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# library(heaps)

Implement heaps in Prolog   [more]

## Predicates

inserts the new Key-Datum pair into the heap
empty_heap(?Heap)
Heap is the empty heap
get_from_heap(+OldHeap, ?Key, ?Datum, -NewHeap)
returns the Key-Datum pair in OldHeap with the smallest Key
heap_size(+Heap, ?Size)
reports the number of elements currently in the heap
heap_to_list(+Heap, -List)
returns the current set of Key-Datum pairs in the Heap as a List.
list_to_heap(+List, -Heap)
takes a list of Key-Datum pairs and forms them into a heap
min_of_heap(+Heap, ?Key, ?Datum)
returns the Key-Datum pair at the top of the heap
min_of_heap(+Heap, ?Key1, ?Datum1, ?Key2, ?Datum2)
returns the smallest and second smallest pairs in the heap
singleton_heap(?Heap, ?Key, ?Datum)
Heap is a heap with single entry Key-Datum

## Description

A heap is a labelled binary tree where the key of each node is less than or equal to the keys of its sons. The point of a heap is that we can keep on adding new elements to the heap and we can keep on taking out the minimum element. If there are N elements total, the total time is O(NlgN). If you know all the elements in advance, you are better off doing a merge-sort, but this file is for when you want to do say a best-first search, and have no idea when you start how many elements there will be, let alone what they are.

A heap is represented as a triple t(N, Free, Tree) where N is the number of elements in the tree, Free is a list of integers which specifies unused positions in the tree, and Tree is a tree made of

```	t			terms for empty subtrees and
t(Key,Datum,Lson,Rson)	terms for the rest
```
The nodes of the tree are notionally numbered like this:
```				    1
2				    3
4               6               5               7
8      12      10     14       9       13      11     15
..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..
```
The idea is that if the maximum number of elements that have been in the heap so far is M, and the tree currently has K elements, the tree is some subtreee of the tree of this form having exactly M elements, and the Free list is a list of K-M integers saying which of the positions in the M-element tree are currently unoccupied. This free list is needed to ensure that the cost of passing N elements through the heap is O(NlgM) instead of O(NlgN). For M say 100 and N say 10^4 this means a factor of two. The cost of the free list is slight. The storage cost of a heap in a copying Prolog (which Dec-10 Prolog is not) is 2K+3M words.