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Solving geometric covering problems by data reduction
 IN PROCEEDINGS OF THE 12TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA ’04), VOLUME 3221 OF LNCS
, 2004
"... We consider a scenario where stops are to be placed along an already existing public transportation network in order to improve its attractiveness for the customers. The core problem is a geometric set covering problem which is N Phard in general. However, if the corresponding covering matrix has ..."
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Cited by 16 (1 self)
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We consider a scenario where stops are to be placed along an already existing public transportation network in order to improve its attractiveness for the customers. The core problem is a geometric set covering problem which is N Phard in general. However, if the corresponding covering matrix has the consecutive ones property, it is solvable in polynomial time. In this paper, we present data reduction techniques for set covering and report on an experimental study considering real world data from railway systems as well as generated instances. The results show that data reduction works very well on instances that are in some sense “close” to having the consecutive ones property. In fact, the real world instances considered could be reduced significantly, in most cases even to triviality. The study also confirms and explains findings on similar techniques for related problems.
An Old SubQuadratic Algorithm for Finding Extremal Sets
, 1994
"... Some previously proposed algorithms are reexamined. They were designed to find all sets in a collection that have no subset in the collection, but are easily modified to find all sets that have no supersets. One is shown to have a worstcase runningtime of O(N 2 = log N ), where N is the sum of ..."
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Cited by 6 (3 self)
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Some previously proposed algorithms are reexamined. They were designed to find all sets in a collection that have no subset in the collection, but are easily modified to find all sets that have no supersets. One is shown to have a worstcase runningtime of O(N 2 = log N ), where N is the sum of the sizes of all the sets. This is lower than the only previously known subquadratic worstcase upper bound for this problem. Key words: Analysis of algorithms, settheoretic algorithms, extremal sets. 1 Introduction Yellin and Jutla [3] tackled the following fundamental problem, for some applications of which see [2]. Given is a collection F = fS 1 ; : : : ; S k g, where each S i is a set over the same domain. A set is a minimal (resp. maximal) set of F iff it has no strict subset (resp. superset) in F . Find the extremal sets of F , i.e., those that are minimal or maximal. With the problem size chosen as N = P i S i , Yellin and Jutla presented an abstract algorithm that requires O(N ...
A Simple SubQuadratic Algorithm for Computing the Subset Partial Order
, 1995
"... A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been presented that compute the partial order (and thereby the minimal and maximal ..."
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Cited by 4 (2 self)
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A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been presented that compute the partial order (and thereby the minimal and maximal sets, i.e., extremal sets) in worstcase time O(N 2 = log N ). This paper develops a simple algorithm that uses only simple data structures, and gives a simple analysis that establishes the above worstcase bound on its running time. The algorithm exploits a variation on lexicographic order that may be of independent interest. 1 Introduction Given is a collection F = fS 1 ; : : : ; S k g, where each S i is a set over the same domain D. Define the size of the collection to be N = P i jS i j. Pritchard [4] presented algorithms for finding those sets in F that have no subset in F . Starting from a naive O(N 2 ) algorithm 1 , an algorithm was obtained that had worstcase complexity O...
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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Cited by 2 (1 self)
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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...
A Fast Bitwise Algorithm for Computing the Subset Partial Order
, 1995
"... A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been discovered that compute the partial order in worstcase time O(N 2 = log N ..."
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Cited by 1 (1 self)
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A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been discovered that compute the partial order in worstcase time O(N 2 = log N ). This paper gives a variant implementation of a previously proposed algorithm which is shown to have a worstcase complexity of O(N 2 (log log N) 2 = log 2 N) operations on a RAM with \Theta(log N) bit words. This is the first known o(N 2 = log N) worstcase running time. 1 Introduction Given is a collection F = fS 1 ; : : : ; S k g, where each S i is a set over the same domain D. Define the size of the collection to be N = P i jS i j. In [5] we presented algorithms for finding those sets in F that have no subset in F , and obtained a fast algorithm for the important special case when all sets in F are small. A particular implementation was later shown [6] to have worstcas...
The Subset Partial Order: . . .
"... Given a family F of k sets with cardinalities s1, s2,..., sk and N = ∑k i=1 si, we show that the size of the partial order graph induced by the subset relation (called the subset graph) is O ( ∑ si≤B 2s ∑ ..."
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Given a family F of k sets with cardinalities s1, s2,..., sk and N = ∑k i=1 si, we show that the size of the partial order graph induced by the subset relation (called the subset graph) is O ( ∑ si≤B 2s ∑
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...
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...
Computing the Subset Partial Order: Progress and Open Problems (Extended Abstract)
"... Paul Pritchard School of Computing and Information Technology Griffith University Queensland, Australia 4111 P.Pritchard@cit.gu.edu.au Abstract A given collection of sets has a natural partial order induced by the subset relation. The problem of efficiently computing this partial order is a fundame ..."
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Paul Pritchard School of Computing and Information Technology Griffith University Queensland, Australia 4111 P.Pritchard@cit.gu.edu.au Abstract A given collection of sets has a natural partial order induced by the subset relation. The problem of efficiently computing this partial order is a fundamental  but, until recently, neglected  one. This paper reviews the published work on the problem, describes recent progress, and highlights the outstanding questions. 1 Introduction Suppose we have information in the form of a number of statements, each of which is a disjunction of atomic propositions. Each statement is naturally represented as the set of (distinct) atomic propositions that it involves. That being so, one statement is stronger than another iff its set is a strict subset of the other's. In [6] we tackled the problem of finding the strongest statements, i.e., of maximally simplifying the information. This problem arose in applying a new approach to constraint satisfactio...