What is the time complexity of Sort Integers by The Number of 1 Bits?

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

What data structures are used to solve Sort Integers by The Number of 1 Bits?

This problem uses Array, Bit Manipulation, Sorting, Counting. Understanding these concepts is essential for solving this Easy problem efficiently.

What are the key insights for solving Sort Integers by The Number of 1 Bits?

**Leveraging Python's `key` for Multi-Criteria Sorting:** The genius of this approach lies in using the `key` argument of the `sorted()` function to return a tuple. When sorting by multiple criteria, creating a tuple `(primary_criterion_value, secondary_criterion_value)` provides a concise and effective way to dictate sorting precedence. Python's tuple comparison naturally handles the lexicographical ordering, perfectly aligning with the problem's 'sort by bit count, then by value' rule. **Efficient Bit Counting with String Methods:** Instead of intricate bit manipulation loops (like Brian Kernighan's algorithm or right-shifting and checking the LSB), this solution opts for a highly readable and often sufficiently performant method: `bin(x).count('1')`. Python's `bin(x)` converts an integer to its binary string representation (e.g., `0b101`), and then the string `count('1')` method efficiently tallies the set bits. This demonstrates that sometimes a clear, high-level approach can be just as effective and much more maintainable than a low-level optimization.

Which companies ask LeetCode 1356. Sort Integers by The Number of 1 Bits in interviews?

This problem is frequently asked at Accenture, J.P. Morgan, Mapbox.

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Sort Integers by The Number of 1 Bits - LeetCode 1356 Solution

Sort Integers by The Number of 1 Bits - Complete Solution Guide

Sort Integers by The Number of 1 Bits is LeetCode problem 1356, a Easy level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given an integer array arr . Sort the integers in the array in ascending order by the number of 1 's in their binary representation and in case of two or more integers have the same number of 1 's you have to sort them in ascending order. Return the array after sorting it . Example 1: Input: arr = [0,1,2,3,4,5,6,7,8] Output: [0,1,2,4,8,3,5,6,7] Explantion: [0] is the only integer with 0 bits. [1,2,4,8] all have 1 bit. [3,5,6] have 2 bits. [7] has 3 bits. The sorted array by bits is [0,1,

Detailed Explanation

This problem challenges us to reorder an array of integers, but not by their typical numerical value alone. Instead, the primary sorting criterion is the *number of '1' bits* present in their binary representation. Imagine translating each number into its binary form – for instance, 5 is '101' (two '1's), and 6 is '110' (two '1's). Both 5 and 6 would group together because they share the same bit count. After grouping, if multiple numbers have an identical count of '1' bits, then a secondary sorting rule kicks in: they should be sorted by their original numerical value in ascending order.

Solution Approach

The provided solution elegantly leverages Python's built-in `sorted()` function with a custom `key`. The core idea is to transform each number into a sortable 'proxy' that encodes both sorting criteria. By returning a tuple `(bin(x).count('1'), x)` for each integer `x`, we create exactly such a proxy. Python's `sorted()` function (and analogous sorting routines in many languages) performs lexicographical comparison on tuples: it first compares the first elements of the tuples. If they are different, that determines the order. If they are identical, it proceeds to compare the second elements, and so on. This precisely implements our two-tiered sorting requirement: first by the count of '1' bits (from `bin(x).count('1')`), then by the original number itself (`x`) as a tie-breaker, all in ascending order by default.

Step-by-Step Algorithm

Count the number of set bits (1s) in the binary representation of each integer in the input array.
Sort the array based on a custom comparison function or key. This function compares the bit counts first; if the bit counts are equal, it compares the integer values.
Return the sorted array.

Key Insights

**Leveraging Python's `key` for Multi-Criteria Sorting:** The genius of this approach lies in using the `key` argument of the `sorted()` function to return a tuple. When sorting by multiple criteria, creating a tuple `(primary_criterion_value, secondary_criterion_value)` provides a concise and effective way to dictate sorting precedence. Python's tuple comparison naturally handles the lexicographical ordering, perfectly aligning with the problem's 'sort by bit count, then by value' rule.
**Efficient Bit Counting with String Methods:** Instead of intricate bit manipulation loops (like Brian Kernighan's algorithm or right-shifting and checking the LSB), this solution opts for a highly readable and often sufficiently performant method: `bin(x).count('1')`. Python's `bin(x)` converts an integer to its binary string representation (e.g., `0b101`), and then the string `count('1')` method efficiently tallies the set bits. This demonstrates that sometimes a clear, high-level approach can be just as effective and much more maintainable than a low-level optimization.
**Implicit Ascending Order and Stability:** Python's `sorted()` function inherently sorts in ascending order and is stable. This means that when a `key` is applied and tuples are generated, they are compared from left to right, and elements that evaluate to 'equal' by the `key` maintain their original relative order. For this problem, since both the bit count and the number itself need to be sorted ascending, the default behavior of `sorted()` and tuple comparison works perfectly without any need for custom comparison logic or `reverse=True` flags.

Complexity Analysis

Time Complexity: O(n log n)

Space Complexity: O(n)

Topics

This problem involves: Array, Bit Manipulation, Sorting, Counting.

Companies

Asked at: Accenture, J.P. Morgan, Mapbox.

