Maximum subarray problem divide and conquer It can be solved in time and space. One way of solving the problem of finding the maximum sum subarray is using the divide & conquer algorithm. Combine subproblem solutions. length <= 105 * -104 <= nums [i] <= 104 Follow up: If you have figured out the O (n) solution, try coding another solution using the divide and conquer approach, which is more subtle. It defines the problem as finding a contiguous subsequence within a given array that has the largest sum. We used divide and conquer method to derive a (n log (n)) worst case time algorithm to solve it. Answer: d Explanation: Dynamic programming, naïve method and Divide and conquer methods can be used to solve the maximum sub array sum problem. In this video, we solve the Maximum Subarray Problem (LeetCode #53) using two powerful approaches in JavaScript: Kadane’s Algorithm (O (n)) – Optimal linear time solution Divide & Conquer Oct 8, 2023 · There are various approach to solve “Maximum Subarray” problem . We can solve ts using Divide and Conquer, what will be the worst case time complexity using Divide and Divide and Conquer The divide-and-conquer design paradigm Divide the problem (instance) into subproblems. Sum: 8 1 Introduction Given a list of integers nd the contiguous sublist with maximum sum and return that maximum sum. The worst case time complexity of the algorithm is O (n*log (n)). The Maximum Subarray problem is a classic example in computer science, used to demonstrate the application of dynamic programming and divide-and-conquer strategies. This problem is also introduced in Chapter 4. high]. To solve this problem, we divide an array A into three subarrays, and ask what is the maximum subarray in each: Problems 1 and 3 are simply this same problem on a smaller array! Divide and conquer lends itself to a recursive approach–we say that we have found a recursive case when the subproblems can be further broken into even smaller subproblems. Aug 13, 2024 · Shamos's Divide and Conquer: Splitting the Problem for O (n log n) Grenander showed the problem to computer scientist Michael Shamos. Here is my solution, but I am gett Divide-and-Conquer Examples Sorting: mergesort and quicksort Binary search (degenerated) Binary tree traversals Multiplication of large integers Matrix multiplication: Strassen’s algorithm Closest-pair algorithm Maximum Subarray Problem Problem: given an array of n numbers, find the (a) contiguous subarray whose sum has the largest value. Conquer: Recursively find the maximum subarrays in the left and right halves. Jun 30, 2024 · Maximum Subarray Sum problem is to find the subarray with maximum sum. Method 1: Standard Divide and Conquer Approach The divide and conquer approach splits the problem into Divide-and-Conquer algorithms { Overview Breaking the problem into subproblems that are themselves smaller instances of the same type of problem ("divide"), Jul 11, 2025 · Given an array arr [] of integers, the task is to find the maximum sum sub-array among all the possible sub-arrays. i* Or overlaps both halfs: i* n/2 j* n/2 n/2 We can compute the best subarray of the first two types with recursive calls on the left and right half. Intuitions, example walk through, and complexity analysis. Combine solutions to sub-problems into overall solution. It explains that divide-and-conquer involves dividing a problem into subproblems, solving the subproblems recursively, and combining the solutions. Constraints: * 1 <= nums. Illustration: Sep 6, 2016 · Excuse me, I have an assignment to solve the Maximum Sub Array Problem using the Brute Force Algorithm O (n^2), Divide and Conquer O (nlogn) and Kadane's Algorithm O (n). utdallas. Brute force will take O(N^2) time while a divide and conquer approach will take O(N log N) time. In this case, the divide function will return the maximum sub-array sum for the provided array. For example, Input: nums [] = [2, -4, 1, 9 Apr 11, 2025 · 3. How Divide-and-Conquer Resolve this Maximum Subarray Problem Suppose we want to calculate the maximum sub-array sum of the array X [l … r], where l and r are the left and right end of the input array. Jun 13, 2025 · Learn how to tackle the Maximum Subarray Problem using the Divide and Conquer approach, a fundamental algorithmic technique. In computer science, the maximum sum subarray problem, also known as the maximum segment sum problem, is the task of finding a contiguous subarray with the largest sum, within a given one-dimensional array A [1n] of numbers. Jan 13, 2019 · The maximum subarray starts at index 6 (value 18) and ends at index 19 (value 8) and the sum is 18+20–7+12–5–22+15–4+1+7–3+11–6+8=45, which is indeed, the maximum sum of contiguous Example 3: Input: nums = [5,4,-1,7,8] Output: 23 Explanation: The subarray [5,4,-1,7,8] has the largest sum 23. Divide and Conquer A subarray A[i*j*] with maximum sum is Either contained entirely in the first half , i. 1 of CLRS, with a presentation of the O (n log n) divide-and-conquer algorithm. ) This is a Python program to solve the maximum subarray problem using divide