'\"
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.TH "struct::disjointset" n 1\&.2 tcllib "Tcl Data Structures"
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.SH NAME
struct::disjointset \- Disjoint set data structure
.SH SYNOPSIS
package require \fBTcl  8\&.6 9\fR
.sp
package require \fBstruct::disjointset  ?1\&.2?\fR
.sp
\fB::struct::disjointset\fR \fIdisjointsetName\fR
.sp
\fIdisjointsetName\fR \fIoption\fR ?\fIarg arg \&.\&.\&.\fR?
.sp
\fIdisjointsetName\fR \fBadd-element\fR \fIitem\fR
.sp
\fIdisjointsetName\fR \fBadd-partition\fR \fIelements\fR
.sp
\fIdisjointsetName\fR \fBpartitions\fR
.sp
\fIdisjointsetName\fR \fBnum-partitions\fR
.sp
\fIdisjointsetName\fR \fBequal\fR \fIa\fR \fIb\fR
.sp
\fIdisjointsetName\fR \fBmerge\fR \fIa\fR \fIb\fR
.sp
\fIdisjointsetName\fR \fBfind\fR \fIe\fR
.sp
\fIdisjointsetName\fR \fBexemplars\fR
.sp
\fIdisjointsetName\fR \fBfind-exemplar\fR \fIe\fR
.sp
\fIdisjointsetName\fR \fBdestroy\fR
.sp
.BE
.SH DESCRIPTION
.PP
This package provides \fIdisjoint sets\fR\&. An alternative name for
this kind of structure is \fImerge-find\fR\&.
.PP
Normally when dealing with sets and their elements the question is "Is
this element E contained in this set S?", with both E and S known\&.
.PP
Here the question is "Which of several sets contains the element
E?"\&. I\&.e\&. while the element is known, the set is not, and we wish to
find it quickly\&. It is not quite the inverse of the original question,
but close\&.
Another operation which is often wanted is that of quickly merging two
sets into one, with the result still fast for finding elements\&. Hence
the alternative term \fImerge-find\fR for this\&.
.PP
Why now is this named a \fIdisjoint-set\fR ?
Because another way of describing the whole situation is that we have
.IP \(bu
a finite \fIset\fR S, containing
.IP \(bu
a number of \fIelements\fR E, split into
.IP \(bu
a set of \fIpartitions\fR P\&. The latter term
applies, because the intersection of each pair P, P' of
partitions is empty, with the union of all partitions
covering the whole set\&.
.IP \(bu
An alternative name for the \fIpartitions\fR would be
\fIequvalence classes\fR, and all elements in the same
class are considered as equal\&.
.PP
Here is a pictorial representation of the concepts listed above:
.CS


	+-----------------+ The outer lines are the boundaries of the set S\&.
	|           /     | The inner regions delineated by the skewed lines
	|  *       /   *  | are the partitions P\&. The *'s denote the elements
	|      *  / \\     | E in the set, each in a single partition, their
	|*       /   \\    | equivalence class\&.
	|       /  *  \\   |
	|      / *   /    |
	| *   /\\  * /     |
	|    /  \\  /      |
	|   /    \\/  *    |
	|  / *    \\       |
	| /     *  \\      |
	+-----------------+

.CE
.PP
For more information see \fIhttp://en\&.wikipedia\&.org/wiki/Disjoint_set_data_structure\fR\&.
.SH API
The package exports a single command, \fB::struct::disjointset\fR\&. All
functionality provided here can be reached through a subcommand of
this command\&.
.PP
.TP
\fB::struct::disjointset\fR \fIdisjointsetName\fR
Creates a new disjoint set object with an associated global Tcl
command whose name is \fIdisjointsetName\fR\&. This command may be used
to invoke various operations on the disjointset\&. It has the following
general form:
.RS
.TP
\fIdisjointsetName\fR \fIoption\fR ?\fIarg arg \&.\&.\&.\fR?
The \fBoption\fR and the \fIarg\fRs determine the exact behavior of
the command\&. The following commands are possible for disjointset
objects:
.RE
.TP
\fIdisjointsetName\fR \fBadd-element\fR \fIitem\fR
Creates a new partition in the specified disjoint set, and fills it
with the single item \fIitem\fR\&.  The command maintains
the integrity of the disjoint set, i\&.e\&. it verifies that none of the
\fIelements\fR are already part of the disjoint set and throws an
error otherwise\&.
.sp
The result of this method is the empty string\&.
