Kimio Kawaguchi

Efficient algorithms for tripartitioning triconnected graphs and 3-edge-connected graphs

The extended k-partition problem is defined as follows. For the following inputs (1)an undirected graph G=(V, E)(n = ¦V¦, m= ¦E¦), (2)a vertex subset V′(⊂- V), (3)distinct vertices ai e V′(1 ≤ i ≤ k) and (4)natural numbers ni(1 ≤ i≤k)(n1 ≤ ... ≤ nk) such that n1 +... + nk =n′ = ¦V′¦, we compute a partition V1∪...∪V k of V and a partition V′1∪...V′ k of V′ such that (a)each V′i is included in Vi, (b)each V′i contains the specified vertex ai, (c)¦V′i¦ = ni and (d)each Vi induces a connected subgraph. If V′ = V, then the problem is called the k-partition problem. In this paper, we show that if the input graph is triconnected the extended tripartition problem can be solved in O(m + (n − n3) · n) time and that the algorithm solves the original tripartition problem in O(m + (n1 + n2) · n) time. Furthermore, we show that for a k-edgeconnected graph G - (V, E) there exists a partition V1 ∪ ... V k of V such that each Vi contains the specified vertex ai, ¦Vi¦ = ni and k subgraphs G1,..., Gk are mutually edge disjoint and each of Gi contains all of elements in Vi(1 ≤ i ≤ k) and the case in which k = 3 can be solved in O(n2) time.

Parallel Algorithms for Partitioning Sorted Sets and Related Problems

We consider the following partition problem: Given a set S of n elements that is organized as k sorted subsets of size n/k each and given a parameter h with 1/k≤h≤n/k, partition S into g=O(n/(hk)) subsets D1D2,..., D g of size Θ(hk) each, such that for any two indices i and j with 1≤i≤j≤g, no element in D1i is bigger than any element in D j . Note that with various combinations of the values of parameters h and k, several fundamental problems, such as merging, sorting,and finding an approximate median, can be formulated as or be reduced to this partition problem. The partition problem also finds applications in solving problems of parallel computing and computational geometry. In this paper, we present efficient parallel algorithms for solving the partition problem and its applications. Our parallel partition algorithm runs in O(log n) time using O(min{(n/h)*max{log h 1},n*max{log(1/h),1}}/log n) processors in the EREW PRAM model.The complexity bounds of our parallel partition algorithm on the respective special cases match those of the optimal EREW PRAM algorithms for merging, sorting, and finding an approximate median. Using our parallel partition algorithm, we are also able to obtain better complexity bounds (even possibly on a weaker parallel model) than the previously best known parallel algorithms for several important problems, including parallel multi-selection, parallel multi-ranking, and parallel sorting of k sorted subsets.

New Results in Graph Routing

Consider a communication network or an undirected graph G in which a limited number of link and/or node faults F might occur. A routing ? for the network (at most one path, called a route, for each ordered pair of nodes) must be chosen without knowing which components might be faulty. A routing is said to be minimal if any route from x to y is assigned to one of the shortest paths from x to y, and is said to be bidirectional if for any ordered pair (x, y) the route from x to y and the route from y to x are assigned to the same path. The diameter of the surviving route graph R(G, ?)/F (denoted by D(R(G, ?)/F)), where two nonfaulty nodes x and y are connected by a directed edge if there are no faults on the route from x to y, could be one of the fault-tolerant measures for the routing ?. In this paper, we show that there exists a bidirectional and minimal routing ?k on a k-dimensional hypercube graph Ck such that D(R(Ck, ?k)F) ? 2 for any set of faults F (|F| < k) in the case that k = 3m and k = 3m + 1 (m ? 1), and that there exists a bidirectional and almost minimal routing ?3m+2 (m ? 0) on C3m+2 such that D(R(C3m+2, ?3m+2)/F) ? 2 for any set of faults F (|F| < 3m + 2). These are solutions for the open problem raised by Dolev et al. (1987, Inform. and Comput.72, 180-196). We also show that we can construct a routing ? for any graph G in some class of (k + 1)-node connected graphs such that D(R(G, ?)/F) ? 2 for any set of faults F (|F| ? k). As long as faults are assumed to occur in a network, the diameter of the surviving route graph for the network is more than one. Thus, the routing shown here is best possible and is said to be optimal.

Efficient fault-tolerant fixed routings on ( k +1)-connected digraphs

Abstract Consider a directed communication network G in which a limited number of directed link and/or node faults F might occur. A routing ϱ for the network (a fixed path for each ordered pair of nodes) must be chosen without knowing which components might become faulty. The diameter of the surviving route graph R(G,ϱ) F (denoted by D( R(G,ϱ) F ) ), where two nonfaulty nodes x and y are connected by a directed edge if there are no faults on the route from x to y, could be one of the fault-tolerant measures for the routing ϱ. In this paper, we show sufficient conditions for classes of (k+1)-connected directed graphs to have routings ϱ3 and ϱ2 on G such that D( R(G,ϱ 3 ) F )≤3 and D( R(G,ϱ 2 ) F ) for any set faults F(|ϱ,≤k). Since the diameter of the surviving route graph is more than 1 provided that faults are assumed to occur in the network, we insist that the routing ϱ2 is optimal. We also show that constant diameter routings (with the diameters of the surviving route graph being 5 and 7) can be constructed for any (k+1)-connected digraph satisfying only a certain size condition.

