Fibonacci Using DP in DSA Javascript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main idea behind using Dynamic Programming (DP) to calculate Fibonacci numbers?

DP stores the results of smaller subproblems (previous Fibonacci numbers) to avoid repeated calculations, making the process faster.

beginner

How does memoization help in Fibonacci calculation?

Memoization saves the result of each Fibonacci number once calculated, so if needed again, it can be returned immediately without recalculating.

intermediate

What is the time complexity of Fibonacci calculation using DP compared to the naive recursive method?

Using DP, the time complexity is O(n), which is much faster than the naive recursive method's exponential time O(2^n).

intermediate

Explain the difference between top-down and bottom-up approaches in Fibonacci DP.

Top-down uses recursion with memoization (solves big problem by breaking down), bottom-up builds the solution from the smallest subproblems up to the desired number iteratively.

intermediate

Why is the bottom-up approach often preferred for Fibonacci DP?

Because it uses simple loops without recursion, it avoids function call overhead and stack overflow risks, making it more efficient and easier to understand.

What does DP store to optimize Fibonacci calculation?

A Previous Fibonacci results

B Random numbers

C Only the last Fibonacci number

D All even Fibonacci numbers

DP stores all previously calculated Fibonacci numbers to avoid repeated work.

Which approach uses recursion with caching in Fibonacci DP?

A Bottom-up

B Top-down

C Iterative

D Brute force

Top-down approach uses recursion with memoization (caching).

What is the time complexity of Fibonacci using DP?

A O(n)

B O(n^2)

C O(2^n)

D O(log n)

DP reduces Fibonacci calculation to linear time O(n).

Why is bottom-up Fibonacci DP safer than top-down?

A It uses recursion

B It is slower

C It avoids stack overflow

D It uses more memory

Bottom-up uses loops, so it avoids recursion stack overflow.

Memoization is an example of which DP technique?

A Divide and conquer

B Bottom-up

C Greedy

D Top-down

Memoization is a top-down DP technique.

Explain how Dynamic Programming improves the calculation of Fibonacci numbers compared to naive recursion.

Think about how remembering past answers helps.

Describe the difference between top-down and bottom-up approaches in Fibonacci DP and when you might prefer each.

Consider recursion vs loops.

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