What is the time complexity of Sum of All Subset XOR Totals?

The optimal solution has a time complexity of O(n * 2^n).

What data structures are used to solve Sum of All Subset XOR Totals?

This problem uses Array, Math, Backtracking, Bit Manipulation, Combinatorics, Enumeration. Understanding these concepts is essential for solving this Easy problem efficiently.

What are the key insights for solving Sum of All Subset XOR Totals?

Insight 1: The problem can be efficiently solved by iterating through all possible subsets using bit manipulation. Each bit in a binary number from 0 to 2^n - 1 can represent whether an element is included in a subset. Insight 2: Bitwise XOR operations are crucial for efficiently calculating the XOR total of each subset. The XOR operation's properties (commutative and associative) simplify the calculation.

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Sum of All Subset XOR Totals - LeetCode 1863 Solution

Sum of All Subset XOR Totals - Complete Solution Guide

Sum of All Subset XOR Totals is LeetCode problem 1863, a Easy level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Framing

Sum of All Subset XOR Totals is a Easy LeetCode problem that rewards careful tracing, edge-case handling, and a clear grasp of Array and Math. The best solutions usually explain why the chosen invariant holds before they optimize for time or space.

Quick Example Mindset

A useful way to test Sum of All Subset XOR Totals is to start with a tiny input that exposes the boundary conditions, then run the same logic on a slightly larger case to verify the array behavior and the math interaction. That second pass is where off-by-one mistakes and missing updates usually appear.

Problem Statement

The XOR total of an array is defined as the bitwise XOR of all its elements , or 0 if the array is empty . For example, the XOR total of the array [2,5,6] is 2 XOR 5 XOR 6 = 1 . Given an array nums , return the sum of all XOR totals for every subset of nums . Note: Subsets with the same elements should be counted multiple times. An array a is a subset of an array b if a can be obtained from b by deleting some (possibly zero) elements of b . Example 1: Input: nums = [1,3] Output: 6 Explanation: T

Detailed Explanation

The problem asks to calculate the sum of XOR totals for all possible subsets of a given array `nums`. The XOR total of a subset is the bitwise XOR of all its elements (or 0 if the subset is empty). The input is an array of integers, and the output is a single integer representing the sum of XOR totals across all subsets. The problem emphasizes that subsets with the same elements should be counted multiple times (meaning we consider all 2^n subsets, where n is the length of `nums`).

Solution Approach

The solution uses a brute-force approach that systematically generates all possible subsets of the input array `nums`. It iterates through numbers from 0 to 2^n - 1 (where n is the length of `nums`), using each number's binary representation to determine which elements to include in a subset. For each subset, it calculates the XOR total and adds it to the running sum. Finally, it returns the total sum of XOR totals.

Step-by-Step Algorithm

Step 1: Initialize `total_xor_sum` to 0. This variable will store the sum of XOR totals.
Step 2: Iterate through all possible subsets using a loop from 0 to 2^n - 1. Each number `i` represents a subset.
Step 3: For each `i`, iterate through the `nums` array. If the `j`-th bit of `i` is set (using the bitwise AND operator `&`), include `nums[j]` in the current subset.
Step 4: Calculate the XOR total of the current subset using bitwise XOR (`^`) operations on the selected elements.
Step 5: Add the current XOR total to `total_xor_sum`.
Step 6: After iterating through all subsets, return `total_xor_sum`.

Key Insights

Insight 1: The problem can be efficiently solved by iterating through all possible subsets using bit manipulation. Each bit in a binary number from 0 to 2^n - 1 can represent whether an element is included in a subset.
Insight 2: Bitwise XOR operations are crucial for efficiently calculating the XOR total of each subset. The XOR operation's properties (commutative and associative) simplify the calculation.
Insight 3: The time complexity is inherently exponential due to the need to consider all subsets. Optimizations beyond the provided approach are unlikely to significantly reduce the complexity for this problem

Complexity Analysis

Time Complexity: O(n * 2^n)

Space Complexity: O(1)

Topics

This problem involves: Array, Math, Backtracking, Bit Manipulation, Combinatorics, Enumeration.

Study Paths

Continue from this problem into the surrounding topic and company clusters to compare how the same pattern appears in other interview settings.

Related topics: Array, Math, Backtracking, Bit Manipulation

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

What should I learn from easy problems?

