Improved Algorithms for Constructing Fault-Tolerant Spanners

Abstract. Let S be a set of n points in a metric space, and let k be a positive integer. Algorithms are given that construct k -fault-tolerant spanners for S . If in such a spanner at most k vertices and/ or edges are removed, then each pair of points in the remaining graph is still connected by a ``short' path. First, an algorithm is given that transforms an arbitrary spanner into a k -fault-tolerant spanner. For the Euclidean metric in R ^d , this leads to an O(n log n + c ^k n) -time algorithm that constructs a k -fault-tolerant spanner of degree O(c ^k ) , whose total edge length is O(c ^k ) times the weight of a minimum spanning tree of S , for some constant c . For constant values of k , this result is optimal. In the second part of the paper, algorithms are presented for the Euclidean metric in R ^d . These algorithms construct (i) in O(n log n + k ² n) time, a k -fault-tolerant spanner with O(k ² n) edges, and (ii) in O(k n log n) time, such a spanner with O(k n log n) edges.     

Fully Dynamic Geometric Spanners

In this paper we present an algorithm for maintaining a spanner over a dynamic set of points in constant doubling dimension metric spaces. For a set S of points in ? ^d , a t-spanner is a sparse graph on the points of S such that there is a path in the spanner between any pair of points whose total length is at most t times the distance between the points. We present the first fully dynamic algorithm for maintaining a spanner whose update time depends solely on the number of points in S. In particular, we show how to maintain a (1+ε)-spanner with O(n/ε ^d ) edges, where points can be inserted to S in an amortized update time of O(log?n) and deleted from S in an amortized update time of $\tilde{O}(n^{1/3})$ . As a by-product of our techniques we obtain a simple incremental algorithm for constructing a (1+ε)-spanner with O(n/ε ^d ) edges in constant doubling dimension metric spaces whose running time is O(nlog?n).     

Computing the Greedy Spanner in Near-Quadratic Time

The greedy algorithm produces high-quality spanners and, therefore, is used in several applications. However, even for points in d-dimensional Euclidean space, the greedy algorithm has near-cubic running time. In this paper, we present an algorithm that computes the greedy spanner for a set of n points in a metric space with bounded doubling dimension in O(n²logn)\ensuremath {\mathcal {O}}(n^{2}\log n) time. Since computing the greedy spanner has an Ω(n ²) lower bound, the time complexity of our algorithm is optimal within a logarithmic factor.     

Finding the k Shortest Paths in Parallel

A concurrent-read exclusive-write PRAM algorithm is developed to find the k shortest paths between pairs of vertices in an edge-weighted directed graph. Repetitions of vertices along the paths are allowed. The algorithm computes an implicit representation of the k shortest paths to a given destination vertex from every vertex of a graph with n vertices and m edges, using O(m+nk log ² k) work and O( log^3k log ^*k+ log n( log log k+ log ^*n)) time, assuming that a shortest path tree rooted at the destination is pre-computed. The paths themselves can be extracted from the implicit representation in O( log k + log n) time, and O(n log n +L) work, where L is the total length of the output. Received July 2, 1997; revised June 18, 1998.     

Proximity problems for points on a rectilinear plane with rectangular obstacles

We consider the following four problems for a setS ofk points on a plane, equipped with the rectilinear metric and containing a setR ofn disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles inR): (a) find aclosest pair of points inS, (b) find anearest neighbor for each point inS, (c) compute the rectilinearVoronoi diagram ofS, and (d) compute a rectilinearminimal spanning tree ofS. We describeO ((n+k) log(n+k))-time sequential algorithms for (a) and (b) based onplane-sweep, and the consideration of geometrically special types of shortest paths, so-calledz-first paths. For (c) we present anO ((n+k) log(n+k) logn)-time sequential algorithm that implements a sophisticateddivide-and-conquer scheme with an addedextension phase. In the extension phase of this scheme we introduce novel geometric structures, in particular so-calledz-diagrams, and techniques associated with the Voronoi diagram. Problem (d) can be reduced to (c) and solved inO ((n+k) log(n+k) logn) time as well. All our algorithms arenear-optimal, as well as easy to implement. An extended abstract appeared inProc. 13th Conf. on the Foundations of Software Technology and Theoretical Computer Science, Bombay, 1993, Springer-Verlag, pp. 218–227. Sumanta Guha was supported in part by a UW-Milwaukee Graduate School Research Committee Award. Ichiro Suzuki was supported in part by the National Science Foundation under Grants CCR-9004346 and IRI-9307506, the Office of Naval Research under Grant N00014-94-1-0284, and an endowed chair supported by Hitachi Ltd. at the Faculty of Engineering Science, Osaka University.     

