Find Maximum Subarray Divide and Conquer in DSA Go - Build from Scratch

Find Maximum Subarray Using Divide and Conquer

📖 Scenario: Imagine you have a list of daily profits and losses from a small business. You want to find the period where the business made the most money continuously. This means finding the subarray with the highest sum.

🎯 Goal: Build a Go program that uses the divide and conquer method to find the maximum sum of a continuous subarray within a given list of integers.

📋 What You'll Learn

Create a slice of integers named profits with the exact values given.

Create a helper function to find the maximum of two integers named max.

Implement the divide and conquer approach with functions named maxCrossingSum and maxSubArraySum.

Print the maximum subarray sum found.

💡 Why This Matters

🌍 Real World

Finding the maximum subarray sum helps in financial analysis to identify the most profitable continuous period.

💼 Career

Understanding divide and conquer algorithms is essential for software engineers working on performance-critical applications and algorithm design.

1

Create the Data Set

Create a slice of integers called profits with these exact values: 13, -3, -25, 20, -3, -16, -23, 18, 20, -7, 12, -5, -22, 15, -4, 7.

DSA Go

# Create the slice profits with the given values
# Your code here

Hint

Use profits := []int{...} to create the slice with the exact values.

2

Add a Helper Function to Find Maximum of Two Integers

Add a function named max that takes two integers a and b and returns the larger one.

DSA Go

package main

func main() {
    profits := []int{13, -3, -25, 20, -3, -16, -23, 18, 20, -7, 12, -5, -22, 15, -4, 7}
}

// Write the max function below
# Your code here

Hint

Use an if statement to compare a and b and return the larger.

3

Implement Divide and Conquer Functions

Add two functions: maxCrossingSum and maxSubArraySum. maxCrossingSum takes the slice profits, and integers left, mid, and right, and returns the maximum sum of a subarray crossing the midpoint. maxSubArraySum takes profits, left, and right, and returns the maximum subarray sum using divide and conquer.

DSA Go

package main

func max(a, b int) int {
    if a > b {
        return a
    }
    return b
}

// Write maxCrossingSum and maxSubArraySum functions below
# Your code here

Hint

Use two loops in maxCrossingSum to find max sums on left and right of mid. Use recursion in maxSubArraySum to divide the problem.

4

Print the Maximum Subarray Sum

In the main function, call maxSubArraySum with profits, 0, and len(profits)-1. Print the returned maximum subarray sum using fmt.Println. Remember to import the fmt package.

DSA Go

package main

import "fmt"

func max(a, b int) int {
    if a > b {
        return a
    }
    return b
}

func maxCrossingSum(profits []int, left, mid, right int) int {
    sum := 0
    leftSum := -1 = left; i-- {
        sum += profits[i]
        if sum > leftSum {
            leftSum = sum
        }
    }

    sum = 0
    rightSum := -1  rightSum {
            rightSum = sum
        }
    }

    return max(leftSum+rightSum, max(leftSum, rightSum))
}

func maxSubArraySum(profits []int, left, right int) int {
    if left == right {
        return profits[left]
    }

    mid := (left + right) / 2

    leftMax := maxSubArraySum(profits, left, mid)
    rightMax := maxSubArraySum(profits, mid+1, right)
    crossMax := maxCrossingSum(profits, left, mid, right)

    return max(leftMax, max(rightMax, crossMax))
}

func main() {
    profits := []int{13, -3, -25, 20, -3, -16, -23, 18, 20, -7, 12, -5, -22, 15, -4, 7}
    // Print the maximum subarray sum below
    # Your code here
}

Hint

Call maxSubArraySum(profits, 0, len(profits)-1) and print the result using fmt.Println.

Practice

1. What is the main idea behind the Divide and Conquer approach to find the maximum subarray sum?

easy

A. Split the array into two halves, find max subarray in left, right, and crossing middle, then combine results

B. Sort the array and pick the largest element as max subarray sum

C. Use a sliding window to find the max sum subarray in one pass

D. Calculate sum of all subarrays and pick the maximum

Solution

1. Step 1: Understand divide and conquer strategy

   The array is split into two halves recursively to simplify the problem.

2. Step 2: Combine results from subproblems

   Find max subarray in left half, right half, and crossing the middle, then take the maximum of these three.

3. Final Answer:

   Split array, find max in left, right, crossing middle, combine -> Option A

4. Quick Check:

   Divide and conquer = split and combine max subarrays [OK]

Hint: Divide array, conquer halves, combine max sums [OK]

Common Mistakes:

• Thinking sorting helps find max subarray sum
• Confusing divide and conquer with sliding window
• Calculating all subarrays sums directly

2. Which of the following Go function signatures correctly represents the divide and conquer function to find maximum subarray sum in an array arr between indices low and high?

easy

A. func maxSubArray(arr []int, low int high int) int

B. func maxSubArray(arr int[], low int, high int) int

C. func maxSubArray(arr []int, low int, high int) (int, int)

D. func maxSubArray(arr []int, low, high int) int

Solution

1. Step 1: Check Go slice and parameter syntax

   In Go, slices are declared as []int, and multiple parameters of same type can be grouped.

