Deterministic broadcasting in ad hoc radio networks

We consider the problem of distributed deterministic broadcasting in radio networks of unknown topology and size. The network is synchronous. If a node u can be reached from two nodes which send messages in the same round, none of the messages is received by u. Such messages block each other and node u either hears the noise of interference of messages, enabling it to detect a collision, or does not hear anything at all, depending on the model. We assume that nodes know neither the topology nor the size of the network, nor even their immediate neighborhood. The initial knowledge of every node is limited to its own label. Such networks are called ad hoc multi-hop networks. We study the time of deterministic broadcasting under this scenario. For the model without collision detection, we develop a linear-time broadcasting algorithm for symmetric graphs, which is optimal, and an algorithm for arbitrary n-node graphs, working in time . Next we show that broadcasting with acknowledgement is not possible in this model at all. For the model with collision detection, we develop efficient algorithms for broadcasting and for acknowledged broadcasting in strongly connected graphs. Received: January 2000 / Accepted: June 2001     

Deterministic Communication in Radio Networks with Large Labels

We study deterministic gossiping in ad hoc radio networks with large node labels. The labels (identifiers) of the nodes come from a domain of size N which may be much larger than the size n of the network (the number of nodes). Most of the work on deterministic communication has been done for the model with small labels which assumes N = O(n). A notable exception is Peleg's paper, where the problem of deterministic communication in ad hoc radio networks with large labels is raised and a deterministic broadcasting algorithm is proposed, which runs in O(n²log n) time for N polynomially large in n. The O(nlog²n)-time deterministic broadcasting algorithm for networks with small labels given by Chrobak et al. implies deterministic O(n log N log n)-time broadcasting and O(n²log²N log n)-time gossiping in networks with large labels. We propose two new deterministic gossiping algorithms for ad hoc radio networks with large labels, which are the first such algorithms with subquadratic time for polynomially large N. More specifically, we propose: a deterministic O(n^3/2log²N log n)-time gossiping algorithm for directed networks; and a deterministic O(n log²N log²n)-time gossiping algorithm for undirected networks.     

Study of General Icomplete Star Interconnection Networks

The star networks,which were originally proposed by Akers and Harel,have suffered from a rigorous restriction on the number of nodes.The general incomplete star networks(GISN) are proposed in this paper to relieve this restriction.An efficient labeling scheme for GISN is given,and routing and broadcasting algorithms are also presented for GIS.The communication diameter of GISN is shown to be bounded by 4n-7.The proposed single node broadcasting algorithm is optimal with respect to time complexity O(nlog2n).     

Improved algorithm for all pairs shortest paths

We present an improved algorithm for all pairs shortest paths. For a graph of n vertices our algorithm runs in O(n³(loglogn/logn)^5/7) time. This improves the best previous result which runs in O(n³(loglogn/logn)^1/2) time.     

Improved one-to-all broadcasting algorithms on faulty SIMD hypercubes

Consider an n-dimensional SIMD hypercube H_n with 3n/2-1 faulty nodes. With , and n+19 steps, this paper presents some one-to-all broadcasting algorithms on the faulty SIMD H_n. The sequence of dimensions used for broadcasting in each algorithm is the same regardless of which node is the source. The proposed one-to-all broadcasting algorithms can tolerate n/2 more faulty nodes than Raghavendra and Sridhar's algorithms (J. Parallel Distrb. Comput. 35 (1996) 57) although 8 extra steps are needed. The fault-tolerance improvement of this paper is about 50%.     

The reverse greedy algorithm for the metric k-median problem

The Reverse Greedy algorithm (RGreedy) for the k-median problem works as follows. It starts by placing facilities on all nodes. At each step, it removes a facility to minimize the total distance to the remaining facilities. It stops when k facilities remain. We prove that, if the distance function is metric, then the approximation ratio of RGreedy is between Ω(logn/loglogn) and O(logn).     