.sp
This method runs in constant time\&.
.TP
\fIdisjointsetName\fR \fBadd-partition\fR \fIelements\fR
Creates a new partition in specified disjoint set, and fills it with
the values found in the set of \fIelements\fR\&. The command maintains
the integrity of the disjoint set, i\&.e\&. it verifies that none of the
\fIelements\fR are already part of the disjoint set and throws an
error otherwise\&.
.sp
The result of the command is the empty string\&.
.sp
This method runs in time proportional to the size of \fIelements\fR]\&.
.TP
\fIdisjointsetName\fR \fBpartitions\fR
Returns the set of partitions the named disjoint set currently
consists of\&. The form of the result is a list of lists; the inner
lists contain the elements of the partitions\&.
.sp
This method runs in time O(N*alpha(N)),
where N is the number of elements in the disjoint set and alpha
is the inverse Ackermann function\&.
.TP
\fIdisjointsetName\fR \fBnum-partitions\fR
Returns the number of partitions the named disjoint set currently
consists of\&.
.sp
This method runs in constant time\&.
.TP
\fIdisjointsetName\fR \fBequal\fR \fIa\fR \fIb\fR
Determines if the two elements \fIa\fR and \fIb\fR of the disjoint set
belong to the same partition\&. The result of the method is a boolean
value, \fBTrue\fR if the two elements are contained in the same
partition, and \fBFalse\fR otherwise\&.
.sp
An error will be thrown if either \fIa\fR or \fIb\fR are not elements
of the disjoint set\&.
.sp
This method runs in amortized time O(alpha(N)), where N is the number of
elements in the larger partition and alpha is the inverse Ackermann function\&.
.TP
\fIdisjointsetName\fR \fBmerge\fR \fIa\fR \fIb\fR
Determines the partitions the elements \fIa\fR and \fIb\fR are
contained in and merges them into a single partition\&.  If the two
elements were already contained in the same partition nothing will
change\&.
.sp
The result of the method is the empty string\&.
.sp
This method runs in amortized time O(alpha(N)), where N is the number of
items in the larger of the partitions being merged\&. The worst case time
is O(N)\&.
.TP
\fIdisjointsetName\fR \fBfind\fR \fIe\fR
Returns a list of the members of the partition of the disjoint set
which contains the element
\fIe\fR\&.
.sp
This method runs in O(N*alpha(N)) time, where N is the total number of
items in the disjoint set and alpha is the inverse Ackermann function,
See \fBfind-exemplar\fR for a faster method, if all that is needed
is a unique identifier for the partition, rather than an enumeration
of all its elements\&.
.TP
\fIdisjointsetName\fR \fBexemplars\fR
Returns a list containing an exemplar of each partition in the disjoint
set\&. The exemplar is a member of the partition, chosen arbitrarily\&.
.sp
This method runs in O(N*alpha(N)) time, where N is the total number of items
in the disjoint set and alpha is the inverse Ackermann function\&.
.TP
\fIdisjointsetName\fR \fBfind-exemplar\fR \fIe\fR
Returns the exemplar of the partition of the disjoint set containing
the element \fIe\fR\&.  Throws an error if \fIe\fR is not found in the
disjoint set\&.  The exemplar is an arbitrarily chosen member of the partition\&.
The only operation that will change the exemplar of any partition is
\fBmerge\fR\&.
.sp
This method runs in O(alpha(N)) time, where N is the number of items in
the partition containing E, and alpha is the inverse Ackermann function\&.
.TP
\fIdisjointsetName\fR \fBdestroy\fR
Destroys the disjoint set object and all associated memory\&.
.PP
.SH "BUGS, IDEAS, FEEDBACK"
This document, and the package it describes, will undoubtedly contain
bugs and other problems\&.
Please report such in the category \fIstruct :: disjointset\fR of the
\fITcllib Trackers\fR [http://core\&.tcl\&.tk/tcllib/reportlist]\&.
Please also report any ideas for enhancements you may have for either
package and/or documentation\&.
.PP
When proposing code changes, please provide \fIunified diffs\fR,
i\&.e the output of \fBdiff -u\fR\&.
.PP
Note further that \fIattachments\fR are strongly preferred over
inlined patches\&. Attachments can be made by going to the \fBEdit\fR
form of the ticket immediately after its creation, and then using the
left-most button in the secondary navigation bar\&.
.SH KEYWORDS
disjoint set, equivalence class, find, merge find, partition, partitioned set, union
.SH CATEGORY
Data structures
