Fibonacci Using DP in DSA C - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Fibonacci DP Array

What is the printed state of the DP array after computing fibonacci(6) using bottom-up dynamic programming?

DSA C

int fibonacci(int n) {
    int dp[7] = {0};
    dp[0] = 0;
    dp[1] = 1;
    for (int i = 2; i <= n; i++) {
        dp[i] = dp[i-1] + dp[i-2];
    }
    for (int i = 0; i <= n; i++) {
        printf("%d ", dp[i]);
    }
    return dp[n];
}

int main() {
    fibonacci(6);
    return 0;
}

A 0 1 2 3 5 8 13

B 0 1 1 3 4 7 11

C 1 1 2 3 5 8 13

D 0 1 1 2 3 5 8

💡 Hint

Remember that fibonacci starts with 0 and 1, then each next number is sum of previous two.

The DP array starts with dp[0]=0 and dp[1]=1. Each next dp[i] is dp[i-1] + dp[i-2]. So dp[2]=1, dp[3]=2, dp[4]=3, dp[5]=5, dp[6]=8.

❓ Predict Output

intermediate

Value of fibonacci(7) with memoization

What is the value returned by fibonacci(7) using top-down memoization with dp array initialized to -1?

DSA C

int dp[8];

int fibonacci(int n) {
    if (dp[n] != -1) return dp[n];
    if (n <= 1) return n;
    dp[n] = fibonacci(n-1) + fibonacci(n-2);
    return dp[n];
}

int main() {
    for (int i = 0; i < 8; i++) dp[i] = -1;
    int result = fibonacci(7);
    printf("%d", result);
    return 0;
}

A 8

B 13

C 21

D 34

💡 Hint

Fibonacci sequence starts 0,1,1,2,3,5,8,13...

fibonacci(7) is the 7th Fibonacci number starting from 0 index: 0,1,1,2,3,5,8,13 so answer is 13.

🔧 Debug

advanced

Identify the error in DP Fibonacci code

What error will this code produce when computing fibonacci(5)?

DSA C

int fibonacci(int n) {
    int dp[5] = {0};
    dp[0] = 0;
    dp[1] = 1;
    for (int i = 2; i <= n; i++) {
        dp[i] = dp[i-1] + dp[i-2];
    }
    return dp[n];
}

int main() {
    int result = fibonacci(5);
    printf("%d", result);
    return 0;
}

A Returns 5 correctly

B Compilation error: dp array size mismatch

C Runtime error: accessing dp[5] out of bounds

D Returns 0 always

💡 Hint

Check the size of dp array and the loop limits carefully.

dp array size is 5, so valid indices are 0 to 4. Accessing dp[5] causes out of bounds runtime error.

🧠 Conceptual

advanced

Time complexity of Fibonacci DP

What is the time complexity of computing fibonacci(n) using bottom-up dynamic programming with a dp array?

A O(n)

B O(2^n)

C O(n^2)

D O(log n)

💡 Hint

Each fibonacci number is computed once in bottom-up DP.

Bottom-up DP computes each fibonacci number from 0 to n once, so time complexity is linear O(n).

🚀 Application

expert

Number of calls in naive vs memoized Fibonacci

How many times is fibonacci(2) computed in naive recursive fibonacci(5) vs memoized fibonacci(5)?

A Naive: 3 times, Memoized: 1 time

B Naive: 1 time, Memoized: 1 time

C Naive: 5 times, Memoized: 2 times

D Naive: 3 times, Memoized: 3 times

💡 Hint

Memoization stores results to avoid repeated calls.

Naive recursion calls fibonacci(2) multiple times (3 times for fibonacci(5)). Memoization computes fibonacci(2) once and reuses it.

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 stores intermediate results to avoid repeated calculations

B. It uses more memory without improving speed

C. It calculates Fibonacci numbers randomly

D. It only works for small Fibonacci numbers

Solution

1. Step 1: Understand recursion vs DP

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

2. Step 2: Role of DP in Fibonacci

   DP saves previously calculated Fibonacci values in a table (memoization or tabulation), avoiding repeated work.

