What is the time complexity of N-Queens II?

The optimal solution has a time complexity of O(n!).

What data structures are used to solve N-Queens II?

This problem uses Backtracking. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving N-Queens II?

Insight 1: Backtracking is essential to explore all possible queen placements systematically. Insight 2: Efficiently track occupied columns and diagonals to avoid invalid placements using sets or arrays.

Which companies ask LeetCode 52. N-Queens II in interviews?

This problem is frequently asked at Amazon, Bloomberg, Google, Liftoff, Microsoft and 2 more companies.

JavaScript is required for the full interactive experience. However, all content is accessible without JavaScript.

N-Queens II - LeetCode 52 Solution

Q: How to solve N-Queens II?

The solution uses a backtracking algorithm to explore all possible queen placements. It iterates through each row, trying to place a queen in each column. Before placing a queen, it checks if the current column and diagonals are safe (not occupied by another queen). If safe, it places the queen, marks the column and diagonals as occupied, and recursively calls the backtracking function for the next row. After the recursive call returns, it unmarks the column and diagonals (backtracks) to explore other possibilities. The algorithm counts the number of times all queens can be placed successfully.

N-Queens II - Complete Solution Guide

N-Queens II is LeetCode problem 52, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

The n-queens puzzle is the problem of placing n queens on an n x n chessboard such that no two queens attack each other. Given an integer n , return the number of distinct solutions to the n-queens puzzle . Example 1: Input: n = 4 Output: 2 Explanation: There are two distinct solutions to the 4-queens puzzle as shown. Example 2: Input: n = 1 Output: 1 Constraints: 1 <= n <= 9

Detailed Explanation

The N-Queens II problem is a variation of the classic N-Queens problem. The goal is to determine the number of distinct ways to place 'n' queens on an 'n x n' chessboard such that no two queens threaten each other. This means no two queens can share the same row, column, or diagonal. The input is an integer 'n' representing the size of the board, and the output is the number of valid arrangements of queens.

Solution Approach

The solution uses a backtracking algorithm to explore all possible queen placements. It iterates through each row, trying to place a queen in each column. Before placing a queen, it checks if the current column and diagonals are safe (not occupied by another queen). If safe, it places the queen, marks the column and diagonals as occupied, and recursively calls the backtracking function for the next row. After the recursive call returns, it unmarks the column and diagonals (backtracks) to explore other possibilities. The algorithm counts the number of times all queens can be placed successfully.

Step-by-Step Algorithm

Step 1: Initialize sets (or arrays in C) to track occupied columns, diagonals, and anti-diagonals.
Step 2: Define a recursive `backtrack` function that takes the current row as input.
Step 3: Base Case: If the current row equals 'n', it means all queens have been placed successfully. Increment the count and return.
Step 4: Iterate through each column in the current row.
Step 5: Calculate the diagonal and anti-diagonal indices based on the row and column.
Step 6: Check if the current column, diagonal, and anti-diagonal are already occupied. If so, continue to the next column.
Step 7: If the current position is safe, mark the column, diagonal, and anti-diagonal as occupied.
Step 8: Recursively call the `backtrack` function for the next row (row + 1).
Step 9: After the recursive call returns, unmark the column, diagonal, and anti-diagonal (backtrack) to explore other possibilities.
Step 10: After exploring all columns, return from the `backtrack` function.
Step 11: After the initial call to the `backtrack` function, return the final count of valid solutions.

Key Insights

Insight 1: Backtracking is essential to explore all possible queen placements systematically.
Insight 2: Efficiently track occupied columns and diagonals to avoid invalid placements using sets or arrays.
Insight 3: Understand that the diagonals and anti-diagonals can be represented using simple arithmetic relationships between row and column indices (row - col for diagonals and row + col for anti-diagonals).

Complexity Analysis

Time Complexity: O(n!)

Space Complexity: O(n)

Topics

This problem involves: Backtracking.

Companies

Asked at: Amazon, Bloomberg, Google, Liftoff, Microsoft, Snowflake, Zenefits.

N-Queens II | LeetCode 52

Hard

Backtracking

Time: O(n!)Space: O(n)

Solution Code

Available in:

import java.util.HashSet;
import java.util.Set;

class Solution {
    public int totalNQueens(int n) {
        Set<Integer> cols = new HashSet<>();
        Set<Integer> diagonals = new HashSet<>();
        Set<Integer> anti_diagonals = new HashSet<>();
        int[] count = {0};

        backtrack(0, n, cols, diagonals, anti_diagonals, count);
        return count[0];
    }

    private void backtrack(int row, int n, Set<Integer> cols, Set<Integer> diagonals, Set<Integer> anti_diagonals, int[] count) {
        if (row == n) {
            count[0]++;
            return;
        }

        for (int col = 0; col < n; col++) {
            int diag = row - col;
            int anti_diag = row + col;

            if (cols.contains(col) || diagonals.contains(diag) || anti_diagonals.contains(anti_diag)) {
                continue;
            }

            cols.add(col);
            diagonals.add(diag);
            anti_diagonals.add(anti_diag);

            backtrack(row + 1, n, cols, diagonals, anti_diagonals, count);

            cols.remove(col);
            diagonals.remove(diag);
            anti_diagonals.remove(anti_diag);
        }
    }
}

View Full Solution on GitHubGitHub Solution View on LeetCodeLeetCode

Problem Info

Description

Problem 52: N-Queens II - Detailed explanation and solution approaches.

