Greedy vs DP How to Know Which to Apply in DSA C++ - Complexity Comparison

Time Complexity: Greedy vs DP How to Know Which to Apply

O(n)

Understanding Time Complexity

We want to understand how to decide between greedy and dynamic programming (DP) approaches by looking at their time costs.

Which method runs faster and when does that matter?

Scenario Under Consideration

Analyze the time complexity of these two approaches solving the same problem.

// Greedy approach example (simplified)
int greedySolution(std::vector<int> &arr) {
  int result = 0;
  for (int i = 0; i < arr.size(); i++) {
    if (canTake(arr[i])) {
      result += arr[i];
    }
  }
  return result;
}

// DP approach example (simplified)
int dpSolution(std::vector<int> &arr) {
  std::vector<int> dp(arr.size() + 1, 0);
  dp[0] = 0;
  dp[1] = arr[0];
  for (int i = 2; i <= arr.size(); i++) {
    dp[i] = std::max(dp[i-1], dp[i-2] + arr[i-1]);
  }
  return dp[arr.size()];
}
    

The greedy code picks items quickly by checking a simple rule once per item. The DP code builds a table considering past choices for each item.

Identify Repeating Operations

Look at what repeats in each approach.

• Greedy primary operation: One loop over all items, checking a simple condition once per item.
• Greedy how many times: Exactly once per item, so n times.
• DP primary operation: One loop over all items, but each step uses previous results stored in dp array.
• DP how many times: Exactly once per item, also n times, but with extra work to compare and add.

How Execution Grows With Input

Both approaches loop through the input list once, but DP does a bit more work per item.

Input Size (n)	Greedy Approx. Operations	DP Approx. Operations
10	~10 checks	~10 comparisons and additions
100	~100 checks	~100 comparisons and additions
1000	~1000 checks	~1000 comparisons and additions

Pattern observation: Both grow linearly with input size, but DP does slightly more work per step.

Final Time Complexity

Time Complexity: O(n)

This means both methods take time proportional to the number of items, but DP has a bit more work inside the loop.

Common Mistake

[X] Wrong: "Greedy is always faster and better than DP because it's simpler."

[OK] Correct: Greedy is faster only when its simple rule always leads to the best answer. If the problem needs to consider past choices, DP is needed despite extra work.

Interview Connect

Knowing when to pick greedy or DP shows you understand problem patterns and efficiency, a key skill in coding challenges and real projects.

Self-Check

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

Practice

1. Which of the following best describes when to use a greedy algorithm instead of dynamic programming?

easy

A. When the problem has no clear optimal substructure

B. When the problem has optimal substructure and no overlapping subproblems

C. When the problem requires checking all possible solutions

D. When the problem has overlapping subproblems and optimal substructure

Solution

1. Step 1: Understand greedy algorithm conditions

   Greedy algorithms work well when the problem has an optimal substructure and no overlapping subproblems, meaning making the best local choice leads to a global optimum.

2. Step 2: Compare with dynamic programming conditions

   Dynamic programming is used when there are overlapping subproblems and optimal substructure, requiring storing intermediate results to avoid recomputation.

3. Final Answer:

   When the problem has optimal substructure and no overlapping subproblems -> Option B

4. Quick Check:

   Greedy = optimal substructure + no overlap [OK]

Hint: Use greedy if no overlapping subproblems exist [OK]

Common Mistakes:

• Confusing overlapping subproblems as a greedy condition
• Using DP when greedy suffices
• Ignoring optimal substructure requirement

2. Which of the following C++ code snippets correctly initializes a DP array for a problem with 5 states?

easy

A. int dp[5];

B. int dp(5, 0);

C. vector<int> dp(5, 0);

D. int dp = new int[5];

Solution

1. Step 1: Check syntax for array initialization

   int dp[5]; leaves elements uninitialized. int dp(5, 0); is invalid syntax for array initialization. int dp = new int[5]; uses incorrect syntax and memory management in C++.

