Given a graph G, the problem is to determine an acyclic orientation of G which minimizes the maximal number of changes of orientation along any shortest path in G. The corresponding value is called the rank of the graph G. The motivation for this graph theoretical problem comes from the design of deadlock-free packet routing protocols [G Tel, Deadlock-free packet switching networks, in: Introduction to Distributed Algorithms, Cambridge University Press, Cambridge, UK, 1994 (Chapter 5)]. This acyclic orientation covering problem on the shortest path systems has been. studied in [J.-C. Bermond, M. Di Ianni, M. Flammini, S. Perennes, Acyclic orientations for deadlock prevention in interconnection networks, in: 23rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), in: Lecture Notes in Computer Science, vol. 1335, SpringerVerlag, 1997, pp. 52-64] where it was shown that the general problem of determining the rank is NP-complete and some upper and lower bounds on the rank were presented for particular topologies, such as grids, tori and hypercubes. The main unresolved problem stated in [J.-C. Bermond, M. Di Ianni, M. Flammini, S. Perennes, Acyclic orientations for deadlock prevention in interconnection networks, in: 23rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), in: Lecture Notes in Computer Science, vol. 1335, Springer-Verlag, 1997, pp. 52-64] was to determine the rank values for other well-known interconnection networks and also for more general classes of graphs. In this paper we give a general lower bound argument for the rank problem and apply it to the class of involution-generated Cayley graphs which among others include hypercubes, star graphs, pancake graphs and transposition-tree based graphs [S.B. Akers, B. Krishnamurthy, A group-theoretic model for symmetric interconnection networks, MEE Transactions on Computers 38 (4) (1989) 555-565]. We also present a large class LCP(T,SP) of graphs with constant rank. This class of graphs is defined as the layered cross product [S. Even, A. Litman, Layered cross product-A technique to construct interconnection networks, Networks 29 (1997) 219-223] of layered trees and series-parallel graphs and includes among others butterflies, Bene, networks, fat-trees and meshes of trees. For some special topologies, improved lower bounds on the rank are alsc presented. We consider some of the modified versions of the rank problem as well. (C) 2007 Elsevier B.V. All rights reserved.

We present deadlock-free packet (wormhole) routing algorithms based on multi-dimensional interval schemes for certain multiprocessor interconnection networks and give their analysis in terms of the compactness and the size (the maximum number of buffers per node (per link)). The issue of a simultaneous reduction of the compactness and the size is fundamental, worth to investigate and of practical importance, as interval routing and wormhole routing have been realized in INMOS Transputer C104 Router chips. In this paper we give an evidence that for some well-known interconnection networks there are efficient deadlock-free multidimensional interval routing schemes (DFMIRS) despite of a provable nonexistence of efficient deterministic shortest path interval routing schemes (IRS). For d-dimensional butterflies we give a d-dimensional DFMIRS with constant compactness and size, while each shortest path IRS is of the compactness at least 2d/2. For d-dimensional cube connected cycles we show a d-dimensional DFMIRS with compactness and size polynomial in d, while each shortest path IRS needs compactness at least 2d/2. For d-dimensional hypercubes (tori) we present a d-dimensional DFMIRS of compactness 1 and size 2 (4), while for shortest path IRS we can achieve the reduction to 2 (5) buffers with compactness 2d−1 (O(n d−1)). We also present a nonconstant lower bound (in the form √d) on the size of deadlock-free packet routing (based on acyclic orientation covering) for a special set of routing paths on d-dimensional hypercubes.

Pebble game on dynamic graphs is studied as an abstract model for the incremental computations. We investigate how the time T and/or the space S is changing according to the number m of insert-edge/delete-edge operations on directed acyclic graphs of size n. Time-space trade-off of the form T= Θ (n2/S+m) is given for standard pebble game on permutation graphs. In the case of the minimal space pebbling an extreme (superpolynomial) explosion of the time is related to the unit change of the graph size. If the space is from a small interval then a superpolynomial time increase is also achieved relative to the small (slowly increasing) change of the graph size.

Given a graph G , the problem is to construct a smallest subset S of vertices whose deletion results in an acyclic subgraph. The set S is called a minimum feedback vertex set for G . Tight upper and lower bounds on the cardinality of minimum feedback vertex sets have been previously obtained for some hypercube-like networks, such as meshes, tori, butterflies, cube-connected cycles and hypercubes. In this paper we construct minimum feedback vertex sets and determine their cardinalities in certain shuffle-based interconnection networks, such as shuffle-exchange, de Bruijn and Kautz networks.

