Overlapping Subproblems and Optimal Substructure in DSA C - Time & Space Complexity

Time Complexity: Overlapping Subproblems and Optimal Substructure

O(n)

Understanding Time Complexity

We want to understand how solving problems with repeated parts affects the time it takes.

How does breaking a problem into smaller parts and reusing answers change the work needed?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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

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

This code calculates Fibonacci numbers using simple recursion and recursion with memoization to avoid repeated work.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls to fib or fib_memo functions.
• How many times: In simple recursion, many calls repeat the same calculations; in memoized version, each unique call happens once.

How Execution Grows With Input

Without memoization, the calls grow very fast because the same problems are solved many times.

Input Size (n)	Approx. Operations (simple recursion)
10	~177 calls
20	~21,891 calls
30	~2,692,537 calls

With memoization, the number of calls grows only linearly with n because each result is stored and reused.

Input Size (n)	Approx. Operations (memoized)
10	10 calls
20	20 calls
30	30 calls

Pattern observation: Simple recursion grows exponentially, memoized recursion grows linearly.

Final Time Complexity

Time Complexity: O(n) with memoization

This means the work grows in a straight line with input size when we reuse answers, avoiding repeated work.

Common Mistake

[X] Wrong: "Recursion always means exponential time because it calls itself many times."

[OK] Correct: When we save results and reuse them, recursion can be very efficient and run in linear time.

Interview Connect

Understanding how to spot repeated work and use memory to save time is a key skill that shows you can write smart, efficient code.

Self-Check

"What if we changed the memo array to a hash map? How would the time complexity change?"

Practice

1. Which of the following best describes the concept of overlapping subproblems in dynamic programming?

easy

A. The problem requires sorting elements before solving.

B. The problem has a unique solution without repeating any subproblems.

C. The problem can be broken down into smaller subproblems which are solved multiple times.

D. The problem can be solved only using recursion without optimization.

Solution

1. Step 1: Understand overlapping subproblems

   Overlapping subproblems means the same smaller problems appear multiple times during the solution process.

2. Step 2: Match with options

   The problem can be broken down into smaller subproblems which are solved multiple times. correctly states that subproblems are solved multiple times, which is the key idea behind overlapping subproblems.

3. Final Answer:

   The problem can be broken down into smaller subproblems which are solved multiple times. -> Option C

4. Quick Check:

   Overlapping subproblems = repeated smaller problems [OK]

Hint: Look for repeated smaller problems solved multiple times [OK]

Common Mistakes:

• Confusing overlapping subproblems with unique subproblems
• Thinking sorting is required for overlapping subproblems
• Assuming recursion alone solves overlapping subproblems

2. Which of the following is the correct syntax to declare a memoization array for storing results of subproblems in C?

easy

A. int memo[] = 0;

B. int memo = new int[1000];

C. int memo(1000);

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

Solution

1. Step 1: Understand C array declaration

   In C, arrays are declared with fixed size and can be initialized with {0} to set all elements to zero.

2. Step 2: Check options for valid C syntax

   int memo[1000] = {0}; uses correct C syntax: int memo[1000] = {0}; initializes an array of 1000 integers to zero.

3. Final Answer:

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

4. Quick Check:

   C array declaration with initialization = int memo[1000] = {0}; [OK]

Hint: Use fixed size array with braces to initialize in C [OK]

Common Mistakes:

• Using C++ style vector or new keyword in C
• Declaring arrays without size
• Incorrect initialization syntax

3. Consider the following C code for Fibonacci with memoization:

int memo[100] = {0};

int fib(int n) {
    if (n <= 1) return n;
    if (memo[n] != 0) return memo[n];
    memo[n] = fib(n-1) + fib(n-2);
    return memo[n];
}

int main() {
    printf("%d", fib(5));
    return 0;
}

What is the output when this code runs?

medium

A. 0

B. 5

C. 3

D. 8

Solution

1. Step 1: Trace fib(5) calls with memoization

   fib(5) calls fib(4) + fib(3). fib(4) calls fib(3) + fib(2), and so on. Base cases fib(0)=0, fib(1)=1.

2. Step 2: Calculate fib(5)

   fib(2)=1, fib(3)=2, fib(4)=3, fib(5)=fib(4)+fib(3)=3+2=5.

3. Final Answer:

   5 -> Option B

4. Quick Check:

   Fibonacci(5) = 5 [OK]

Hint: Recall Fibonacci sequence: 0,1,1,2,3,5... [OK]

Common Mistakes:

• Confusing index with Fibonacci value
• Ignoring memoization effect
• Miscomputing base cases

4. The following code attempts to compute the nth Fibonacci number using memoization but has a bug:

int memo[100] = {0};

int fib(int n) {
    if (n == 0) return 0;
    if (n == 1) return 1;
    if (memo[n] != 0) return memo[n];
    memo[n] = fib(n-1) + fib(n-2);
    return memo[n];
}

int main() {
    printf("%d", fib(0));
    return 0;
}

What is the bug in this code?

medium

A. fib(0) returns 0 but memo[0] is never set, causing repeated calls.

B. The memo array is not initialized to zero.

C. The base case for fib(1) is missing.

D. The memoization check uses wrong index.

Solution

1. Step 1: Analyze base cases and memo usage

   fib(0) returns 0 but does not store it in memo[0]. So memo[0] remains 0, which is the default value.

2. Step 2: Understand effect on memoization

   Because the base case results are not stored in memo[n], fib(0) (and fib(1)) will be called repeatedly in the recursion tree for larger n, instead of returning a cached value.

3. Final Answer:

   fib(0) returns 0 but memo[0] is never set, causing repeated calls. -> Option A

4. Quick Check:

   Base case not stored in memo causes repeated calls [OK]

Hint: Store base case results in memo to avoid repeated calls [OK]

Common Mistakes:

• Assuming memo array auto-initializes for base cases
• Missing base case for fib(1)
• Incorrect memo check condition

5. You want to find the minimum cost to reach the top of stairs where each step has a cost. You can climb 1 or 2 steps at a time. The cost array is cost[] = {10, 15, 20}. Using dynamic programming with overlapping subproblems and optimal substructure, what is the minimum cost to reach the top (beyond last step)?

hard

A. 15

B. 20

C. 25

D. 10

Solution

1. Step 1: Define subproblems and recurrence

   Minimum cost to reach step i: dp[i] = cost[i] + min(dp[i-1], dp[i-2]). dp[0]=10, dp[1]=15.

2. Step 2: Calculate minimum cost step by step

   dp[2] = 20 + min(10,15) = 30. To reach top (beyond last step), min(dp[1], dp[2]) = min(15,30) = 15.

3. Final Answer:

   15 -> Option A

4. Quick Check:

   Minimum cost to top = 15 [OK]

Hint: Use min cost of last two steps for final answer [OK]

Common Mistakes:

• Adding all costs instead of choosing min path
• Ignoring that you can start from step 0 or 1
• Confusing cost to step vs cost beyond last step