An efficient distributed algorithm for constructing small dominating sets

The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem. We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in rounds with high probability, where n is the number of nodes, is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within of the optimal in expectation and within of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements. Received: January 2002 / Accepted: August 2002 RID="*" ID="*" Supported by NSF CAREER award NSF CCR-9983901 RID="*" ID="*" Supported by NSF CAREER award NSF CCR-9983901     

An algorithm for the asynchronous Write-All problem based on process collision

Summary. The problem of using P processes to write a given value to all positions of a shared array of size N is called the Write-All problem. We present and analyze an asynchronous algorithm with work complexity , where (assuming and ). Our algorithm is a generalization of the naive two-processor algorithm where the two processes each start at one side of the array and walk towards each other until they collide. Received: October 1999 / Accepted: September 2000     

Improved Bounds for Wireless Localization

We consider a novel class of art gallery problems inspired by wireless localization that has recently been introduced by Eppstein, Goodrich, and Sitchinava. Given a simple polygon P, place and orient guards each of which broadcasts a unique key within a fixed angular range. In contrast to the classical art gallery setting, broadcasts are not blocked by the edges of P. At any point in the plane one must be able to tell whether or not one is located inside P only by looking at the set of keys received. In other words, the interior of the polygon must be described by a monotone Boolean formula composed from the keys. We improve both upper and lower bounds for the general problem where guards may be placed anywhere by showing that the maximum number of guards to describe any simple polygon on n vertices is between roughly \frac35n\frac{3}{5}n and \frac45n\frac{4}{5}n . A guarding that uses at most \frac45n\frac{4}{5}n guards can be obtained in O(nlog n) time. For the natural setting where guards may be placed aligned to one edge or two consecutive edges of P only, we prove that n−2 guards are always sufficient and sometimes necessary.     

ATM layouts with bounded hop count and congestion

Summary. In this paper we introduce and analyze two new cost measures related to the communication overhead and the space requirements associated with virtual path layouts in ATM networks, that is the edge congestion and the node congestion. Informally, the edge congestion of a given edge e at an incident node u is defined as the number of VPs terminating at or starting from u and using e, while the node congestion of a node v is defined as the number of VPs having v as an endpoint. We investigate the problem of constructing virtual path layouts allowing to connect a specified root node to all the others in at most h hops and with maximum edge or node congestion c, for two given integers h and c. We first give tight results concerning the time complexity of the construction of such layouts for both the two congestion measures, that is we exactly determine all the tractable and intractable cases. Then, we provide some combinatorial bounds for arbitrary networks, together with optimal layouts for specific topologies such as chains, rings and grids. Received: December 1997 / Accepted: August 2000     

Parameterized algorithm for eternal vertex cover

In this paper we initiate the study of a “dynamic” variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem introduced by Klostermeyer and Mynhardt, from the perspective of parameterized algorithms. This problem consists in placing a minimum number of guards on the vertices of a graph such that these guards can protect the graph from any sequence of attacks on its edges. In response to an attack, each guard is allowed either to stay in his vertex, or to move to a neighboring vertex. However, at least one guard has to fix the attacked edge by moving along it. The other guards may move to reconfigure and prepare for the next attack. Thus at every step the vertices occupied by guards form a vertex cover. We show that the problem admits a kernel of size k⁴(k+1)+2k, which shows that the problem is fixed parameter tractable when parameterized by the number of available guards k. Finally, we also provide an algorithm with running time O(²O(k²)+nm) for Eternal Vertex Cover, where n is the number of vertices and m the number of edges of the input graph. In passing we also observe that Eternal Vertex Cover is NP-hard, yet it has a polynomial time 2-approximation algorithm.     

Adaptive and efficient mutual exclusion

Summary. This paper presents adaptive algorithms for mutual exclusion using only read and write operations; the performance of the algorithms depends only on the point contention, i.e., the number of processes that are concurrently active during the algorithm execution (and not on n, the total number of processes). Our algorithm has O(k) remote step complexity and O(logk) system response time, wherek is the point contention. The remote step complexity is the maximal number of steps performed by a process where a wait is counted as one step. The system response time is the time interval between subsequent entries to the critical section, where one time unit is the minimal interval in which every active process performs at least one step. The space complexity of this algorithm is O(N logn), where N is the range of processes' names. We show how to make the space complexity of our algorithm depend solely on n, while preserving the other performance measures of the algorithm. Received: March 2001 / Accepted: November 2001     

