Memoization to Optimize Recursion in DSA C - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Fibonacci with Memoization

What is the output of the following C code that uses memoization to compute Fibonacci numbers?

DSA C

#include <stdio.h>

int fib(int n, int memo[]) {
    if (n <= 1) return n;
    if (memo[n] != -1) return memo[n];
    memo[n] = fib(n-1, memo) + fib(n-2, memo);
    return memo[n];
}

int main() {
    int n = 5;
    int memo[6];
    for (int i = 0; i <= n; i++) memo[i] = -1;
    int result = fib(n, memo);
    printf("%d\n", result);
    return 0;
}

A 5

B 8

C 13

D 3

💡 Hint

Recall that Fibonacci sequence starts with 0, 1, 1, 2, 3, 5...

The 5th Fibonacci number (0-based index) is 5. The memo array stores intermediate results to avoid repeated calculations.

🧠 Conceptual

intermediate

Memoization Purpose in Recursion

What is the main purpose of using memoization in recursive functions?

A To store intermediate results and avoid repeated calculations

B To make the code shorter and easier to read

C To increase the number of recursive calls

D To convert recursion into iteration automatically

💡 Hint

Think about how recursion can repeat the same work multiple times.

Memoization saves results of expensive function calls and returns the cached result when the same inputs occur again, improving efficiency.

🔧 Debug

advanced

Identify the Error in Memoized Fibonacci

What error will occur when running this code snippet?

DSA C

#include <stdio.h>

int fib(int n, int memo[]) {
    if (n <= 1) return n;
    if (memo[n] != 0) return memo[n];
    memo[n] = fib(n-1, memo) + fib(n-2, memo);
    return memo[n];
}

int main() {
    int n = 5;
    int memo[6] = {0};
    int result = fib(n, memo);
    printf("%d\n", result);
    return 0;
}

A The code prints 0 because memo array is initialized with zeros

B The code runs correctly and prints 5

C The code causes a segmentation fault

D The code prints 1 instead of 5

💡 Hint

Check how memo array is initialized and how memo[n] is checked before recursion.

Since Fibonacci numbers are positive and fib(0)=0, using 0 as uninitialized marker works. The code correctly computes fib(5)=5.

❓ Predict Output

advanced

Output of Factorial with Memoization

What is the output of this C program that uses memoization to compute factorial?

DSA C

#include <stdio.h>

int fact(int n, int memo[]) {
    if (n <= 1) return 1;
    if (memo[n] != 0) return memo[n];
    memo[n] = n * fact(n-1, memo);
    return memo[n];
}

int main() {
    int n = 4;
    int memo[5] = {0};
    int result = fact(n, memo);
    printf("%d\n", result);
    return 0;
}

A 120

B 10

C 0

D 24

💡 Hint

Recall factorial 4! = 4*3*2*1

The factorial of 4 is 24. Memoization stores intermediate factorial values to avoid repeated calculations.

🚀 Application

expert

Memoization Impact on Time Complexity

Consider a naive recursive function to compute Fibonacci numbers with exponential time complexity. After adding memoization, what is the new time complexity?

A O(2^n)

B O(n^2)

C O(n)

D O(log n)

💡 Hint

Memoization avoids repeated calls by storing results.

Memoization reduces the exponential time complexity O(2^n) of naive recursion to linear O(n) by caching results.

Practice

1. What is the main purpose of using memoization in recursive functions?

easy

A. To avoid using any variables inside the function

B. To store results of expensive function calls and reuse them

C. To convert recursion into iteration automatically

D. To make the function run slower by adding extra steps

Solution

1. Step 1: Understand what memoization does

   Memoization saves the results of function calls so they don't need to be recalculated.

2. Step 2: Identify the benefit in recursion

   This reduces repeated work and speeds up recursive functions.

3. Final Answer:

   To store results of expensive function calls and reuse them -> Option B

4. Quick Check:

   Memoization = Store and reuse results [OK]

Hint: Memoization saves repeated results to speed recursion [OK]

Common Mistakes:

• Thinking memoization slows down the function
• Confusing memoization with converting recursion to loops
• Believing memoization removes variables

2. Which of the following is the correct way to declare a memoization array for Fibonacci numbers in C?

easy

A. int memo[] = {1, 1};

B. int memo = 100;

C. int memo[100] = {0};

D. int memo(100);

Solution

1. Step 1: Check valid C array declaration

   In C, arrays are declared with type, name, and size in square brackets.