Efficient algorithms for a mixed K -partition problem of graphs without specifying bases

Abstract This paper describes efficient algorithms for partitioning a k-edge-connected graph into k edge-disjoint connected subgraphs, each of which has a specified number of elements (vertices and edges). If each subgraph contains the specified element (called base), we call this problem the mixed k-partition problem with bases (called k-PART-WB), otherwise we call it the mixed k-partition problem without bases (called k-PART-WOB). In this paper, we show that k-PART-WB always has a solution for every k-edge-connected graph and we consider the problem without bases and we obtain the following results: (1) for any k⩾2, k-PART-WOB can be solved in O(¦V¦ √¦V¦ log 2 ¦V¦ + ¦E¦) time for every 4-edge-connected graph G = (V,E), (2) 3-PART-WOB can be solved in O(¦V¦ 2 ) for every 2-edge-connected graph G = (V,E) and (3) 4-PART-WOB can be solved in O(¦E¦ 2 ) for every 3-edge-connected graph G = (V,E).

Highly Fault-Tolerant Routings and Fault-Induced Diameter for Generalized Hypercube Graphs

Consider a communication networkGin which a limited number of link and/or node faultsFmight occur. A routing ? for the network (a fixed path between each pair of nodes) must be chosen without knowing which components might become faulty. The diameter of the surviving route graphR(G, ?)/F, where the surviving route graphR(G, ?)/Fis a directed graph consisting of all nonfaulty nodes inGwith a directed edge fromxtoyiff there are no faults on the route fromxtoy, could be one of the fault-tolerant measures for the routing ?. In this paper, we show that we can construct efficient and highly fault-tolerant routings on ak-dimensional generalizedd-hypercubeC(d,k) such that the diameter of the surviving route graph is bounded by constant for the case that the number of faults exceeds the connectivity ofC(d,k).

A parallel method for finding the convex hull of discs

We present a parallel method for finding the convex hull of a set of discs in the CREW PRAM model. We show that the convex hull of n discs can be computed in O(log/sup 1+/spl epsiv// n) time using O(n/log/sup /spl epsiv// n) processors, where /spl epsiv/ is any positive constant. We also show that it can be constructed in O(log n loglog n) time using O(n log n) processors. The first result achieves cost optimal and the second one runs faster. The main technique which we used in the algorithm is a complex divide-and-conquer technique.<<ETX>>

Parallel robust algorithms for constructing strongly convex hulls

Parallel Robust Algorithms for Constructing Strongly Convex Hulls* Wei Chen~ Koichi Wadat Kimio Kawaguchit Given a set S of n points in the plane, an e-strongly convex 6-hull of S is defined as a convex polygon P with the vertices taken from S such that no point of S lies farther than d outside P and such that even if the vertices of .F’ arc perturbed by as much as e, P remains convex (e.g., the convex hull of S is a O-strongly convex O-hull of S). This paper presents the first parallel robust method for the generalized convex hull problem. Wc show that an c-strongly convex O (e + ~)-hull of S can be constructed in 0(log3 n) time using n processors with imprecise computations, where /3 is the error unit of the primitive operations. The rmult implies an improved sequential algorithm for t hc problem. Our method consists of three algorithms: finding an approximate common tangent of two sets of n points each in O (log2 n) time lining n processors, computing a convex O(e + /?)-hull of n points in 0(log3 n) time using n processors, and constructing an e-strongly convex O(e + ~)-hull of a convex n-gon in O(log n) time with n processors using imprecise computations. Wc use the CREW PRAM computational model in the paper. * This work is partly supported by The Hori Information Science Promotion Foundation(1995). t Department of Electrica.land Computer Engineering, Nagoya Institute of Technology, Showa, Nagoya 466, .Japan. E-mail: (then, wada, kawaguchi)@elcom. nitecll.ac.jp Permission to makedtgital/hard copies of all or part of thk material for personal or classrmm use is grantad without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication and its data appear, and notice is given that copyright ia by permission of the ACM, Jnc. To copy otherwise, to republish, to post on servers or to dkibutc to lists, requires specific permiaaion andlor fee. Computational Geometry’96, Philadelphia PA, USA

Efficient Algorithms for a Mixed k-Partition Problem of Graphs without Specifying Bases

This paper describes efficient algorithms for partitioning a k-edge-connected graph into k edge-disjoint connected subgraphs, each of which has a specified number of elements(vertices and edges). If each subgraph contains the specified element (called base), we call this problem the mixed k-partition problem with bases(called k-PART-WB), otherwise we call it the mixed k-partition problem without bases (called k-PART-WOB). In this paper, we show that k-PART-WB always has a solution for every k-edge-connected graph and we consider the problem without bases and we obtain the following results: (1)for any k≥2, k-PART-WOB can be solved in O(∥V∥√∥V∥log2∥V∥+∥E∥) time for every 4-edge-connected graph G=(V,E), (2)3-PART-WOB can be solved in O(∥V∥2) for every 2-edge-connected graph G=(V,E) and (3)4-PART-WOB can be solved in O(∥E∥2) for every 3-edge-connected graph G=(V,E).

Robust algorithms for constructing strongly convex hulls in parallel

Given a set S of n points in the plane, an e-strongly convex δ-hull of S is defined as a convex polygon P with the vertices taken from S such that no point of S lies farther than δ outside P and such that even if the vertices of P are perturbed by as much as e, P remains convex. This paper presents the first parallel robust method for this generalized convex hull problem (note that the convex hull of S is the 0-strongly convex 0-hull of S). We show that an e-strongly convex O(e + β)-hull of S can be constructed in O(log3n) time using n processors with imprecise computations, where β is the error unit of primitive operations. This result also implies an improved sequential algorithm. Our algorithm consists of two parts: (1) computing a convex O(e + β) -hull of n points, in O(log3n) time using n processors, and (2) constructing an e-strongly convex O(e + β)-hull of a convex polygon with n vertices, in O(log2n) time with n processors. We also find an approximate bridge of two sets with n points each, in O(log2n) time using n processors, which we use as a subroutine. All these algorithms are fundamental and have their own applications. The parallel computational model in this paper is the EREW PRAM.