Networks on which hot-potato routing does not livelock

Summary. Hot-potato routing is a form of synchronous routing which makes no use of buffers at intermediate nodes. Packets must move at every time step, until they reach their destination. If contention prevents a packet from taking its preferred outgoing edge, it is deflected on a different edge. Two simple design principles for hot potato routing algorithms are minimum advance, that advances at least one packet towards its destination from every nonempty node (and possibly deflects all other packets), and maximum advance, that advances the maximum possible number of packets. Livelock is a situation in which packets keep moving indefinitely in the network without any packet ever reaching its destination. It is known that even maximum advance algorithms might livelock on some networks. We show that minimum advance algorithms never livelock on tree networks, and that maximum advance algorithms never livelock on triangulated networks. Received: March 1999 / Accepted: August 1999     

Nearly optimal perfectly periodic schedules

We consider the problem of scheduling a set of pages on a single broadcast channel using time-multiplexing. In a perfectly periodic schedule, time is divided into equal size slots, and each page is transmitted in a time slot precisely every fixed interval of time (the period of the page). We study the case in which each page i has a given demand probability , and the goal is to design a perfectly periodic schedule that minimizes the average time a random client waits until its page is transmitted. We seek approximate polynomial solutions. Approximation bounds are obtained by comparing the costs of a solution provided by an algorithm and a solution to a relaxed (non-integral) version of the problem. A key quantity in our methodology is a fraction we denote by , that depends on the maximum demand probability: . The best known polynomial algorithm to date guarantees an approximation of . In this paper, we develop a tree-based methodology for perfectly periodic scheduling, and using new techniques, we derive algorithms with better bounds. For small values, our best algorithm guarantees approximation of . On the other hand, we show that the integrality gap between the cost of any perfectly periodic schedule and the cost of the fractional problem is at least . We also provide algorithms with good performance guarantees for large values of . Received: December 2001 / Accepted: September 2002     

Necessary and sufficient conditions on information for causal message ordering and their optimal implementation

Summary. This paper formulates necessary and sufficient conditions on the information required for enforcing causal ordering in a distributed system with asynchronous communication. The paper then presents an algorithm for enforcing causal message ordering. The algorithm allows a process to multicast to arbitrary and dynamically changing process groups. We show that the algorithm is optimal in the space complexity of the overhead of control information in both messages and message logs. The algorithm achieves optimality by transmitting the bare minimum causal dependency information specified by the necessity conditions, and using an encoding scheme to represent and transmit this information. We show that, in general, the space complexity of causal 0message ordering in an asynchronous system is , where is the number of nodes in the system. Although the upper bound on space complexity of the overhead of control information in the algorithm is , the overhead is likely to be much smaller on the average, and is always the least possible. Received: January 1996 / Accepted: February 1998     

Using extended feature objects for partial similarity retrieval

In this paper, we introduce the concept of extended feature objects for similarity retrieval. Conventional approaches for similarity search in databases map each object in the database to a point in some high-dimensional feature space and define similarity as some distance measure in this space. For many similarity search problems, this feature-based approach is not sufficient. When retrieving partially similar polygons, for example, the search cannot be restricted to edge sequences, since similar polygon sections may start and end anywhere on the edges of the polygons. In general, inherently continuous problems such as the partial similarity search cannot be solved by using point objects in feature space. In our solution, we therefore introduce extended feature objects consisting of an infinite set of feature points. For an efficient storage and retrieval of the extended feature objects, we determine the minimal bounding boxes of the feature objects in multidimensional space and store these boxes using a spatial access structure. In our concrete polygon problem, sets of polygon sections are mapped to 2D feature objects in high-dimensional space which are then approximated by minimal bounding boxes and stored in an R-tree. The selectivity of the index is improved by using an adaptive decomposition of very large feature objects and a dynamic joining of small feature objects. For the polygon problem, translation, rotation, and scaling invariance is achieved by using the Fourier-transformed curvature of the normalized polygon sections. In contrast to vertex-based algorithms, our algorithm guarantees that no false dismissals may occur and additionally provides fast search times for realistic database sizes. We evaluate our method using real polygon data of a supplier for the car manufacturing industry. Edited by R. Güting. Received October 7, 1996 / Accepted March 28, 1997     