2. Step 2: Identify correct function signature

   func maxSubArray(arr []int, low, high int) int uses correct slice type and groups low, high as int parameters, returning int sum.

3. Final Answer:

   func maxSubArray(arr []int, low, high int) int -> Option D

4. Quick Check:

   Go slice syntax and grouped params = func maxSubArray(arr []int, low, high int) int [OK]

Hint: Use []int for slices and group same-type params [OK]

Common Mistakes:

• Using int[] instead of []int for slices
• Returning multiple values when only sum needed
• Separating same-type parameters unnecessarily

3. What is the output of the following Go code snippet that uses divide and conquer to find maximum subarray sum?

package main
import "fmt"

func maxCrossingSum(arr []int, low, mid, high int) int {
    leftSum := -1 << 31
    sum := 0
    for i := mid; i >= low; i-- {
        sum += arr[i]
        if sum > leftSum {
            leftSum = sum
        }
    }
    rightSum := -1 << 31
    sum = 0
    for j := mid + 1; j <= high; j++ {
        sum += arr[j]
        if sum > rightSum {
            rightSum = sum
        }
    }
    return leftSum + rightSum
}

func maxSubArray(arr []int, low, high int) int {
    if low == high {
        return arr[low]
    }
    mid := (low + high) / 2
    leftMax := maxSubArray(arr, low, mid)
    rightMax := maxSubArray(arr, mid+1, high)
    crossMax := maxCrossingSum(arr, low, mid, high)
    if leftMax >= rightMax && leftMax >= crossMax {
        return leftMax
    } else if rightMax >= leftMax && rightMax >= crossMax {
        return rightMax
    } else {
        return crossMax
    }
}

func main() {
    arr := []int{-2, 1, -3, 4, -1, 2, 1, -5, 4}
    fmt.Println(maxSubArray(arr, 0, len(arr)-1))
}

medium

A. 5

B. 7

C. 6

D. 4

Solution

1. Step 1: Understand the input array and max subarray

   The array is [-2,1,-3,4,-1,2,1,-5,4]. The max subarray is [4,-1,2,1] with sum 6.

2. Step 2: Trace divide and conquer calls

   The function recursively finds max in left, right, and crossing middle, correctly returning 6.

3. Final Answer:

   6 -> Option C

4. Quick Check:

   Max subarray sum = 6 [OK]

Hint: Max subarray for example array is 6 [OK]

Common Mistakes:

• Choosing sum including -5 which lowers total
• Confusing max crossing sum with max overall
• Miscomputing indices for mid and crossing sums

4. Identify the bug in this Go function for finding max crossing sum and select the fix:

func maxCrossingSum(arr []int, low, mid, high int) int {
    leftSum := 0
    sum := 0
    for i := mid; i >= low; i-- {
        sum += arr[i]
        if sum > leftSum {
            leftSum = sum
        }
    }
    rightSum := 0
    sum = 0
    for j := mid + 1; j <= high; j++ {
        sum += arr[j]
        if sum > rightSum {
            rightSum = sum
        }
    }
    return leftSum + rightSum
}

medium

A. Initialize leftSum and rightSum to very small number (e.g. -1<<31) instead of 0

B. Change loops to run from low to mid and mid+1 to high

C. Return max of leftSum and rightSum instead of sum

D. Add 1 to mid when calculating leftSum loop

Solution

1. Step 1: Analyze initialization of leftSum and rightSum

   They start at 0, which fails if all numbers are negative, missing correct max sums.

2. Step 2: Fix initialization to very small number

   Set leftSum and rightSum to -1<<31 (minimum int) to correctly track max sums even if negative.

3. Final Answer:

   Initialize leftSum and rightSum to very small number (e.g. -1<<31) -> Option A

4. Quick Check:

   Proper initialization avoids missing negative max sums [OK]

Hint: Initialize max sums to very small number, not zero [OK]

Common Mistakes:

• Starting max sums at zero causes wrong results with negatives
• Changing loop directions incorrectly
• Returning sum instead of combined max sums

5. Given an array arr of integers, how would you modify the divide and conquer approach to also return the start and end indices of the maximum subarray along with its sum?

hard

A. Store indices globally and update them only when crossing sum is calculated

B. Change functions to return a struct with sum, start, and end; update recursive calls to track indices

C. Return only sum and find indices by scanning array after result

D. Use a separate loop after divide and conquer to find indices of max sum

Solution

1. Step 1: Modify function return types

   Change functions to return a struct containing sum, start index, and end index to track subarray boundaries.

2. Step 2: Update recursive calls and combine logic

   When combining left, right, and crossing results, also combine their indices accordingly to keep track of max subarray range.

3. Final Answer:

   Return struct with sum, start, end; update recursive calls to track indices -> Option B

4. Quick Check:

   Return sum and indices together for full max subarray info [OK]

Hint: Return sum plus indices in struct, update recursively [OK]

Common Mistakes:

• Using global variables causing bugs in recursion
• Finding indices after sum calculation separately
• Ignoring indices in crossing sum calculation