Priority Queue Introduction and Concept in DSA C++ - Time & Space Complexity

Time Complexity: Priority Queue Introduction and Concept

O(n log n)

Understanding Time Complexity

We want to understand how the time to add or remove items changes as the priority queue grows.

How does the number of steps grow when we insert or remove elements?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

#include <iostream>
#include <queue>

int main() {
    std::priority_queue<int> pq;
    int n = 1000;
    for (int i = 0; i < n; ++i) {
        pq.push(i);  // Insert element
    }
    while (!pq.empty()) {
        pq.pop();    // Remove highest priority element
    }
    return 0;
}
    

This code inserts n elements into a priority queue and then removes all of them one by one.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Inserting (push) and removing (pop) elements in the priority queue.
• How many times: Each operation happens n times, once per element.

How Execution Grows With Input

Each insertion or removal takes more steps as the queue grows, but not in a simple linear way.

Input Size (n)	Approx. Operations
10	About 60 steps (each push/pop about 3 steps)
100	About 1400 steps (each push/pop about 7 steps)
1000	About 20000 steps (each push/pop about 10 steps)

Pattern observation: The steps grow roughly by multiplying n with the logarithm of n, not just n alone.

Final Time Complexity

Time Complexity: O(n log n)

This means the total time grows a bit faster than just the number of elements, because each insert or remove takes extra steps to keep order.

Common Mistake

[X] Wrong: "Each insertion or removal takes constant time, so total is O(n)."

[OK] Correct: The priority queue must reorder elements to keep the highest priority on top, which takes extra steps growing with the size.

Interview Connect

Understanding how priority queues work and their time costs helps you explain choices in real problems where order matters, like task scheduling or event handling.

Self-Check

"What if we used a simple array and sorted it after all insertions? How would the time complexity change?"

Practice

1. What is the main property of a priority queue in data structures?

easy

A. Elements are always served in the order they were inserted

B. Elements are served based on their priority, not just insertion order

C. It stores elements in sorted order permanently

D. It only allows insertion of unique elements

Solution

1. Step 1: Understand priority queue behavior

   A priority queue serves elements based on priority, not just the order they arrive. Options A, C, and D describe other data structures or incorrect properties.

2. Final Answer:

   Elements are served based on their priority, not just insertion order -> Option B

3. Quick Check:

   Priority queue = priority-based serving [OK]

Hint: Remember: priority queue serves highest priority first [OK]

Common Mistakes:

• Confusing priority queue with regular queue
• Thinking elements are always sorted inside
• Assuming only unique elements allowed

2. Which of the following is the correct way to declare a max priority queue of integers in C++ STL?

easy

A. std::priority_queue<int> pq;

B. std::priority_queue pq;

C. std::priority_queue pq;

D. std::priority_queue pq;

Solution

1. Step 1: Recall default priority queue type

   By default, std::priority_queue is a max heap using std::vector and std::less. std::priority_queue<int> pq; uses default syntax. B creates min priority queue. C and D use invalid container type std::list.

2. Final Answer:

   std::priority_queue<int> pq; -> Option A

3. Quick Check:

   Default priority queue is max heap [OK]

Hint: Default priority_queue<int> is max heap in C++ STL [OK]

Common Mistakes:

• Using std::greater<int> for max heap (it's for min heap)
• Using unsupported container types like std::list
• Overcomplicating declaration when default works

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

std::priority_queue<int> pq;
pq.push(10);
pq.push(5);
pq.push(20);
std::cout << pq.top() << std::endl;
pq.pop();
std::cout << pq.top() << std::endl;

medium

A. 20\n10

B. 10\n5

C. 5\n10

D. 20\n5

Solution

1. Step 1: Insert elements into max priority queue

   Elements 10, 5, 20 pushed; max 20 at top, print 20. After pop, 20 removed, next max 10, print 10.

2. Final Answer:

   20\n10 -> Option A

3. Quick Check:

   Max heap top changes from 20 to 10 after pop [OK]

Hint: Top always shows current max in max priority queue [OK]

Common Mistakes:

• Confusing priority queue with FIFO queue
• Assuming pop removes smallest element
• Mixing up order of outputs

4. Identify the error in the following C++ code using priority queue:

std::priority_queue<int> pq;
pq.push(15);
pq.push(30);
pq.pop();
std::cout << pq.top() << std::endl;
pq.pop();
pq.pop();

medium

A. Missing semicolon after pq.push(30)

B. Using std::cout without including <iostream>

C. Calling pop() on empty priority queue causes runtime error

D. Priority queue cannot store integers

Solution

1. Step 1: Trace push and pop operations

   Push 15, 30; first pop removes 30, top now 15 printed; second pop removes 15; third pop on empty queue causes runtime error.

2. Final Answer:

   Calling pop() on empty priority queue causes runtime error -> Option C

3. Quick Check:

   Pop on empty queue = error [OK]

Hint: Never pop from empty priority queue [OK]

Common Mistakes:

• Ignoring empty check before pop
• Assuming pop returns value
• Confusing syntax errors with runtime errors

5. You want to maintain a priority queue that always gives the smallest element first (a min priority queue). Which of the following is the correct way to declare it in C++ STL?

hard

A. std::priority_queue pq;

B. std::priority_queue<int> pq;

C. std::priority_queue pq;

D. std::priority_queue pq;

Solution

1. Step 1: Understand default and min heap declaration

   Default std::priority_queue is max heap (std::less). For min heap (smallest first), use std::greater with std::vector. A and B are max heaps; D uses unsupported std::list.

2. Final Answer:

   std::priority_queue<int, std::vector<int>, std::greater<int>> pq; -> Option D

3. Quick Check:

   Min heap = priority_queue with std::greater [OK]

Hint: Use std::greater for min priority queue in C++ STL [OK]

Common Mistakes:

• Using std::less for min heap
• Using unsupported containers like std::list
• Forgetting to specify comparator