Property Testing on k-Vertex-Connectivity of?Graphs

We present an algorithm for testing the k-vertex-connectivity of graphs with the given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. Fixed degree bound d, a graph G with n vertices and a maximum degree at most d is called ε-far from k-vertex-connectivity when at least $\frac{\epsilon dn}{2}$ edges must be added to or removed from G to obtain a k-vertex-connected graph with a maximum degree at most d. The algorithm always accepts every graph that is k-vertex-connected and rejects every graph that is ε-far from k-vertex-connectivity with a probability of at least 2/3. The algorithm runs in $O(d(\frac{c}{\epsilon d})^{k}\log\frac {1}{\epsilon d})$ time (c>1 is a constant) for (k?1)-vertex-connected graphs, and in $O(d(\frac{ck}{\epsilon d})^{k}\log\frac{k}{\epsilon d})$ time (c>1 is a constant) for general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k≥4.     

Fast Parallel Reordering and Isomorphism Testing of k -Trees

In this paper two problems on the class of k -trees, a subclass of the class of chordal graphs, are considered: the fast reordering problem and the isomorphism problem. An O(log ² n) time parallel algorithm for the fast reordering problem is described that uses O(nk(n-k)/\kern -1ptlog n) processors on a CRCW PRAM proving membership in the class NC for fixed k . An O(nk(k+1)!) time sequential algorithm for the isomorphism problem is obtained representing an improvement over the O(n ² k(k+1)!) algorithm of Sekharan (the second author) [10]. A parallel version of this sequential algorithm is presented that runs in O(log ² n) time using O((nk((k+1)!+n-k))/log n) processors improving on a parallel algorithm of Sekharan for the isomorphism problem [10]. Both the sequential and parallel algorithms use a concept introduced in this paper called the kernel of a k -tree.     

Minimizing Makespan in Batch Machine Scheduling

We study the scheduling of a set of n jobs, each characterized by a release (arrival) time and a processing time, for a batch processing machine capable of running at most B jobs at a time. We obtain an O(n log n)-time algorithm when B is unbounded. When there are only m distinct release times and the inputs are integers, we obtain an O(n(BR_max)^m-1(2/m)^m-3)-time algorithm where R_max is the difference between the maximum and minimum release times. When there are k distinct processing times and m release times, we obtain an O(n log m + k^k+2 B^k+1 m² log m)-time algorithm. We obtain even better algorithms for m=2 and for k=1. These algorithms improve most of the corresponding previous algorithms for the respective special cases and lead to improved approximation schemes for the general problem.     

The Computational Complexity of Some Julia Sets

Numerous computer programs have been written to compute sets of points which approximate Julia sets [4]. Usually, no error estimations are added so that it remains unclear, how good such approximations are. Furthermore, high precision pictures are unreliable because of rounding errors, since the realizing computer programs use fixed length floating point numbers. Computable error estimation w.r.t. the Hausdorff metric d_H means that the set is recursive [10]. Many Julia sets J are recursive [11]. Recursive compact subsets of the Euclidean plane have a computable Turing machine time complexity [10]. In this paper we prove that the Julia set of a complex function f(z) = z² + c for c < 1/4 can be computed locally in time O(k²M(k)) (where M(k) is a time bound for multiplication of k-bit integers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide for a single point whether it belongs to a grid K_k (2^−k · )² such that d_H(K_k,J) ≤ = 2^−k.     

