Heap and Priority Queue

Table of Contents

• 1. Introduction to Heap and Priority Queue
• 2. What are the applications of Heap and Priority Queue?
• 3. Heap in C++ standard library
• 4. Priority Queue in C++ standard library
  • 4.1. Minimum Priority Queue
• 5. The Applications of Priority Queues
• 6. References

1 Introduction to Heap and Priority Queue

A heap is a tree-based data structure that adheres to the heap property: in a max heap, any parent node's key (value) is greater than or equal to the keys of its children. Conversely, in a min heap, the key of a parent node is less than or equal to the keys of its children.

For more details, refer to Wikipedia: Heap (data structure).

A priority queue is an abstract data type similar to a regular queue or stack, where each element has an associated priority. Elements with higher priority are served before those with lower priority.

For more details, refer to Wikipedia: Priority queue.

Although they are conceptually different, priority queues are often referred to as heaps because they are commonly implemented using heaps.

The binary heap is a data structure that efficiently supports the basic priority queue operations. It allows O(1) performance for accessing the root, O(log n) for insertions and deletions, and O(n) for initially building the heap from a set of n elements.

Note that the time complexity to build a heap is O(n log n) using a top-down approach, whereas a bottom-up approach achieves O(n).

Monotonic Stack

Table of Contents

• 1. Introduction to monotonic stack
• 2. What can monotonically increasing stacks do?
  • 2.1. To find the previous less element of each element in a vector with O(n) time.
  • 2.2. To find the next less element of each element in a vector with O(n) time.
• 3. What can monotonically decreasing stacks do?
• 4. When are monotonic stacks used?
• 5. The applications of monotonic stacks
• 6. References

1 Introduction to monotonic stack

A monotonic stack is a stack whose elements are monotonically increasing or decreasing.

If we pop greater elements from the stack before pushing a new element, the stack will increase from bottom to top. If we pop lesser elements, the stack will decrease.

The typical paradigm for an increasing monotonic stack is like:

for (int i = 0; i < A.size(); i++) {
    while (!stk.empty() && stk.top() > A[i]) {
        stk.pop();
    }
    stk.push(A[i]);
}

2 What can monotonically increasing stacks do?

Monotonically increasing stacks can be used to solve a variety of algorithmic problems. One common use case is for finding the nearest greater element in an array. By maintaining a stack of elements in increasing order, we can quickly determine the next greater element for each element in the array. Monotonically increasing stacks can also be used for finding the maximum area of a histogram, as well as for solving problems related to dynamic programming and graph algorithms.

2.1 To find the previous less element of each element in a vector with O(n) time.

What is the previous less element of an element in a vector? Suppose there is a vector [3, 8, 9, 6], for each element 3, 8, 9, and 6:

• There is no previous less element for 3.
• The previous less element of 8 is 3.
• The previous less element of 9 is 8.
• The previous less element of 6 is 3.