We consider a well-known distributed colouring game played on a simple connected graph: initially, each vertex is coloured black or white; at each round, each vertex simultaneously recolours itself by the colour of the simple (strong) majority of its neighbours. A set of vertices M is said to be a dynamo, if starting the game with only the vertices of M coloured black, the computation eventually reaches an all-black configuration. The importance of this game follows from the fact that it models the spread of faults in point-to-point systems with majority-based voting; in particular, dynamos correspond to those sets of initial failures which will lead the entire system to fail. Investigations on dynamos have been extensive but restricted to establishing tight bounds on the size (i.e., how small a dynamic monopoly might be). In this paper we start to study dynamos systematically with respect to both the size and the time (i.e., how many rounds are needed to reach all-black configuration) in various models and topologies. We derive tight tradeoffs between the size and the time for a number of regular graphs, including rings, complete d-ary trees, tori, wrapped butterflies, cube connected cycles and hypercubes. In addition, we determine optimal size bounds of irreversible dynamos for butterflies and shuffle-exchange using simple majority and for DeBruijn using strong majority rules. Finally, we make some observations concerning irreversible versus reversible monotone models and slow complete computations from minimal dynamos.

In this paper we present a general and still flexible modular technique for the design of efficient leader election algorithms in N -node networks. Our approach can be viewed as a generalization of the previous method introduced by Korach, Kutten and Moran [7]. We show how well-known O(N) message leader election algorithms in oriented hypercubes and tori [12,11,15,16] can be derived by our technique. This is in contrast with Ω( N log N ) message lower bound for the approach in [7]. Moreover, our technique can be used to design new linear leader election algorithms for unoriented butterflies and cube connected cycles, thus demonstrating its usefulness. This is an improvement over the O ( N log N ) solutions obtained from the general leader election algorithm [5]. These results are of interest, since tori and corresponding chordal rings were the only known symmetric topologies for which linear leader election algorithms in unoriented case were known [11,15].

We give a survey of recent theoretical results for communication problems in point to point networks. This survey is based on the previous surveys in [57,25]. Communication problems are studied as routing path systems satisfying given communication patterns in a network. Efficiency parameters of path systems such as congestion, dilation, stretch factor, compactness and buffer size are considered. We focus on the current research directions and the various techniques that are used. Open problems related to this line of research and an overview of several related research directions are given.

Mobile agents operating in networked environments face threats from other agents as well as from the hosts (i.e., network sites) they visit. A black hole is a harmful host that destroys incoming agents without leaving any trace. To determine the location of such a harmful host is a dangerous but crucial task, called black hole search. The most important parameter for a solution strategy is the number of agents it requires (the size); the other parameter of interest is the total number of moves performed by the agents (the cost). Any solution requires moves in general networks; the same lower bound holds for rings. In this paper we show that this lower bound does not hold for hypercubes and related networks. In fact, we present a general strategy which allows two agents to locate the black hole with moves in hypercubes, cube-connected cycles, star graphs, wrapped butterflies, chordal rings, as well as in multidimensional meshes and tori of restricted diameter.

We present deadlock-free packet/wormhole routing algorithms ba\-sed on multidimensional interval schemes for certain hypercube related multiprocessor interconnection networks and give their analysis in terms of the compactness (i.e.~the maximum number of intervals per link) and the buffer-size (i.e.~the ma\-xi\-mum number of buffers per node/link). The issue of a simultaneous reduction of the compactness and the buffer-size is fundamental, worth to investigate and of practical importance, since the interval routing and wormhole routing have been industrially realized in INMOS Transputer C104 Router chips. In this paper we give an evidence that for some well-known interconnection networks there are efficient deadlock-free multidimensional interval routing schemes (DFMIRS) despite of a provable non-existence of efficient deterministic shortest path interval routing schemes (IRS). For $d$-dimensional hypercubes (tori) we present a $d$-dimensional DFMIRS of compactness $1$ and size $2$ (of compactness $1$ and size $4$), while for shortest path IRS we can achieve the reduction to $2$ (to at most $5$) buffers per node with compactness $2^{d-1}$ (with compactness $O(n^{d-1})$). For $d$-dimensional generalized butterflies we give a $d$-dimensional DFMIRS with compactness $2$ and size $3$, while each shortest path IRS is of the compactness at least superpolynomial in $d$. For $d$-dimensional cube-connected cycles we show a $d$-dimensional DFMIRS with compactness and size polynomial in $d$, while each shortest path IRS needs compactness at least $2^{d/2}$. We also present a nonconstant lower bound (in the form $\sqrt{d}$) on the size of deadlock-free packet routing (based on acyclic orientation covering) for a set of monotone routing paths on $d$-dimensional hypercubes.

Immunity index of a graph is the least integer c1 such that each configuration of sizec1 is immune. Catastrophic index of a graph is the least integer c2 such that each configuration of size c2 is catastrophic. This paper contains the first systematic study of immunity indices on a variety of interconnection networks and their distance from catastrophic indices.

