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

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Fibonacci with Memoization

What is the output of the following JavaScript code that uses memoization to calculate Fibonacci numbers?

DSA Javascript

function fib(n, memo = {}) {
  if (n <= 1) return n;
  if (memo[n]) return memo[n];
  memo[n] = fib(n - 1, memo) + fib(n - 2, memo);
  return memo[n];
}
console.log(fib(6));

A 8

B 21

C 15

D 13

💡 Hint

Recall that Fibonacci sequence starts with 0, 1, 1, 2, 3, 5, 8, 13...

The function calculates Fibonacci numbers using memoization. fib(6) returns the 6th Fibonacci number which is 13.

❓ Predict Output

intermediate

Memoized Factorial Output

What will be printed after running this memoized factorial function?

DSA Javascript

function factorial(n, memo = {}) {
  if (n <= 1) return 1;
  if (memo[n]) return memo[n];
  memo[n] = n * factorial(n - 1, memo);
  return memo[n];
}
console.log(factorial(5));

A 120

B 24

C 60

D 720

💡 Hint

Factorial of 5 is 5 * 4 * 3 * 2 * 1.

The function uses memoization to calculate factorial efficiently. factorial(5) = 120.

🔧 Debug

advanced

Identify the Error in Memoized Fibonacci

What error will this code produce when run?

DSA Javascript

function fib(n, memo = {}) {
  if (n <= 1) return n;
  if (memo[n]) return memo[n];
  memo[n] = fib(n - 1) + fib(n - 2, memo);
  return memo[n];
}
console.log(fib(5));

A Outputs 5

B TypeError: Cannot read property '5' of undefined

C StackOverflowError due to infinite recursion

D ReferenceError: memo is not defined

💡 Hint

Check how the memo object is passed in recursive calls.

The first recursive call fib(n - 1) does not pass memo, so memo is undefined inside that call, causing a TypeError when accessing memo[n].

❓ Predict Output

advanced

Output of Memoized Climbing Stairs Problem

What is the output of this memoized function that counts ways to climb stairs?

DSA Javascript

function climbStairs(n, memo = {}) {
  if (n <= 2) return n;
  if (memo[n]) return memo[n];
  memo[n] = climbStairs(n - 1, memo) + climbStairs(n - 2, memo);
  return memo[n];
}
console.log(climbStairs(5));

A 8

B 13

C 5

D 10

💡 Hint

Number of ways to climb 5 stairs with 1 or 2 steps at a time.

The function counts ways to climb stairs using memoization. climbStairs(5) = 8 ways.

🧠 Conceptual

expert

Memoization Impact on Recursive Time Complexity

Which statement best describes how memoization changes the time complexity of a naive recursive Fibonacci function?

A It does not affect time complexity but reduces space complexity.

B It increases time complexity because of extra memory usage.

C It reduces time complexity from exponential to linear by storing intermediate results.

D It changes time complexity from linear to logarithmic by dividing the problem.

💡 Hint

Think about how many times each Fibonacci number is calculated with and without memoization.

Memoization stores results of subproblems, so each is computed once, reducing exponential calls to linear time.

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 store results of expensive function calls and reuse them later

C. To avoid using recursion altogether

D. To increase the number of recursive calls

Solution

1. Step 1: Understand what memoization does

   Memoization saves the results of function calls so that if the same inputs occur again, the saved result is used instead of recalculating.

2. Step 2: Identify the effect on recursion

   This reduces repeated work and speeds up recursive functions by avoiding duplicate calculations.

3. Final Answer:

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

4. Quick Check:

   Memoization = Store and reuse results [OK]

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

Common Mistakes:

• Thinking memoization makes recursion slower
• Confusing memoization with iteration
• Believing memoization removes recursion

2. Which of the following is the correct way to declare a memo object inside a recursive function in JavaScript?

easy

A. const memo = {}; inside the function, then pass it as a parameter

B. var memo = []; inside the function without passing it

C. let memo = 0; inside the function

D. memo = new Map(); outside the function and never pass it

Solution

1. Step 1: Understand memo object usage

   The memo object should be an object or map to store key-value pairs of inputs and results.

2. Step 2: Correct placement and passing

   Declaring const memo = {} inside the function and passing it as a parameter ensures the same memo is used across recursive calls.

3. Final Answer:

   const memo = {}; inside the function, then pass it as a parameter -> Option A

4. Quick Check:

   Memo object declared and passed = const memo = {}; inside the function, then pass it as a parameter [OK]

Hint: Declare memo as object and pass it to recursive calls [OK]

Common Mistakes:

• Using array instead of object for memo
• Not passing memo to recursive calls
• Declaring memo as a number

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

function fib(n, memo = {}) {
  if (n <= 1) return n;
  if (memo[n]) return memo[n];
  memo[n] = fib(n - 1, memo) + fib(n - 2, memo);
  return memo[n];
}
console.log(fib(5));

medium

A. 5

B. 8

C. 3

D. Undefined

Solution

1. Step 1: Understand Fibonacci calculation with memoization

   fib(5) calculates Fibonacci number at position 5 using memo to avoid repeated calls.

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: Remember Fibonacci sequence: 0,1,1,2,3,5... [OK]

Common Mistakes:

• Confusing fib(5) with fib(6)
• Ignoring memo and recalculating
• Assuming output is 8

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

function factorial(n, memo = {}) {
  if (n === 0) return 1;
  if (memo[n]) return memo[n];
  memo[n] = n * factorial(n - 1);
  return memo[n];
}

medium

A. Remove memo object completely

B. Change base case to n === 1

C. Pass memo in recursive call: factorial(n - 1, memo)

D. Initialize memo as an array instead of object

Solution

1. Step 1: Check recursive call parameters

   The recursive call to factorial misses passing the memo object, so memo is reset each time.

2. Step 2: Fix by passing memo in recursion

   Change call to factorial(n - 1, memo) to keep memo consistent across calls.

3. Final Answer:

   Pass memo in recursive call: factorial(n - 1, memo) -> Option C

4. Quick Check:

   Pass memo to recursive calls to fix [OK]

Hint: Always pass memo object in recursive calls [OK]

Common Mistakes:

• Forgetting to pass memo in recursion
• Changing base case unnecessarily
• Removing memo defeats optimization

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 improves performance?

hard

A. Store results only for even stair numbers in memo

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

C. Rewrite the function to use a loop without memoization

D. Use a memo object passed through recursive calls to store results for each stair number

Solution

1. Step 1: Understand the problem and recursion

   The function counts ways to climb stairs by summing ways(n-1) and ways(n-2).

2. Step 2: Apply memoization correctly

   Passing a memo object through recursive calls stores results for each stair number, avoiding repeated calculations.

3. Step 3: Evaluate other options

   Global counters don't store results, loops change approach, and storing only even results misses odd cases.

4. Final Answer:

   Use a memo object passed through recursive calls to store results for each stair number -> Option D

5. Quick Check:

   Memo object for all inputs optimizes recursion [OK]

Hint: Memoize all inputs to avoid repeated stair count calculations [OK]

Common Mistakes:

• Not passing memo object through recursion
• Partial memoization missing some inputs
• Confusing memoization with iteration