Yefim Dinitz

Two-dimensional mapping for wireless OFDMA systems

The recent Orthogonal Frequency Division Multiple Access (OFDMA) transmission technique is gaining popularity as a preferred technology in the Broadband Wireless Access (BWA) emerging standards. In standards 802.16-2004 and 802.16e, the basic allocation units are comprised of sub-channels and OFDMA time symbols; each sub-channel is a group of sub-carriers, so that all the sub-channels are considered equally adequate to all users. We study the naturally arising new approach of two-dimensional mapping of incoming requests into the matrix that represents the system resources, where each allocation is of an arbitrary multi-rectangular shape (to the best of our knowledge, this approach has not been discussed elsewhere). We define a cost model and constraints related to practical OFDMA systems, which depend on the spatial shape of the two-dimensional allocation; the main objective function is the spatial efficiency. We show that the arising problem, even in its simplest form, is NP-hard . We present run-time efficient heuristic solutions for various mapping problems, taking into account the above QoS and OFDMA related constraints. In particular, a novel solution for two-dimensional mapping under priority constraints is suggested. Extensive simulations with parameters of real systems were used to investigate the performance of the proposed solutions in terms of throughput, delay and system load. The results show that high throughput can be achieved with relatively simple mapping algorithms. We believe that the proposed two-dimensional mapping approach is prospective, due to its fitness to modern standards

Dinitz' algorithm: the original version and even's version

This paper is devoted to the max-flow algorithm of the author: to its original version, which turned out to be unknown to non-Russian readers, and to its modification created by Shimon Even and Alon Itai; the latter became known worldwide as “Dinics algorithm”. It also presents the origins of the Soviet school of algorithms, which remain unknown to the Western Computer Science community, and the substantial influence of Shimon Even on the fortune of this algorithm.

A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph

The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G=(V,E) is considered. For k?2, this problem is known to be NP-hard. Combining properties of inclusion-minimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive polynomial time algorithm for finding a (?k/2?+1)-connected subgraph with a weight at most twice the optimum to the original problem. In particular, we obtain a 2-approximation algorithm for the case k=3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(|V|3|E|)=O(|V|5).

On the single-source unsplittable flow problem

Let G=(V,E) be a capacitated directed graph with a source s and k terminals t/sub i/ with demands d/sub i/, 1/spl les/i/spl les/k. We would like to concurrently route every demand on a single path from s to the corresponding terminal without violating the capacities. There are several interesting and important variations of this unsplittable flow problem. If the necessary cut condition is satisfied, we show how to compute an unsplittable flow satisfying the demands such that the total flow through any edge exceeds its capacity by at most the maximum demand. For graphs in which all capacities are at least the maximum demand, we therefore obtain an unsplittable flow with congestion at most 2, and this result is best possible. Furthermore, we show that all demands can be routed unsplittable in 5 rounds, i.e., all demands can be collectively satisfied by the union of 5 unsplittable flows. Finally, we show that 22.6% of the total demand can be satisfied unsplittably. These results are extended to the case when the cut condition is not necessarily satisfied. We derive a 2-approximation algorithm for congestion, a 5-approximation algorithm for the number of rounds and a 4.43=1/0.226-approximation algorithm for the maximum routable demand.

On an algorithm of Zemlyachenko for subtree isomorphism

Zemlyachenkos linear time algorithm for free tree isomorphism is unique in that it also partitions the set of rooted subtrees of a given rooted tree into isomorphism equivalence classes. Unfortunately, his algorithm is very hard to follow. In this note, we use modern data structures to explain and implement Zemlyachenkos scheme. We give a full description of a free rendition of his method using some of his ideas and adding some new ones; in particular, the usage of the data structures is new.

A 3-Approximation Algorithm for Finding Optimum 4,5-Vertex-Connected Spanning Subgraphs

The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G=(V,E) is considered. For k?2, this problem is known to be NP-hard. Based on the paper of Auletta, Dinitz, Nutov, and Parente in this issue, we derive a 3-approximation algorithm for k?{4,5}. This improves the best previously known approximation ratios 416 and 41730, respectively. The complexity of the suggested algorithm is O(|V|5) for the deterministic and O(|V|4log|V|) for the randomized version. The way of solution is as follows. Analyzing a subgraph constructed by the algorithm of the aforementioned paper, we prove that all its “small” cuts have exactly two sides and separate a certain fixed pair of vertices. Such a subgraph is augmented to a k-connected one (optimally) by at most four executions of a min-cost k-flow algorithm.

