Divide and Conquer Strategy and Recurrence Relations in DSA C# - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

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

DSA C#

int mid = (low + [1]) / 2;

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

Alow

Bhigh

Cmid

Dlength

Attempts:

3 left

💡 Hint

Common Mistakes

Using low twice instead of high.

Using length 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 correctly call the recursive function on the left half.

DSA C#

MergeSort(arr, low, [1]);

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

Amid + 1

Bhigh

Cmid

Dlow

Attempts:

3 left

💡 Hint

Common Mistakes

Using mid + 1 instead of mid.

Using high instead of mid.

The left half is from low to mid.

3fill in blank

hard

Fix the error in the base case condition to stop recursion correctly.

DSA C#

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 > without equality.

The recursion stops when low is greater than or equal to high.

4fill in blank

hard

Fill both blanks to correctly express the recurrence relation for Merge Sort.

DSA C#

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

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

A/

B+

C-

D*

Attempts:

3 left

💡 Hint

Common Mistakes

Using multiplication instead of division for problem size.

Using subtraction instead of addition for merging cost.

The recurrence relation splits the problem in half and adds linear work.

5fill in blank

hard

Fill all three blanks to complete the Master Theorem case for Merge Sort's recurrence.

DSA C#

If a = 2, b = 2, and f(n) = [1], then T(n) = [2]([3]).

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

An log n

Bcn

CΘ

DO

Attempts:

3 left

💡 Hint

Common Mistakes

Confusing f(n) with n log n.

Using O instead of Θ for tight bound.

Merge Sort's recurrence solves to Θ(n log n) using Master Theorem with f(n) = cn.

Practice

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

easy

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

B. Solve the problem by trying all possible solutions one by one

C. Use a single loop to process all elements sequentially

D. Store all intermediate results to avoid repeated calculations

Solution

1. Step 1: Understand the divide and conquer approach

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

2. Step 2: Combine the 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 A

4. Quick Check:

   Divide and conquer = split, 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 processes elements sequentially only
• Mixing it up with dynamic programming

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

easy

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

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

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

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

Solution

1. Step 1: Recall Merge Sort divides array into two halves

   It splits the array into two parts, sorts each recursively.

2. Step 2: Merge step takes linear time O(n)

   After sorting halves, merging takes O(n) time.

3. Final Answer:

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

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)
• Confusing merge cost as constant time
• Choosing wrong number of recursive calls

3. What is the output of the following C# code that uses a divide and conquer approach to sum an array?

int SumArray(int[] arr, int start, int end) {
    if (start == end) return arr[start];
    int mid = (start + end) / 2;
    return SumArray(arr, start, mid) + SumArray(arr, mid + 1, end);
}

int[] nums = {2, 4, 6, 8};
Console.WriteLine(SumArray(nums, 0, nums.Length - 1));

medium

A. 20

B. 10

C. 0

D. Throws runtime error

Solution

1. Step 1: Understand the recursive sum function

   The function splits the array into halves until one element remains, then sums back up.

2. Step 2: Calculate sum of array elements

   Sum = 2 + 4 + 6 + 8 = 20.

3. Final Answer:

   20 -> Option A

4. Quick Check:

   Sum of {2,4,6,8} = 20 [OK]

Hint: Divide array, sum halves recursively, add results [OK]

Common Mistakes:

• Confusing mid calculation causing infinite recursion
• Forgetting base case when start == end
• Miscounting sum of elements

4. Identify the error in this C# code implementing a divide and conquer algorithm to find the maximum element:

int MaxElement(int[] arr, int start, int end) {
    if (start == end) return arr[start];
    int mid = (start + end) / 2;
    int leftMax = MaxElement(arr, start, mid);
    int rightMax = MaxElement(arr, mid, end);
    return Math.Max(leftMax, rightMax);
}

medium

A. The base case should check if start > end

B. Math.Max should be replaced with Math.Min

C. The second recursive call should be MaxElement(arr, mid + 1, end)

D. The mid calculation should be (start + end + 1) / 2

Solution

1. Step 1: Analyze recursive calls for correct subarray ranges

   The first call covers start to mid, so the second must cover mid+1 to end to avoid overlap.

2. Step 2: Identify the off-by-one error

   Using mid again in second call causes overlap and possible infinite recursion.

3. Final Answer:

   The second recursive call should be MaxElement(arr, mid + 1, end) -> Option C

4. Quick Check:

   Correct subarray split avoids overlap [OK]

Hint: Second half starts at mid+1, not mid [OK]

Common Mistakes:

• Using mid instead of mid+1 causing infinite recursion
• Changing base case incorrectly
• Replacing max with min mistakenly

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(\log n)

D. O(n)

Solution

1. Step 1: Identify parameters a, b, and f(n)

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

2. Step 2: Calculate log_b(a) and compare with f(n)

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

3. Step 3: Apply Master Theorem case 2

   Since f(n) = Theta(n^{log_b(a)}), time complexity is O(n).

4. Final Answer:

   O(n) -> Option D

5. Quick Check:

   Master theorem case 2 = O(n) [OK]

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

Common Mistakes:

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