What is the time complexity of Number of Sub-arrays With Odd Sum?

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

What data structures are used to solve Number of Sub-arrays With Odd Sum?

This problem uses Array, Math, Dynamic Programming, Prefix Sum. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Number of Sub-arrays With Odd Sum?

Insight 1: A subarray sum is odd if and only if it contains an odd number of odd elements. This is because adding an even number doesn't change the parity (oddness or evenness) of the sum, while adding an odd number flips it. Insight 2: We can efficiently solve this problem by tracking the counts of even and odd prefix sums. A prefix sum at index 'i' is the sum of all elements from index 0 to 'i'.

Which companies ask LeetCode 1524. Number of Sub-arrays With Odd Sum in interviews?

This problem is frequently asked at Directi.

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Number of Sub-arrays With Odd Sum - LeetCode 1524 Solution

Number of Sub-arrays With Odd Sum - Complete Solution Guide

Number of Sub-arrays With Odd Sum is LeetCode problem 1524, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given an array of integers arr , return the number of subarrays with an odd sum . Since the answer can be very large, return it modulo 10 9 + 7 . Example 1: Input: arr = [1,3,5] Output: 4 Explanation: All subarrays are [[1],[1,3],[1,3,5],[3],[3,5],[5]] All sub-arrays sum are [1,4,9,3,8,5]. Odd sums are [1,9,3,5] so the answer is 4. Example 2: Input: arr = [2,4,6] Output: 0 Explanation: All subarrays are [[2],[2,4],[2,4,6],[4],[4,6],[6]] All sub-arrays sum are [2,6,12,4,10,6]. All sub-arrays have

Detailed Explanation

The problem asks us to find the number of subarrays within a given integer array that have an odd sum. The final answer needs to be returned modulo 10^9 + 7 due to potential overflow. A subarray is a contiguous sequence of elements within the array. For instance, given the array [1, 2, 3], possible subarrays are [1], [2], [3], [1, 2], [2, 3], and [1, 2, 3]. The goal is to count how many of these subarrays have an odd sum.

Solution Approach

The solution calculates the number of subarrays with odd sums by keeping track of the counts of even and odd prefix sums. For each element in the input array, the current prefix sum is updated. The parity of the prefix sum is then checked. If the current prefix sum is odd, the number of odd prefixes is incremented. If it's even, the number of even prefixes is incremented. The final result is the product of the total even prefixes and odd prefixes, taken modulo 10^9 + 7.

Step-by-Step Algorithm

Step 1: Initialize `odd_prefixes` to 0 and `even_prefixes` to 1. We start with one even prefix sum (0, the sum before the array begins).
Step 2: Initialize `current_sum_parity` to 0, representing an even sum.
Step 3: Iterate through the input array `arr`.
Step 4: For each number `num` in `arr`, update the `current_sum_parity` by adding `num` and taking the modulo 2. This determines whether the current prefix sum is even or odd.
Step 5: If `current_sum_parity` is 1 (odd), increment `odd_prefixes`. Otherwise, increment `even_prefixes`.
Step 6: After iterating through the entire array, calculate the product of `odd_prefixes` and `even_prefixes`, and take the modulo 10^9 + 7. This product represents the number of subarrays with an odd sum.
Step 7: Return the result.

Key Insights

Insight 1: A subarray sum is odd if and only if it contains an odd number of odd elements. This is because adding an even number doesn't change the parity (oddness or evenness) of the sum, while adding an odd number flips it.
Insight 2: We can efficiently solve this problem by tracking the counts of even and odd prefix sums. A prefix sum at index 'i' is the sum of all elements from index 0 to 'i'.
Insight 3: To determine if a subarray's sum is odd, we only need to know the parity (odd/even) of the prefix sums at the start and end of the subarray.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(1)

Topics

This problem involves: Array, Math, Dynamic Programming, Prefix Sum.

Companies

Asked at: Directi.

Number of Sub-arrays With Odd Sum | LeetCode 1524

Medium

Array Math Dynamic Programming Prefix Sum

Time: O(n)Space: O(1)

Solution Code

Available in:

class Solution {
    public int numOfSubarrays(int[] arr) {
        int MOD = (int)1e9 + 7;
        
        // We need to find the number of pairs of prefix sums (P_i, P_j) 
        // with i < j such that (P_j - P_i) is odd.
        // This occurs if one prefix sum is even and the other is odd.
        // So, the answer is (number of even prefix sums) * (number of odd prefix sums).
        
