swi-prolog-nox 7.6.4+dfsg-1build1 / usr / lib / swi-prolog / library / heaps.pl

/*  Part of SWI-Prolog

    Author:        Lars Buitinck
    E-mail:        larsmans@gmail.com
    WWW:           http://www.swi-prolog.org
    Copyright (c)  2006-2015, Lars Buitinck
    All rights reserved.

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:- module(heaps,
          [ add_to_heap/4,              % +Heap0, +Priority, ?Key, -Heap
            delete_from_heap/4,         % +Heap0, -Priority, +Key, -Heap
            empty_heap/1,               % +Heap
            get_from_heap/4,            % ?Heap0, ?Priority, ?Key, -Heap
            heap_size/2,                % +Heap, -Size:int
            heap_to_list/2,             % +Heap, -List:list
            is_heap/1,                  % +Term
            list_to_heap/2,             % +List:list, -Heap
            merge_heaps/3,              % +Heap0, +Heap1, -Heap
            min_of_heap/3,              % +Heap, ?Priority, ?Key
            min_of_heap/5,              % +Heap, ?Priority1, ?Key1,
                                        %        ?Priority2, ?Key2
            singleton_heap/3            % ?Heap, ?Priority, ?Key
          ]).

/** <module> heaps/priority queues
 *
 * 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/A* search and Kruskal's minimum-spanning-tree algorithm.
 *
 * This module implements min-heaps, meaning that items are retrieved in
 * ascending order of key/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 data items can be arbitrary Prolog data, keys/priorities must
 * be ordered by @=</2. Be careful when using variables as keys, since binding
 * them in between heap operations may change the ordering.
 *
 * 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).
 *
 * @author Lars Buitinck
 */

/*
 * Heaps are represented as heap(H,Size) terms, where H is a pairing heap and
 * Size is an integer. A pairing heap is either nil or a term
 * t(X,PrioX,Sub) where Sub is a list of pairing heaps t(Y,PrioY,Sub) s.t.
 * PrioX @< PrioY. See predicate is_heap/2, below.
 */

%!  add_to_heap(+Heap0, +Priority, ?Key, -Heap) is semidet.
%
%   Adds Key with priority Priority  to   Heap0,  constructing a new
%   heap in Heap.

add_to_heap(heap(Q0,M),P,X,heap(Q1,N)) :-
    meld(Q0,t(X,P,[]),Q1),
    N is M+1.

%!  delete_from_heap(+Heap0, -Priority, +Key, -Heap) is semidet.
%
%   Deletes Key from Heap0, leaving its priority in Priority and the
%   resulting data structure in Heap.   Fails if Key is not found in
%   Heap0.
%
%   @bug This predicate is extremely inefficient and exists only for
%        SICStus compatibility.

delete_from_heap(Q0,P,X,Q) :-
    get_from_heap(Q0,P,X,Q),
    !.
delete_from_heap(Q0,Px,X,Q) :-
    get_from_heap(Q0,Py,Y,Q1),
    delete_from_heap(Q1,Px,X,Q2),
    add_to_heap(Q2,Py,Y,Q).

%!  empty_heap(?Heap) is semidet.
%
%   True if Heap is an empty heap. Complexity: constant.

empty_heap(heap(nil,0)).

%!  singleton_heap(?Heap, ?Priority, ?Key) is semidet.
%
%   True if Heap is a heap with the single element Priority-Key.
%
%   Complexity: constant.

singleton_heap(heap(t(X,P,[]), 1), P, X).

%!  get_from_heap(?Heap0, ?Priority, ?Key, -Heap) is semidet.
%
%   Retrieves the minimum-priority  pair   Priority-Key  from Heap0.
%   Heap is Heap0 with that pair removed.   Complexity:  logarithmic
%   (amortized), linear in the worst case.

get_from_heap(heap(t(X,P,Sub),M), P, X, heap(Q,N)) :-
    pairing(Sub,Q),
    N is M-1.

%!  heap_size(+Heap, -Size:int) is det.
%
%   Determines the number of elements in Heap. Complexity: constant.

heap_size(heap(_,N),N).