2. Step 2: Identify correct vector initialization

   vector<int> dp(5, 0); correctly creates a vector of size 5 with all elements initialized to 0, which is the preferred way to initialize DP arrays in modern C++.

3. Final Answer:

   vector<int> dp(5, 0); -> Option C

4. Quick Check:

   Use vector<int> dp(size, 0) for DP arrays [OK]

Hint: Use vector with size and initial value for DP arrays [OK]

Common Mistakes:

• Using incorrect syntax for array initialization
• Not initializing all elements to zero
• Using raw pointers without delete

3. Consider the following C++ code snippet for a problem that can be solved by either greedy or DP. What is the output?

int main() {
    int arr[] = {1, 2, 3, 4, 5};
    int n = 5;
    int sum = 0;
    for (int i = 0; i < n; i++) {
        if (arr[i] % 2 == 0) {
            sum += arr[i];
        }
    }
    std::cout << sum << std::endl;
    return 0;
}

medium

A. 6

B. 9

C. 15

D. 10

Solution

1. Step 1: Identify even numbers in the array

   The array is {1, 2, 3, 4, 5}. Even numbers are 2 and 4.

2. Step 2: Calculate sum of even numbers

   Sum = 2 + 4 = 6.

3. Final Answer:

   6 -> Option A

4. Quick Check:

   Sum of even numbers = 6 [OK]

Hint: Sum only even numbers using modulo check [OK]

Common Mistakes:

• Summing all numbers instead of only even
• Off-by-one error in loop
• Misunderstanding modulo operator

4. The following code attempts to solve the coin change problem using a greedy approach. What is the main issue?

int coinChange(int amount) {
    int coins[] = {25, 10, 5, 1};
    int count = 0;
    for (int i = 0; i < 4; i++) {
        while (amount >= coins[i]) {
            amount -= coins[i];
            count++;
        }
    }
    return count;
}

medium

A. Greedy approach fails for some coin sets, not always optimal

B. The loop condition is incorrect and causes infinite loop

C. Coins array is not sorted in ascending order

D. The function does not return the count

Solution

1. Step 1: Understand greedy approach for coin change

   The greedy approach picks the largest coin repeatedly. It works only if coin denominations are canonical (like US coins).

2. Step 2: Identify problem with greedy for arbitrary coins

   For some coin sets, greedy does not give minimum coins. Dynamic programming is needed for guaranteed optimality.

3. Final Answer:

   Greedy approach fails for some coin sets, not always optimal -> Option A

4. Quick Check:

   Greedy coin change not always optimal [OK]

Hint: Greedy coin change fails if coins not canonical [OK]

Common Mistakes:

• Assuming greedy always gives minimum coins
• Ignoring coin set properties
• Confusing loop correctness with algorithm correctness

5. You have a set of tasks each with a deadline and profit. You want to schedule tasks to maximize profit without missing deadlines. Which approach is best and why?

struct Task {
    int deadline;
    int profit;
};

hard

A. Use greedy algorithm by sorting tasks by deadline only

B. Use dynamic programming to try all subsets of tasks and pick max profit

C. Use brute force to check all permutations of tasks

D. Use greedy algorithm by sorting tasks by profit and scheduling latest available slot

Solution

1. Step 1: Understand task scheduling problem

   This is a classic greedy problem where tasks are sorted by profit descending, then scheduled in the latest free slot before deadline.

2. Step 2: Compare approaches

   Dynamic programming or brute force is too slow. Sorting by deadline only does not maximize profit. Greedy by profit with latest slot is efficient and optimal.

3. Final Answer:

   Use greedy algorithm by sorting tasks by profit and scheduling latest available slot -> Option D

4. Quick Check:

   Greedy by profit + latest slot = max profit [OK]

Hint: Sort tasks by profit, schedule latest free slot [OK]

Common Mistakes:

• Sorting only by deadline misses max profit
• Trying DP or brute force unnecessarily
• Scheduling tasks without checking deadlines