On constructing the relative neighborhood graphs in EuclideanK-dimensional spaces

In this paper, a new algorithm for constructing the relative neighborhood graph(RNG) of ann points set in Euclideank-dimensional space is presented, for fixedk≥3. The worst case running time of the algorithm isO(n ^2?a(k) (logn)^1?a(k) ), fora(k)=2^?(k+1), which is under the assumption that no three input points form an isosceles triangle. Previous algorithms needO(n ²) time. Our algorithm proceeds in two phases. First, a supergraph ofRNG withO(n) edges is constructed and then those edges which do not belong toRNG are eliminated.     

Homogeneous Formulas and Symmetric Polynomials

We investigate the arithmetic formula complexity of the elementary symmetric polynomials S^k_n{S^k_n} . We show that every multilinear homogeneous formula computing S^k_n{S^k_n} has size at least k^W(logk)n{k^{\Omega(\log k)}n} , and that product-depth d multilinear homogeneous formulas for S^k_n{S^k_n} have size at least 2^W(k1/d)n{2^{\Omega(k^{1/d})}n} . Since Sⁿ_2n{S^{n}_{2n}} has a multilinear formula of size O(n ²), we obtain a superpolynomial separation between multilinear and multilinear homogeneous formulas. We also show that S^k_n{S^k_n} can be computed by homogeneous formulas of size k^O(logk)n{k^{O(\log k)}n} , answering a question of Nisan and Wigderson. Finally, we present a superpolynomial separation between monotone and non-monotone formulas in the noncommutative setting, answering a question of Nisan.     

The densest hemisphere problem

Given a set K of n points on the unit sphere S^d in d-dimensional Euclidean space, a hemisphere of S^d is densest if it contains a largest subset of K. In this paper we consider the problem of determining a densest hemisphere and present the following complementary results: (i) a discretized version of the original problem, restated as a feasibility question, is NP-complete when both n and d are arbitrary; (ii) when the number d of dimensions is fixed, there exists a polynomial time algorithm which solves the problem in time O(n_d?1 log n) on a random access machine with unit cost arithmetic operations.     

A faster algorithm for finding edge-disjoint branchings

An O(k²mn) algorithm is proposed for finding k edge-disjoint branchings in a directed multigraph with m edges and n vertices. With appropriate preprocessing the time bound can be reduced to O(kmn + k³n²). Our proposed algorithm runs faster than any previously known algorithm and provides yet another constructive proof of Edmonds' Branchings Theorem.     

Maintaining the Classes of 4-Edge-Connectivity in a Graph On-Line

Two vertices of an undirected graph are called k -edge-connected if there exist k edge-disjoint paths between them (equivalently, they cannot be disconnected by the removal of less than k edges from the graph). Equivalence classes of this relation are called classes of k -edge-connectivity or k -edge-connected components. This paper describes graph structures relevant to classes of 4 -edge-connectivity and traces their transformations as new edges are inserted into the graph. Data structures and an algorithm to maintain these classes incrementally are given. Starting with the empty graph, any sequence of q Same-4-Class? queries and n Insert-Vertex and m Insert-Edge updates can be performed in O(q + m + n log n) total time. Each individual query requires O(1) time in the worst-case. In addition, an algorithm for maintaining the classes of k -edge-connectivity (k arbitrary) in a (k-1) -edge-connected graph is presented. Its complexity is O(q + m + n) , with O(M +k ² n log (n/k)) preprocessing, where M is the number of edges initially in the graph and n is the number of its vertices. Received July 5, 1995; revised October 21, 1996.     

Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization

We show that the NP-complete Feedback Vertex Set problem, which asks for the smallest set of vertices to remove from a graph to destroy all cycles, is deterministically solvable in O(ck⋅m) time. Here, m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, and c is a constant. We extend this to an algorithm enumerating all solutions in O(dk⋅m) time for a (larger) constant d. As a further result, we present a fixed-parameter algorithm with runtime O(k²⋅m²) for the NP-complete Edge Bipartization problem, which asks for at most k edges to remove from a graph to make it bipartite.     