Sort Integers by The Number of 1 Bits | LeetCode 1356

Easy

Array Bit Manipulation Sorting Counting

Time: O(n log n)Space: O(n)

Solution Code

Available in:

import java.util.Arrays;

class Solution {
    public int[] sortByBits(int[] arr) {
        Integer[] boxedArr = Arrays.stream(arr).boxed().toArray(Integer[]::new);

        Arrays.sort(boxedArr, (a, b) -> {
            int bitCountA = Integer.bitCount(a);
            int bitCountB = Integer.bitCount(b);

            if (bitCountA != bitCountB) {
                return bitCountA - bitCountB;
            } else {
                return a - b;
            }
        });

        return Arrays.stream(boxedArr).mapToInt(Integer::intValue).toArray();
    }
}

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Problem Info

Description

Problem 1356: Sort Integers by The Number of 1 Bits - Detailed explanation and solution approaches.

Topics

Array Bit Manipulation Sorting Counting

Companies

AccentureJ.P. MorganMapbox

Stats

Difficulty:Easy

Languages:4

Companies:3

Problem #:1356

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

3 steps

1
Count the number of set bits (1s) in the binary representation of each integer in the input array.
2
Sort the array based on a custom comparison function or key. This function compares the bit counts first; if the bit counts are equal, it compares the integer values.
3
Return the sorted array.

💡

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 log n)

✨ Log-linear - Great for sorting algorithms!

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The dominant operation in all solutions except C is the sorting process. Python's `sorted()` and Java's `Arrays.sort()` use optimized comparison-based sorting algorithms (usually variations of merge sort or quicksort) with O(n log n) time complexity. C++'s `sort()` also has O(n log n) time complexity. C's solution uses bubble sort, which is O(n^2), making it less efficient than other solutions. The space complexity comes from auxiliary storage needed during sorting or the creation of intermediate arrays.

📖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

Radix Sort

Radix sort could be used if the range of input integers is limited, offering a linear time complexity (O(n)) solution. This approach would require careful handling of the dual sorting criteria.

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

Trade-offs: Radix sort is faster than comparison-based sorts for limited input ranges but involves more complex implementation.

Counting Sort

If the range of numbers and the maximum number of set bits are known, a counting sort variation could be used to potentially achieve linear time complexity.

Time: O(n + k), where k is the maximum number of set bitsSpace: O(k)

Trade-offs: This approach would require more space if k is large but could be more efficient than comparison-based sorts if the number of set bits is small and the input numbers are in a specific range.

✨

Key Insights

1**Leveraging Python's `key` for Multi-Criteria Sorting:** The genius of this approach lies in using the `key` argument of the `sorted()` function to return a tuple. When sorting by multiple criteria, creating a tuple `(primary_criterion_value, secondary_criterion_value)` provides a concise and effective way to dictate sorting precedence. Python's tuple comparison naturally handles the lexicographical ordering, perfectly aligning with the problem's 'sort by bit count, then by value' rule.
2**Efficient Bit Counting with String Methods:** Instead of intricate bit manipulation loops (like Brian Kernighan's algorithm or right-shifting and checking the LSB), this solution opts for a highly readable and often sufficiently performant method: `bin(x).count('1')`. Python's `bin(x)` converts an integer to its binary string representation (e.g., `0b101`), and then the string `count('1')` method efficiently tallies the set bits. This demonstrates that sometimes a clear, high-level approach can be just as effective and much more maintainable than a low-level optimization.
3**Implicit Ascending Order and Stability:** Python's `sorted()` function inherently sorts in ascending order and is stable. This means that when a `key` is applied and tuples are generated, they are compared from left to right, and elements that evaluate to 'equal' by the `key` maintain their original relative order. For this problem, since both the bit count and the number itself need to be sorted ascending, the default behavior of `sorted()` and tuple comparison works perfectly without any need for custom comparison logic or `reverse=True` flags.

📚

Key Terms Explained

For Students

Array

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

Sorting

Arranging elements in a specific order (ascending/descending).

Edge Cases

An array with only one element: The solution should correctly handle this trivial case. All solutions do.
An array containing duplicate integers: The solution should maintain the original relative order of duplicates (since they have the same number of 1s and the same numerical value). This is guaranteed by the stability of the sorting algorithms used in the Python, Java, and C++ solutions.

⚠️

Common Mistakes to Avoid

✗Incorrectly handling the tie-breaker condition: Failing to sort based on numerical value when the number of set bits is the same leads to incorrect results.

✗Using inefficient bit counting methods: Iterating through the binary representation as a string (or other less-efficient methods) instead of using built-in bit manipulation functions increases the time complexity.

✅

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?

💼Data analysis: Sorting data based on multiple attributes (e.g., sorting files by creation date and then file size).

🏦Network optimization: Sorting network packets based on priority levels (represented in bits) and timestamps.

💡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: September 5, 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

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 13

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

@InterviewProJul 6

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

How should I approach easy-level algorithm problems?

Easy problems build foundational skills. Focus on: 1) Understanding the problem completely before coding, 2) Writing clean, readable code, 3) Handling all edge cases (empty input, single element, etc.), 4) Analyzing time and space complexity. Don't skip complexity analysis even for simple solutions.

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 "Sort Integers by The Number of 1 Bits"?

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 bit manipulation techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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