The BG distributed simulation algorithm

We present a shared memory algorithm that allows a set of f+1 processes to wait-free “simulate” a larger system of n processes, that may also exhibit up to f stopping failures. Applying this simulation algorithm to the k-set-agreement problem enables conversion of an arbitrary k-fault-tolerant{\it n}-process solution for the k-set-agreement problem into a wait-free k+1-process solution for the same problem. Since the k+1-processk-set-agreement problem has been shown to have no wait-free solution [5,18,26], this transformation implies that there is no k-fault-tolerant solution to the n-process k-set-agreement problem, for any n. More generally, the algorithm satisfies the requirements of a fault-tolerant distributed simulation.\/ The distributed simulation implements a notion of fault-tolerant reducibility\/ between decision problems. This paper defines these notions and gives examples of their application to fundamental distributed computing problems. The algorithm is presented and verified in terms of I/O automata. The presentation has a great deal of interesting modularity, expressed by I/O automaton composition and both forward and backward simulation relations. Composition is used to include a safe agreement\/ module as a subroutine. Forward and backward simulation relations are used to view the algorithm as implementing a multi-try snapshot\/ strategy. The main algorithm works in snapshot shared memory systems; a simple modification of the algorithm that works in read/write shared memory systems is also presented. Received: February 2001 / Accepted: February 2001     

Storing continuous-media data on a compact disc

We consider the problem of storing a given set of files containing continuous-media data on a compact disc by interleaving them so that the required total area is minimal. We show that the problem is NP-hard and consider a number of special cases. The fact that these special cases are related to well-known combinatorial optimization problems has been used in solution techniques developed to handle such cases. We propose an approximation algorithm based on these techniques for the general case.     

Long lived adaptive splitter and applications

Summary. Long-lived and adaptive implementations of mutual exclusion and renaming in the read/write shared memory model are presented. An implementation of a task is adaptive if the step complexity of any operation in the implementation is a function of the number of processes that take steps concurrently with the operation. The renaming algorithm assigns a new unique id in the range to any process whose initial unique name is taken from a set of size N, for an arbitrary N and where k is the number of processes that actually take steps or hold a name while the new name is being acquired. The step complexity of acquiring a new name is , while the step complexity of releasing a name is 1. The space complexity of the algorithm is where n is an upper bound on the number of processes that may be active at the same time (acquiring or holding new names), which could be N in the worst case. Both the system response time and the worst case number of operations per process in the presented mutual-exclusion algorithm are adaptive. Both algorithms rely on the basic building block of a long-lived and adaptive splitter. While the adaptive-splitter satisfies a slightly different set of properties than the Moir-Anderson splitter [MA95], it is adaptive and long-lived. In addition, the new splitter properties enable the construction of a non-blocking long-lived (2k-1)-renaming algorithm (which is optimal in the size of the new name space). We believe that the mechanisms introduced in our splitter implementation are interesting on their own, and might be used in other adaptive and long-lived constructions. Received: March 2000 / Accepted July 2001     

Smooth multiple B-spline surface fitting with Catmull%ndash;Clark subdivision surfaces for extraordinary corner patches

This paper presents an algorithm for simultaneously fitting smoothly connected multiple surfaces from unorganized measured data. A hybrid mathematical model of B-spline surfaces and Catmull–Clark subdivision surfaces is introduced to represent objects with general quadrilateral topology. The interconnected multiple surfaces are G ² continuous across all surface boundaries except at a finite number of extraordinary corner points where G ¹ continuity is obtained. The algorithm is purely a linear least-squares fitting procedure without any constraint for maintaining the required geometric continuity. In case of general uniform knots for all surfaces, the final fitted multiple surfaces can also be exported as a set of Catmull–Clark subdivision surfaces with global C ² continuity and local C ¹ continuity at extraordinary corner points. Published online: 14 May 2002 Correspondence to: W. Ma     

Detection of global predicates: Techniques and their limitations

Summary. We show that the problem of predicate detection in distributed systems is NP-complete. In the past, efficient algorithms have been developed for special classes of predicates such as stable predicates, observer independent predicates, and conjunctive predicates. We introduce a class of predicates, semi-linear predicates, which properly contains all of the above classes. We first discuss stable, observer independent and semi-linear classes of predicates and their relationships with each other. We also study closure properties of these classes with respect to conjunction and disjunction. Finally, we discuss algorithms for detection of predicates in these classes. We provide a non-deterministic detection algorithm for each class of predicate. We show that each class can be equivalently characterized by the degree of non-determinism present in the algorithm. Stable predicates are defined as those that can be detected by an algorithm with the most non-determinism. All other classes can be derived by appropriately constraining the non-determinism in this algorithm.     

