323 lines
10 KiB
Prolog
323 lines
10 KiB
Prolog
% This file has been included as an YAP library by Vitor Santos Costa, 1999
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% File : HEAPS.PL
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% Author : R.A.O'Keefe
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% Updated: 29 November 1983
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% Purpose: Implement heaps in Prolog.
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/** @defgroup Heaps Heaps
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@ingroup YAPLibrary
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@{
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A heap is a labelled binary tree where the key of each node is less than
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or equal to the keys of its sons. The point of a heap is that we can
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keep on adding new elements to the heap and we can keep on taking out
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the minimum element. If there are N elements total, the total time is
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O(NlgN). If you know all the elements in advance, you are better off
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doing a merge-sort, but this file is for when you want to do say a
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best-first search, and have no idea when you start how many elements
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there will be, let alone what they are.
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The following heap manipulation routines are available once included
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with the `use_module(library(heaps))` command.
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*/
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/* A heap is a labelled binary tree where the key of each node is less
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than or equal to the keys of its sons. The point of a heap is that
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we can keep on adding new elements to the heap and we can keep on
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taking out the minimum element. If there are N elements total, the
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total time is O(NlgN). If you know all the elements in advance, you
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are better off doing a merge-sort, but this file is for when you
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want to do say a best-first search, and have no idea when you start
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how many elements there will be, let alone what they are.
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A heap is represented as a triple t(N, Free, Tree) where N is the
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number of elements in the tree, Free is a list of integers which
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specifies unused positions in the tree, and Tree is a tree made of
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t terms for empty subtrees and
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t(Key,Datum,Lson,Rson) terms for the rest
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The nodes of the tree are notionally numbered like this:
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1
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2 3
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4 6 5 7
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8 12 10 14 9 13 11 15
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.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
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The idea is that if the maximum number of elements that have been in
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the heap so far is M, and the tree currently has K elements, the tree
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is some subtreee of the tree of this form having exactly M elements,
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and the Free list is a list of K-M integers saying which of the
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positions in the M-element tree are currently unoccupied. This free
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list is needed to ensure that the cost of passing N elements through
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the heap is O(NlgM) instead of O(NlgN). For M say 100 and N say 10^4
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this means a factor of two. The cost of the free list is slight.
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The storage cost of a heap in a copying Prolog (which Dec-10 Prolog is
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not) is 2K+3M words.
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*/
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/**
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@pred add_to_heap(+ _Heap_,+ _key_,+ _Datum_,- _NewHeap_)
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Inserts the new _Key-Datum_ pair into the heap. The insertion is not
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stable, that is, if you insert several pairs with the same _Key_ it
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is not defined which of them will come out first, and it is possible for
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any of them to come out first depending on the history of the heap.
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*/
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/** @pred empty_heap(? _Heap_)
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Succeeds if _Heap_ is an empty heap.
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*/
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/** @pred get_from_heap(+ _Heap_,- _key_,- _Datum_,- _Heap_)
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Returns the _Key-Datum_ pair in _OldHeap_ with the smallest
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_Key_, and also a _Heap_ which is the _OldHeap_ with that
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pair deleted.
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*/
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/** @pred heap_size(+ _Heap_, - _Size_)
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Reports the number of elements currently in the heap.
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*/
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/** @pred heap_to_list(+ _Heap_, - _List_)
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Returns the current set of _Key-Datum_ pairs in the _Heap_ as a
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_List_, sorted into ascending order of _Keys_.
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*/
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/** @pred list_to_heap(+ _List_, - _Heap_)
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Takes a list of _Key-Datum_ pairs (such as keysort could be used to sort)
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and forms them into a heap.
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*/
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/** @pred min_of_heap(+ _Heap_, - _Key1_, - _Datum1_,
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- _Key2_, - _Datum2_)
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Returns the smallest (Key1) and second smallest (Key2) pairs in the
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heap, without deleting them.
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*/
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/** @pred min_of_heap(+ _Heap_, - _Key_, - _Datum_)
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Returns the Key-Datum pair at the top of the heap (which is of course
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the pair with the smallest Key), but does not remove it from the heap.
