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 adding memoization changes the speed of a recursive function.

How does remembering past answers help us do less work?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public int Fibonacci(int n, Dictionary<int, int> memo)
{
    if (n <= 1) return n;
    if (memo.ContainsKey(n)) return memo[n];
    memo[n] = Fibonacci(n - 1, memo) + Fibonacci(n - 2, memo);
    return memo[n];
}
    

This code calculates the nth Fibonacci number using recursion with memoization to store results.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls to Fibonacci with smaller n.
• How many times: Each unique n is computed once and then reused from memo.

How Execution Grows With Input

Each number from 0 up to n is calculated once, then reused.

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

Pattern observation: The work grows roughly in a straight line as n grows.

Final Time Complexity

Time Complexity: O(n)

This means the function does work proportional to n, much faster than without memoization.

Common Mistake

[X] Wrong: "Recursion with memoization still does all calls like normal recursion."

[OK] Correct: Memoization saves results so repeated calls don't redo work, cutting down the total calls drastically.

Interview Connect

Understanding how memoization changes recursion speed shows you can improve slow code by remembering past answers, a key skill in problem solving.

Self-Check

"What if we removed the memo dictionary? How would the time complexity change?"

Practice

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

easy

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

B. To avoid using any variables inside the function

C. To convert recursion into iteration automatically

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

Solution

1. Step 1: Understand recursion inefficiency

   Recursive functions often repeat the same calculations multiple times, which wastes time.

2. Step 2: Role of memoization

   Memoization saves the results of these repeated calculations so they can be reused instead of recalculated.

3. Final Answer:

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

4. Quick Check:

   Memoization = Store and reuse results [OK]

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

Common Mistakes:

• Thinking memoization slows down recursion
• Confusing memoization with iteration
• Believing memoization removes recursion

2. Which of the following is the correct way to declare a memo dictionary for integer keys and values in C#?

easy

A. HashSet memo = new HashSet();

B. List memo = new List();

C. Dictionary memo = new Dictionary();

D. Dictionary memo = new Dictionary();

Solution

1. Step 1: Identify correct generic dictionary syntax

   In C#, a dictionary with integer keys and values is declared as Dictionary.

2. Step 2: Check options for correct syntax

   Dictionary memo = new Dictionary(); uses Dictionary with proper initialization, others use wrong types or missing generics.

3. Final Answer:

   Dictionary memo = new Dictionary(); -> Option C

4. Quick Check:

   Dictionary = Correct memo declaration [OK]

Hint: Use Dictionary for integer key-value memo [OK]

Common Mistakes:

• Using List or HashSet instead of Dictionary
• Missing generic type parameters
• Using untyped Dictionary without generics

3. What will be the output of the following C# code?

using System;
using System.Collections.Generic;

class Program {
    static Dictionary memo = new Dictionary();

    static int Fib(int n) {
        if (n <= 1) return n;
        if (memo.ContainsKey(n)) return memo[n];
        memo[n] = Fib(n - 1) + Fib(n - 2);
        return memo[n];
    }

    static void Main() {
        Console.WriteLine(Fib(5));
    }
}

medium

A. 5

B. 8

C. 3

D. 13

Solution

1. Step 1: Understand Fibonacci calculation

   Fib(5) = Fib(4) + Fib(3). Calculating recursively with memoization avoids repeated work.

2. Step 2: Calculate Fib(5) step-by-step

   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: Recall Fibonacci sequence values or trace memoized calls [OK]

Common Mistakes:

• Confusing Fib(5) with Fib(6)
• Ignoring memoization effect
• Miscounting base cases

4. Identify the error in this memoized factorial function and select the fix:

static Dictionary memo = new Dictionary();

static int Factorial(int n) {
    if (n == 0) return 1;
    if (memo.ContainsKey(n)) return memo[n];
    memo[n] = n * Factorial(n - 1);
    return n;
}

medium

A. Use memo[n] = Factorial(n - 1) * (n - 1);

B. Return memo[n] instead of n after recursive call

C. Remove memo dictionary completely

D. Change base case to if (n == 1) return 1;

Solution

1. Step 1: Analyze return statement after recursion

   The function returns 'n' instead of the computed factorial stored in memo[n], which is incorrect.

2. Step 2: Correct return value

   Return memo[n] to return the calculated factorial value after storing it.

3. Final Answer:

   Return memo[n] instead of n after recursive call -> Option B

4. Quick Check:

   Return memo[n] fixes incorrect output [OK]

Hint: Return stored memo value, not input parameter [OK]

Common Mistakes:

• Returning input parameter instead of memoized result
• Changing base case unnecessarily
• Misusing memo dictionary keys

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 implements this in C#?

hard

A. Use a Dictionary to store results; base cases for 0 and 1 stairs return 1; recursive call sums ways(n-1) and ways(n-2).

B. Use a List<int> to store results; base case returns 0 for 0 stairs; recursive call multiplies ways(n-1) and ways(n-2).

C. Use a Dictionary but no base cases; recursive call sums ways(n-1) and ways(n-2).

D. Use an array of size n; base cases return n; recursive call subtracts ways(n-1) from ways(n-2).

Solution

1. Step 1: Understand problem and base cases

   Number of ways to climb stairs with 1 or 2 steps: ways(0)=1 (one way to stand still), ways(1)=1.

2. Step 2: Correct memoization approach

   Use Dictionary to store results; recursive call sums ways(n-1) and ways(n-2) to count all paths.

3. Final Answer:

   Use a Dictionary to store results; base cases for 0 and 1 stairs return 1; recursive call sums ways(n-1) and ways(n-2). -> Option A

4. Quick Check:

   Memo + base cases + sum recursion = Use a Dictionary to store results; base cases for 0 and 1 stairs return 1; recursive call sums ways(n-1) and ways(n-2). [OK]

Hint: Base cases 0,1 = 1; sum ways(n-1) + ways(n-2) with memo [OK]

Common Mistakes:

• Wrong base cases values
• Using multiplication instead of addition
• Missing base cases causing infinite recursion