Easy problems introduce core patterns that appear in harder problems. Master basic operations (iteration, conditionals, simple data structures), recognize common patterns (counting, searching, basic transformations), and practice explaining your thought process clearly.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

Sum of All Subset XOR Totals | LeetCode 1863

Easy

Array Math Backtracking Bit Manipulation Combinatorics Enumeration

Study Path

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Array Math Backtracking Bit Manipulation Combinatorics Enumeration

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Time: O(n * 2^n)Space: O(1)

Solution Code

Available in:

class Solution {
    public int subsetXORSum(int[] nums) {
        int n = nums.length;
        int sum = 0;
        for (int i = 0; i < (1 << n); i++) {
            int xor = 0;
            for (int j = 0; j < n; j++) {
                if ((i >> j) % 2 == 1) {
                    xor ^= nums[j];
                }
            }
            sum += xor;
        }
        return sum;
    }
}

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Problem Info

Description

Problem 1863: Sum of All Subset XOR Totals - Detailed explanation and solution approaches.

Topics

Array Math Backtracking Bit Manipulation Combinatorics Enumeration

Stats

Difficulty:Easy

Languages:4

Problem #:1863

Problem Explanation

🧭

Next Steps

Use this problem as a starting point, then move into adjacent patterns and company sets to reinforce the same idea in different contexts.

Topic hub

Study neighboring problems by pattern and difficulty.

Company hub

See where this problem appears in interview loops.

Pattern guides

Read deeper walkthroughs of the algorithm family.

💡

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

6 steps

1
Step 1: Initialize `total_xor_sum` to 0. This variable will store the sum of XOR totals.
2
Step 2: Iterate through all possible subsets using a loop from 0 to 2^n - 1. Each number `i` represents a subset.
3
Step 3: For each `i`, iterate through the `nums` array. If the `j`-th bit of `i` is set (using the bitwise AND operator `&`), include `nums[j]` in the current subset.
4
Step 4: Calculate the XOR total of the current subset using bitwise XOR (`^`) operations on the selected elements.
5
Step 5: Add the current XOR total to `total_xor_sum`.
6
Step 6: After iterating through all subsets, return `total_xor_sum`.

💡

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 * 2^n)

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The outer loop iterates 2^n times (number of subsets). The inner loop iterates n times (length of `nums`). Therefore, the time complexity is O(n * 2^n). The algorithm uses a constant amount of extra space, independent of the input size, resulting in O(1) space complexity.

📖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

Recursive Approach

A recursive function could generate all subsets. For each subset, it recursively calculates the XOR sum.

Time: O(n * 2^n)Space: O(n) due to recursive call stack

Trade-offs: The recursive approach might be less efficient due to the overhead of function calls and stack management compared to the iterative bit manipulation approach.

✨

Key Insights

1Insight 1: The problem can be efficiently solved by iterating through all possible subsets using bit manipulation. Each bit in a binary number from 0 to 2^n - 1 can represent whether an element is included in a subset.
2Insight 2: Bitwise XOR operations are crucial for efficiently calculating the XOR total of each subset. The XOR operation's properties (commutative and associative) simplify the calculation.
3Insight 3: The time complexity is inherently exponential due to the need to consider all subsets. Optimizations beyond the provided approach are unlikely to significantly reduce the complexity for this problem

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Edge Cases

Edge case 1: Empty input array `nums`. The solution correctly handles this by producing an XOR total of 0.
Edge case 2: Input array with a single element. The solution correctly calculates the XOR sum as the single element (since the subsets are just the empty set and the set itself).

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly using bit manipulation to select subset elements. A common error is failing to properly check the bits using the right-shift (`>>`) and bitwise AND (`&`) operators.

✗Common mistake 2: Off-by-one errors in the loop limits. Ensuring that the loop correctly covers all subsets (0 to 2^n -1) is essential.

✅

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: Combinatorial problems in areas such as network design, where you need to calculate aggregate properties of all possible connections or paths.

🏦Real-world scenario 2: Cryptography and security, where bitwise operations and subset analysis are frequently used in algorithms for generating keys, checksums, or error detection.

💡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

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Even for easy problems, discuss time and space complexity. Interviewers want to see you think analytically.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

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: August 3, 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

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

@CodeNinjaJul 21

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 14

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 7

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Frequently Asked Questions

When should I use in-place modification vs creating a new array?

What should I learn from easy problems?

What is the importance of time and space complexity analysis?

What is the optimal approach for "Sum of All Subset XOR Totals"?

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

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