Communication problems are studied as simple directed path systems satisfying given communication requests in point-to-point networks. Efficiency measures of these path system,, such Lis congestion, dilation, compactness and buffer-size are analyzed. We focus on some recent algorithmic developments and novel techniques for the design of efficient communication schemes. Related open problems and an overview of several related research directions axe also given.

Multi-dimensional interval routing schemes (MIRS) have been introduced in Flammini et al. [5] as an extension of interval routing schemes (IRS) defined by Santoro and Khatib [21] and van Leeuwen and Tan [16]. In this paper we study the efficiency of multi-dimensional interval routing schemes with respect to the space complexity and the congestion. We bring certain comparative complexity results for general graphs as well as for specific networks.In the first part of the paper, we present three main contributions to the complexity of shortest path MIRS.For certain hypercube-like graphs there are efficient multi-dimensional interval routing schemes despite of provable nonexistence of efficient deterministic interval routing schemes.We compare the DIS-MIRS and the CON-MIRS models introduced in [5] and prove that the DIS-MIRS model is asymptotically stronger than the CON-MIRS model when considering space requirements of the full-information shortest path routing schemes.We introduce a powerful lower bound technique on the CON-MIRS model and prove that not for every Cayley graph there exists space-efficient shortest path MIRS. (On the contrary, there does not exist a powerful space lower bound technique for DIS-MIRS model,)The congestion is a common phenomenon in networks which can completely degrade their performance. Therefore, it is reasonable to study routing schemes which are not necessarily shortest path, but allow high network throughput. In the second part of the paper we present congestion results for MIRS.We show that there exists a multipath < 2, n + 2 >-DIS-MIRS of n-dimensional cube-connected cycles with asymptotically optimal congestion.We give a tradeoff between the congestion and the space complexity of multipath MIRS on general graphs. For any graph G and given 1 less than or equal to s less than or equal to \V\ there exists a multipath [2 + [\V\/(2s)], 1]-MIRS with congestion pi + \V\(.)Delta(.)s, where n is the forwarding index of the graph G and Delta is the maximum degree of vertices in the graph G.For planar graphs of constant bounded degree there exist a multipath [O(root\V\, 1)-MIRS and a (deterministic) O(root\V\log\V\)-IRS, both with asymptotically optimal congestion. (C) 2000 Elsevier Science B.V. All rights reserved.

Interval routing is a popular compact routing method for point-to-point networks which found industrial applications in novel transputer routing technology (May and Thompson, Transputers and Routers: Components for Concurrent Machines, Inmos, 1991).Recently much effort is devoted to relate the efficiency (measured by the dilation or the stretch factor) to space requirements (measured by the compactness or the total number of memory bits) in a variety of compact routing methods (Eilam, Moran and Zaks, 10th International Workshop on Distributed Algorithms (WDAG), Lecture Notes in Computer Science, vol. 1151, Springer, Berlin, 1996, pp. 191-205; Fraigniaud and Gavoille, 8th Annual ACM Symp. on Parallel Algorithms and Architectures (SPAA), ACM Press, New York, 1996; Gavoille and Perennes, Proc. SIROCCO'96, Carleton Press, 1996, pp. 88-103; Kranakis and Krizanc, 13th Annual Symp. on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Computer Science, vol. 1046, Springer, Berlin, 1996, pp. 529-540; Meyer auf der Heide and Scheideler, Proc. 37th Symp. on Foundations of Computer Science (FOCS), November 1996; Peleg and Upfal, J. ACM 36 (1989) 510-530; Tse and Lau, Proc. SIROCCO'95, Carleton Press, 1995, pp. 123-134). We add new results in this direction for interval routing.For the shortest path interval routing we apply a technique from Flammini, van Leeuwen and Marchetti-Spaccamela (MFCS'95, Lecture Notes in Computer Science, vol. 969, Springer, Berlin, 1995, pp. 37-49) to some interconnection networks (shuffle exchange (SE), cube connected cycles (CCC), butterfly (BF) and star (S)) and get improved lower bounds on compactness in the form Omega(n(1/2-epsilon)), any epsilon > 0, for SE, Omega(root n/log n) for CCC and BF, and Omega(n(log log n/log n)(5)) for S, where n is the number of nodes in the corresponding network. Previous lower bounds for these networks were only constant (Fraigniaud and Gavoille, CONPAR'94, Lecture Notes in Computer Science, vol. 854, Springer, Berlin, 1994, pp. 785-796).For the dilation bounded interval routing we give a routing algorithm with the dilation [1.5D] and the compactness O(root n log n) on n-node networks with the diameter D. It is the first nontrivial upper bound on the dilation bounded interval routing on general networks. Moreover. we construct a network on which each interval routing with the dilation 1.5D - 3 needs the compactness at least Omega(root n). It is an asymptotical improvement over the previous lower bounds in Tse and Lau (Proc. SIROCCO'95. Carleton Press. 1995, pp. 123-134) and it is also better than independently obtained lower bounds in Tse and Lau (Proc. Computing: The Australasian Theory Symp. (CATS'97), Sydney, Australia, February 1997). (C) 2000 Elsevier Science B.V. All rights reserved.

