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An Efficient Representation for Sparse Sets
 ACM Letters on Programming Languages and Systems
, 1993
"... this paper, we have described a representation suitable for sets with a fixedsize universe. The representation supports constanttime implementations of clearset, member, addmember, deletemember, cardinality, and chooseone. Based on the efficiency of these operations, the new representation wi ..."
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Cited by 30 (4 self)
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this paper, we have described a representation suitable for sets with a fixedsize universe. The representation supports constanttime implementations of clearset, member, addmember, deletemember, cardinality, and chooseone. Based on the efficiency of these operations, the new representation will often be superior to alternatives such as bit vectors, balanced binary trees, hash tables, linked lists, etc. Additionally, the new representation supports enumeration of the members in O(n) time, making it a competitive choice for relatively sparse sets requiring operations like forall, setcopy, setunion, and setdifference.
HimML: Standard ML with Fast Sets and Maps
 In 5th ACM SIGPLAN Workshop on ML and its Applications
, 1994
"... We propose to add sets and maps to Standard ML. Our implementation uses hashtries to code them, yields fast generalpurpose settheoretic operations, and is based on a runtime where all equal objects are shared. We present evidence that this systematic use of hashconsing, and the use of hashtrie ..."
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Cited by 7 (2 self)
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We propose to add sets and maps to Standard ML. Our implementation uses hashtries to code them, yields fast generalpurpose settheoretic operations, and is based on a runtime where all equal objects are shared. We present evidence that this systematic use of hashconsing, and the use of hashtries to code sets, provide good performance. 1 Introduction Sets have been an adequate foundation for mathematics for nearly a century, and are also an important conceptual medium in computer science. Modern specification languages like VDM [18] and Z [30] are based on sets. But few programming languages provide generalpurpose sets and maps: although they could be adequate for prototyping, it is feared that they would be too slow for real applications. We have designed and implemented an extension of Standard ML [17], called HimML 1 [12] providing fast general (polymorphic) settheoretic datastructures, and a comprehensive set of efficient operations on them. After mentioning related work...
Efficient Data Structures for Maintaining Set Partitions (Extended Abstract)
 Proceedings of Seventh Scandinavian Workshop on Algorithm Theory
, 1999
"... ) Michael Bender Saurabh Sethia Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 117944400 fbendersaurabhskienag@cs.sunysb.edu April 22, 1999 1 Introduction Each test or feature in a classification system defines a set partition on a class of object ..."
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Cited by 3 (1 self)
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) Michael Bender Saurabh Sethia Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 117944400 fbendersaurabhskienag@cs.sunysb.edu April 22, 1999 1 Introduction Each test or feature in a classification system defines a set partition on a class of objects. Adding new features refines the classification, whereas deleting features may result in merging previously distinguished classes. As an illustration, consider the set of automobile types f VW Beetle, Toyota, Lexus, Cadillac g. The feature size partitions the cars into sets of small and large cars, ff VW Beetle, Toyotag, f Lexus, Cadillac gg. The feature domesticorigin partitions the cars into ff VW Beetle, Toyota, Lexus g, f Cadillac gg. The feature uglyshape distinguishes f VW Beetle, Cadillac g from f Toyota, Lexus g. Incorporating both size and origin induces the refined partition ff VW Beetle, Toyotag, f Lexus g, f Cadillac gg, whereas the union of all three features completely di...
Fully Dynamic Algorithms for Maintaining Extremal Sets in A Family of Sets
, 1995
"... The extremal sets of a family F of sets consist of all minimal and maximal sets of F that have no subset and superset in F respectively. We consider the problem of efficiently maintaining all extremal sets in F when it undergoes dynamic updates including set insertion, deletion and setcontents upda ..."
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The extremal sets of a family F of sets consist of all minimal and maximal sets of F that have no subset and superset in F respectively. We consider the problem of efficiently maintaining all extremal sets in F when it undergoes dynamic updates including set insertion, deletion and setcontents update (insertion, deletion and value update of elements). Given F containing k sets with N elements and domain (the union of these sets) size n, where clearly k; n N for any F , we present a set of algorithms that, requiring a space of O(N + kn log N + k 2 ) words, process in O(1) time a query on whether a set of F is minimal and/or maximal, and maintain all extremal sets of F in O(N ) time per set insertion in the worst case, deletion and setcontents update. Both time bounds are tight. Our algorithms are the first known fully dynamic algorithms that answer an extremal set query in constant time and maintain extremal sets in linear time for any set insertion and deletion. Keywords: Dy...
