Fibonacci Using DP in DSA Typescript - Time & Space Complexity

Time Complexity: Fibonacci Using DP

O(n)

Understanding Time Complexity

We want to understand how fast the Fibonacci sequence calculation runs when using dynamic programming.

How does the time needed grow as we ask for bigger Fibonacci numbers?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function fibonacciDP(n: number): number {
  const dp: number[] = [0, 1];
  for (let i = 2; i <= n; i++) {
    dp[i] = dp[i - 1] + dp[i - 2];
  }
  return dp[n];
}
    

This code calculates the nth Fibonacci number by building up from smaller results and storing them.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The for-loop that calculates each Fibonacci number once.
• How many times: The loop runs from 2 up to n, so about n-1 times.

How Execution Grows With Input

Each new Fibonacci number is found by adding two previous numbers, done once per step.

Input Size (n)	Approx. Operations
10	About 8 additions
100	About 98 additions
1000	About 998 additions

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

Final Time Complexity

Time Complexity: O(n)

This means the time needed grows directly in proportion to the size of n.

Common Mistake

[X] Wrong: "Calculating Fibonacci with DP is as slow as the simple recursive method because it still calls itself many times."

[OK] Correct: DP stores results and avoids repeated work, so it only does each calculation once, making it much faster.

Interview Connect

Understanding this helps you explain how saving past results speeds up problems that repeat work, a key skill in many coding challenges.

Self-Check

"What if we changed the DP array to only keep the last two Fibonacci numbers instead of all? How would the time complexity change?"

Practice

1. What is the main advantage of using Dynamic Programming (DP) to calculate Fibonacci numbers compared to a simple recursive approach?

easy

A. It uses less memory by not storing any results.

B. It avoids repeated calculations by storing intermediate results.

C. It calculates Fibonacci numbers randomly.

D. It only works for small Fibonacci numbers.

Solution

1. Step 1: Understand recursive Fibonacci calculation

   Simple recursion recalculates the same Fibonacci numbers many times, causing inefficiency.

2. Step 2: Understand DP optimization

   DP stores previously calculated Fibonacci numbers in a table (memoization or tabulation) to avoid repeated work.

3. Final Answer:

   It avoids repeated calculations by storing intermediate results. -> Option B

4. Quick Check:

   DP = stores results to save time [OK]

Hint: DP saves time by remembering past answers [OK]

Common Mistakes:

• Thinking DP uses less memory than recursion
• Believing DP calculates randomly
• Assuming DP only works for small inputs

2. Which of the following TypeScript code snippets correctly initializes the DP array for Fibonacci calculation up to n?

easy

A. const dp: number[] = new Array(n + 1).fill(0); dp[1] = 1;

B. const dp: number[] = new Array(n).fill(1);

C. const dp: number[] = []; dp[0] = 1; dp[1] = 1;

D. const dp: number[] = new Array(n + 1); dp[1] = 1;

Solution

1. Step 1: Check array size and initialization

   We need an array of size n+1 to store Fibonacci numbers from 0 to n. Filling with 0 and setting dp[1] = 1 is correct.

2. Step 2: Validate each option

   const dp: number[] = new Array(n + 1).fill(0); dp[1] = 1; correctly creates dp with size n+1, fills with 0, and sets dp[1] = 1. Others have wrong sizes or initial values.

3. Final Answer:

   const dp: number[] = new Array(n + 1).fill(0); dp[1] = 1; -> Option A

4. Quick Check:

   Array size = n+1, dp[1] = 1 [OK]

Hint: DP array size must be n+1 with dp[1] = 1 [OK]

Common Mistakes:

• Using array size n instead of n+1
• Initializing dp[0] or dp[1] incorrectly
• Leaving dp array uninitialized

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

function fib(n: number): number {
  const dp: number[] = new Array(n + 1).fill(0);
  dp[1] = 1;
  for (let i = 2; i <= n; i++) {
    dp[i] = dp[i - 1] + dp[i - 2];
  }
  return dp[n];
}
console.log(fib(5));

medium

A. 13

B. 8

C. 3

D. 5

Solution

1. Step 1: Trace DP array values for n=5

   dp[0]=0, dp[1]=1; then dp[2]=1, dp[3]=2, dp[4]=3, dp[5]=5.

2. Step 2: Return dp[5]

   The function returns dp[5] which is 5.

3. Final Answer:

   5 -> Option D

4. Quick Check:

   Fibonacci(5) = 5 [OK]

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

Common Mistakes:

• Confusing index starting point
• Returning dp[n-1] instead of dp[n]
• Miscomputing dp values in loop

4. Identify the error in this TypeScript DP Fibonacci code and select the fix:

function fib(n: number): number {
  const dp: number[] = new Array(n).fill(0);
  dp[0] = 0;
  dp[1] = 1;
  for (let i = 2; i <= n; i++) {
    dp[i] = dp[i - 1] + dp[i - 2];
  }
  return dp[n];
}

medium

A. Change array size to n+1.

B. Initialize dp[2] to 1 before the loop.

C. Start loop from i = 1 instead of 2.

D. Remove dp[0] initialization.

Solution

1. Step 1: Analyze array size and access

   Array dp has size n, but loop accesses dp[n] beyond the initial length.

2. Step 2: Fix array size

   Increase dp size to n+1 to safely access dp[n].

3. Final Answer:

   Change array size to n+1 to avoid out-of-bounds error. -> Option A

4. Quick Check:

   Array size must cover dp[n] [OK]

Hint: Array size must be n+1 for dp[n] access [OK]

Common Mistakes:

• Ignoring array size mismatch
• Starting loop incorrectly
• Unnecessary dp initialization changes

5. You want to optimize the space used by the DP Fibonacci function. Which modification correctly reduces space complexity from O(n) to O(1)?

hard

A. Use a recursive function without memoization.

B. Store all Fibonacci numbers in a global array.

C. Use two variables to store only the last two Fibonacci numbers instead of an array.

D. Initialize the dp array with zeros and fill it completely every call.

Solution

1. Step 1: Understand space usage in DP Fibonacci

   DP uses an array of size n+1, so space is O(n).

2. Step 2: Optimize space by storing only needed values

   Since each Fibonacci number depends only on the previous two, storing just two variables suffices.

3. Step 3: Compare options

   Use two variables to store only the last two Fibonacci numbers instead of an array. uses two variables, reducing space to O(1). Others do not reduce space or are inefficient.

4. Final Answer:

   Use two variables to store only the last two Fibonacci numbers instead of an array. -> Option C

5. Quick Check:

   Two vars = O(1) space [OK]

Hint: Keep only last two Fibonacci numbers to save space [OK]

Common Mistakes:

• Using recursion without memoization increases time
• Storing all numbers globally wastes space
• Filling full dp array every call wastes memory