Interval routing is an attractive space-efficient routing method for point-to-point networks (introduced in [13] and [15]) which has found industrial applications in the INMOS T9000 transputer design. Surveys of the principal theoretical results as well as recent trends in the area of interval routing can be found in [16, 5, 11]. Interval routing is based on compact routing tables, in which the set of nodes reachable via outgoing links is compactly represented in the form of intervals. The space efficiency can be measured by compactness, that is the maximum number of intervals per link. Previous research mostly concentrated on shortest path interval routing schemes (IRS for short). Shortest path IRS of compactness 1 have been designed for a number of well-known interconnection networks including trees, rings, complete bipartite graphs, meshes, tori, and hypercubes. However, there are interconnection networks having provably large [8] compactness for shortest path IRS, for example shuffle-exchange, De Bruijn, cube-connected cycles, butterfly, pancake, and star graphs. Several generalizations of IRS were therefore proposed. Multidimensional interval routing schemes (MIRS for short) were introduced in [3] and were used to represent the information on all shortest paths. MIRS with low memory requirements were proposed in [3] for hypercubes, grids, tori and certain types of chordal rings. Other efficient MIRS were designed in [12]. Certain graph operators have been found interesting in the design of communication networks. The impact of some graph operators on the compactness of interval routing has been previously studied in [6, 4, 10]. These results characterize the effect of the cartesian product, the composition, and the join of graphs on the minimum number of linear intervals needed for the optimal deterministic routing. We present the study of another graph-theoretic operation, namely the layered cross product of graphs. The Layered Cross Product (LCP in short) was introduced in [2] as a technique for constructing some complex interconnection networks on the basis of structurally simple multiplicands. Certain useful properties of networks decomposable as the layered cross product of simple graphs have been exploited. In [1], an efficient compact routing protocol was introducedv for the LCP of trees. In [7], efficient deadlock-free packet and wormhole routing protocols have been proposed for interconnection networks constructed as the layered cross product of trees and series parallel graphs. In this paper we consider the class of networks constructed as the layered cross product of trees and cycles.

We consider broadcasting on asynchronous anonymous totally unoriented n × m torus, where N=n·m is the number of nodes. We present a broadcasting algorithm with message complexity 1.43 N+O(n+m) and prove the lower bound in the form 1.14 N – O(1). This is an improvement over the previous 2N+O( N ) 2N+O(√(N)) upper bound and 1.04N #x2212;O( N ) 1.04 N – O(√(N)) lower bound achieved by Diks, Kranakis and Pelc [DKP96]. Unlike the algorithm from [DKP96], our algorithm works also on non-square tori, does not require the knowledge of sizes n and m and uses only messages of size O(1) bits. This is the first known broadcasting algorithm on unoriented tori that does not use all edges.

In this paper we introduce the notion of time optimality in the strong sense and we investigate the communication complexity of strong time-optimal distributed algorithms. We show that a strong time-optimal algorithm solving the MST problem on networks with n nodes and m links interchanges at least ( m-n ) 2 messages. We also present an Θ( m 2 ) bound on communication complexity for strong time-optimal algorithms solving the gossip problem. As a consequence, similar results can be shown for other interesting problems on arbitrary networks, such as electing a leader, determining the median and the center, computing the diameter and others.

We consider broadcasting on asynchronous anonymous totally unoriented N node hypercubes. First we generalize a technique, introduced in [3], for partial broadcasting and orientation. Using this technique we develop a broadcasting algorithm on unoriented hypercubes that uses only linear number of bits and runs in optimal time. This gives a positive answer to the question raised in [7] whether O(N) bits are su cient for broadcasting on unoriented N -node hypercubes. It is also an improvement over the previous algorithms from [3, 1] both in time and bit complexities. As an application of broadcasting, we develop an algorithm for computing identities of all nodes in unoriented hypercubes with linear number of messages. (The question was stated in [7]). This allows every subset of nodes (such as covers, independent sets, etc) to be determined in O(N) messages.

In this paper, we provide eecient broadcasting and election algorithms in unoriented hypercubes. First, O(N) broadcasting and traversing algorithms are introduced, where N is the number of hy-percube vertices. This answers aarmatively the open question stated in Tel95a] whether linear-message broadcasting and traversing can be achieved on hypercubes without sense of direction. Moreover, by exploiting new techniques we designed an O(N log log N) leader election algorithm. This is the rst known solution being able to exploit graph-theoretic properties of unoriented hypercubes such that it outperforms algorithms designed for general graphs GHS83].