        // odd_prefixes will count prefix sums with odd value.
        // even_prefixes will count prefix sums with even value.
        // We start with a prefix sum of 0 before the array, which is even.
        int odd_prefixes = 0;
        int even_prefixes = 1;
        
        int current_sum_parity = 0;  // 0 for even, 1 for odd
        
        for (int num : arr) {
            current_sum_parity = (current_sum_parity + num) % 2;
            
            if (current_sum_parity == 1) {
                odd_prefixes++;
            } else {
                even_prefixes++;
            }
        }
        
        return (int)((long)odd_prefixes * even_prefixes % MOD);
    }
}

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Problem Info

Description

Problem 1524: Number of Sub-arrays With Odd Sum - Detailed explanation and solution approaches.

Topics

Array Math Dynamic Programming Prefix Sum

Companies

Directi

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:1524

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

7 steps

1
Step 1: Initialize `odd_prefixes` to 0 and `even_prefixes` to 1. We start with one even prefix sum (0, the sum before the array begins).
2
Step 2: Initialize `current_sum_parity` to 0, representing an even sum.
3
Step 3: Iterate through the input array `arr`.
4
Step 4: For each number `num` in `arr`, update the `current_sum_parity` by adding `num` and taking the modulo 2. This determines whether the current prefix sum is even or odd.
5
Step 5: If `current_sum_parity` is 1 (odd), increment `odd_prefixes`. Otherwise, increment `even_prefixes`.
6
Step 6: After iterating through the entire array, calculate the product of `odd_prefixes` and `even_prefixes`, and take the modulo 10^9 + 7. This product represents the number of subarrays with an odd sum.
7
Step 7: Return the result.

💡

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)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is O(n) as we iterate through the array once. The space complexity is O(1) because we only store a few integer variables, which do not depend on the input size.

📖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

Brute Force Approach

Iterate through all possible subarrays and calculate their sums. Count the number of subarrays with an odd sum.

Time: O(n^2), where n is the length of the input array.Space: O(1)

Trade-offs: The brute force approach is simple to understand but is inefficient for large arrays due to its quadratic time complexity. The given solution is far more efficient.

✨

Key Insights

1Insight 1: A subarray sum is odd if and only if it contains an odd number of odd elements. This is because adding an even number doesn't change the parity (oddness or evenness) of the sum, while adding an odd number flips it.
2Insight 2: We can efficiently solve this problem by tracking the counts of even and odd prefix sums. A prefix sum at index 'i' is the sum of all elements from index 0 to 'i'.
3Insight 3: To determine if a subarray's sum is odd, we only need to know the parity (odd/even) of the prefix sums at the start and end of the subarray.

📚

Key Terms Explained

For Students

Array

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

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: Empty array. The code correctly handles this because the outer loop is skipped, and the initial even_prefixes (1) and odd_prefixes (0) lead to a result of 0 * 1 = 0, which is correct as there are no sub-arrays.
Edge case 2: Array with all even numbers. In this case, odd_prefixes will remain 0, and the result will be 0.
Edge case 3: Array with all odd numbers. Let the length be n. The even_prefixes will be 1 for prefix sum 0, and n-1 or n even prefixes depending on whether n is even or odd. The odd_prefixes will be n or n-1 odd prefixes.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Integer overflow. Multiplying `odd_prefixes` and `even_prefixes` can result in a value exceeding the maximum integer value, leading to incorrect results. The modulo operation (`% MOD`) is necessary to prevent this.

✗Common mistake 2: Incorrect initialization. Failing to initialize `even_prefixes` to 1 will lead to an incorrect count. We start with a prefix sum of 0 before the array which is even.

✗Common mistake 3: Forgetting the Modulo. Not applying the modulo operation after each arithmetic operation on the prefix sums can lead to overflow and incorrect results.

✅

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: Analyzing financial data. Identifying sequences of transactions that result in an odd balance, which might be relevant for specific auditing or security protocols.

🏦Real-world scenario 2: Signal processing. Determining patterns in a signal where the parity of the accumulated signal strength is significant.

💡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

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ 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 24, 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

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

❓

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.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

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 "Number of Sub-arrays With Odd Sum"?

For this medium-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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