Towards optimal range medians

We consider the following problem: Given an unsorted array of n elements, and a sequence of intervals in the array, compute the median in each of the subarrays defined by the intervals. We describe a simple algorithm which needs O(nlogk+klogn) time to answer k such median queries. This improves previous algorithms by a logarithmic factor and matches a comparison lower bound for k=O(n). The space complexity of our simple algorithm is O(nlogn) in the pointer machine model, and O(n) in the RAM model. In the latter model, a more involved O(n) space data structure can be constructed in O(nlogn) time where the time per query is reduced to O(logn/loglogn). We also give efficient dynamic variants of both data structures, achieving O(log²n) query time using O(nlogn) space in the comparison model and O((logn/loglogn)²) query time using O(nlogn/loglogn) space in the RAM model, and show that in the cell-probe model, any data structure which supports updates in O(log^O(1)n) time must have Ω(logn/loglogn) query time.Our approach naturally generalizes to higher-dimensional range median problems, where element positions and query ranges are multidimensional—it reduces a range median query to a logarithmic number of range counting queries.     

Broadcasting in undirected ad hoc radio networks

We consider distributed broadcasting in radio networks, modeled as undirected graphs, whose nodes have no information on the topology of the network, nor even on their immediate neighborhood. For randomized broadcasting, we give an algorithm working in expected time in n-node radio networks of diameter D, which is optimal, as it matches the lower bounds of Alon et al. [1] and Kushilevitz and Mansour [16]. Our algorithm improves the best previously known randomized broadcasting algorithm of Bar-Yehuda, Goldreich and Itai [3], running in expected time . (In fact, our result holds also in the setting of n-node directed radio networks of radius D.) For deterministic broadcasting, we show the lower bound on broadcasting time in n-node radio networks of diameter D. This implies previously known lower bounds of Bar-Yehuda, Goldreich and Itai [3] and Bruschi and Del Pinto [5], and is sharper than any of them in many cases. We also give an algorithm working in time , thus shrinking - for the first time - the gap between the upper and the lower bound on deterministic broadcasting time to a logarithmic factor. Received: 1 August 2003, Accepted: 18 March 2005, Published online: 15 June 2005 Dariusz R. Kowalski: This work was done during the stay of Dariusz Kowalski at the Research Chair in Distributed Computing of the Université du Québec en Outaouais, as a postdoctoral fellow. Research supported in part by KBN grant 4T11C04425. Andrzej Pelc: Research of Andrzej Pelc was supported in part by NSERC discovery grant and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais.     

All-port total exchange in cartesian product networks

We present a general solution to the total exchange (TE) communication problem for any homogeneous multidimensional network under the all-port assumption. More specifically, we consider cartesian product networks where every dimension is the same graph (e.g. hypercubes, square meshes, n-ary d-cubes) and where each node is able to communicate simultaneously with all its neighbors. We show that if we are given an algorithm for a single n-node dimension which requires T steps, we can construct an algorithm for d-dimensions and running time of n^d−1T steps, which is provably optimal for many popular topologies. Our scheme, in effect, generalizes the TE algorithm given by Bertsekas et al. (J. Parallel Distrib. Comput. 11 (1991) 263–275) for the hypercubes and complements our theory (IEEE Trans. Parallel Distrib. Systems 9(7) (1998) 639) for the single-port model.     

Optimal Broadcasting in Faulty Trees

We consider broadcasting a message from one node of a tree to all other nodes. In the presence of up to k link failures the tree becomes disconnected, and only nodes in the connected component C containing the source can be informed. The maximum ratio between the time used by a broadcasting scheme B to inform C and the optimal time to inform C, taken over all components C yielded by configurations of at most k faults, is the k-vulnerability of B. This is the maximum slowdown incurred by B due to the lack of a priori knowledge of fault location, for at most k faults. This measure of fault tolerance is similar to the competitive factor of on-line algorithms: in both cases, the performance of an algorithm lacking some crucial information is compared to the performance of an “off-line” algorithm, one that is given this information as input. It is also the first known tool to measure and compare fault tolerance of broadcasting schemes in trees. We seek broadcasting schemes with low vulnerability, working for tree networks. It turns out that schemes that give the best broadcasting time in a fault-free environment may have very high vulnerability, i.e., poor fault tolerance, for some trees. The main result of this paper is an algorithm that, given an arbitrary tree T and an integer k, computes a broadcasting scheme B with lowest possible k-vulnerability among all schemes working for T. Our algorithm has running time O(kn²+n² log n), where n is the size of the tree. We also give an algorithm to find a “universally fault-tolerant” broadcasting scheme in a tree T: one that approximates the lowest possible k-vulnerability, for all k simultaneously.     

