3023035
Computing the first few Betti numbers of semi-algebraic sets in single exponential time
Abstract In this paper we describe an algorithm that takes as input a description of a semi-algebraic set S ⊂ R k , defined by a Boolean formula with atoms of the form P > 0 , P 0 , P = 0 for P ∈ P ⊂ R [ X 1 , … , X k ] , and outputs the first l + 1 Betti numbers of S , b 0 ( S ) , … , b l ( S ) . The complexity of the algorithm is ( s d ) k O ( l ) , where s = # ( P ) and d = max P ∈ P deg ( P ) , which is singly exponential in k for l any fixed constant. Previously, singly exponential time algorithms were known only for computing the Euler–Poincare characteristic, the zeroth and the first Betti numbers.
18808745
An improved algorithm for quantifier elimination over real closed fields
We give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms the combinatorial part of the complexity of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depend only on a constant number of the free variables, the algebraic part of the complexity can also be made independent of the number of free variables. This new feature of our algorithm allows us to obtain a new algorithm for a variant of the quantifier elimination problem. We give an almost optimal algorithm for this new problem, which we call the uniform quantifier elimination problem and apply it to solve a problem arising in the field of constraint databases. No algorithm with reasonable complexity bound was known for this latter problem till now. We also point out interesting logical consequences of this algorithmic result, concerning the expressive power of a constraint query language over the reals. Moreover, our improved algorithm for performing quantifier elimination immediately leads to improved algorithms for several problems for which quantifier elimination is a basic step, for example, the problem of computing the closure of a given semi-algebraic set.
8618090
Computing the first Betti number and the connected components of semi-algebraic sets
In this paper we describe the first singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. We also describe algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one.
33624885
On the combinatorial and topological complexity of a single cell
The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of a geometric objects in R/sup k/, each object of constant description complexity, is an important problem in computational geometry which has attracted much attention over the past decade. It has been conjectured that the combinatorial complexity of a single cell is bounded by a function much closer to O(n/sup k-1/) rather than O(n/sup k/) which is the bound for the combinatorial complexity of the whole arrangement. Till now, this was known to be rule only for k/spl les/3 and only for some special cases in higher dimensions. A classic result in real algebraic geometry due to Oleinik-Petrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. However, till now no better bounds were known if we restricted attention to a single connected component of a basic semi-algebraic set. In this paper, we show how these two problems are related. We prove a new bound on the sum of the Betti numbers of one connected component of a basic semi-algebraic set which is an improvement over the Oleinik-Petrovsky-Thom-Milnor bound. This also implies that the topological complexity of a single cell, measured by the sum of the Betti numbers, is bounded by O(n/sup k-1/).
8143891
Different bounds on the different Betti numbers of semi-algebraic sets
A classic result in real algebraic geometry due to Oleinik-Petrovsky, Thom and Milnor, bounds the {\em topological complexity} (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers.
In this paper we prove separate bounds on the different Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. These are the first results in this direction.
24764914
Computing the Betti numbers of arrangements via spectral sequences
In this paper, we consider the problem of computing the Betti numbers of an arrangement of n compact semi-algebraic sets, S1,...,Sn ⊂ Rk, where each Si is described using a constant number of polynomials with degrees bounded by a constant. Such arrangements are ubiquitous in computational geometry. We give an algorithm for computing lth Betti number, β(∪i=1n Si), 0 ≤ l ≤ k - 1 using O(nl+2) algebraic operations. Additionally, one has to perform linear algebra on integer matrices of size bounded by O(nl+2). All previous algorithms for computing the Betti numbers of arrangements, triangulated the whole arrangement giving rise to a complex of size O(n2k) in the worst case. Thus, the complexity of computing the Betti numbers (other than the zeroth one) for these algorithms was O(n2k). To our knowledge this is the first algorithm for computing βl(∪i=1n Si) that does not rely on such a global triangulation, and has a graded complexity which depends on l.
