TY - JOUR

T1 - Union-find with constant time deletions

AU - Alstrup, Stephen

AU - Gørtz, Inge Li

AU - Rauhe, Theis

AU - Thorup, Mikkel

AU - Zwick, Uri

PY - 2005

Y1 - 2005

N2 - A union-find data structure maintains a collection of disjoint sets under makeset, union and find operations. Kaplan, Shafrir and Tarjan [SODA 2002] designed data structures for an extension of the unionfind problem in which elements of the sets maintained may be deleted. The cost of a delete operation in their implementations is the same as the cost of a find operation. They left open the question whether delete operations can be implemented more efficiently than find operations. We resolve this open problem by presenting a relatively simple modification of the classical union-find data structure that supports delete, as well as makeset and union, operations in constant time, while still supporting find operations in O(log n) worst-case time and O(α(n)) amortized time, where n is the number of elements in the set returned by the find operation, and a(n) is a functional inverse of Ackermann's function.

AB - A union-find data structure maintains a collection of disjoint sets under makeset, union and find operations. Kaplan, Shafrir and Tarjan [SODA 2002] designed data structures for an extension of the unionfind problem in which elements of the sets maintained may be deleted. The cost of a delete operation in their implementations is the same as the cost of a find operation. They left open the question whether delete operations can be implemented more efficiently than find operations. We resolve this open problem by presenting a relatively simple modification of the classical union-find data structure that supports delete, as well as makeset and union, operations in constant time, while still supporting find operations in O(log n) worst-case time and O(α(n)) amortized time, where n is the number of elements in the set returned by the find operation, and a(n) is a functional inverse of Ackermann's function.

UR - http://www.scopus.com/inward/record.url?scp=26444459800&partnerID=8YFLogxK

U2 - 10.1007/11523468_7

DO - 10.1007/11523468_7

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AN - SCOPUS:26444459800

SN - 0302-9743

VL - 3580

SP - 78

EP - 89

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

T2 - 32nd International Colloquium on Automata, Languages and Programming, ICALP 2005

Y2 - 11 July 2005 through 15 July 2005

ER -