Memoization to Optimize Recursion in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is memoization in the context of recursion?

Memoization is a technique to store the results of expensive function calls and reuse them when the same inputs occur again, avoiding repeated calculations.

intermediate

How does memoization improve recursive function performance?

By saving results of subproblems, memoization prevents the function from recalculating the same values multiple times, reducing time complexity from exponential to polynomial in many cases.

beginner

In C#, which data structure is commonly used to implement memoization?

A Dictionary is commonly used to store computed results with input parameters as keys and results as values.

intermediate

What is the difference between top-down memoization and bottom-up dynamic programming?

Top-down memoization uses recursion with caching results, while bottom-up dynamic programming solves smaller subproblems first iteratively and builds up to the solution.

beginner

Why is memoization especially useful for problems like Fibonacci sequence calculation?

Because naive recursion recalculates the same Fibonacci numbers many times, memoization stores these results once computed, drastically reducing redundant work.

What does memoization store to optimize recursion?

A Results of previous function calls

B Input parameters only

C All possible outputs in advance

D Random values to speed up

Memoization stores the results of previous function calls to avoid repeated calculations.

Which C# data structure is best suited for memoization?

A List<T>

B Dictionary<TKey, TValue>

C Queue<T>

D Stack<T>

Dictionary allows fast lookup of stored results by input keys.

Memoization changes the time complexity of naive Fibonacci recursion from:

A O(2^n) to O(n)

B O(n) to O(1)

C O(n^2) to O(n^3)

D O(n!) to O(n^2)

Memoization reduces exponential time O(2^n) to linear time O(n) for Fibonacci.

Which statement is true about memoization?

A It always uses iteration instead of recursion.

B It only works for sorting algorithms.

C It increases the number of recursive calls.

D It stores intermediate results to avoid recomputation.

Memoization stores intermediate results to avoid recomputation in recursion.

What is a key benefit of memoization in recursive algorithms?

A Uses less memory than recursion alone.

B Simplifies code by removing recursion.

C Prevents repeated calculations of the same inputs.

D Eliminates the need for base cases.

Memoization prevents repeated calculations by caching results.

Explain how memoization works to optimize a recursive function like Fibonacci calculation.

Think about saving answers to smaller problems to reuse later.

Describe the difference between memoization and bottom-up dynamic programming.

Consider how the problem is solved step-by-step in each method.

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