8152994
Localizing Program Errors for Cimple Debugging
We present automated techniques for the explanation of counter-examples, where a counter-example should be understood as a sequence of program statements. Our approach is based on variable dependency analysis and is applicable to programs written in Cimple, an expressive subset of the C programming language. Central to our approach is the derivation of a focus-statement sequence (FSS) from a given counter-example: a subsequence of the counter-example containing only those program statements that directly or indirectly affect the variable valuation leading to the program error behind the counter-example. We develop a ranking procedure for FSSs where FSSs of higher rank are conceptually easier to understand and correct than those of lower rank. We also analyze constraints over uninitialized variables in order to localize program errors to specific program segments; this often allows the user to subsequently take appropriate debugging measures. We have implemented our techniques in the FocusCheck model checker, which efficiently checks for assertion violations in Cimple programs on a per-procedure basis. The practical utility of our approach is illustrated by its successful application to a fast, linear-time median identification algorithm commonly used in statistical analysis and in the Resolution Advisory module of the Traffic Collision Avoidance System.
17912290
An environment for enabling interactive grids
Traditional use of grid computing allows a user to submit batch jobs in a grid environment. We believe, next generation grids will extend the application domain to include interactive graphical sessions. We term such grids interactive grids. In this paper, we describe some of the challenges involved in building interactive grids. These include fine grain access control, QoS guarantees, and dynamic account management. In order to architect interactive grids, we propose and describe I-GENV, an environment for enabling interactive grids. I-GENV consists of GISH-'Grid Interactive Shell', Controlled Desktop, SAC-'Session Admission Control' module, GMMA-'Grid Monitoring and Management Agents', System Policies, and Dynamic Account Manager. We also present our testbed implementation of I-GENV using and extending Globus Toolkit 2.0 for the Grid middleware infrastructure, and VNC as the remote display technology.
12026284
mmGrid: distributed resource management infrastructure for multimedia applications
We are developing mmGrid (Multimedia Grid) as an extensible middleware architecture supporting multimedia applications in a grid computing environment. Our vision is to provide support for interactive applications from the following domains: graphics, visualization, streaming media and tele-immersion. The initial deployment will be within an enterprise as a mechanism for provisioning computing resources. However the scheduling system of mmGrid will be flexible, and will allow interactive and batch jobs to use the grid-computing paradigm. This paper presents our argument for using remote display technology in this environment. We also report on the use cases we support at this point, the system architecture for mmGrid and our research directions.
2261271
Selecting and Composing Web Services through Iterative Reformulation of Functional Specifications
We propose a specification-driven approach to Web service composition. The proposed framework allows users to start with a high-level, possibly incomplete specification of a desired (goal) service that is to be realized using a subset of the available component services. These services are represented by the system using transition systems augmented with guards over variables with infinite domains and are used to determine a strategy for their composition that would realize the goal service. In the event that the goal service cannot be realized using the available services, the system identifies the cause(s) for such failure which can then be used by the developer to reformulate the goal specification. Thus, the system supports Web service composition through iterative refinement of the functional specifications. We present a prototype implementation in tabled-logic programming environment that illustrates the key features of the proposed approach
7011829
QoS aware job scheduling in a cluster-based Web server for multimedia applications
We propose a cluster-based Web server where a few computing nodes are separately reserved for high-performance computing applications, such as multimedia, SSL, and CGI. As an example application, we consider a multimedia server that dynamically generates video units to satisfy the bit rate and bandwidth requirements of a variety of clients. To perform QoS aware scheduling of multiple multimedia jobs on the computing servers, a two-step algorithm is proposed. The first step is to fairly schedule multimedia streams to satisfy each stream's QoS requirement; and the second step is to balance the workload among heterogeneous computing nodes in the cluster. We propose a new quota-based adaptive coscheduling (QACS) algorithm that greatly reduces delay jitter by eliminating the out-of-order departure for outgoing streams, as well as achieves high throughput in a heterogeneous cluster. Experimental results show that the proposed scheduling technique gives adequate QoS guarantees to multiple streams.