Why Dynamic Programming and When Recursion Alone Fails in DSA C - Test Your Knowledge

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate Fibonacci numbers using simple recursion.

DSA C

int fib(int n) {
    if (n <= 1) return n;
    return fib(n-1) [1] fib(n-2);
}

Drag options to blanks, or click blank then click option'

A*

B-

C+

D/

Attempts:

3 left

💡 Hint

Common Mistakes

Using subtraction or multiplication instead of addition.

Forgetting to return the sum.

The Fibonacci sequence is defined as fib(n) = fib(n-1) + fib(n-2). So we add the two previous results.

2fill in blank

medium

Complete the code to add memoization to the recursive Fibonacci function.

DSA C

int fibMemo(int n, int memo[]) {
    if (n <= 1) return n;
    if (memo[n] != -1) return memo[n];
    memo[n] = fibMemo(n-1, memo) [1] fibMemo(n-2, memo);
    return memo[n];
}

Drag options to blanks, or click blank then click option'

A/

B+

C*

D-

Attempts:

3 left

💡 Hint

Common Mistakes

Not checking if memo[n] is already computed.

Using wrong operator instead of '+'.

Memoization stores results to avoid repeated calculations. The addition operator combines the two previous Fibonacci numbers.

3fill in blank

hard

Fix the error in the recursive factorial function to correctly compute factorial.

DSA C

int factorial(int n) {
    if (n == 0) return 1;
    return n [1] factorial(n-1);
}

Drag options to blanks, or click blank then click option'

A*

B-

C+

D/

Attempts:

3 left

💡 Hint

Common Mistakes

Using '+' or '-' instead of '*'.

Missing base case for n == 0.

Factorial is the product of all positive integers up to n, so we multiply n by factorial(n-1).

4fill in blank

hard

Fill both blanks to create a bottom-up dynamic programming solution for Fibonacci.

DSA C

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

Drag options to blanks, or click blank then click option'

A-

B+

D*

Attempts:

3 left

💡 Hint

Common Mistakes

Using addition instead of subtraction for index.

Using multiplication instead of addition for dp values.

The loop calculates each Fibonacci number by adding the two previous numbers: dp[i-1] + dp[i-2].

5fill in blank

hard

Fill all three blanks to implement memoized Fibonacci with initialization.

DSA C

int fibMemoInit(int n) {
    int memo[n+1];
    for (int i = 0; i <= n; i++) {
        memo[i] = [1];
    }
    return fibHelper(n, memo);
}

int fibHelper(int n, int memo[]) {
    if (n <= 1) return n;
    if (memo[n] != [2]) return memo[n];
    memo[n] = fibHelper(n-1, memo) [3] fibHelper(n-2, memo);
    return memo[n];
}

Drag options to blanks, or click blank then click option'

A-1

B0

C+

D*

Attempts:

3 left

💡 Hint

Common Mistakes

Initializing memo with 0 instead of -1.

Checking memo[n] != 0 instead of != -1.

Using multiplication instead of addition.

Initialize memo array with -1 to mark uncomputed values. Check if memo[n] != -1 to reuse. Add results of recursive calls.

Practice

1. Why do we use dynamic programming instead of just recursion for some problems?

easy

A. Because recursion alone can repeat the same calculations many times, making it slow.

B. Because recursion cannot solve problems with loops.

C. Because dynamic programming uses less memory than recursion always.

D. Because recursion is only for sorting problems.

Solution

1. Step 1: Understand recursion's limitation

   Recursion solves problems by calling itself but often recalculates the same results multiple times.

2. Step 2: Role of dynamic programming

   Dynamic programming stores these repeated results to avoid recalculations, making the solution faster.