Exact and Approximation Algorithms for Clustering

In this paper we present an n^ O(k ^1-1/d ) -time algorithm for solving the k -center problem in \reals ^d , under L _∈ fty - and L ₂ -metrics. The algorithm extends to other metrics, and to the discrete k -center problem. We also describe a simple (1+ɛ) -approximation algorithm for the k -center problem, with running time O(nlog k) + (k/ɛ)^ O(k ^1-1/d ) . Finally, we present an n^ O(k ^1-1/d ) -time algorithm for solving the L -capacitated k -center problem, provided that L=Ω(n/k ^1-1/d ) or L=O(1) . Received July 25, 2000; revised April 6, 2001.     

An O(2<Superscript>O(k)</Superscript>n<Superscript>3</Superscript>) FPT Algorithm for the Undirected Feedback Vertex Set Problem

We describe an algorithm for the Feedback Vertex Set problem on undirected graphs, parameterized by the size k of the feedback vertex set, that runs in time O(c^kn³) where c = 10.567 and n is the number of vertices in the graph. The best previous algorithms were based on the method of bounded search trees, branching on short cycles. The best previous running time of an FPT algorithm for this problem, due to Raman, Saurabh and Subramanian, has a parameter function of the form 2^{O(k log k /log log k)}. Whether an exponentially linear in k FPT algorithm for this problem is possible has been previously noted as a significant challenge. Our algorithm is based on the new FPT technique of iterative compression. Our result holds for a more general form of the problem, where a subset of the vertices may be marked as forbidden to belong to the feedback set. We also establish "exponential optimality" for our algorithm by proving that no FPT algorithm with a parameter function of the form O(2^o(k)) is possible, unless there is an unlikely collapse of parameterized complexity classes, namely FPT = M[1].     

Fast Parallel Reordering and Isomorphism Testing of k -Trees

Abstract. In this paper two problems on the class of k -trees, a subclass of the class of chordal graphs, are considered: the fast reordering problem and the isomorphism problem. An O(log ² n) time parallel algorithm for the fast reordering problem is described that uses O(nk(n-k)/\kern -1ptlog n) processors on a CRCW PRAM proving membership in the class NC for fixed k . An O(nk(k+1)!) time sequential algorithm for the isomorphism problem is obtained representing an improvement over the O(n ² k(k+1)!) algorithm of Sekharan (the second author) [10]. A parallel version of this sequential algorithm is presented that runs in O(log ² n) time using O((nk((k+1)!+n-k))/log n) processors improving on a parallel algorithm of Sekharan for the isomorphism problem [10]. Both the sequential and parallel algorithms use a concept introduced in this paper called the kernel of a k -tree.     

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

An axis-parallel k-dimensional box is a Cartesian product R ₁×R ₂×???×R _k where R _i (for 1≤i≤k) is a closed interval of the form [a _i ,b _i ] on the real line. For a graph G, its boxicity box?(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a $\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}An axis-parallel k-dimensional box is a Cartesian product R ₁×R ₂×⋅⋅⋅×R _k where R _i (for 1≤i≤k) is a closed interval of the form [a _i ,b _i ] on the real line. For a graph G, its boxicity box (G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a ?1+\frac1clogn?^d-1\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1} approximation ratio for any constant c≥1 when d≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard.     

Metric concepts and implications in describing compositional changes for world river's water chemistry

The paper is designed to give the reader an outline that is useful for understanding the importance of distance, as a metric concept, and its implications when compositional (geochemical) data are managed from a statistical point of view in a given sample space. Application examples are shown by considering the construction of confidence regions and mixing models. The analyzed data are related to the chemistry of the most important rivers of the world as referring to the GEMS/WATER Global Register of River Inputs when each sample (river) is represented as a composition. A compositional vector of d parts, x=[x₁,x₂,…,x_d], is defined as a vector in which the only relevant information is contained in the ratios between its components. All the components of the vector are assumed positive and are called parts (variables), while the whole compositional vector, with the sum of the parts equal to a constant, represents the composition. In this case data are not represented by variables free to vary from −∞ to +∞ within a Euclidean space but occupy a restricted part of it called the simplex. The d-part simplex, S^d, is a subset of a d-dimensional real space. In this context the metric of the R space, with the definition of basic algebraic operations and of inner product, norm and distance, thus giving an Euclidean vector space structure, cannot be applied since the scale is relative and not absolute.