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+# PairingHeap
+
+`PairingHeap` is a purely functional heap with excellent empirical
+performance. It supports O(1) `add` and `combine` (merge) operations,
+making it particularly useful when heaps need to be frequently merged.
+
+See the [Wikipedia article](https://en.wikipedia.org/wiki/Pairing_heap)
+for more details on the data structure.
+
+## Supported Operations
+
+- `add`:              insert an element: O(1)
+- `remove`:           remove the minimum element: O(log N) amortized
+- `combine`:          merge two heaps: O(1)
+- `minimumOption`:    get the minimum element without removing: O(1)
+- `pop`:              remove and return the minimum element: O(log N) amortized
+- `toList`:           sorted list of all elements: O(N log N)
+- `size`:             number of elements: O(1)
+- `isEmpty`:          check if the heap is empty: O(1)
+- `exists`:           check if any element satisfies a predicate: O(N)
+- `forall`:           check if all elements satisfy a predicate: O(N)
+- `takeLargest`:      efficiently find the k largest elements: O(N log k)
+
+## Comparison with Heap
+
+Unlike the binary `Heap`, `PairingHeap` supports an efficient O(1)
+`combine` (merge) operation, making it the better choice when merging
+heaps is needed. The binary `Heap` can be constructed in O(N) time
+via `heapify`, while `PairingHeap` builds in O(N) time via
+`fromIterable`.
+
+Both `Heap` and `PairingHeap` implement the `PartiallyOrderedSet`
+typeclass.
+
+## Example Usage
+
+Start by creating an empty `PairingHeap`:
+
+```scala mdoc
+import cats._, cats.implicits._, cats.collections._
+
+val empty = PairingHeap.empty[Int]
+
+empty.isEmpty
+
+empty.size
+```
+
+Add some elements:
+
+```scala mdoc
+val heap = PairingHeap.empty[Int] + 5 + 3 + 8 + 1
+
+heap.size
+
+heap.minimumOption
+
+heap.show
+```
+
+Remove the minimum element:
+
+```scala mdoc
+val heap2 = heap.remove
+
+heap2.minimumOption
+
+heap2.show
+```
+
+Build a heap from an existing collection:
+
+```scala mdoc
+val fromList = PairingHeap.fromIterable(List(10, 4, 7, 2, 9))
+
+fromList.minimumOption
+
+fromList.toList
+```
+
+Merge two heaps efficiently with `combine`:
+
+```scala mdoc
+val left = PairingHeap.fromIterable(List(1, 3, 5))
+val right = PairingHeap.fromIterable(List(2, 4, 6))
+
+val merged = left.combine(right)
+
+merged.toList
+```
+
+Find the 3 largest values from a large collection:
+
+```scala mdoc
+val items = List(3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5)
+
+val top3 = PairingHeap.takeLargest(items, 3)
+
+top3.toList
+```