1. The UNION-FIND Abstract Data Type
Variation on the mathematical set
Supports managing a dynamic equivalence relation
Can be implemented with lists, trees, etc.
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

2. CSE 373, Copyright S. Tanimoto, 2002 UNION-FIND ADT -
A Set of Elements
Let S be a set of elements.
We can assume that there is a way to create the set and insert an element into it.
{ a, b, c, d, e}
For example...
s = newSet();
s.insert(a); s.insert(b); ...
s.member(a)  true
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

3. CSE 373, Copyright S. Tanimoto, 2002 UNION-FIND ADT -
Equivalence Relation
Let R be a binary relation on S.
Then R is an equivalence relation on S iff
R is reflexive on S.
R is symmetric
R is transitive
Example. Suppose S is the set of all people living at midnight on New Year’s Eve 2000.
(x, y) is in R iff x and y have the same first name.
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

4. Another Equivalence Relation
Let S = { a, b, c, d, e}
Let R = { (a,a), (b,b), (c,c), (d,d), (e,e) (a,b), (b,a),
(a,c), (c,a), (b,c), (c,b), (d,e), (e,d) }
S can be partitioned into two subsets, such that within each subset, all the elements are related in both directions.
P = { {a, b, c}, {d, e}}
The two subsets are called equivalence classes.
They are "induced" by R.
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

5. CSE 373, Copyright S. Tanimoto, 2002 UNION-FIND ADT -
The FIND Operation
Let’s assume that we have an equivalence relation R, and a set of induced equivalence classes {E1, E2 , E3 , ... , Ek}
Each Ei will have a representative element.
FIND(x) for x in S, returns the representative element for the equivalence class containing x.
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

6. CSE 373, Copyright S. Tanimoto, 2002 UNION-FIND ADT -
FIND -- an Example
Let S = { a, b, c, d, e}
P = { {a, b, c}, {d, e}}
E1 = {a, b, c}, E2 = {d, e}
Let a and d be the representative elements of E1 and E2, respectively.
FIND(c)  a
FIND(d)  d
FIND(e)  d
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

7. CSE 373, Copyright S. Tanimoto, 2002 UNION-FIND ADT -
The UNION Operation
Let’s assume that we have an equivalence relation R, and a set of induced equivalence classes {E1, E2 , E3 , ... , Ek}
UNION( Ei, Ej ) changes the partition so that Ei and Ej have been merged into one subset.
For example, after UNION(E2, E3) we have {E1, E2 U E3 , ... , Ek }
Note: Each Ei can be represented by its represent. element. So it’s OK to write UNION(a, d)
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

8. Application of UNION-FIND
In an electric circuit, finding the buses and other common signal points.
The set S is given as a list of nodes.
The relation R is (partially) given as a wire list.
We want to find a minimal partition of S such that within each subset of nodes, they are all electrically connected by wire.
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

9. Electric Circuit (Cont)
Start with an initial partition into singleton sets.
S = { a, b, c, d, e }
Wire list:
WIRE(a, b)
WIRE(a, c)
WIRE(d, e).
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

10. Electric Circuit Procedure
Start with an initial partition into singleton sets.
P = { {a}, {b}, {c}, {d}, {e} }
Process one wire at a time. WIRE(a, b)
Perform FIND operations on each element, then UNION on the results: UNION(FIND(a),FIND(b))
Continue until all wires have been processed.
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

11. An Array Implementation1 2 3 4
index a b c d e
element 1 2 3 4
representative elt. index
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

12. CSE 373, Copyright S. Tanimoto, 2002 UNION-FIND ADT -
UNION(a,b) 1 2 3 4
index a b c d e
element 2 3 4
representative elt. index
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

13. CSE 373, Copyright S. Tanimoto, 2002 UNION-FIND ADT -
UNION(a,c) 1 2 3 4
index a b c d e
element 3 4
representative elt. index
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

14. CSE 373, Copyright S. Tanimoto, 2002 UNION-FIND ADT -
UNION(d,e) 1 2 3 4
index a b c d e
element 3 3
representative elt. index
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -

15. CSE 373, Copyright S. Tanimoto, 2002 UNION-FIND ADT -
Time Complexity
In this method FIND takes O(1) and UNION takes O(n)
Later, we’ll see that using Up-trees, we can make the time complexity almost O(n) for a full sequence of n FIND and n UNION operations.
CSE 373, Copyright S. Tanimoto, UNION-FIND ADT -