Improved algorithm for the symmetry number problem on trees

The symmetry number of a tree is defined as the number of nodes of the maximum subtree of the tree that exhibits axial symmetry. The best previous algorithm for computing the symmetry number for an unrooted unordered tree is due to [P.P. Mitra, M.A.U. Abedin, M.A. Kashem, Algorithms for solving the symmetry number problem on trees, Inform. Process. Lett. 91 (2004) 163-169] and runs in O(n³) time. In this paper we show an improvement on this time complexity by encoding small trees. Our algorithm runs in time O(n³²(loglogn/logn)).     

A note on constructing binary heaps with periodic networks

We consider the problem of constructing binary heaps on constant degree networks performing compare-exchange operations only. The heap data structure, introduced by William and Williams [Comm. ACM 7 (6) (1964) 347-348], has many applications and, therefore, has been intensively studied in sequential and parallel context. In particular, Brodal and Pinotti [Theoret. Comput. Sci. 250 (2001) 235-245] have recently presented two families of comparator networks: the first of depth 4logN and the second of size O(NloglogN) for constructing binary heaps of size N. In this note, we give an new construction of such a network with the running time improved to 3logN. Moreover, the network has a novel property of being 3-periodic, that is, for each unit of time i the same sets of operations are performed in units i and i+3. Then we argue that our construction is optimal with respect to the length of the period, that is, we prove that there is no 2-periodic network that is able to build a binary heap in sublinear time. Finally, we show that our construction can be used to decrease also the depth of the networks with O(NloglogN) size.     

An improved algorithm for the maximum agreement subtree problem

In this paper, we solve the maximum agreement subtree problem for a set T of k rooted leaf-labeled evolutionary trees on n leaves where T contains a binary tree. We show that the O(kn³)-time dynamic-programming algorithm proposed by Bryant [Building trees, hunting for trees, and comparing trees: theory and methods in phylogenetic analysis, Ph.D. thesis, Dept. Math., University of Canterbury, 1997, pp. 174-182] can be implemented in O(kn²+n²logk−2nloglogn) and O(kn3−1/(k−1)) time by using multidimensional range search related data structures proposed by Gabow et al. [Scaling and related techniques for geometry problems, in: Proc. 16th Annual ACM Symp. on Theory of Computing, 1984, pp. 135-143] and Bentley [Multidimensional binary search trees in database applications, IEEE Trans. Softw. Eng. SE-5 (4) (1979) 333-340], respectively. When k<2+(logn−logloglogn)/(loglogn), the first implementation will be significantly faster than Bryant's algorithm. For k=3, it yields the best known algorithm which runs in O(n²lognloglogn)-time.     

Upper and lower bounds for deterministic broadcast in powerline communication networks

Powerline communication networks assume an interesting position in the communication network space: similarly to wireless networks, powerline networks are based on a shared broadcast medium; unlike wireless networks, however, the signal propagation is constrained to the power lines of the electrical infrastructure, which is essentially a graph. This article presents an algorithmic model to study the design of communication services over powerline communication networks. As a case study, we focus on the fundamental broadcast problem, and present and analyze a distributed algorithm \(\textsc {Color}\textsc {Cast}\) which terminates in at most n communication rounds, where n denotes the network size, even in a model where link qualities are unpredictable and time-varying. For comparison, the achieved broadcast time is lower than what can be achieved by any unknown-topology algorithm (lower bounds \(\varOmega (n\log n / \log (n/D))\) and \(\varOmega (n \log D)\) are proved in Kowalski and Pelc (Distrib Comput 18(1):43–57, 2005) resp. Clementi et al. (2001) where D is the network diameter). Moreover, existing known-topology broadcast algorithms often fail to deliver the broadcast message entirely in this model. This article also presents a general broadcast lower bound for the powerline model.     