and conquer technique. Similarly for maxRightSum. The maxSum () function is recursively called only to search for the maximum sum in the left and right subarrays, respectively. Divide & Conquer-2 The Maximum Subarray Sum Problem. It is based on the observation that the maximum subarray sum will either be: In the left half of the This project implements and compares three approaches to solving the Maximum Subarray Problem in C++: Brute Force (O (n²)) Divide and Conquer / Recursive (O (n log n)) Hybrid approach using recursion with a brute-force base case for small inputs The program also times each method to compare their practical runtime efficiency. Jul 23, 2025 · The divide and conquer is an algorithmic approach to solve problems by dividing them into smaller sub-problems, solving them, and then constructing the complete solution using the solution of smaller sub-problems. Welcome to CodeCraft with Lucky! In this video, we dive into the fascinating world of algorithms with a focus on the Maximum Subarray Sum problem using the Divide and Conquer approach. Divide-and-conquer suggests that we divide the subarray into two subarrays of as equal size as possible. This algorithm implements the divide and conquer approach to find the sub-array having a maximum sum. For instance, in the below array, the highlighted subarray has the maximum sum (6): In this tutorial, we’ll take a look at two solutions for finding the maximum subarray in an array. The naive solution for ts problem is to calculate sum of all subarrays starting with every element and return the maximum of all. Note: Max subarray sum is an excellent problem to learn problem-solving using the divide and conquer approach, dynamic programming, and single loop (kadane's algorithm). Jan 24, 2020 · See the code for more idea of CSS: • LeetCode 53 Maximum Subarray using Divide In this video I am going to show how to use divide and conquer to find the maximum value of sum of a contiguous Oct 21, 2019 · Steps − Divide the array into two parts Find the maximum of the following three Maximum subarray sum of left subarray Maximum subarray sum of right subarray Maximum subarray sum such that subarray crosses the midpoint Example #include <iostream> using namespace std; int max(int a, int b) { return (a > b)? a : b; } int max(int a, int b, int c) { May 16, 2023 · Divide and Conquer Approach (O (nlogn)) What is the Maximum Subarray problem? The problem statement is straightforward. Time complexity analysis through masters theorem is also explained. divide the array into two subarrays, then find the maximum subarray recursively. 1-2 Write pseudocode for the brute-force method of solving the maximum-subarray problem. Divide and conquer problems 2008/5/9 Applications Multiplication and division as we see in lectures. We look at the problem differently: let's find the nonempty, contiguous subarray of our array whose values have the largest sum. The k -maximum subarrays problem is to find k such subarrays with the largest sums. Maximum Subarray OR Largest Sum Contiguous Subarray Problem – Divide and Conquer Objective: The maximum subarray problem is the task of finding the contiguous subarray within a one-dimensional array of numbers which has the largest sum. 1-5. Dec 30, 2022 · Divide and conquer is a general algorithmic strategy that involves dividing a problem into smaller Tagged with python, codenewbie, programming, competativeprogramming. Implement a brute force algorithm to find the maximum subarray. . Example 3: Input: nums = [5,4,-1,7,8] Output: 23 Explanation: The subarray [5,4,-1,7,8] has the largest sum 23. Given an array of n elements, write a program to find the maximum subarray sum. In this chapter, we shall see more algorithms based on divide-and-conquer. The idea is to divide the array into two halves, then recursively find the maximum subarray sum for each half as well as the subarray Sep 14, 2025 · Given an integer array, find the minimum and maximum element present in it by making minimum comparisons by using the divide-and-conquer technique. Nov 3, 2021 · Time and space complexity analysis of divide and conquer approach For simplifying the analysis, we assume that the original problem size is a power of 2, and T (n) is the time complexity of the Overview The largest subarray sum problem can be effectively solved using Kadane's Algorithm, which operates in O (n) time and O (1) space complexity by maintaining a running sum and updating the maximum found as it iterates through the array. Lecture 4 Divide and Conquer II: Counting inversions, counting intersections, max subarray, maxima set CS 161 Design and Analysis of Algorithms Ioannis Panageas Steps of method: Divide input into parts (smaller problems) Conquer (solve) each part recursively Jan 19, 2020 · We looked at solving maximum subarray problem using brute force which would solve in theta (n^3) time and solved maximum subarray problem using divide and conquer approach in theta (n lgn) time. There are two methods to solve the maximum subarray problem: Kadane’s algorithm and the divide-and-conquer approach. Problem Description 1. find the maximum by comparing two subarrays, and the Lecture 3: The Maximum Subarray Problem Divide-and-Conquer Divide-and-conquer. Then we shall see two divide-and-conquer algorithms for multiplying n n matri- ces. Then we recursively solve the problem for both parts and finally merge these solutions in a solution of the whole array. [Python code] Use the following ideas to develop a non-recursive, linear-time algorithm for the maximum-subarray problem. 