Disjoint set / Union Find Data Structure

Muhammad Tariq Khan
Log(n) Academy

 Topic of Discrete Mathematics
Muhammad Ahmed

 Suppose we have a database of people.
 We want to figure out who is related to whom.
 Initially, we only have a list of people, and information about their relations. For
example “Muhammad is related to Ahmed”.
 Further info: Muhammad is brother of Ahmed. This means that they belongs to
same family. Mathematically they belong to same SET.
Muhammad Ahmed

 If Muhammad is related to Ahmed, and Ahmed is related to Saam, then
Muhammad is related to Saam.
 Once we have relationships information, we would like to answer queries like “Is
Muhammad related to Saam?”
Muhammad SaamAhmed

 A relation R over a set S is called an Equivalence Relation if it has following
properties
1. Reflexivity: for all element x is in set S, x R x
2. Symmetry: for all elements x and y, x R y if and only if y R x
3. Transitivity: for all elements x, y and z, if x R y and y R z then x R z
 The relation “is related to” is an equivalence relation over the set of people.
Muhammad SaamAhmed

 The ≤ relationship is not an equivalence relation.
 It is Reflexive, since x ≤ x,
 and Transitive, since x ≤ y and y ≤ z implies x ≤ z,
 it is Not Symmetric since x ≤ y does not imply y ≤ x.

 Electrical connectivity, where all connections are by metal wires, is an
equivalence relation.
 It is clearly Reflexive, since any component is connected to itself.
 It is Symmetric because if component a is connected to component b then b must
be electrically connected to a.
 It is Transitive, since if component a is connected to component b and b is
connected to c then a is connected to c.

Jack Jamy Omar Osman Ali
Jack
Jamy
Omar
Osman
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Jack Jamy
Omar
Osman
Ali

 What if we have 1000 elements/objects/people
 Will this be efficient in-terms of time, space and memory to create a matrix of
1000 x 1000

 As an example, suppose the equivalence relation is defined over the five-element
set {a1, a2, a3, a4, a5 }.
 There are 25 pairs of elements, each of which is related or not. (30 pairs – 5 self-
pairs=25).

 However, given the information
• a1 R a2,
• a5 R a1,
• a4 R a2,
• a3 R a4,
Implies that all pairs are related.
 We would like to be able to infer this information
quickly.

 Union find operation keeps track of elements which are split into one or more
disjoint set.
 It has 2 primary operation: find and union

 To apply Union Find, first construct a mapping between your objects and integers
in the range (0, n).
 NOTE: This step is not necessary in general, but it will allow us to construct an
array-based union find which is very efficient and easy to work with.
E FAB D
0 1 2 3 4 5 6 7
H C G

E B A F D H C G
2 0 2 2 6 7 2 0
0 1 2 3 4 5 6 7
 Instructions
 Find(B) = A
 Find(D) = A
 Find(H) = A
 Find(E) = A
Indexes
E FAB D
0 1 2 3 4 5 6 7
H C G

 FIND: To find which component a particular element belongs to, first find the root
of that component by following the parent nodes until a self loop is reached ( a
node who’s parent is its self)
 UNION: To unify 2 elements find which are the root nodes of each component and
if the root nodes are different make one of the root node to be the parent of the
other.

 The input is initially a collection of n sets, each with one element.
 This initial representation is that all relations (except reflexive relations) are
false.
 Each set has a different element so that Si ∩ Sj = null ; this makes the sets
Disjoint.
 Find returns the name of the SET that contains a given element, i.e., Si = find(a)
 Union merges two sets to create a new set Sk = Si U Sj.

 If we want to add the relation a R b, then we first see if a and b are already
related.
 This is done by performing Finds on both a and b to check whether they are in the
same set.
 If they are not, we apply Union which merges the two sets a and b belong to.
 The algorithm to do this is frequently known as Union/Find for this reason.
 Example : Union ( D,G )

 Kruskal’s minimum spanning tree
 Grid percolation
 Network connectivity
 Least Common Ancestor in tree
 Image processing
 Maze Generation

 Typical tree traversal not required, so no need for pointers to children, instead we
need a pointer to parent – an up-tree.
 Union is clearly a constant time operation.
 Running time of find( i ) is proportional to the height of the tree containing node i.
 This can be proportional to n in the worst case (but not always)
 Improvement: Modify Union to ensure that heights stay small.

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Disjoint set / Union Find Data Structure

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