Amortized Complexity of Data Structures
, 1991
"... This thesis investigates the amortized complexity of some fundamental data structure problems and introduces interesting ideas for proving lower bounds on amortized complexity and for performing amortized analysis. The problems are as follows: ffl Dictionary Problem: A dictionary is a dynamic set t ..."
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This thesis investigates the amortized complexity of some fundamental data structure problems and introduces interesting ideas for proving lower bounds on amortized complexity and for performing amortized analysis. The problems are as follows: ffl Dictionary Problem: A dictionary is a dynamic set that supports searches of elements and changes under insertions and deletions of elements. It is open whether there exists a dictionary data structure that takes constant amortized time per operation and uses space polynomial in the dictionary size. We prove that dictionary operations require loglogarithmic amortized time under a multilevel hashing model that is based on Yao's cell probe model. ffl Splay Algorithm's Analysis: Splay is a simple, efficient algorithm for searching binary search trees, devised by Sleator and Tarjan, that uses rotations to reorganize the tree. Tarjan conjectured that Splay takes linear time to process deque operation sequences on a binary tree and proved a speci...
Parallel Algorithms for Fully Dynamic Maintenance of Extremal Sets in
"... Let F be a family of sets containing N elements. The extremal sets of F are those that have no subset or superset in F and are hence minimal or maximal respectively. We consider the problem of maintaining all extremal sets in F in parallel when F undergoes dynamic updates including set insertion, d ..."
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Let F be a family of sets containing N elements. The extremal sets of F are those that have no subset or superset in F and are hence minimal or maximal respectively. We consider the problem of maintaining all extremal sets in F in parallel when F undergoes dynamic updates including set insertion, deletion and setcontents update (insertion, deletion and value update of elements). We present a set of parallel algorithms that, using O( N log N ) processors on a CREW PRAM, maintain all extremal sets of F in O(logN ) time per set insertion, deletion and setcontents update in the worst case. We also show that a batch of q queries on whether a set of F is minimal and/or maximal can be answered in O(1) time using q CREW processors. With a cost matching the time complexity of the optimal sequential algorithm [7], our algorithms are the first known NC algorithms that use a sublinear number of processors for fully dynamic maintenance of extremal sets of F . Keywords: CREW PRAM, dynamic a...
Lazy Structure Sharing for Query Optimization
, 1993
"... We study lazy structure sharing as a tool for optimizing equivalence testing on complex data types. We investigate a number of strategies for a restricted case of the problem and provide upper and lower bounds on their performance (how quickly they effect ideal configurations of our data structure). ..."
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We study lazy structure sharing as a tool for optimizing equivalence testing on complex data types. We investigate a number of strategies for a restricted case of the problem and provide upper and lower bounds on their performance (how quickly they effect ideal configurations of our data structure). In most cases, the bounds provide nontrivial improvements over the naive lineartime equivalencetesting strategy that employs no optimization. 1 Supported by a Fannie and John Hertz Foundation fellowship, National Science Foundation Grant No. CCR8920505, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) under NSFSTC8809648. 2 Dept. of Computer Science and Automation, Indian Institute of Science, Bangalore 560012, India. Work completed while at Princeton University and DIMACS and supported by DIMACS under NSFSTC8809648. 3 Also affiliated with NEC Research Institute, 4 Independence Way, Princeton, NJ 08540. Research at Princeton University partially sup...
Finding Extremal Sets of A Normal Family of Sets in O(N²/(log²N)) Time and O(N²/(log³N)) Space
, 1995
"... Yellin and Jutla [7] proposed an algorithm for the problem of finding the extremal sets in a family of sets containing N elements that can be implemented in O( N 2 log N ) time and O( N 2 log N ) space due to Pritchard [3] who also showed that an earlier algorithm can be adapted to solve the p ..."
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Yellin and Jutla [7] proposed an algorithm for the problem of finding the extremal sets in a family of sets containing N elements that can be implemented in O( N 2 log N ) time and O( N 2 log N ) space due to Pritchard [3] who also showed that an earlier algorithm can be adapted to solve the problem in O( N 2 log N ) time and O( N 2 log 2 N ) space. We show that this problem can be solved in O( N 2 log 2 N ) time and O( N 2 log 3 N ) space in the worst case when F is normal, thus present the first algorithm that reaches the lower bound both in time and space complexity for this case. Keywords: Complexity analysis, extremal set, partial order, set inclusion. 1 Introduction In a given family of sets F = fS 1 ; S 2 ; : : : ; S k g, where elements of S i are drawn from some finite domain, a set S i is said minimal (resp. maximal) if S j 6ae S i (resp. S i 6ae S j ) for all 1 j k [5]. The extremal sets of F consist of all the minimal and maximal sets of F . The proble...