6K Oct 5, 2023 · Finding maximum subarray by using divide and conquer method. There are many ways of finding the maximum sum subarray of n integers. 2. We call this the maximum subarray. This means that maxEnding at index i = max (maxEnding at index (i - 1) + arr [i], arr [i]) and the maximum value of maxEnding at any index will be our answer. Start at the left end of the array, and progress toward the right, keeping track of the maximum subarray seen so far. 1-1 What does FIND-MAXIMUM-SUBARRAY FIND-MAXIMUM-SUBARRAY return when all elements of A A are negative? It will return the greatest element of A A. Unlike subsequences, subarrays are required to occupy consecutive positions within the original array. Solution 2: Using Divide and Conquer approach Apr 7, 2020 · Compute the maximum sum of the contiguous subarray is an interesting problem that allows us to compare different ways to solve a problem. It provides you with a wide range of questions from easy level to hard level. See source codes here:O(n^3) algori LeetCode 53 | Maximum Subarray | Divide and Conquer | Hindi Next Gen App Tech 350 subscribers Subscribed Another divide and conquer Maximum subarray sum Given: an array of integers (positive and negative), find the indices that give the maximum contiguous subarray sum. we discuss three main approach which are optimaized – Kedane algo , Dp , Divide and conquer and excluded the bruteforce approach which cost about O (n^2). Array may contain negative and positive numbers which makes this a difficult problem. Example: input: 1 4 -9 8 1 3 3 1 -1 -4 -6 2 8 19 -10 -11 suba Nov 14, 2019 · Given an array A of n real numbers, the maximum subarray problem is to find a contiguous subarray which has the largest sum. CS596 Machine Learning, Spring 2021Yang Xu, Assistant Professor of Computer ScienceCollege of SciencesSan Diego State UniversityWebsite: clcs. And there are more. j* Or contained entirely in the right half , i. The article "Most Asked Divide and Conquer Coding Problems" covers all the important coding problems. Divide-and-Conquer algorithms { Overview Breaking the problem into subproblems that are themselves smaller instances of the same type of problem ("divide"), Feb 8, 2024 · The video explains the maximum Sum Sub array problem using Divide and Conquer. In this assignment, we would like you to solve this problem in (n) time. pptx Cannot retrieve latest commit at this time. Maximum Subarray in Python, Java, C++ and more. For the 1−maximum subarray the well known divide-and-conquer algorithm, presented in most textbooks, although suboptimal, is easy to implement and can be made optimal with a simple Aug 13, 2022 · Finding a maximum and minimum element from a given array is the application of the Divide and Conquer algorithm. Nov 13, 2021 · This Might HelpMaximum Subarray Problem Using Divide and Conquer Algorithm Hard 222. Nov 21, 2019 · Using Divide and Conquer approach, we can find the maximum sub array sum in O (nlogn) time. 4. The divide and conquer approach is a powerful problem-solving technique that breaks down a problem into smaller subproblems, solves them recursively, and then combines the results to solve the original problem. Conquer the subproblems by solving them recursively. Jul 22, 2025 · If the maximum subarray sum ending at the previous index is negative, it is always better to start a new subarray from the current element. Therefore, the way to calculate maxLeftSum is to call divide on the left sub-array. For instance, given an input array [-2, 1, -3, 4, -1, 2, 1, -5, 4], the maximum subarray is [4, -1, 2, 1], with a desired output sum of 6. We partition the array into two parts (left and right) of roughly the same size. Divide the array into left and right halves around the middle element. If we have to answer the problem "Sum of minimum of all subarrays" then we will use the segment tree to answer rangeMin () queries. Jan 8, 2024 · The maximum subarray problem is a task to find the series of contiguous elements with the maximum sum in any given array. The document discusses the divide-and-conquer technique and its application to the maximum subarray problem. Follow up: If you have figured out the O(n) solution, try coding another solution using the divide and conquer approach, which is more subtle. 