Memoization Top Down DP in DSA Typescript - Time & Space Complexity

Time Complexity: Memoization Top Down DP

O(n)

Understanding Time Complexity

We want to understand how using memoization in top-down dynamic programming changes the time it takes to solve problems.

Specifically, how does remembering past answers speed up repeated calculations?

Scenario Under Consideration

Analyze the time complexity of the following memoized Fibonacci function.

function fib(n: number, memo: number[] = []): number {
  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 the nth Fibonacci number using recursion and stores results to avoid repeated work.

Identify Repeating Operations

Look for repeated calculations and how memoization affects them.

• Primary operation: Recursive calls to fib for smaller n.
• How many times: Without memo, calls grow exponentially; with memo, each n is computed once.

How Execution Grows With Input

With memoization, each number from 0 up to n is calculated once and stored.

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

Pattern observation: The number of operations grows roughly in a straight line with n.

Final Time Complexity

Time Complexity: O(n)

This means the time to compute fib(n) grows linearly with n because each value is computed once and reused.

Common Mistake

[X] Wrong: "Memoization still makes the function exponential because of recursion."

[OK] Correct: Memoization stores results, so each input is solved once, preventing repeated exponential calls.

Interview Connect

Understanding memoization shows you can optimize recursive solutions by saving work, a key skill in many coding problems.

Self-Check

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

Practice

1. What is the main purpose of memoization in top-down dynamic programming?

easy

A. To reduce the memory usage by deleting variables

B. To sort the input data before processing

C. To convert recursive code into iterative code

D. To store results of subproblems to avoid repeated calculations

Solution

1. Step 1: Understand memoization concept

   Memoization saves the results of subproblems so they are not recalculated.

2. Step 2: Compare options

   Only To store results of subproblems to avoid repeated calculations correctly describes this purpose; others describe unrelated actions.

3. Final Answer:

   To store results of subproblems to avoid repeated calculations -> Option D

4. Quick Check:

   Memoization = store subproblem results [OK]

Hint: Memoization caches answers to reuse later [OK]

Common Mistakes:

• Confusing memoization with sorting
• Thinking memoization reduces memory by deleting variables
• Believing memoization converts recursion to iteration

2. Which of the following is the correct TypeScript syntax to declare a memo object for caching number results indexed by number keys?

easy

A. const memo: Record = {};

B. const memo: Array<number> = [];

C. const memo: Map = new Map();

D. const memo: Set<number> = new Set();

Solution

1. Step 1: Identify memo type for number keys and values

   Memo should map number keys to number values, so Record fits.

2. Step 2: Check other options

   Array is indexed by number but not key-value pairs; Map uses string keys here; Set stores unique values, not key-value pairs.

3. Final Answer:

   const memo: Record = {}; -> Option A

4. Quick Check:

   Memo object with number keys and values = Record [OK]

Hint: Use Record for number-keyed memo objects [OK]

Common Mistakes:

• Using Map with wrong key/value types
• Using Set which stores only values, not key-value pairs
• Using Array when keys are not continuous indexes

3. Consider the following TypeScript code using memoization for Fibonacci numbers:

function fib(n: number, memo: Record = {}): number {
  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));

What is the output of this code?

medium

A. Undefined

B. 5

C. 3

D. 8

Solution

1. Step 1: Trace fib(5) calls with memoization

   fib(5) = fib(4) + fib(3), fib(4) = fib(3) + fib(2), fib(3) = fib(2) + fib(1), fib(2) = fib(1) + fib(0). Values: fib(0)=0, fib(1)=1.

2. Step 2: Calculate values bottom-up

   fib(2)=1, fib(3)=2, fib(4)=3, fib(5)=5.

