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Finding and Maintaining Rigid Components

Audrey Lee, Ileana Streinu, Louis Theran
2005 Canadian Conference on Computational Geometry  
To this end, we introduce a new data structure problem called union pairfind, which maintains disjoint edge sets and supports pair-find queries of whether two vertices are spanned by a set.  ...  We give the first complete analysis that the complexity of finding and maintaining rigid components of planar bar-and-joint frameworks and arbitrary d-dimensional body-and-bar frameworks, using a family  ...  Conclusion We have presented a new data structure problem called union pair-find.  ... 
dblp:conf/cccg/LeeST05 fatcat:525rnfy7czekphrmumxvz5ggie

Data structures and algorithms for disjoint set union problems

Zvi Galil, Giuseppe F. Italiano
1991 ACM Computing Surveys  
Special attention is devoted to recent extensions of the original set union problem, and an attempt is made to provide a unifying theoretical framework for this growing body of algorithms.  ...  This paper surveys algorithmic techniques and data structures that have been proposed tosolve thesetunion problem and its variants, Thediscovery of these data structures required anew set ofalgorithmic  ...  Blum [1986] proposed a data structure for the set union problem that supports each union and find in O(log n/log log n) time in the worst case.  ... 
doi:10.1145/116873.116878 fatcat:vlfyjncu6rhezdk7qvxrlivayu

Worst-case analysis of the set-union problem with extended backtracking

Giorgio Gambosi, Giuseppe F. Italiano, Maurizio Talamo
1989 Theoretical Computer Science  
A data structure is presented which maintains a partition of zn n-item set making it possible to perform each Union in O(1g lg n) time, each Find in O(lg n) time and allows backtracking over the Unions  ...  Moreover, it is shown that the data structure can be slightly modified as to present an O(k i-in lg n) time complexity on a sequence of k Unions and Backtracks and m Finds.  ...  In Section 4 a data structure for the union-@d-backtrack problem is introduced and its worst case time and space complexities are derived.  ... 
doi:10.1016/0304-3975(89)90119-9 fatcat:ezn6iw2rkrfhnkpzjbeidfzkfu

Union Find [article]

Yu Geng
2021 figshare.com  
In computer science, a disjoint-set data structure, also called a unionfind data structure or merge–find set, is a data structure that stores a collection of disjoint (non-overlapping) sets.  ...  It provides operations for adding new sets, merging sets (replacing them by their union), and finding a representative member of a set.  ...  Data structure. • Integer array id[] of size N. 11 network connectivity quick find quick union improvements applications Quick-union [lazy approach]Data structure. a truism (roughly) since 1950  ... 
doi:10.6084/m9.figshare.13669103.v1 fatcat:3asgmadjsvggrldlfne6crukjm

A partially persistent data structure for the set-union problem

C. Gaibisso, G. Gambosi, M. Talamo
1990 RAIRO - Theoretical Informatics and Applications  
The remainder of this paper is organized as follows: in section 2 we introducé a data structure for the Union-PFind problem, in section 3 we analyse its worst-case time and space complexity.  ...  In this paper we consider an extension of the classical Set-Union problem, introducing a new kind of Find, referred as PFind, defined as follows: PFind(x, k)\ given an item xeS, return the name of the  ... 
doi:10.1051/ita/1990240201891 fatcat:lgjbtvvojvfsveczirvxu5evbe

Union-Find with Constant Time Deletions [chapter]

Stephen Alstrup, Inge Li Gørtz, Theis Rauhe, Mikkel Thorup, Uri Zwick
2005 Lecture Notes in Computer Science  
A union-find data structure maintains a collection of disjoint sets under makeset, union and 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  ...  ) time, as it forms the basis of our new data structure for the union-find-delete problem.  ... 
doi:10.1007/11523468_7 fatcat:xkel5ldv6bcxlkh7mlxancpony

Union-Find with Constant Time Deletions

Stephen Alstrup, Mikkel Thorup, Inge Li Gørtz, Theis Rauhe, Uri Zwick
2014 ACM Transactions on Algorithms  
A union-find data structure maintains a collection of disjoint sets under makeset, union and 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  ...  ) time, as it forms the basis of our new data structure for the union-find-delete problem.  ... 
doi:10.1145/2636922 fatcat:2wvjhpdm75cjda775ikz2twryu

A persistent union-find data structure

Sylvain Conchon, Jean-Christophe Filliâtre
2007 Proceedings of the 2007 workshop on Workshop on ML - ML '07  
The problem of disjoint sets, also known as union-find, consists in maintaining a partition of a finite set within a data structure.  ...  This paper details the implementation of a persistent union-find data structure as efficient as its imperative counterpart.  ...  We also thank the members of the Proval project for many discussions related to the persistent union-find problem. In particular, we are grateful to Claude Marché for mentioning T.-R.  ... 
doi:10.1145/1292535.1292541 dblp:conf/ml/ConchonF07 fatcat:f2dqeya3t5ag5e4mnay5oxzzva

