Fibonacci Using DP in DSA Javascript - 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 fibonacci(n) {
  const dp = [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 storing previous results to avoid repeated work.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

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

How Execution Grows With Input

Each new Fibonacci number requires one addition operation using two stored values.

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

Pattern observation: The number of operations grows directly with n, so doubling n roughly doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the nth Fibonacci number grows in a straight line as n gets bigger.

Common Mistake

[X] Wrong: "Calculating Fibonacci with recursion is always fast because it just calls itself."

[OK] Correct: Simple recursion repeats many calculations, making it much slower than this dynamic programming approach.

Interview Connect

Understanding how dynamic programming improves Fibonacci calculation shows you can spot and fix repeated work, a key skill in many coding problems.

Self-Check

"What if we used recursion with memoization instead of a loop? 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 calculates Fibonacci numbers using only recursion without loops

C. It stores intermediate results to avoid repeated calculations

D. It always returns the Fibonacci sequence in reverse order

Solution

1. Step 1: Understand recursion vs DP

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

2. Step 2: Role of DP in Fibonacci

   DP stores previously calculated Fibonacci values in a table (memoization or tabulation) to reuse them, avoiding repeated work.

3. Final Answer:

   It stores intermediate results to avoid repeated calculations -> Option C

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 does not use loops or arrays
• Confusing DP with simple recursion

2. Which of the following JavaScript code snippets correctly initializes an array for bottom-up DP to compute Fibonacci numbers up to n?

easy

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

B. const fib = []; fib[0] = 1; fib[1] = 1;

C. const fib = new Array(n).fill(0); fib[1] = 1;

D. const fib = new Array(n+1).fill(1); fib[0] = 0;

Solution

1. Step 1: Array size for DP

   We need an array of size n+1 to store Fibonacci numbers from 0 to n.

2. Step 2: Initial values for Fibonacci

   Fibonacci sequence starts with fib[0] = 0 and fib[1] = 1, so these must be set correctly.

3. Final Answer:

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

4. Quick Check:

   Array size n+1, fib[0]=0, fib[1]=1 [OK]

Hint: Array size n+1 with fib[0]=0 and fib[1]=1 [OK]

Common Mistakes:

• Using array size n instead of n+1
• Setting fib[0] = 1 incorrectly
• Filling array with 1 instead of 0

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

function fibDP(n) {
  const fib = [0, 1];
  for (let i = 2; i <= n; i++) {
    fib[i] = fib[i - 1] + fib[i - 2];
  }
  return fib[n];
}
console.log(fibDP(6));

medium

A. 8

B. 13

C. 21

D. 34

Solution

1. Step 1: Trace Fibonacci values up to n=6

   fib[0]=0, fib[1]=1, fib[2]=1, fib[3]=2, fib[4]=3, fib[5]=5, fib[6]=8.

2. Step 2: Return fib[6]

   The function returns fib[6], which is 8.

3. Final Answer:

   8 -> Option A

4. Quick Check:

   fib(6) = 8 [OK]

Hint: fib(6) = 8 by adding previous two numbers [OK]

Common Mistakes:

• Confusing index and value (e.g., fib[6] = 13)
• Starting array with wrong initial values
• Off-by-one errors in loop

4. Identify the error in this JavaScript DP Fibonacci function and choose the correct fix:

function fibDP(n) {
  const fib = [0, 1];
  for (let i = 1; i <= n; i++) {
    fib[i] = fib[i - 1] + fib[i - 2];
  }
  return fib[n];
}

medium

A. Remove the loop and use recursion instead

B. Initialize fib array with zeros instead of [0,1]

C. Change return to fib[n-1] to fix off-by-one

D. Change loop start to i = 2 to avoid undefined fib[-1]

Solution

1. Step 1: Analyze loop start index

   The loop starts at i=1, so fib[i-2] becomes fib[-1] which is undefined and causes error.

2. Step 2: Correct loop start

   Starting loop at i=2 ensures fib[i-2] is fib[0], which is defined, fixing the error.

3. Final Answer:

   Change loop start to i = 2 to avoid undefined fib[-1] -> Option D

4. Quick Check:

   Loop from 2 avoids negative index [OK]

Hint: Start loop at 2 to avoid negative indices [OK]

Common Mistakes:

• Starting loop at 1 causing undefined access
• Changing return index incorrectly
• Removing DP and using recursion unnecessarily

5. Given the following code to compute Fibonacci numbers using DP, what will be the output of fibDP(0) and fibDP(1)? How does the function handle these edge cases?

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

hard

A. fibDP(0) = 1, fibDP(1) = 1; initializes fib array incorrectly

B. fibDP(0) = 0, fibDP(1) = 1; returns n directly for n ≤ 1

C. fibDP(0) = 0, fibDP(1) = 0; misses setting fib[1] properly

D. fibDP(0) and fibDP(1) cause errors due to array size

Solution

1. Step 1: Check base case handling

   The function returns n directly if n ≤ 1, so fibDP(0) returns 0 and fibDP(1) returns 1 immediately.

2. Step 2: Confirm array initialization and loop

   For n > 1, the array is initialized and filled correctly, but for n=0 or 1, the function returns early, handling edge cases properly.

3. Final Answer:

   fibDP(0) = 0, fibDP(1) = 1; returns n directly for n ≤ 1 -> Option B

4. Quick Check:

   Edge cases return n directly [OK]

Hint: Returns n directly for n ≤ 1 to handle edges [OK]

Common Mistakes:

• Not handling n=0 or n=1 separately
• Incorrect array initialization for small n
• Assuming loop runs for n=0 or n=1