Memoization to Optimize Recursion in DSA Javascript - 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 memoized Fibonacci function.

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

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

Identify Repeating Operations

Look for repeated calls and stored results.

• Primary operation: Recursive calls to fib with smaller n.
• How many times: Each unique n is computed once, 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 the input size, not more.

Common Mistake

[X] Wrong: "Memoization makes recursion instantly fast for all problems."

[OK] Correct: Memoization only helps if the problem has overlapping subproblems; otherwise, it won't reduce repeated work.

Interview Connect

Understanding memoization shows you can improve slow recursive solutions by saving past results, a key skill in coding challenges.

Self-Check

"What if we removed memoization? 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 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