Bounds for the Element Distinctness Problem on one-tape Turing machines

We disprove a conjecture of López-Ortiz by showing that the Element Distinctness Problem for n numbers of size O(logn) can be solved in O(n²(logn)^3/2(loglogn)^1/2) steps by a nondeterministic one-tape Turing machine. Further we give a simplified algorithm for solving the problem for shorter numbers in time O(n²logn) on a deterministic one-tape Turing machine and a new proof of the matching lower bound.     

PARALLEL INCREMENTAL ALGORITHMS FOR ANALYZING ACTIVITY NETWORKS

This paper presents parallel incremental algorithms for analyzing activity networks. The start-over algorithm used for this problem is a modified version of an algorithm due to Chaudhuri and Ghosh (BIT 26 (1986), 418-429). The computational model used is a shared memory single-instruction stream, multiple-data stream computer that allows both read and write conflicts. It is shown that the incremental algorithms for the event and activity insertion problems both require only O(loglogn) parallel time, in contrast to O(logn log logn) parallel time for the corresponding start-over algorithm.     

Faster communication in known topology radio networks

This paper concerns the communication primitives of broadcasting (one-to-all communication) and gossiping (all-to-all communication) in known topology radio networks, i.e., where for each primitive the schedule of transmissions is precomputed in advance based on full knowledge about the size and the topology of the network. The first part of the paper examines the two communication primitives in arbitrary graphs. In particular, for the broadcast task we deliver two new results: a deterministic efficient algorithm for computing a radio schedule of length D + O(log³ n), and a randomized algorithm for computing a radio schedule of length D + O(log² n). These results improve on the best currently known D + O(log⁴ n) time schedule due to Elkin and Kortsarz (Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 222–231, 2005). Later we propose a new (efficiently computable) deterministic schedule that uses 2D + Δlog n + O(log³ n) time units to complete the gossiping task in any radio network with size n, diameter D and max-degree Δ. Our new schedule improves and simplifies the currently best known gossiping schedule, requiring time , for any network with the diameter D = Ω(log ⁱ⁺⁴ n), where i is an arbitrary integer constant i ≥ 0, see Gąsieniec et al. (Proceedings of the 11th International Colloquium on Structural Information and Communication Complexity, vol. 3104, pp. 173–184, 2004). The second part of the paper focuses on radio communication in planar graphs, devising a new broadcasting schedule using fewer than 3D time slots. This result improves, for small values of D, on the currently best known D + O(log³ n) time schedule proposed by Elkin and Kortsarz (Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 222–231, 2005). Our new algorithm should be also seen as a separation result between planar and general graphs with small diameter due to the polylogarithmic inapproximability result for general graphs by Elkin and Kortsarz (Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, vol. 3122, pp. 105–116, 2004; J. Algorithms 52(1), 8–25, 2004). The second author is supported in part by a grant from the Israel Science Foundation and by the Royal Academy of Engineering. Part of this research was performed while this author (Q. Xin) was a PhD student at The University of Liverpool.     

Computing rank-width exactly

We prove that the rank-width of an n-vertex graph can be computed exactly in time O(n²n³log²nloglogn). To improve over a trivial O(n³logn)-time algorithm, we develop a general framework for decompositions on which an optimal decomposition can be computed efficiently. This framework may be used for other width parameters, including the branch-width of matroids and the carving-width of graphs.     

A randomized sublinear time parallel GCD algorithm for the EREW PRAM

We present a randomized parallel algorithm that computes the greatest common divisor of two integers of n bits in length with probability 1−o(1) that takes O(nloglogn/logn) time using O(n6+?) processors for any ?>0 on the EREW PRAM parallel model of computation. The algorithm either gives a correct answer or reports failure.We believe this to be the first randomized sublinear time algorithm on the EREW PRAM for this problem.     

Range mode and range median queries in constant time and sub-quadratic space

Given a list of n items and a function defined over sub-lists, we study the space required for computing the function for arbitrary sub-lists in constant time.For the function mode we improve the previously known space bound O(n²/logn) to O(n²loglogn/log²n) words.For median the space bound is improved to O(n²loglog²n/log²n) words from O(n²⋅log(k)n/logn), where k is an arbitrary constant and log(k) is the iterated logarithm.