prioritizing lexicographical order.
Candidate can handle recursive exploration with attention to efficiency.

Common Pitfalls or Variants

Common pitfalls

Not pruning invalid sequences early enough, leading to excessive computation.
Failing to optimize the sequence construction for lexicographical order, which may result in incorrect or suboptimal solutions.
Overcomplicating the backtracking process without utilizing pruning to reduce the search space.

Follow-up variants

Use of a heuristic algorithm to improve efficiency when n is large.
Modification to generate the smallest valid lexicographical sequence.
Exploring dynamic programming or memoization for optimizing repeated subproblems in backtracking.

FAQ

What is the main challenge in constructing the lexicographically largest valid sequence?

The main challenge lies in efficiently exploring the sequence space with backtracking and ensuring the lexicographically largest sequence is selected.

How does pruning improve the backtracking process in this problem?

Pruning eliminates invalid sequences early, reducing the number of paths the backtracking algorithm needs to explore, making the solution more efficient.

How can heuristic algorithms help with this problem?

Heuristic algorithms can help prioritize certain branches in the backtracking process, leading to quicker identification of valid sequences, improving overall efficiency.

What is the time complexity of the solution?

The time complexity is O(n!) due to the backtracking nature of the solution, as each recursive call explores n possibilities.

Can dynamic programming be used to optimize the solution?

While dynamic programming is not traditionally used for this type of problem, it may be useful for optimizing certain repeated subproblems within the backtracking approach.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def constructDistancedSequence(self, n: int) -> List[int]:
        def dfs(u):
            if u == n * 2:
                return True
            if path[u]:
                return dfs(u + 1)
            for i in range(n, 1, -1):
                if cnt[i] and u + i < n * 2 and path[u + i] == 0:
                    cnt[i] = 0
                    path[u] = path[u + i] = i
                    if dfs(u + 1):
                        return True
                    path[u] = path[u + i] = 0
                    cnt[i] = 2
            if cnt[1]:
                cnt[1], path[u] = 0, 1
                if dfs(u + 1):
                    return True
                path[u], cnt[1] = 0, 1
            return False

        path = [0] * (n * 2)
        cnt = [2] * (n * 2)
        cnt[1] = 1
        dfs(1)
        return path[1:]

Continue Practicing

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