## table of contents

struct::disjointset(3tcl) | Tcl Data Structures | struct::disjointset(3tcl) |

# 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*arg*s 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*
[http://core.tcl.tk/tcllib/reportlist]. 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

1.1 | tcllib |