3. Final Answer:

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

4. Quick Check:

   DP = stores results to save time [OK]

Hint: DP saves past answers to speed up Fibonacci [OK]

Common Mistakes:

• Thinking DP uses more memory without speed benefit
• Believing DP calculates numbers randomly
• Assuming DP only works for small inputs

2. Which of the following is the correct way to initialize the base cases in a bottom-up DP approach for Fibonacci in C?

easy

A. int fib[2] = {0, 0};

B. int fib[2] = {1, 1};

C. int fib[2] = {0, 1};

D. int fib[2] = {1, 0};

Solution

1. Step 1: Identify Fibonacci base cases

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

2. Step 2: Match initialization with base cases

   Array fib[2] = {0, 1} correctly sets fib(0) = 0 and fib(1) = 1.

3. Final Answer:

   int fib[2] = {0, 1}; -> Option C

4. Quick Check:

   Base fib = 0,1 [OK]

Hint: Start fib with 0 and 1 for base cases [OK]

Common Mistakes:

• Swapping the order of 0 and 1
• Setting both base cases to 1
• Initializing both to zero

3. What is the output of the following C code snippet using bottom-up DP to find fib(5)?

int fib[6];
fib[0] = 0;
fib[1] = 1;
for(int i = 2; i <= 5; i++) {
    fib[i] = fib[i-1] + fib[i-2];
}
printf("%d", fib[5]);

medium

A. 5

B. 8

C. 3

D. 13

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 fib[5]

   fib[5] is 5 after the loop completes.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   fib(5) = 5 [OK]

Hint: Calculate fib stepwise from 0 and 1 [OK]

Common Mistakes:

• Confusing fib(5) with fib(6)
• Starting loop from wrong index
• Off-by-one errors in array indexing

4. Identify the error in this DP Fibonacci code snippet and select 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];
}

medium

A. Initialize fib[2] before loop

B. Change loop to start from i = 2

C. Change loop condition to i <= 10

D. No error, code is correct

Solution

1. Step 1: Analyze loop start index

   The loop starts at i=1, but fib[1] is already initialized, so fib[1] gets overwritten incorrectly.

2. Step 2: Correct loop start

   Loop should start at i=2 to use fib[0] and fib[1] for calculation without overwriting base cases.

3. Final Answer:

   Change loop to start from i = 2 -> Option B

4. Quick Check:

   Loop start = 2 fixes overwrite [OK]

Hint: Start loop after base cases to avoid overwrite [OK]

Common Mistakes:

• Starting loop at 1 overwrites fib[1]
• Not initializing base cases properly
• Using wrong loop end condition

5. You want to find the 10th Fibonacci number using DP but with optimized space using only two variables instead of an array. Which code snippet correctly implements this?

hard

A. int a = 0, b = 1, c, n = 10;
for(int i = 0; i < n; i++) {
c = a + b;
a = c;
b = a;
}
printf("%d", c);

B. int a = 0, b = 1, c, n = 10;
for(int i = 1; i < n; i++) {
c = a + b;
b = a;
a = c;
}
printf("%d", a);

C. int a = 1, b = 1, c, n = 10;
for(int i = 2; i <= n; i++) {
c = a + b;
a = b;
b = c;
}
printf("%d", c);

D. int a = 0, b = 1, c, n = 10;
for(int i = 2; i <= n; i++) {
c = a + b;
a = b;
b = c;
}
printf("%d", b);

Solution

1. Step 1: Understand space optimization

   Only two variables (a, b) track last two Fibonacci numbers; c stores current sum.

2. Step 2: Check loop and updates

   Loop from 2 to n, update c = a + b, then shift a = b, b = c correctly to move forward.

3. Final Answer:

   Code snippet A correctly implements optimized Fibonacci calculation -> Option D

4. Quick Check:

   Two vars + sum update = correct [OK]

Hint: Use two vars and update in loop for space optimization [OK]

Common Mistakes:

• Swapping updates of a and b incorrectly
• Wrong loop range causing off-by-one errors
• Initializing variables incorrectly