Memoization to Optimize Recursion in DSA C - Time & Space Complexity

Time Complexity: Memoization to Optimize Recursion

O(n)

Understanding Time Complexity

We want to see how using memoization changes the speed of recursive functions.

How does remembering past answers help us do less work?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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];
}

// memo array initialized with -1 for all indices

This code calculates Fibonacci numbers using recursion and stores results to avoid repeated work.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls to fib with memoization check.
• How many times: Each unique fib(n) is computed once; repeated calls return stored result immediately.

How Execution Grows With Input

Without memoization, calls grow very fast because many repeat. With memoization, each number is calculated once.

Input Size (n)	Approx. Operations
10	About 10 calls
100	About 100 calls
1000	About 1000 calls

Pattern observation: The work grows roughly in a straight line with n, not exponentially.

Final Time Complexity

Time Complexity: O(n)

This means the function does work proportional to n, calculating each Fibonacci number once.

Common Mistake

[X] Wrong: "Memoization doesn't change the time complexity; recursion is always slow."

[OK] Correct: Memoization saves results so repeated calls are fast, reducing exponential calls to linear.

Interview Connect

Understanding memoization shows you can improve slow recursive solutions by saving work, a key skill in problem solving.

Self-Check

"What if we removed the memo array and just used plain recursion? How would the time complexity change?"

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