2. Step 2: Identify correct syntax

   int memo[100] = {0}; uses int memo[100] = {0}; which initializes all elements to zero correctly.

3. Final Answer:

   int memo[100] = {0}; -> Option C

4. Quick Check:

   Array declaration with size and initialization = int memo[100] = {0}; [OK]

Hint: Use int array[size] = {0} to initialize memo array [OK]

Common Mistakes:

• Using parentheses instead of brackets for arrays
• Assigning a single int instead of an array
• Not initializing the array to zero

3. What will be the output of the following C code snippet using memoization for Fibonacci?

int memo[100] = {0};

int fib(int n) {
  if (n <= 1) return n;
  if (memo[n] != 0) return memo[n];
  memo[n] = fib(n-1) + fib(n-2);
  return memo[n];
}

int main() {
  printf("%d", fib(5));
  return 0;
}

medium

A. 5

B. 8

C. 3

D. 0

Solution

1. Step 1: Trace fib(5) with memoization

   fib(5) = fib(4) + fib(3), and so on, with memo storing results to avoid repeats.

2. Step 2: Calculate Fibonacci values

   fib(0)=0, fib(1)=1, fib(2)=1, fib(3)=2, fib(4)=3, fib(5)=5.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   fib(5) = 5 [OK]

Hint: Remember fib(5) = 5 in Fibonacci sequence [OK]

Common Mistakes:

• Confusing fib(5) with fib(6)
• Ignoring memoization effect on output
• Mistaking base cases

4. Identify the error in this memoized factorial function in C:

int memo[100] = {0};

int fact(int n) {
  if (n == 0) return 1;
  if (memo[n] != 0) return memo[n];
  memo[n] = n * fact(n - 1);
  return memo[n];
}

int main() {
  printf("%d", fact(5));
  return 0;
}

medium

A. Memo array is not initialized to -1 to detect uncalculated values

B. Base case should return 0 instead of 1

C. Function should use iteration, not recursion

D. Memo array size is too small

Solution

1. Step 1: Understand memo check condition

   The code checks if memo[n] != 0 to decide if value is cached.

2. Step 2: Identify problem with zero initialization

   Since factorial(0) = 1, memo[1] will be 0 initially, but 0 is a valid value for uninitialized. This causes confusion.

3. Step 3: Correct approach

   Initialize memo array with -1 to mark uncalculated values, so 0 is a valid cached result.

4. Final Answer:

   Memo array is not initialized to -1 to detect uncalculated values -> Option A

5. Quick Check:

   Use -1 to mark uncalculated memo entries [OK]

Hint: Initialize memo with -1 to detect uncalculated values [OK]

Common Mistakes:

• Assuming zero means uncalculated without initialization
• Changing base case incorrectly
• Thinking recursion must be replaced by iteration

5. You want to optimize a recursive function that calculates the number of ways to climb n stairs taking 1 or 2 steps at a time. Which memoization approach below correctly stores intermediate results to avoid repeated calculations?

hard

A. Use a global variable to count calls without storing results

B. Recalculate all values each time without storing

C. Use a linked list to store all previous results dynamically

D. Use an array int memo[n+1] initialized with -1, store results at memo[i] for steps i

Solution

1. Step 1: Understand problem and memoization need

   Number of ways to climb stairs is a classic recursion problem with overlapping subproblems.

2. Step 2: Choose correct memoization method

   Using an array sized n+1 initialized with -1 allows storing results for each step count to reuse.

3. Final Answer:

   Use an array int memo[n+1] initialized with -1, store results at memo[i] for steps i -> Option D

4. Quick Check:

   Array memo with -1 initialization stores intermediate results [OK]

Hint: Use array memo[n+1] with -1 to store step results [OK]

Common Mistakes:

• Not initializing memo array properly
• Using global counters without memoization
• Using complex data structures unnecessarily