A lower bound for the complexity of the Union-Split-Find problem [chapter]

K. Mehlhorn, S. Näher, H. Alt
1987 Lecture Notes in Computer Science  
We prove a 8{loglogn) (i.e. matching upper and lower) bound on the complexity of the Union-Split-Find problem, a variant of the Union-Find problem .  ...  We complement this with a 9(logn) bound for the Split-Find problem under the separation assumption. This shows that the separation assumption can imply an exponential loss in efficiency.  ...  This is true for t he worst-case complexity and for the amortized complexity Concl usions and Open Problems In this paper we proved several new lower hounds for the Union-Split-Find problem .  ... 
doi:10.1007/3-540-18088-5_41 fatcat:i5zsjvppkzcepnnvvjtnpru4ou

Semi-dynamic Connectivity in the Plane [chapter]

Sergio Cabello, Michael Kerber
2015 Lecture Notes in Computer Science  
Motivated by a path planning problem we consider the following procedure. Assume that we have two points s and t in the plane and take K = ∅.  ...  We show how to add one set to K in O(1 + kα(n)) amortized time plus the time needed to find all sets of K intersecting the newly added set, where n is the cardinality of K, k is the number of sets in K  ...  Acknowledgments We thank Chee Yap for posing to us the problem about connectivity under subdivisions.  ... 
doi:10.1007/978-3-319-21840-3_10 fatcat:i73ungmkwzhhzat6pdgn7t22de

Page 133 of Administrative Science Quarterly Vol. 4, Issue 1 [page]

1959 Administrative Science Quarterly  
Disregarding acknowledged weaknesses in the data, we find here a rather stimulating and promising discussion of the nature and consequences of different organizational control structures Among other problems  ...  While the reader may not necessarily find in the authors’ new tech-  ... 

Semi-dynamic connectivity in the plane [article]

Sergio Cabello, Michael Kerber
2015 arXiv   pre-print
Motivated by a path planning problem we consider the following procedure. Assume that we have two points s and t in the plane and take K=∅.  ...  We show how to add one set to K in O(1+kα(n)) amortized time plus the time needed to find all sets of K intersecting the newly added set, where n is the cardinality of K, k is the number of sets in K intersecting  ...  Acknowledgments We thank Chee Yap for posing to us the problem about connectivity under subdivisions.  ... 
arXiv:1502.03690v1 fatcat:6vm7a36oevd2pde3blrhydysu4

Don't rush into a union

Mihai Pătraşcu, Mikkel Thorup
2011 Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11  
For t U = Ω(t q ), the problem is equivalent to the well-understood union-find problem: INSERTEDGE(s, t) can be implemented by UNION(FIND(s), FIND(t)).  ...  We present a new threshold phenomenon in data structure lower bounds where slightly reduced update times lead to exploding query times.  ...  Concerning finds in the union-find data structure, we get one for each original find on a non-free node.  ... 
doi:10.1145/1993636.1993711 dblp:conf/stoc/PatrascuT11a fatcat:fd6g22oi6rd3tnmp6h3w22z6wm

Don't Rush into a Union: Take Time to Find Your Roots [article]

Mihai Patrascu, Mikkel Thorup
2011 arXiv   pre-print
For t_u = Omega(t_q), the problem is equivalent to the well-understood union-find problem: InsertEdge(s,t) can be implemented by Union(Find(s), Find(t)).  ...  We present a new threshold phenomenon in data structure lower bounds where slightly reduced update times lead to exploding query times.  ...  Concerning finds in the union-find data structure, we get one for each original find on a non-free node.  ... 
arXiv:1102.1783v2 fatcat:n34xpbua45a3po2sx6tnskkjo4

Efficient union-find for planar graphs and other sparse graph classes

Jens Gustedt
1998 Theoretical Computer Science  
We solve the Union-Find Problem (UF) efficiently for the case the input is restricted to several graph classes, namely partial k-trees for any fixed k, d-dimensional grids for any fixed dimension d and  ...  By efficiency we do not only mean linear time in a theoretical setting but also a practical reorganization of memory such that a dynamic data structures for UF is allocated consecutively.  ...  The real world bottleneck for Union-Find is the use of a dynamic data structure.  ... 
doi:10.1016/s0304-3975(97)00291-0 fatcat:zc2ae36f7zgqpjfh7a5ic3j5wm
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