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*/
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:- module(heaps,[
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add_to_heap/4, % Heap x Key x Datum -> Heap
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get_from_heap/4, % Heap -> Key x Datum x Heap
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empty_heap/1, % Heap
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heap_size/2, % Heap -> Size
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heap_to_list/2, % Heap -> List
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list_to_heap/2, % List -> Heap
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min_of_heap/3, % Heap -> Key x Datum
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min_of_heap/5 % Heap -> (Key x Datum) x (Key x Datum)
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]).
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/*
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:- mode
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add_to_heap(+, +, +, -),
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add_to_heap(+, +, +, +, -),
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add_to_heap(+, +, +, +, +, +, -, -),
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sort2(+, +, +, +, -, -, -, -),
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get_from_heap(+, ?, ?, -),
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repair_heap(+, +, +, -),
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heap_size(+, ?),
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heap_to_list(+, -),
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heap_tree_to_list(+, -),
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heap_tree_to_list(+, +, -),
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list_to_heap(+, -),
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list_to_heap(+, +, +, -),
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min_of_heap(+, ?, ?),
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min_of_heap(+, ?, ?, ?, ?),
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min_of_heap(+, +, ?, ?).
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*/
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% add_to_heap(OldHeap, Key, Datum, NewHeap)
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% inserts the new Key-Datum pair into the heap. The insertion is
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% not stable, that is, if you insert several pairs with the same
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% Key it is not defined which of them will come out first, and it
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% is possible for any of them to come out first depending on the
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% history of the heap. If you need a stable heap, you could add
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% a counter to the heap and include the counter at the time of
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% insertion in the key. If the free list is empty, the tree will
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% be grown, otherwise one of the empty slots will be re-used. (I
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% use imperative programming language, but the heap code is as
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% pure as the trees code, you can create any number of variants
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% starting from the same heap, and they will share what common
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% structure they can without interfering with each other.)
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add_to_heap(t(M,[],OldTree), Key, Datum, t(N,[],NewTree)) :- !,
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N is M+1,
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add_to_heap(N, Key, Datum, OldTree, NewTree).
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add_to_heap(t(M,[H|T],OldTree), Key, Datum, t(N,T,NewTree)) :-
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N is M+1,
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add_to_heap(H, Key, Datum, OldTree, NewTree).
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add_to_heap(1, Key, Datum, _, t(Key,Datum,t,t)) :- !.
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add_to_heap(N, Key, Datum, t(K1,D1,L1,R1), t(K2,D2,L2,R2)) :-
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E is N mod 2,
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M is N//2,
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% M > 0, % only called from list_to_heap/4,add_to_heap/4
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sort2(Key, Datum, K1, D1, K2, D2, K3, D3),
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add_to_heap(E, M, K3, D3, L1, R1, L2, R2).
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add_to_heap(0, N, Key, Datum, L1, R, L2, R) :- !,
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add_to_heap(N, Key, Datum, L1, L2).
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add_to_heap(1, N, Key, Datum, L, R1, L, R2) :- !,
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add_to_heap(N, Key, Datum, R1, R2).
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sort2(Key1, Datum1, Key2, Datum2, Key1, Datum1, Key2, Datum2) :-
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Key1 @< Key2,
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!.
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sort2(Key1, Datum1, Key2, Datum2, Key2, Datum2, Key1, Datum1).
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% get_from_heap(OldHeap, Key, Datum, NewHeap)
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% returns the Key-Datum pair in OldHeap with the smallest Key, and
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% also a New Heap which is the Old Heap with that pair deleted.
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% The easy part is picking off the smallest element. The hard part
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% is repairing the heap structure. repair_heap/4 takes a pair of
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% heaps and returns a new heap built from their elements, and the
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% position number of the gap in the new tree. Note that repair_heap
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% is *not* tail-recursive.
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get_from_heap(t(N,Free,t(Key,Datum,L,R)), Key, Datum, t(M,[Hole|Free],Tree)) :-
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M is N-1,
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repair_heap(L, R, Tree, Hole).