3. Final Answer:

   5 -> Option B

4. Quick Check:

   fib(5) = 5 [OK]

Hint: Remember fib(5) = 5 in Fibonacci sequence [OK]

Common Mistakes:

• Confusing fib(5) with fib(6)
• Forgetting base cases fib(0)=0 and fib(1)=1
• Assuming memo[n] is undefined for n=1

4. The following TypeScript function attempts to use memoization for factorial calculation but has a bug:

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

What is the bug in this code?

medium

A. It incorrectly checks memo[n] which fails for factorial(0)

B. It does not handle the base case correctly

C. It uses the wrong multiplication operator

D. It does not pass memo object in recursive calls

Solution

1. Step 1: Analyze memo[n] check for zero value

   When n=0, factorial is 1, but memo[0] is undefined initially. The check if(memo[n]) fails for memo[0] = 0 or 1 because 0 or 1 are falsy/truthy but memo[0] might be undefined.

2. Step 2: Identify correct memo check

   Checking if(memo[n]) fails if memo[n] is 0 or falsy. Should check if(n in memo) instead to detect cached values correctly.

3. Final Answer:

   It incorrectly checks memo[n] which fails for factorial(0) -> Option A

4. Quick Check:

   Check memo keys with 'in' operator, not truthy check [OK]

Hint: Use 'n in memo' to check cache, not 'if(memo[n])' [OK]

Common Mistakes:

• Assuming memo[n] check works for zero or falsy values
• Missing base case for n=0
• Not passing memo in recursive calls

5. You want to optimize a function that counts the number of ways to climb n stairs taking 1 or 2 steps at a time using top-down memoization. Which of the following TypeScript implementations correctly applies memoization to avoid repeated calculations?

hard

A. function climbStairs(n: number, memo: Record = {}): number {
if (n <= 1) return 1;
if (memo[n]) return memo[n];
memo[n] = climbStairs(n - 1, memo) + climbStairs(n - 2, memo);
return memo[n];
}

B. function climbStairs(n: number): number {
if (n <= 1) return 1;
return climbStairs(n - 1) + climbStairs(n - 2);
}

C. function climbStairs(n: number, memo: Record = {}): number {
if (n <= 1) return 1;
if (memo[n] !== undefined) return memo[n];
memo[n] = climbStairs(n - 1, memo) + climbStairs(n - 2, memo);
return memo[n];
}

D. function climbStairs(n: number, memo: Record = {}): number {
if (n <= 1) return 1;
if (memo[n] === 0) return memo[n];
memo[n] = climbStairs(n - 1, memo) + climbStairs(n - 2, memo);
return memo[n];
}

Solution

1. Step 1: Check memoization condition correctness

   function climbStairs(n: number, memo: Record = {}): number { if (n <= 1) return 1; if (memo[n] !== undefined) return memo[n]; memo[n] = climbStairs(n - 1, memo) + climbStairs(n - 2, memo); return memo[n]; } uses 'memo[n] !== undefined' to check cache, which correctly handles zero or falsy values.

2. Step 2: Compare other options

   function climbStairs(n: number, memo: Record = {}): number { if (n <= 1) return 1; if (memo[n]) return memo[n]; memo[n] = climbStairs(n - 1, memo) + climbStairs(n - 2, memo); return memo[n]; } uses 'if(memo[n])' which fails if memo[n] is 0; function climbStairs(n: number): number { if (n <= 1) return 1; return climbStairs(n - 1) + climbStairs(n - 2); } has no memoization; function climbStairs(n: number, memo: Record = {}): number { if (n <= 1) return 1; if (memo[n] === 0) return memo[n]; memo[n] = climbStairs(n - 1, memo) + climbStairs(n - 2, memo); return memo[n]; } checks 'memo[n] === 0' which misses other cached values.

3. Final Answer:

   Option C correctly applies memoization with proper cache check -> Option C

4. Quick Check:

   Use '!== undefined' to check memo cache [OK]

Hint: Check cache with '!== undefined' to handle all values [OK]

Common Mistakes:

• Using 'if(memo[n])' which fails for zero values
• Not passing memo in recursive calls
• Not using memoization at all