Bit complexity of breaking and achieving symmetry in chains and rings

We consider a failure-free, asynchronous message passing network with <i>n</i> links, where the processors are arranged on a ring or a chain. The processors are identically programmed but have distinct identities, taken from {0, 1,… ,<i>M</i> − 1}. We investigate the communication costs of three well studied tasks: Consensus, Leader, and MaxF (finding the maximum identity). We show that in chain and ring topologies, the message complexities of all three tasks are the same. Hence, we study a finer measure of complexity: the number of transmitted <i>bits</i> required to solve a task <i>T</i>, denoted <i>BitC</i>(<i>T</i>). We prove several new lower bounds (and some simple upper bounds) that imply the following results: For the two processors case, <i>BitC</i>(Consensus) = 2 and <i>BitC</i>(Leader) = <i>BitC</i>(MaxF) = 2log₂ <i>M</i> ± <i>O</i>(1), where the gap between the lower and upper bounds is almost always 1. For a chain, <i>BitC</i>(Consensus) = Θ(<i>n</i>), <i>BitC</i>(Leader) = Θ(<i>n</i> + log <i>M</i>), and <i>BitC</i>(MaxF) = Θ(<i>n</i> log <i>M</i>). For the ring topology, we prove the lower bound of Ω(<i>n</i> log <i>M</i>) for Leader, and (hence) MaxF. We consider also a chain where the intermediate processors have no identities. We prove that <i>BitC</i>(Leader) = Θ(<i>n</i> log <i>M</i>), which is equal to <i>n</i> times the bit complexity of the problem for two processors. For the specific case when the chain length is even, we prove that <i>BitC</i>(Leader) = Θ(<i>n</i>), for both above settings. In addition, we show that for any algorithm solving MaxF, there exists an input, for which <i>every</i> execution has the bit complexity Ω(<i>n</i> log <i>M</i>) (this is not the case for Leader). In our proofs, we use both methods of distributed computing and of communication complexity theory, establishing new links between the two areas.

A 2-level cactus model for the system of minimum and minimum+1 edge-cuts in a graph and its incremental maintenance

Let A denote the cardinality of a minimum edgecut of a Multigraph G. The known cactus model represents the A-cuts of G in a clear and compact way and is used in related studies. We suggest new tools for modeling connectivity structures, and, using them, generalize the cactus model to represent all the X and (~+ I)-cuts. Representations obtained, different for A odd and even, have properties similar to those of the cactus model. In particular, they provide algorithms for the maintenance of the (A + 2)-connectivity classes of vertices (called also “(J + 2)-components”) in an arbitrary graph undergoing insertions of edges; the complexity, for A odd and even, is the same as achieved for the cases ~ = 1 and 2, respectively. As a metaresult, we give also a simple characterization of families of cuts that can be modeled by a cactus.

The connectivity carcass of a vertex subset in a graph and its incremental maintenance

Let G = (V, E) be an undirected graph, S be a subset of its vertices, ts be the set of minimum edge-cuts partitioning S. A data structure representing both cuts in I.Z,S and the partition of V by all these cuts is suggested. One can build it in ISI-1 max-flow computations in G. It can be maintained, for an arbitrary sequence of u edge insertions, in O(min{]V]. Il?l, klV12 +wa(u, IVI)}) time, where k is the size of a cut in C.g. For two vertices of G, queries asking whether they are separated by a cut in C.S are answered in O (a (q, IV t)) amortized time per query, where q is the number of queries; such a cut itself is shown in O (IVI) amortized time. The dag representation of all cuts in C,S separating two given vertices in S is obtained in O(min{lEl, klVl}) amortized time.

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

Abstract. 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 qSame-4-Class? queries and nInsert-Vertex and mInsert-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 +k2 n log (n/k)) preprocessing, where M is the number of edges initially in the graph and n is the number of its vertices.