1 The maximum-subarray problem 4. Your procedure should run in Θ(n²) time. 1. Key Steps Divide: Split the array into two halves. Formally, the task is to find indices and with , such that the sum Oct 27, 2023 · It calculates the maximum sum subarray ending at a particular position by using the maximum sum subarray ending at the previous position. You usually have an array filled with integers (positive and negatives Apr 23, 2020 · 00:00 Introduction 0:48 Divide And Conquer 06:12 Greedy 07:48 Kadane 09:33 Comparison of all Maximum Subarray Problem has been one of the most frequently asked questions in the top technology Jun 13, 2025 · Discover the power of the Divide and Conquer technique in solving the Maximum Subarray Problem, a fundamental problem in computer science. For the maximum subarray problem, the array is divided at the midpoint, the maximum subarrays within the left/right halves and crossing the The maximum subarray problem is the task of finding the contiguous subarray within a one-dimensional array, a [1n], of numbers which has the largest sum, where, Apr 19, 2021 · Learn how to solve the maximum subarray problem using the divide and conquer approach in Python with step-by-step explanations and code examples. Jul 23, 2025 · Please refer to Maximum Subarray Sum for implementation. Examples: Input: arr [] = {-2, 1, -3, 4, -1, 2, 1, -5, 4} Output: 6 {4, -1, 2, 1} is the required sub-array. For example, given an array {12, -13, -5, 25, -20, 30, 10), the maximum subarray sum is 45. Next we will try to solve the problem in theta (n) time. The problem can be simply stated as follows: given an array of integers, find the contiguous subarray (containing at least one number) which has the largest sum and return that sum. Divide-and-Conquer Approach (Efficient Solution) The Divide-and-Conquer strategy breaks the problem into smaller subproblems, solves them recursively, and combines their solutions. A subarray of array X[] is a contiguous segment from X[i] to X[j], where 0 <= i <= j <= n-1. 63K subscribers Subscribed Feb 19, 2025 · Divide and Conquer is the technique where all the main problems are divided into subproblems and after that subproblems will be solved and merged into a single solution. sdsu. On the other hand, a base case is reached when the subproblems are small enough that we can no longer deconstruct the problem into smaller ones, and recursion stops. Jul 11, 2025 · Thus, the time complexity of our divide and conquer algorithm will O (Nlog (N)). It presents several approaches: brute force of evaluating all possible subarrays in O (n^2) time; divide and conquer by recursively dividing the array in half and finding maximum crossing and non-crossing subarrays in O (n log n) time; and a transformation Apr 8, 2025 · What is the Maximum Subarray Problem? The maximum subarray problem, also known as the maximum segment sum problem, is the task of finding a contiguous subarray with the largest sum within a given one-dimensional array of numbers. Shamos thought about it for one night and came up with a divide and conquer method which is O (n log n). Problem Analysis: Sep 5, 2016 · I am learning Algorithms from a book 'An Introduction To Algorthms'. So, there are four steps of the divide and conquer method: divide, conquer, combine and base case. It presents a brute force solution with O (n2) time complexity and a more efficient divide and conquer solution with O (nlogn) time complexity. A solution using divide-and-conquer: To find a maximum subarray of the subarray A [low. Feb 8, 2019 · I created a recursive function that takes in an array of ints, and returns the sum of the continuous subarray with the largest sum. com/@varunainashots Data Structure:https://www. 99K subscribers 1. (The linear time algorithm is the subject of Exercise 4. Solve each part recursively. Input: arr [] = {2, 2, -2} Output: 4 Approach: Till now we are only aware of Kadane's Algorithm which solves this problem in O (n) using dynamic programming. The article emphasizes this approach's efficiency and practical applications across various fields, demonstrating its significance in algorithmic In data structures and algorithms, Divide and Conquer is a recursive problem-solving approach that divides the problem into smaller subproblems, recursively solves each subproblem, and combines the subproblem's solutions to get the solution of the original problem. Very naïve algorithm: Brute force algorithm: At best, θ(n2) time complexity Can we do divide and conquer? Want to use answers from left and right half subarrays. Maximum Sub Array Sum Problem using Divide and Conquer Technique Sreelakshmi@csit 1. It's quite clever. This script solves the max subarray problem via divide-and-conquer with O (nlogn) time complexity. Maximum subarray sum in right half. Algorithm / 3. While Kadane is faster, knowing the recursive approach shows true depth. Applying “divide and conquer” approach One way to attack problems is to use a “divide and conquer” approach. Question: USE Java. Algorithms Lecture 13: Maximum Sub-array Problem using Divide-and-Conquer Ghassan Shobaki Computer Science Lectures 8. Example: int [] A = {−2, 1, −3, 4, −1, 2, 1, −5, 4}; Output: contiguous subarray with the largest sum is 4, −1 79. Problem 1: Max Subarray Problem Recall the max subarray problem presented in class. Your procedure should run in Θ (n 