3. Final Answer:

   Because recursion alone can repeat the same calculations many times, making it slow. -> Option A

4. Quick Check:

   Recursion repeats work, DP saves results [OK]

Hint: DP saves repeated results to speed up recursion [OK]

Common Mistakes:

• Thinking recursion can't handle loops
• Believing DP always uses less memory
• Assuming recursion is only for sorting

2. Which of the following is the correct way to declare a memoization array in C for dynamic programming?

easy

A. memo = new int[1000];

B. int memo[] = 0;

C. int memo[1000] = {0};

D. int memo(1000);

Solution

1. Step 1: Check C array declaration syntax

   In C, arrays are declared with size and optional initialization like int memo[1000] = {0};

2. Step 2: Analyze options

   int memo[1000] = {0}; uses correct syntax. int memo[] = 0; is invalid syntax. int memo(1000); uses parentheses which is wrong in C. memo = new int[1000]; uses 'new' which is C++ syntax.

3. Final Answer:

   int memo[1000] = {0}; -> Option C

4. Quick Check:

   C arrays use brackets [] [OK]

Hint: Use brackets [] to declare arrays in C [OK]

Common Mistakes:

• Using C++ 'new' in C code
• Using parentheses instead of brackets
• Not initializing memo array properly

3. What will be the output of the following C code snippet?

int fib(int n, int memo[]) {
  if (n <= 1) return n;
  if (memo[n] != -1) return memo[n];
  memo[n] = fib(n-1, memo) + fib(n-2, memo);
  return memo[n];
}

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

medium

A. 5

B. 8

C. 13

D. Runtime error

Solution

1. Step 1: Understand the Fibonacci function with memoization

   fib(5) calculates Fibonacci of 5 using memo array initialized with -1 to store results.

2. Step 2: Calculate fib(5)

   Fibonacci sequence: 0,1,1,2,3,5. fib(5) = 5.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   fib(5) = 5 [OK]

Hint: Fibonacci(5) = 5, memo avoids repeated calls [OK]

Common Mistakes:

• Confusing fib(5) with fib(6)
• Not initializing memo causing wrong results
• Assuming runtime error due to memo usage

4. Identify the error in this recursive Fibonacci function without memoization:

int fib(int n) {
  if (n == 0) return 0;
  if (n == 1) return 1;
  return fib(n-1) + fib(n-3);
}

medium

A. Base cases are missing.

B. It should call fib(n-2) instead of fib(n-3).

C. Function will cause infinite recursion.

D. No error, code is correct.

Solution

1. Step 1: Check recursive calls in Fibonacci

   Fibonacci is defined as fib(n) = fib(n-1) + fib(n-2).

2. Step 2: Identify incorrect call

   Code calls fib(n-3) instead of fib(n-2), which is wrong and breaks Fibonacci definition.

3. Final Answer:

   It should call fib(n-2) instead of fib(n-3). -> Option B

4. Quick Check:

   Fibonacci uses n-1 and n-2 calls [OK]

Hint: Fibonacci sums fib(n-1) and fib(n-2) only [OK]

Common Mistakes:

• Using wrong recursive calls
• Missing base cases
• Thinking code is correct as is

5. You want to find the minimum number of coins to make a value using coins of denominations 1, 3, and 4. Why is recursion alone inefficient here, and how does dynamic programming help?

hard

A. Recursion cannot handle multiple coin types; dynamic programming can.

B. Recursion always gives wrong answers; dynamic programming is always correct.

C. Recursion uses loops which slow it down; dynamic programming uses no loops.

D. Recursion repeats subproblems; dynamic programming stores results to avoid repeats.

Solution

1. Step 1: Understand recursion inefficiency

   Recursion tries all combinations repeatedly, recalculating same subproblems many times.

2. Step 2: How dynamic programming improves

   Dynamic programming saves results of subproblems (like minimum coins for smaller values) to reuse, reducing time drastically.

3. Final Answer:

   Recursion repeats subproblems; dynamic programming stores results to avoid repeats. -> Option D

4. Quick Check:

   DP saves repeated work, recursion repeats it [OK]

Hint: DP stores sub-results; recursion repeats work [OK]

Common Mistakes:

• Thinking recursion can't handle multiple coins
• Believing recursion uses loops
• Assuming recursion is always wrong