%!  heap_to_list(+Heap, -List:list) is det.
%
%   Constructs a list List  of   Priority-Element  terms, ordered by
%   (ascending) priority. Complexity: $O(n \log n)$.

heap_to_list(Q,L) :-
    to_list(Q,L).
to_list(heap(nil,0),[]) :- !.
to_list(Q0,[P-X|Xs]) :-
    get_from_heap(Q0,P,X,Q),
    heap_to_list(Q,Xs).

%!  is_heap(+X) is semidet.
%
%   Returns true if X is a heap.  Validates the consistency of the
%   entire heap. Complexity: linear.

is_heap(V) :-
    var(V), !, fail.
is_heap(heap(Q,N)) :-
    integer(N),
    nonvar(Q),
    (   Q == nil
    ->  N == 0
    ;   N > 0,
        Q = t(_,MinP,Sub),
        are_pairing_heaps(Sub, MinP)
    ).

% True iff 1st arg is a pairing heap with min key @=< 2nd arg,
% where min key of nil is logically @> any term.
is_pairing_heap(V, _) :-
    var(V),
    !,
    fail.
is_pairing_heap(nil, _).
is_pairing_heap(t(_,P,Sub), MinP) :-
    MinP @=< P,
    are_pairing_heaps(Sub, P).

% True iff 1st arg is a list of pairing heaps, each with min key @=< 2nd arg.
are_pairing_heaps(V, _) :-
    var(V),
    !,
    fail.
are_pairing_heaps([], _).
are_pairing_heaps([Q|Qs], MinP) :-
    is_pairing_heap(Q, MinP),
    are_pairing_heaps(Qs, MinP).

%!  list_to_heap(+List:list, -Heap) is det.
%
%   If List is a list of  Priority-Element  terms, constructs a heap
%   out of List. Complexity: linear.

list_to_heap(Xs,Q) :-
    empty_heap(Empty),
    list_to_heap(Xs,Empty,Q).

list_to_heap([],Q,Q).
list_to_heap([P-X|Xs],Q0,Q) :-
    add_to_heap(Q0,P,X,Q1),
    list_to_heap(Xs,Q1,Q).

%!  min_of_heap(+Heap, ?Priority, ?Key) is semidet.
%
%   Unifies Key with  the  minimum-priority   element  of  Heap  and
%   Priority with its priority value. Complexity: constant.

min_of_heap(heap(t(X,P,_),_), P, X).

%!  min_of_heap(+Heap, ?Priority1, ?Key1, ?Priority2, ?Key2) is semidet.
%
%   Gets the two minimum-priority elements from Heap. Complexity: logarithmic
%   (amortized).
%
%   Do not use this predicate; it exists for compatibility with earlier
%   implementations of this library and the SICStus counterpart. It performs
%   a linear amount of work in the worst case that a following get_from_heap
%   has to re-do.

min_of_heap(Q,Px,X,Py,Y) :-
    get_from_heap(Q,Px,X,Q0),
    min_of_heap(Q0,Py,Y).

%!  merge_heaps(+Heap0, +Heap1, -Heap) is det.
%
%   Merge the two heaps Heap0 and Heap1 in Heap. Complexity: constant.

merge_heaps(heap(L,K),heap(R,M),heap(Q,N)) :-
    meld(L,R,Q),
    N is K+M.


% Merge two pairing heaps according to the pairing heap definition.
meld(nil,Q,Q) :- !.
meld(Q,nil,Q) :- !.
meld(L,R,Q) :-
    L = t(X,Px,SubL),
    R = t(Y,Py,SubR),
    (   Px @< Py
    ->  Q = t(X,Px,[R|SubL])
    ;   Q = t(Y,Py,[L|SubR])
    ).

% "Pair up" (recursively meld) a list of pairing heaps.
pairing([], nil).
pairing([Q], Q) :- !.
pairing([Q0,Q1|Qs], Q) :-
    meld(Q0, Q1, Q2),
    pairing(Qs, Q3),
    meld(Q2, Q3, Q).