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repair_heap(t(K1,D1,L1,R1), t(K2,D2,L2,R2), t(K2,D2,t(K1,D1,L1,R1),R3), N) :-
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K2 @< K1,
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!,
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repair_heap(L2, R2, R3, M),
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N is 2*M+1.
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repair_heap(t(K1,D1,L1,R1), t(K2,D2,L2,R2), t(K1,D1,L3,t(K2,D2,L2,R2)), N) :- !,
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repair_heap(L1, R1, L3, M),
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N is 2*M.
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repair_heap(t(K1,D1,L1,R1), t, t(K1,D1,L3,t), N) :- !,
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repair_heap(L1, R1, L3, M),
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N is 2*M.
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repair_heap(t, t(K2,D2,L2,R2), t(K2,D2,t,R3), N) :- !,
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repair_heap(L2, R2, R3, M),
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N is 2*M+1.
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repair_heap(t, t, t, 1) :- !.
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% heap_size(Heap, Size)
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% reports the number of elements currently in the heap.
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heap_size(t(Size,_,_), Size).
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% heap_to_list(Heap, List)
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% returns the current set of Key-Datum pairs in the Heap as a
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% List, sorted into ascending order of Keys. This is included
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% simply because I think every data structure foo ought to have
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% a foo_to_list and list_to_foo relation (where, of course, it
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% makes sense!) so that conversion between arbitrary data
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% structures is as easy as possible. This predicate is basically
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% just a merge sort, where we can exploit the fact that the tops
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% of the subtrees are smaller than their descendants.
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heap_to_list(t(_,_,Tree), List) :-
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heap_tree_to_list(Tree, List).
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heap_tree_to_list(t, []) :- !.
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heap_tree_to_list(t(Key,Datum,Lson,Rson), [Key-Datum|Merged]) :-
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heap_tree_to_list(Lson, Llist),
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heap_tree_to_list(Rson, Rlist),
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heap_tree_to_list(Llist, Rlist, Merged).
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heap_tree_to_list([H1|T1], [H2|T2], [H2|T3]) :-
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H2 @< H1,
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!,
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heap_tree_to_list([H1|T1], T2, T3).
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heap_tree_to_list([H1|T1], T2, [H1|T3]) :- !,
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heap_tree_to_list(T1, T2, T3).
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heap_tree_to_list([], T, T) :- !.
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heap_tree_to_list(T, [], T).
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% list_to_heap(List, Heap)
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% takes a list of Key-Datum pairs (such as keysort could be used to
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% sort) and forms them into a heap. We could do that a wee bit
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% faster by keysorting the list and building the tree directly, but
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% this algorithm makes it obvious that the result is a heap, and
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% could be adapted for use when the ordering predicate is not @<
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% and hence keysort is inapplicable.
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list_to_heap(List, Heap) :-
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list_to_heap(List, 0, t, Heap).
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list_to_heap([], N, Tree, t(N,[],Tree)) :- !.
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list_to_heap([Key-Datum|Rest], M, OldTree, Heap) :-
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N is M+1,
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add_to_heap(N, Key, Datum, OldTree, MidTree),
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list_to_heap(Rest, N, MidTree, Heap).
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% min_of_heap(Heap, Key, Datum)
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% returns the Key-Datum pair at the top of the heap (which is of
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% course the pair with the smallest Key), but does not remove it
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% from the heap. It fails if the heap is empty.
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% min_of_heap(Heap, Key1, Datum1, Key2, Datum2)
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% returns the smallest (Key1) and second smallest (Key2) pairs in
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% the heap, without deleting them. It fails if the heap does not
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% have at least two elements.
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min_of_heap(t(_,_,t(Key,Datum,_,_)), Key, Datum).
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min_of_heap(t(_,_,t(Key1,Datum1,Lson,Rson)), Key1, Datum1, Key2, Datum2) :-
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min_of_heap(Lson, Rson, Key2, Datum2).
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min_of_heap(t(Ka,_Da,_,_), t(Kb,Db,_,_), Kb, Db) :-
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Kb @< Ka, !.
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min_of_heap(t(Ka,Da,_,_), _, Ka, Da).
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min_of_heap(t, t(Kb,Db,_,_), Kb, Db).
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empty_heap(t(0,[],t)).
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