2) Θ(n2) time. Jul 29, 2018 · Finding a maximum-subarray is quite popular problem around, and it has multiple solutions: brute-force, divide-and-conquer and kadane. Divide–and–conquer is a very powerful use of recursion Maximum sub array problem In the “Maximum Subarray Sum using Divide and Conquer” problem we have given an array of both positive and negative integers. Mar 6, 2024 · Problem Formulation: The Maximum Subarray problem aims to find a contiguous subarray within a one-dimensional array of numbers which has the largest sum. e. edu Learn about the Maximum Subarray Sum problem and how to solve it using the Divide and Conquer approach with step-by-step explanation, examples, code, and visualizations. To solve the problem, one needs to divide the array recursively into 2 parts, find the maximum subarray for the left part, the right part, and the part that crosses the midpoint for each of the divided parts, and About The maximum subarray problem is the task of finding the contiguous subarray within a one-dimensional array, a [1n], of numbers which has the largest sum. The first one solves the maximum-subarray problem: it takes as input an array of num- bers, and it determines the contiguous subarray whose values have the greatest sum. edu 👉Subscribe to our new channel:https://www. Implement the divide-and-conquer algorithm of maximum subarray introduced in Chapter 4. [Better Approach] Using Divide and Conquer - O (n*logn) time and O (n) space Divide the given array in two halves and return the maximum of following three: Maximum subarray sum in left half. Divide and Conquer Solution ¶ The divide and conquer approach to the Max Sum Subarray problem is a recursive strategy that breaks down the problem into smaller subproblems, solves these subproblems individually, and combines their solutions to get the final result for the original problem. The problem differs from the problem of finding the maximum subsequence sum. Better than official and forum solutions. Jun 23, 2014 · In this lesson, we have solved another famous programming interview question - finding maximum sub-array sum in an array. It divides the The document discusses the maximum subarray problem and different solutions to solve it. We had also Feb 1, 2013 · Edit: When thinking about the problem recursively, realize the purpose of the function, and leverage it. Break up problem into several parts. Closet pair Problem: Given n points (xi,yi) 1≤i≤n, find two points with the closet distance Tower of Hanoi Maximum Sum Subarray Think, then see the solutions. I want to implement a divide and conquer algorithm to find the maximum sub-array of an array. Combine: Find the maximum subarray crossing the midpoint and compare it with the See full list on personal. In-depth solution and explanation for LeetCode 53. Explanation of Maximum sum subarray problem using divide and conquer approach and its running timeHere we discussed brute force solution and improved running Jul 3, 2024 · Conclusion The “Maximum Subarray” problem provides a profound insight into how dynamic programming and divide-and-conquer strategies can be used to solve complex problems efficiently Dec 9, 2020 · I am aware that this is one of the most common coding questions when it comes to integral arrays. We are interested in to find maximum subarray sum (contiguous) of given 1-D array. com/playlist?list=PLxCzCOWd7aiEwaANNt3OqJPVIxwp2eb There is a log bound for comparison-based algorithms Can get log without divide and conquer (DAC) DAC solves more general "dominance counting" problems If the points are already sorted then we can solve this in time and dominance counting in log Mar 24, 2025 · Both Kadane’s and Divide & Conquer are must-know strategies for mastering array problems in interviews. youtube. Brute Force Algorithm The maximum-subarray problem Given an array of integers, find a contiguous subarray with the maximum sum. Sep 17, 2025 · Maximum Subarray Sum using Divide and Conquer Given an integer array, find the maximum sum among all subarrays possible. The document discusses the maximum subarray problem of finding the contiguous subarray with the largest sum within an array of numbers. Steps to Find Maximum Subarray Sum using Divide and Conquer The steps for maximum subarray sum are given below using divide and conquer algorithm: Find the middle index of the array. For both problems, the array is a random array, and please output the original array, the maximum subarray and the sum of the maximum subarray. Maximum subarray sum such that the subarray crosses the midpoint. I am looking for a solution to the problem of finding the longest contiguous subArray product withi Divide and Conquer Merge sort is an example of a divide-and-conquer al-gorithm Recall the three steps (at each level) to solve a divide-and-conquer problem recursively [Python code] Write pseudocode for the brute-force method of solving the maximum-subarray problem. One of which we’ll design with O (n) time and space complexity. ylrnr ivkcr kxmme hphoi ufusqq owxr kfkddf twtq ujnmmp yslig xnqypezm fknqh docnm uzi wmsvx