struct::disjointset - Tcl Data Structures

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# struct::disjointset(n) 1.1 tcllib "Tcl Data Structures"

## Name

struct::disjointset - Disjoint set data structure

## Synopsis

- package require Tcl 8.6
- package require struct::disjointset ?1.1?

- ::struct::disjointset disjointsetName
- disjointsetName option ?arg arg ...?
- disjointsetName add-element item
- disjointsetName add-partition elements
- disjointsetName partitions
- disjointsetName num-partitions
- disjointsetName equal a b
- disjointsetName merge a b
- disjointsetName find e
- disjointsetName exemplars
- disjointsetName find-exemplar e
- disjointsetName destroy

## Description

This package provides disjoint sets. An alternative name for this kind of structure is merge-find.

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.

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 merge-find for this.

Why now is this named a disjoint-set ? Because another way of describing the whole situation is that we have

- a finite set S, containing

- a number of elements E, split into

- a set of partitions 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.

- An alternative name for the partitions would be equvalence classes, and all elements in the same class are considered as equal.

Here is a pictorial representation of the concepts listed above:

    	+-----------------+ 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.
    	|       /  *  \   |
    	|      / *   /    |
    	| *   /\  * /     |
    	|    /  \  /      |
    	|   /    \/  *    |
    	|  / *    \       |
    	| /     *  \      |
    	+-----------------+

For more information see http://en.wikipedia.org/wiki/Disjoint_set_data_structure.

## API

The package exports a single command, ::struct::disjointset. All functionality provided here can be reached through a subcommand of this command.

::struct::disjointset disjointsetName

  Creates a new disjoint set object with an associated global Tcl command whose name is disjointsetName. This command may be used to invoke various operations on the disjointset. It has the following general form:

  disjointsetName option ?arg arg ...?

    The option and the args determine the exact behavior of the command. The following commands are possible for disjointset objects:

disjointsetName add-element item

  Creates a new partition in the specified disjoint set, and fills it with the single item item. The command maintains the integrity of the disjoint set, i.e. it verifies that none of the elements are already part of the disjoint set and throws an error otherwise.

  The result of this method is the empty string.

  This method runs in constant time.

disjointsetName add-partition elements

  Creates a new partition in specified disjoint set, and fills it with the values found in the set of elements. The command maintains the integrity of the disjoint set, i.e. it verifies that none of the elements are already part of the disjoint set and throws an error otherwise.

  The result of the command is the empty string.

  This method runs in time proportional to the size of elements].

disjointsetName partitions

  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.

  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.

disjointsetName num-partitions

  Returns the number of partitions the named disjoint set currently consists of.

  This method runs in constant time.

disjointsetName equal a b

  Determines if the two elements a and b of the disjoint set belong to the same partition. The result of the method is a boolean value, True if the two elements are contained in the same partition, and False otherwise.

  An error will be thrown if either a or b are not elements of the disjoint set.

  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.

disjointsetName merge a b

  Determines the partitions the elements a and b are contained in and merges them into a single partition. If the two elements were already contained in the same partition nothing will change.

  The result of the method is the empty string.

  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).

disjointsetName find e

  Returns a list of the members of the partition of the disjoint set which contains the element e.

  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 find-exemplar for a faster method, if all that is needed is a unique identifier for the partition, rather than an enumeration of all its elements.

disjointsetName exemplars

  Returns a list containing an exemplar of each partition in the disjoint set. The exemplar is a member of the partition, chosen arbitrarily.

  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.

disjointsetName find-exemplar e

  Returns the exemplar of the partition of the disjoint set containing the element e. Throws an error if e 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 merge.

  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.

disjointsetName destroy

  Destroys the disjoint set object and all associated memory.

## Bugs, Ideas, Feedback

This document, and the package it describes, will undoubtedly contain bugs and other problems. Please report such in the category struct :: disjointset of the Tcllib Trackers. Please also report any ideas for enhancements you may have for either package and/or documentation.

When proposing code changes, please provide unified diffs, i.e the output of diff -u.

Note further that attachments are strongly preferred over inlined patches. Attachments can be made by going to the Edit form of the ticket immediately after its creation, and then using the left-most button in the secondary navigation bar.

## Keywords

disjoint set, equivalence class, find, merge find, partition, partitioned set, union

## Category

Data structures