## A.20 library(heaps): heaps/priority queues

author
  Lars Buitinck

Heaps are data structures that return the entries inserted into them in an ordered fashion, based on a priority. This makes them the data structure of choice for implementing priority queues, a central element of algorithms such as best-first or A* search and Kruskal's minimum-spanning-tree algorithm.

This module implements min-heaps, meaning that key-value items are retrieved in ascending order of key. In other words, the key determines the priority. It was designed to be compatible with the SICStus Prolog library module of the same name. merge_heaps/3 and singleton_heap/3 are SWI-specific extension. The portray_heap/1 predicate is not implemented.

Although the values can be arbitrary Prolog terms, the keys determine the priority, so keys must be ordered by @=</2. This means that variables can be used as keys, but binding them in between heap operations may change the ordering. It also means that rational terms (cyclic trees), for which standard order is not well-defined, cannot be used as keys.

The current version implements pairing heaps. These support insertion and merging both in constant time, deletion of the minimum in logarithmic amortized time (though delete-min, i.e., get_from_heap/3, takes linear time in the worst case).

[semidet]add_to_heap(+Heap0, +Key, ?Value, -Heap)
  Adds Value with priority Key to Heap0, constructing a new heap in Heap.
[semidet]delete_from_heap(+Heap0, -Key, +Value, -Heap)
  Deletes Value from Heap0, leaving its priority in Key and the resulting data structure in Heap. Fails if Value is not found in Heap0.
  bug
    This predicate is extremely inefficient and exists only for SICStus compatibility.
[semidet]empty_heap(?Heap)
  True if Heap is an empty heap. Complexity: constant.
[semidet]singleton_heap(?Heap, ?Key, ?Value)
  True if Heap is a heap with the single element Key-Value.

  Complexity: constant.

[semidet]get_from_heap(?Heap0, ?Key, ?Value, -Heap)
  Retrieves the minimum-priority pair Key-Value from Heap0. Heap is Heap0 with that pair removed. Complexity: logarithmic (amortized), linear in the worst case.
[det]heap_size(+Heap, -Size:int)
  Determines the number of elements in Heap. Complexity: constant.
[det]heap_to_list(+Heap, -List:list)
  Constructs a list List of Key-Value terms, ordered by (ascending) priority. Complexity: O(N log N).
[semidet]is_heap(+X)
  Returns true if X is a heap. Validates the consistency of the entire heap. Complexity: linear.
[det]list_to_heap(+List:list, -Heap)
  If List is a list of Key-Value terms, constructs a heap out of List. Complexity: linear.
[semidet]min_of_heap(+Heap, ?Key, ?Value)
  Unifies Value with the minimum-priority element of Heap and Key with its priority value. Complexity: constant.
[semidet]min_of_heap(+Heap, ?Key1, ?Value1, ?Key2, ?Value2)
  Gets the two minimum-priority elements from Heap. Complexity: logarithmic (amortized).
  bug
    This predicate is extremely inefficient and exists for compatibility with earlier implementations of this library and SICStus compatibility. It performs a linear amount of work in the worst case that a following get_from_heap has to re-do.
[det]merge_heaps(+Heap0, +Heap1, -Heap)
  Merge the two heaps Heap0 and Heap1 in Heap. Complexity: constant.