Fibonacci Using DP in DSA Javascript - Build from Scratch

Fibonacci Using DP

📖 Scenario: You are helping a friend who wants to find the nth number in the Fibonacci sequence quickly. The Fibonacci sequence starts with 0 and 1, and each next number is the sum of the two before it.Calculating Fibonacci numbers one by one can be slow for big numbers. So, you will use a smart way called Dynamic Programming (DP) to remember past results and avoid repeating work.

🎯 Goal: Build a program that uses a list to store Fibonacci numbers as you calculate them. This way, you can find the nth Fibonacci number efficiently by reusing stored answers.

📋 What You'll Learn

Create a list called fib with the first two Fibonacci numbers: 0 and 1

Create a variable called n and set it to 7

Use a for loop with variable i from 2 to n (inclusive) to fill the fib list using the rule: fib[i] = fib[i-1] + fib[i-2]

Print the fib list after the loop finishes

💡 Why This Matters

🌍 Real World

Fibonacci numbers appear in nature, computer algorithms, and financial models. Using dynamic programming helps calculate them quickly without repeating work.

💼 Career

Understanding dynamic programming and efficient loops is important for software developers, data scientists, and anyone working with algorithms.

1

Create the Fibonacci list

Create a list called fib with the first two Fibonacci numbers: 0 and 1

DSA Javascript

# Your code here

Hint

Remember, the first two Fibonacci numbers are 0 and 1. Use square brackets to create a list.

2

Set the target Fibonacci position

Create a variable called n and set it to 7

DSA Javascript

const fib = [0, 1];
# Your code here

Hint

Use const to create a variable named n and assign it the value 7.

3

Fill the Fibonacci list using a loop

Use a for loop with variable i from 2 to n (inclusive) to fill the fib list using the rule: fib[i] = fib[i - 1] + fib[i - 2]

DSA Javascript

const fib = [0, 1];
const n = 7;
# Your code here

Hint

Start the loop at 2 and go up to n. Inside the loop, calculate the current Fibonacci number by adding the two previous numbers.

4

Print the Fibonacci list

Print the fib list after the loop finishes using console.log(fib)

DSA Javascript

const fib = [0, 1];
const n = 7;
for (let i = 2; i <= n; i++) {
    fib[i] = fib[i - 1] + fib[i - 2];
}
# Your code here

Hint

Use console.log(fib) to show the full list of Fibonacci numbers.

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