Fibonacci Using DP in DSA C++ - Cheat Sheet & Quick Revision

Recall & Review

beginner

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

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

beginner

In Fibonacci DP, what does the 'memoization' technique mean?

Memoization means saving the results of Fibonacci calculations in a table or array so we can reuse them instead of recalculating.

intermediate

Why is the DP approach to Fibonacci more efficient than the simple recursive approach?

Because DP avoids repeated work by storing results, reducing the time from exponential to linear.

beginner

What is the time complexity of Fibonacci calculation using DP?

The time complexity is O(n), where n is the Fibonacci number index.

beginner

Show the base cases used in Fibonacci DP and explain why they are important.

Base cases: fib(0) = 0 and fib(1) = 1. They are important because they stop the recursion and start the building of the solution.

What does DP store to optimize Fibonacci calculation?

A No values, it recalculates every time

B Random numbers

C Only the last Fibonacci number

D Previous Fibonacci values

DP stores previous Fibonacci values to avoid recalculating them.

What is the time complexity of Fibonacci using DP?

A O(log n)

B O(n)

C O(2^n)

D O(n^2)

DP reduces the time complexity to O(n) by storing intermediate results.

Which technique is used in DP to save results of subproblems?

A Brute force

B Backtracking

C Memoization

D Divide and conquer

Memoization saves results to avoid repeated calculations.

What are the base cases for Fibonacci DP?

A fib(0) = 0, fib(1) = 1

B fib(1) = 0, fib(2) = 1

C fib(0) = 1, fib(1) = 1

D fib(2) = 0, fib(3) = 1

Base cases fib(0)=0 and fib(1)=1 start the calculation.

Why is simple recursion inefficient for Fibonacci?

A It recalculates the same values many times

B It uses too little memory

C It only works for small numbers

D It uses loops instead of recursion

Simple recursion repeats calculations, causing exponential time.

Explain how Dynamic Programming improves Fibonacci number calculation compared to simple recursion.

Think about how saving answers helps avoid doing the same work twice.

Describe the role of base cases in the Fibonacci DP approach and why they are necessary.

Base cases are like the starting point or foundation for the calculation.

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 using only addition.

D. It uses a recursive function without any loops.

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 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 = store results to avoid repeats [OK]

Hint: DP saves results to skip repeated work [OK]

Common Mistakes:

• Thinking DP uses less memory than recursion
• Confusing DP with simple recursion
• Believing DP avoids addition operations

2. Which of the following C++ code snippets correctly initializes the base cases for Fibonacci using bottom-up DP?

easy

A. int fib[10]; fib[0] = 0; fib[1] = 1;

B. int fib[10]; fib[1] = 0; fib[2] = 1;

C. int fib[10]; fib[0] = 1; fib[1] = 1;

D. int fib[10]; fib[0] = 1; fib[2] = 0;

Solution

1. Step 1: Identify Fibonacci base cases

   Fibonacci sequence starts with fib(0) = 0 and fib(1) = 1.

2. Step 2: Check initialization correctness

   int fib[10]; fib[0] = 0; fib[1] = 1; correctly sets fib[0] = 0 and fib[1] = 1, matching the base cases.

3. Final Answer:

   int fib[10]; fib[0] = 0; fib[1] = 1; -> Option A

4. Quick Check:

   fib[0]=0, fib[1]=1 base cases [OK]

Hint: Base cases: fib[0]=0 and fib[1]=1 [OK]

Common Mistakes:

• Starting fib from index 1 instead of 0
• Swapping values of fib[0] and fib[1]
• Initializing wrong indices

3. What is the output of the following C++ code snippet?

int fib[6];
fib[0] = 0; fib[1] = 1;
for(int i = 2; i < 6; i++) {
    fib[i] = fib[i-1] + fib[i-2];
}
std::cout << fib[5];

medium

A. 5

B. 8

C. 13

D. 21

Solution

1. Step 1: Calculate Fibonacci values step-by-step

   fib[0]=0, fib[1]=1 fib[2]=1, fib[3]=2, fib[4]=3, fib[5]=5

2. Step 2: Identify output

   The code prints fib[5], which is 5.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   fib[5]=5 [OK]

Hint: Calculate fib stepwise from base cases [OK]

Common Mistakes:

• Confusing fib[5] with fib[6]
• Off-by-one errors in loop
• Incorrect addition order

4. Identify the error in the following Fibonacci DP code and choose the correct fix:

int fib[10];
fib[0] = 0;
fib[1] = 1;
for(int i = 1; i < 10; i++) {
    fib[i] = fib[i-1] + fib[i-2];
}
std::cout << fib[9];

medium

A. Change loop condition to i <= 10

B. Initialize fib[2] before the loop

C. Change fib[0] to 1

D. Change loop start to i = 2

Solution

1. Step 1: Analyze loop start index

   The loop starts at i=1, but fib[1] is already initialized. Calculating fib[1] again causes fib[-1] access (invalid).

2. Step 2: Correct loop start

   Loop should start at i=2 to use fib[i-1] and fib[i-2] safely.

3. Final Answer:

   Change loop start to i = 2 -> Option D

4. Quick Check:

   Loop from i=2 avoids invalid index [OK]

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

Common Mistakes:

• Starting loop at 1 causing invalid access
• Changing base case values incorrectly
• Extending loop beyond array size

5. You want to find the 50th Fibonacci number using DP in C++. Which approach is best to handle large Fibonacci numbers efficiently and why?

hard

A. Use bottom-up DP with int array since int can hold large numbers.

B. Use simple recursion without memoization for clarity.

C. Use a vector of unsigned long long and bottom-up DP to avoid recursion overhead.

D. Use recursion with memoization but store results in int array.

Solution

1. Step 1: Consider data type size

   50th Fibonacci number is very large; int cannot hold it, unsigned long long can hold bigger values.

2. Step 2: Choose DP approach

   Bottom-up DP avoids recursion overhead and stack issues, making it efficient for large n.

3. Final Answer:

   Use a vector of unsigned long long and bottom-up DP to avoid recursion overhead. -> Option C

4. Quick Check:

   Big number + bottom-up DP = efficient and safe [OK]

Hint: Use big data type and bottom-up DP for large n [OK]

Common Mistakes:

• Using int causing overflow
• Using recursion causing stack overflow
• Ignoring data type limits