====== Using Priority Queue in Algorithms ======

Priority Queue is a data structure that maintains elements in a partial order, based on their priority. It provides quick access to the highest/lowest priority element and efficient insertion and extraction operations.

===== Applications and Motivations =====

==== 1. Graph Search Algorithms ====

Dijkstra: Always selects the node with minimum distance for exploration, guaranteeing optimal path finding.

A*: Expands nodes in order of estimated total cost (f = g + h), prioritizing promising paths.

Prim: Always chooses the edge with minimum cost to build the minimum spanning tree.

==== 2. Resource Scheduling ====

CPU Scheduling: Organizes processes by priority, ensuring execution of the most important tasks.

Event Simulations: Processes events in chronological order or by importance.

Load Balancing: Distributes tasks based on priority and availability.

==== 3. Data Compression ====

Huffman Algorithm: Combines nodes with lowest frequencies to build an optimal encoding tree, resulting in efficient compression.

==== 4. Selection Problems ====

Top K elements: Efficiently maintains the largest/smallest k elements from a collection without sorting the entire set.

Median of a stream: Keeps two heaps to calculate the median in real-time.

K-Nearest Neighbors: Identifies the closest k neighbors in a multidimensional space.

==== 5. Greedy Algorithms ====

Interval Scheduling: Selects activities that finish earliest.

Huffman Coding: Builds optimal encoding trees.

Minimum Spanning Tree: Constructs minimum cost trees.

==== 6. Data Stream Processing ====

Streaming Analytics: Identifies trends and anomalies in real-time.

Sliding Window: Maintains the most relevant k elements in a sliding window.

Rate Limiting: Controls event frequency in distributed systems.

==== 7. Optimization Problems ====

Branch and Bound: Explores the most promising partial solutions first.

Knapsack Problem: Selects items with maximum value for a limited capacity.

Job Sequencing: Organizes tasks to maximize profit or minimize delays.

==== 8. Advanced Graph Algorithms ====

Kruskal with Union-Find: Builds minimum spanning trees.

Johnson's Algorithm: Calculates shortest paths between all pairs of nodes.

Network Flow: Optimizes flow in a network with capacities.

==== 9. Interval Problems ====

Merge Overlapping Intervals: Efficiently combines overlapping intervals.

Meeting Rooms: Determines the minimum number of rooms needed for scheduling meetings.

Skyline Problem: Calculates the outline formed by buildings.

==== 10. Advanced Data Structures ====

Fibonacci Heaps: Implements decrease-key operations in amortized O(1) time.

Treap: Combines binary search trees with heaps for efficient operations.

Segment Trees: Supports range queries and updates.

===== Key Advantages =====

Quick access to the extreme element (O(1))

Efficient insertion and extraction (O(log n))

Memory efficient compared to maintaining a sorted list

Flexibility through custom comparators

Simple implementation using binary heaps

===== Java Implementation =====

<code java>
// Min-heap (minimum element at top)
PriorityQueue<Integer> minHeap = new PriorityQueue<>();

// Max-heap (maximum element at top)
PriorityQueue<Integer> maxHeap = new PriorityQueue<>(Collections.reverseOrder());

// With custom comparator
PriorityQueue<Task> taskQueue = new PriorityQueue<>((t1, t2) ->
t1.getPriority() - t2.getPriority());

// Basic operations
minHeap.offer(element);  // Add element
int top = minHeap.peek();  // Access top
int removed = minHeap.poll();  // Extract top
</code>

===== Complexity =====

^  Operation  ^  Complexity  ^
|  peek()  |  O(1)  |
|  offer() / add()  |  O(log n)  |
|  poll() / remove()  |  O(log n)  |
|  contains()  |  O(n)  |
|  remove(Object)  |  O(n)  |

===== Conclusions =====

Priority Queue is a fundamental data structure in algorithms, offering an excellent balance between efficiency and simplicity. Its appropriate use can significantly improve the performance of algorithms that require repeated access to extreme elements from a dynamic collection.