Fibonacci Using DP in DSA C++ - Time & Space Complexity

Time Complexity: Fibonacci Using DP

O(n)

Understanding Time Complexity

We want to understand how fast the Fibonacci numbers can be calculated using dynamic programming.

How does the time needed grow as we ask for bigger Fibonacci numbers?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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

This code calculates the nth Fibonacci number by building up answers from smaller numbers using a table.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The for-loop that fills the dp array from 2 to n.
• How many times: Exactly n-1 times (from 2 up to n).

How Execution Grows With Input

Each new Fibonacci number requires one addition and one assignment inside the loop.

Input Size (n)	Approx. Operations
10	About 9 additions and assignments
100	About 99 additions and assignments
1000	About 999 additions and assignments

Pattern observation: The number of operations grows roughly in a straight line as n increases.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the nth Fibonacci number grows directly in proportion to n.

Common Mistake

[X] Wrong: "Calculating Fibonacci with recursion is always fast because it just calls itself."

[OK] Correct: Simple recursion repeats many calculations, making it much slower than this dynamic programming approach.

Interview Connect

Knowing how to use dynamic programming to speed up Fibonacci calculations shows you understand how to avoid repeated work, a key skill in many coding problems.

Self-Check

"What if we used recursion with memoization instead of a loop? How would the time complexity change?"

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