Topics

Backtracking

Companies

AmazonBloombergGoogleLiftoffMicrosoftSnowflakeZenefits

Stats

Difficulty:Hard

Languages:4

Companies:7

Problem #:52

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

11 steps

1
Step 1: Initialize sets (or arrays in C) to track occupied columns, diagonals, and anti-diagonals.
2
Step 2: Define a recursive `backtrack` function that takes the current row as input.
3
Step 3: Base Case: If the current row equals 'n', it means all queens have been placed successfully. Increment the count and return.
4
Step 4: Iterate through each column in the current row.
5
Step 5: Calculate the diagonal and anti-diagonal indices based on the row and column.
6
Step 6: Check if the current column, diagonal, and anti-diagonal are already occupied. If so, continue to the next column.
7
Step 7: If the current position is safe, mark the column, diagonal, and anti-diagonal as occupied.
8
Step 8: Recursively call the `backtrack` function for the next row (row + 1).
9
Step 9: After the recursive call returns, unmark the column, diagonal, and anti-diagonal (backtrack) to explore other possibilities.
10
Step 10: After exploring all columns, return from the `backtrack` function.
11
Step 11: After the initial call to the `backtrack` function, return the final count of valid solutions.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n!)

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity is O(n!) because, at each row, we potentially explore 'n' columns. Although the constraints on columns and diagonals cut down some paths, we still explore a number of branches that grow factorially with 'n'. The space complexity is O(n) because the sets/arrays used to track the occupied columns, diagonals, and anti-diagonals each store at most a number of elements proportional to 'n'.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Bit Manipulation

Instead of using sets or arrays, we can use bit manipulation to represent the occupied columns and diagonals. This can potentially improve performance slightly.

Time: O(n!)Space: O(n)

Trade-offs: Bit manipulation can be slightly faster due to its low-level nature, but it may be less readable and maintainable than using sets or arrays. The space complexity remains the same because we still need to track the occupied states recursively.

✨

Key Insights

1Insight 1: Backtracking is essential to explore all possible queen placements systematically.
2Insight 2: Efficiently track occupied columns and diagonals to avoid invalid placements using sets or arrays.
3Insight 3: Understand that the diagonals and anti-diagonals can be represented using simple arithmetic relationships between row and column indices (row - col for diagonals and row + col for anti-diagonals).

📚

Key Terms Explained

For Students

Edge Cases

Edge case 1: n = 1. The solution should correctly return 1, as a single queen can be placed on a 1x1 board.
Edge case 2: n = 2 or n = 3. The solution should correctly return 0, as no two queens can be placed on a 2x2 or 3x3 board without attacking each other. The backtracking algorithm correctly explores and rejects these invalid cases.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to backtrack (unmark) the column and diagonals after exploring a branch. This will lead to incorrect counts, as it prevents exploration of other valid arrangements.

✗Common mistake 2: Incorrectly calculating the indices for diagonals and anti-diagonals. It's crucial to get the arithmetic right to accurately track occupied diagonals.

✗Common mistake 3: Not handling the base case (row == n) correctly, leading to incorrect counting of valid solutions.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Constraint satisfaction problems in scheduling or resource allocation. The N-Queens problem is a classic example of a constraint satisfaction problem, where the goal is to find a configuration that satisfies a set of constraints.

🏦Real-world scenario 2: AI planning and search algorithms. Backtracking is a fundamental technique used in AI for planning and search, where the goal is to find a sequence of actions that leads to a desired state, while satisfying certain constraints.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

Backtracking = DFS + pruning. The key is identifying when to backtrack. Always restore state after recursion.

🎯 Interview Tip

It's okay to not finish hard problems in interviews. Show your problem-solving process.

⚠️ Watch Out

Always test your solution with edge cases: empty input, single element, and boundary values.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: July 23, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 13

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 6

❓

Frequently Asked Questions

What is the difference between backtracking and brute force?

Backtracking prunes invalid paths early using constraints, while brute force explores all possibilities. In backtracking, you "choose" an option, "explore" it recursively, then "unchoose" (backtrack) if it doesn't lead to a solution. This eliminates entire branches of the search tree.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "N-Queens II"?

For this hard-level Backtracking problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether standard techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?