10508326
The Combinatorial and Topological Complexity of a Single Cell 


Abstract. The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of n geometric objects in Rk, 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(nk-1) rather than O(nk) which is the bound for the combinatorial complexity of the whole arrangement. Until now, this was known to be true only for k ≤ 3 and only for some special cases in higher dimensions. A classic result in real algebraic geometry due to Oleinik and Petrovsky [15], Thom [18], and Milnor [14], bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. However, until 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(nk-1) . The techniques used for proving this topological result combined with those developed by Halperin and Sharir for the single cell problem in three dimensions allow us to prove a bound of O(nk-1+ε) on the combinatorial complexity of a single cell. Finally, we show that under a certain natural geometric assumption on the objects (namely, that whenever they intersect, the intersection is robustly transversal) it is possible to prove a bound of O(nk-1) on the combinatorial complexity of a single cell. 

9536962
On the combinatorial and algebraic complexity of quantifier elimination
In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the imput polynomials) and the combinatorial part (the dependence on the number of polynomials) are sparated. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that is output, are independent of the number of input polynomials. As special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.
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.
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.
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.
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.
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
8225775
A global lower bound on parameter estimation error with periodic distortion functions
We present a global Ziv-Zakai (1969) type lower bound on the mean square error for estimation of signal parameter vectors, where some components of the distortion function may be periodic. Periodic distortion functions arise naturally in the context of direction of arrival or phase estimation problems. The bound is applied to an image registration problem, and compared to the performance of the maximum-likelihood estimator (MLE).
12323199
On Context-Specific Substitutability of Web Services
Web service substitution refers to the problem of identifying a service that can replace another service in the context of a composition with a specified functionality. Existing solutions to this problem rely on detecting the functional and behavioral equivalence of a particular service to be replaced and candidate services that could replace it. We introduce the notion of context-specific substitutability, where context refers to the overall functionality of the composition that is required to be maintained after replacement of its constituents. Using the context information, we investigate two variants of the substitution problem, namely environment-independent and environment- dependent, where environment refers to the constituents of a composition and show how the substitutability criteria can be relaxed within this model. We provide a logical formulation of the resulting criteria based on model checking techniques as well as prove the soundness and completeness of the proposed approach.
17954434
Fast proxy delivery of multiple streaming sessions in shared running buffers
With the falling price of memory, an increasing number of multimedia servers and proxies are now equipped with a large memory space. Caching media objects in the memory of a proxy helps to reduce the network traffic, the disk I/O bandwidth requirement, and the data delivery latency. The running buffer approach and its alternatives are representative techniques to caching streaming data in the memory. There are two limits in the existing techniques. First, although multiple running buffers for the same media object co-exist in a given processing period, data sharing among multiple buffers is not considered. Second, user access patterns are not insightfully considered in the buffer management. In this paper, we propose two techniques based on shared running buffers in the proxy to address these limits. Considering user access patterns and characteristics of the requested media objects, our techniques adaptively allocate memory buffers to fully utilize the currently buffered data of streaming sessions, with the aim to reduce both the server load and the network traffic. Experimentally comparing with several existing techniques, we show that the proposed techniques achieve significant performance improvement by effectively using the shared running buffers.
17164095
Local and Symbolic Bisimulation Using Tabled Constraint Logic Programming
Bisimulation is a fundamental notion that characterizes behavioral equivalence of concurrent systems. In this paper, we study the problem of encoding efficient bisimulation checkers for finite- as well as infinite-state systems as logic programs. We begin with a straightforward andshort (less than 10 lines) encoding of finite-state bisimulation checker as a tabledlogic program. In a goal-directedsystem like XSB, this encoding yields a local bisimulation checker: one where state space exploration is done only until a dissimilarity is revealed. More importantly, the logic programming formulation of local bisimulation can be extended to do symbolic bisimulation for checking the equivalence of infinite-state concurrent systems representedb y symbolic transition systems. We show how the two variants of symbolic bisimulation (late andearly equivalences) can be formulatedas a tabledconstrain t logic program in a way that precisely brings out their differences. Finally, we show that our symbolic bisimulation checker actually outperforms non-symbolic checkers even for relatively small finite-state systems.