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14
The tidy set: A minimal simplicial set for computing homology of clique complexes
 In Proc. ACM Symposium of Computational Geometry
, 2010
"... We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are pop ..."
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Cited by 7 (1 self)
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We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing the clique complex. We give algorithms for constructing tidy sets, implement them, and present experiments. Our preliminary results show that tidy sets are orders of magnitude smaller than clique complexes, giving us a homology engine with small memory requirements.
Scalable Inference in Hierarchical Generative Models
 In Proceedings of the Ninth International Symposium on Artificial Intelligence and Mathematics
, 2006
"... Borrowing insights from computational neuroscience, we present a family of inference algorithms for a class of generative statistical models specifically designed to run on commonlyavailable distributedcomputing hardware. ..."
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Cited by 6 (1 self)
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Borrowing insights from computational neuroscience, we present a family of inference algorithms for a class of generative statistical models specifically designed to run on commonlyavailable distributedcomputing hardware.
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...
Hierarchical Expectation Refinement for Learning Generative Perception Models
, 2005
"... We present a class of generative models well suited to modeling perceptual processes and an algorithm for learning their parameters that promises to scale to learning very large models. The models are hierarchical, composed of multiple levels, and allow input only at the lowest level, the base of ..."
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Cited by 1 (0 self)
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We present a class of generative models well suited to modeling perceptual processes and an algorithm for learning their parameters that promises to scale to learning very large models. The models are hierarchical, composed of multiple levels, and allow input only at the lowest level, the base of the hierarchy. Connections within a level are generally local and may or may not be directed. Connections between levels are directed and generally do not span multiple levels. The learning algorithm falls within the general family of expectation maximization algorithms. Parameter estimation proceeds levelbylevel starting with components in the lowest level and moving up the hierarchy.
Abstract Implementing Partial Persistence in ObjectOriented Languages
"... A partially persistent data structure is a data structure which preserves previous versions of itself when it is modified. General theoretical schemes are known (e.g. the fat node method) for making any data structure partially persistent. To our knowledge however no general implementation of these ..."
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A partially persistent data structure is a data structure which preserves previous versions of itself when it is modified. General theoretical schemes are known (e.g. the fat node method) for making any data structure partially persistent. To our knowledge however no general implementation of these theoretical methods exists to date. This paper evaluates different methods to achieve this goal and presents the first working implementation of partial persistence in the objectoriented language Java. Our approach is transparent, i.e., it allows any existing data structures to become persistent without changing its implementation where all previous solutions require an extensive modification of the code by hand. This transparent property is important in view of the large number of algorithmic results that rely on persistence. Our implementation uses aspectoriented programming, a modularization technique which allows us to instrument the existing code with the needed hooks for the persistence implementation. The implementation is then validated by running benchmarks to analyze both the cost of persistence and of the aspect oriented approach. We also illustrate its applicability by implementing a random binary search tree and making it persistent, and then using the resulting structure to implement a point location data structure in just a few lines. 1
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...
Planbased Axiom Absorption for Tableaubased Description Logics Reasoning
, 2008
"... This is to certify that the thesis prepared ..."
An Indexing System for Mobilityaware Information Management
"... We analyze the problem of minimizing the cost for satisfying information demand in a network of moving nodes, where the communication between nodes is subject to distance constraints and a costly communication link with an information source (central server or Internet). This problem has application ..."
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We analyze the problem of minimizing the cost for satisfying information demand in a network of moving nodes, where the communication between nodes is subject to distance constraints and a costly communication link with an information source (central server or Internet). This problem has applications in several fields ranging from vehicular networks to space exploration, and it is NPhard. We propose a novel indexing system that is able to reduce the search space and show that in practical cases, our system is able to find an optimal solution in a reasonable length of time. An extensive experimental analysis on large real and synthetic datasets shows that the proposed method responds in less than 10 seconds on datasets with millions of events and thousands of information requests, with an improvement of up to 100 times compared to the nonindexed solution. 1.