Divide and Conquer Strategy and Recurrence Relations in DSA Go - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate the midpoint for dividing the array.

DSA Go

mid := (low + [1]) / 2

Drag options to blanks, or click blank then click option'

Alow

Bhigh

Cmid

Dlen(arr)

Attempts:

3 left

💡 Hint

Common Mistakes

Using low twice instead of high.

Using length of array instead of high index.

The midpoint is calculated by adding low and high indices and dividing by 2.

2fill in blank

medium

Complete the code to recursively call the divide and conquer function on the left half.

DSA Go

divideAndConquer(arr, low, [1])

Drag options to blanks, or click blank then click option'

Amid + 1

Bhigh

Cmid - 1

Dlow

Attempts:

3 left

💡 Hint

Common Mistakes

Using mid + 1 which is for the right half.

Using high which is the end of the whole array.

For the left half, the high index is mid - 1.

3fill in blank

hard

Fix the error in the base case condition to stop recursion when the segment is invalid.

DSA Go

if low [1] high {
    return
}

Drag options to blanks, or click blank then click option'

A>

B<

C>=

D<=

Attempts:

3 left

💡 Hint

Common Mistakes

Using < or <= which causes infinite recursion.

Using >= which stops too early.

The recursion stops when low is greater than high, meaning no elements to process.

4fill in blank

hard

Fill both blanks to complete the recurrence relation for merge sort time complexity.

DSA Go

T(n) = 2T([1]) + [2]n

Drag options to blanks, or click blank then click option'

An/2

B1

Cn/3

Dlog n

Attempts:

3 left

💡 Hint

Common Mistakes

Using n-1 instead of n/2 for subproblem size.

Using log n for merge cost which is incorrect.

Merge sort divides the problem into two halves of size n/2 and merges in linear time n.

5fill in blank

hard

Fill all three blanks to complete the recurrence relation for binary search time complexity.

DSA Go

T(n) = T([1]) + [2] + [3]

Drag options to blanks, or click blank then click option'

An/2

B1

Clog n

Dn

Attempts:

3 left

💡 Hint

Common Mistakes

Using n or log n for subproblem size instead of n/2.

Using more than constant time for work per step.

Binary search halves the problem size each time (n/2), does constant work (1), and no extra cost beyond that.

Practice

1. What is the main idea behind the Divide and Conquer strategy in algorithms?

easy

A. Store all intermediate results to avoid repeated calculations

B. Solve the problem by checking every possible solution one by one

C. Use a single loop to solve the problem directly without recursion

D. Break the problem into smaller subproblems, solve them independently, then combine results

Solution

1. Step 1: Understand Divide and Conquer concept

   It splits a big problem into smaller parts that are easier to solve.

2. Step 2: Combine solutions of subproblems

   After solving smaller parts, their results are combined to get the final answer.

3. Final Answer:

   Break the problem into smaller subproblems, solve them independently, then combine results -> Option D

4. Quick Check:

   Divide and Conquer = Break, Solve, Combine [OK]

Hint: Divide big problem into smaller parts, solve, then combine [OK]

Common Mistakes:

• Confusing divide and conquer with brute force
• Thinking it solves without breaking problem
• Mixing with dynamic programming concepts

2. Which of the following is the correct recurrence relation for the Merge Sort algorithm?

easy

A. T(n) = 3T(n/3) + O(n^2)

B. T(n) = T(n-1) + O(1)

C. T(n) = 2T(n/2) + O(n)

D. T(n) = T(n/2) + O(1)

Solution

1. Step 1: Recall Merge Sort process

   Merge Sort divides array into two halves, sorts each half, then merges.

2. Step 2: Write recurrence relation

   It calls itself twice on half size (2T(n/2)) plus merging cost O(n).

3. Final Answer:

   T(n) = 2T(n/2) + O(n) -> Option C

4. Quick Check:

   Merge Sort recurrence = 2T(n/2) + n [OK]

Hint: Merge Sort splits in two, merges in linear time [OK]

Common Mistakes:

• Using T(n-1) instead of T(n/2)
• Ignoring merge cost O(n)
• Confusing with other divide and conquer algorithms

3. Consider the following Go function implementing a divide and conquer approach:

func sum(arr []int) int {
    if len(arr) == 1 {
        return arr[0]
    }
    mid := len(arr) / 2
    left := sum(arr[:mid])
    right := sum(arr[mid:])
    return left + right
}

fmt.Println(sum([]int{2, 4, 6, 8}))

What is the output of this code?

medium

A. 20

B. 10

C. 0

D. Runtime error

Solution

1. Step 1: Understand the function behavior

   The function splits the array into halves recursively and sums elements.

2. Step 2: Trace the calls for input [2,4,6,8]

   Sum of 2+4=6, sum of 6+8=14, total 6+14=20.

3. Final Answer:

   20 -> Option A

4. Quick Check:

   2+4+6+8 = 20 [OK]

Hint: Sum all elements by splitting and adding halves [OK]

Common Mistakes:

• Confusing mid index slicing
• Assuming sum returns average
• Thinking it causes runtime error

4. The following Go code is intended to implement a divide and conquer algorithm to find the maximum element in an array. Identify the error:

func max(arr []int) int {
    if len(arr) == 1 {
        return arr[0]
    }
    mid := len(arr) / 2
    left := max(arr[:mid])
    right := max(arr[mid:])
    if left > right {
        return left
    } else {
        return right
    }
}

fmt.Println(max([]int{}))

medium

A. The base case should check for len(arr) == 0

B. The code will panic due to empty slice input

C. The mid calculation is incorrect

D. The comparison between left and right is wrong

Solution

1. Step 1: Analyze base case

   The base case handles only len(arr) == 1, missing handling for empty slices (len(arr) == 0).

2. Step 2: Trace execution on empty input

   len(arr) == 0 skips base case, mid = 0, recurses on arr[:0] and arr[0:], both empty, causing infinite recursion and stack overflow panic.

3. Final Answer:

   The code will panic due to empty slice input -> Option B

4. Quick Check:

   Infinite recursion on empty input causes panic [OK]

Hint: Handle empty input in recursive divide-and-conquer base cases [OK]

Common Mistakes:

• Ignoring empty slice input
• Assuming base case handles empty slice
• Thinking mid calculation causes error

5. Given the recurrence relation T(n) = 3T(n/3) + n, which of the following is the correct time complexity using the Master Theorem?

hard

A. O(n log n)

B. O(n^2)

C. O(n)

D. O(log n)

Solution

1. Step 1: Identify parameters for Master Theorem

   Here, a=3, b=3, f(n)=n.

2. Step 2: Calculate log_b(a)

   log_3(3) = 1, so n^{log_b(a)} = n^1 = n.

3. Step 3: Compare f(n) with n^{log_b(a)}

   f(n) = n matches n^{log_b(a)} = n, so case 2 applies.

4. Step 4: Apply Master Theorem case 2

   Time complexity is O(n log n).

5. Final Answer:

   O(n log n) -> Option A

6. Quick Check:

   Master Theorem case 2 = O(n log n) [OK]

Hint: Compare f(n) with n^{log_b(a)} for Master Theorem [OK]

Common Mistakes:

• Confusing log base or a and b values
• Choosing wrong Master Theorem case
• Ignoring the extra n factor in recurrence