An optimal bandwidth allocation strategy for the delivery of compressed prerecorded video

The transportation of prerecorded, compressed video data without loss of picture quality requires the network and video servers to support large fluctuations in bandwidth requirements. Fully utilizing a client-side buffer for smoothing bandwidth requirements can limit the fluctuations in bandwidth required from the underlying network and the video-on-demand servers. This paper shows that, for a fixed-size buffer constraint, the critical bandwidth allocation technique results in plans for continuous playback of stored video that have (1) the minimum number of bandwidth increases, (2) the smallest peak bandwidth requirements, and (3) the largest minimum bandwidth requirements. In addition, this paper introduces an optimal bandwidth allocation algorithm which, in addition to the three critical bandwidth allocation properties, minimizes the total number of bandwidth changes necessary for continuous playback. A comparison between the optimal bandwidth allocation algorithm and other critical bandwidth-based algorithms using 17 full-length movie videos and 3 seminar videos is also presented.     

Asynchronous group mutual exclusion

Abstract. Mutual exclusion and concurrency are two fundamental and essentially opposite features in distributed systems. However, in some applications such as Computer Supported Cooperative Work (CSCW) we have found it necessary to impose mutual exclusion on different groups of processes in accessing a resource, while allowing processes of the same group to share the resource. To our knowledge, no such design issue has been previously raised in the literature. In this paper we address this issue by presenting a new problem, called Congenial Talking Philosophers, to model group mutual exclusion. We also propose several criteria to evaluate solutions of the problem and to measure their performance. Finally, we provide an efficient and highly concurrent distributed algorithm for the problem in a shared-memory model where processes communicate by reading from and writing to shared variables. The distributed algorithm meets the proposed criteria, and has performance similar to some naive but centralized solutions to the problem. Received: November 1998 / Accepted: April 2000     

Guard games on graphs: Keep the intruder out!

A team of mobile agents, called guards, tries to keep an intruder out of an assigned area by blocking all possible attacks. In a graph model for this setting, the guards and the intruder are located on the vertices of a graph, and they move from node to node via connecting edges. The area protected by the guards is an induced subgraph of the given graph. We investigate the algorithmic aspects of the guarding problem, which is to find the minimum number of guards sufficient to patrol the area. We show that the guarding problem is PSPACE-hard and provide a set of approximation algorithms. All approximation algorithms are based on the study of a variant of the game where the intruder must reach the guarded area in a single step in order to win. This variant of the game appears to be a 2-approximation for the guarding problem, and for graphs without cycles of length 5 the minimum number of required guards in both games coincides. We give a polynomial time algorithm for solving the one-step guarding problem in graphs of bounded treewidth, and complement this result by showing that the problem is W[1]-hard parameterized by the treewidth of the input graph. We also show that the problem is fixed parameter tractable (FPT) parameterized by the treewidth and maximum degree of the input graph. Finally, we turn our attention to a large class of sparse graphs, including planar graphs and graphs of bounded genus, namely apex-minor-free graphs. We prove that the one-step guarding problem is FPT and possess a PTAS on apex-minor-free graphs.     

Model checking for a probabilistic branching time logic with fairness

We consider concurrent probabilistic systems, based on probabilistic automata of Segala & Lynch [55], which allow non-deterministic choice between probability distributions. These systems can be decomposed into a collection of “computation trees” which arise by resolving the non-deterministic, but not probabilistic, choices. The presence of non-determinism means that certain liveness properties cannot be established unless fairness is assumed. We introduce a probabilistic branching time logic PBTL, based on the logic TPCTL of Hansson [30] and the logic PCTL of [55], resp. pCTL [14]. The formulas of the logic express properties such as “every request is eventually granted with probability at least p”. We give three interpretations for PBTL on concurrent probabilistic processes: the first is standard, while in the remaining two interpretations the branching time quantifiers are taken to range over a certain kind of fair computation trees. We then present a model checking algorithm for verifying whether a concurrent probabilistic process satisfies a PBTL formula assuming fairness constraints. We also propose adaptations of existing model checking algorithms for pCTL [4, 14] to obtain procedures for PBTL under fairness constraints. The techniques developed in this paper have applications in automatic verification of randomized distributed systems. Received: June 1997 / Accepted: May 1998