Fibonacci Using DP in DSA Typescript - 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 by repeating the same work can be slow. So, you will use a method called Dynamic Programming (DP) to remember past results and speed things up.

🎯 Goal: Build a TypeScript program that uses Dynamic Programming to find the nth Fibonacci number efficiently.

📋 What You'll Learn

Create an array to store Fibonacci numbers up to n.

Use a loop to fill the array with Fibonacci numbers.

Use Dynamic Programming to avoid repeated calculations.

Print the nth Fibonacci number.

💡 Why This Matters

🌍 Real World

Calculating Fibonacci numbers quickly is useful in computer science problems, financial models, and nature simulations where repeated calculations can slow down programs.

💼 Career

Understanding Dynamic Programming and efficient algorithms is important for software development, especially in roles involving optimization and problem solving.

1

Create the input variable n

Create a variable called n and set it to 10 to represent the position in the Fibonacci sequence you want to find.

DSA Typescript

# Your code here
// Create a variable called n and set it to 10

Hint

Use const n = 10; to create the variable.

2

Create the array fib to store Fibonacci numbers

Create an array called fib with size n + 1. Initialize fib[0] to 0 and fib[1] to 1.

DSA Typescript

const n = 10;
// Create an array fib with size n + 1
// Initialize fib[0] = 0 and fib[1] = 1
// Your code here

Hint

Use new Array(n + 1) to create the array and assign the first two Fibonacci numbers.

3

Fill the fib array using a loop and Dynamic Programming

Use a for loop with variable i starting from 2 up to n. Inside the loop, set fib[i] to fib[i - 1] + fib[i - 2].

DSA Typescript

const n = 10;
const fib: number[] = new Array(n + 1);
fib[0] = 0;
fib[1] = 1;
// Use a for loop from i = 2 to n
// Calculate fib[i] = fib[i - 1] + fib[i - 2]
// Your code here

Hint

Use a for loop and update fib[i] inside it.

4

Print the nth Fibonacci number

Write a console.log statement to print the value of fib[n].

DSA Typescript

const n = 10;
const fib: number[] = new Array(n + 1);
fib[0] = 0;
fib[1] = 1;
for (let i = 2; i <= n; i++) {
    fib[i] = fib[i - 1] + fib[i - 2];
}
// Print fib[n] below
// Your code here

Hint

Use console.log(fib[n]); to print the result.

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