General compact labeling schemes for dynamic trees

Let F be a function on pairs of vertices. An F-labeling scheme is composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute F(u, v) for any two vertices u and v directly from their labels. As applications for labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. This paper investigates labeling schemes for dynamic trees. We consider two dynamic tree models, namely, the leaf-dynamic tree model in which at each step a leaf can be added to or removed from the tree and the leaf-increasing tree model in which the only topological event that may occur is that a leaf joins the tree. A general method for constructing labeling schemes for dynamic trees (under the above mentioned dynamic tree models) was previously developed in Korman et al. (Theory Comput Syst 37:49–75, 2004). This method is based on extending an existing static tree labeling scheme to the dynamic setting. This approach fits many natural functions on trees, such as distance, separation level, ancestry relation, routing (in both the adversary and the designer port models), nearest common ancestor etc.. Their resulting dynamic schemes incur overheads (over the static scheme) on the label size and on the communication complexity. In particular, all their schemes yield a multiplicative overhead factor of Ω(log n) on the label sizes of the static schemes. Following (Korman et al., Theory Comput Syst 37:49–75, 2004), we develop a different general method for extending static labeling schemes to the dynamic tree settings. Our method fits the same class of tree functions. In contrast to the above paper, our trade-off is designed to minimize the label size, sometimes at the expense of communication. Informally, for any function k(n) and any static F-labeling scheme on trees, we present an F-labeling scheme on dynamic trees incurring multiplicative overhead factors (over the static scheme) of on the label size and on the amortized message complexity. In particular, by setting for any , we obtain dynamic labeling schemes with asymptotically optimal label sizes and sublinear amortized message complexity for the ancestry relation, the id-based and label-based nearest common ancestor relation and the routing function. Supported in part at the Technion by an Aly Kaufman fellowship.     

Labeling schemes for weighted dynamic trees

A Distance labeling scheme is a type of localized network representation in which short labels are assigned to the vertices, allowing one to infer the distance between any two vertices directly from their labels, without using any additional information sources. As most applications for network representations in general, and distance labeling schemes in particular, concern large and dynamically changing networks, it is of interest to focus on distributed dynamic labeling schemes. The paper considers dynamic weighted trees where the vertices of the trees are fixed but the (positive integral) weights of the edges may change. The two models considered are the edge-dynamic model, where from time to time some edge changes its weight by a fixed quanta, and the increasing-dynamic model in which edge weights can only grow. The paper presents distributed approximate distance labeling schemes for the two dynamic models, which are efficient in terms of the required label size and communication complexity involved in updating the labels following the weight changes.     

A note on models for graph representations

This paper is intended more to ask questions than give answers. Specifically, we consider models for labeling schemes, and discuss issues regarding the number of labels consulted vs. the sizes of the labels.     

Compact and localized distributed data structures

This survey concerns the role of data structures for compactly storing and representing various types of information in a localized and distributed fashion. Traditional approaches to data representation are based on global data structures, which require access to the entire structure even if the sought information involves only a small and local set of entities. In contrast, localized data representation schemes are based on breaking the information into small local pieces, or labels , selected in a way that allows one to infer information regarding a small set of entities directly from their labels, without using any additional (global) information. The survey concentrates mainly on combinatorial and algorithmic techniques, such as adjacency and distance labeling schemes and interval schemes for routing, and covers complexity results on various applications, focusing on compact localized schemes for message routing in communication networks.Received: August 2001, Accepted: May 2002, Supported in part by a grant from the Israel Science Foundation.     

A note on exact distance labeling

We show that the vertices of an edge-weighted undirected graph can be labeled with labels of size O(n) such that the exact distance between any two vertices can be inferred from their labels alone in time. This improves the previous best exact distance labeling scheme that also requires O(n)-sized labels but time to compute the distance. Our scheme is almost optimal as exact distance labeling is known to require labels of length Ω(n).     

Distributed verification of minimum spanning trees

The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own state and label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper, we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (as long as W > (log n)^1+ε for some fixed ε > 0). Both our bounds improve previously known bounds for the problem. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings. A preliminary version of this work was presented in ACM PODC 2006. A. Korman was supported in part at the Technion by an Aly Kaufman fellowship. S. Kutten was supported in part by a grant from the Israeli Ministry for Science and Technology.     

Fully dynamic biconnectivity in graphs

We present an algorithm for maintaining the biconnected components of a graph during a sequence of edge insertions and deletions. It requires linear storage and preprocessing time. The amortized running time for insertions and for deletions isO(m ^2/3 ), wherem is the number of edges in the graph. Any query of the form ‘Are the verticesu andv biconnected?’ can be answered in timeO(1). This is the first sublinear algorithm for this problem. We can also output all articulation points separating any two vertices efficiently. If the input is a plane graph, the amortized running time for insertions and deletions drops toO(√n logn) and the query time isO(log² n), wheren is the number of vertices in the graph. The best previously known solution takes timeO(n ^2/3 ) per update or query.     

An efficient distributed algorithm for finding all hinge vertices in networks

Let G=(V, E) be a graph with vertex set V of size n and edge set E of size m. A vertex v∈V is called a hinge vertex if there exist two vertices in V\{v} such that their distance becomes longer when v is removed. In this paper, we present a distributed algorithm that finds all hinge vertices on an arbitrary graph. The proposed algorithm works for named static asynchronous networks and achieves O(n ²) time complexity and O(m) message complexity. In particular, the total messages exchanged during the algorithm are at most 2m(log n+nlog n+1) bits.     

An anonymous self-stabilizing algorithm for 1-maximal independent set in trees

We present an anonymous, constant-space, self-stabilizing algorithm for finding a 1-maximal independent set in tree graphs (and some rings). We show that the algorithm converges in O(n²) moves under any central daemon (one that at each time-step selects one of the privileged nodes to move).     

Distance estimation and object location via rings of neighbors

We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: low-stretch routing schemes [48], distance labeling [25], searchable small worlds [31], and triangulation-based distance estimation [34]. Focusing on metrics of low doubling dimension, we approach these problems with a common technique called rings of neighbors, which refers to a sparse distributed data structure that underlies all our constructions. Apart from improving the previously known bounds for these problems, our contributions include extending Kleinberg’s small world model to doubling metrics, and a short proof of the main result in Chan et al. [15]. Doubling dimension is a notion of dimensionality for general metrics that has recently become a useful algorithmic concept in the theoretical computer science literature. This work was done when A. Slivkins was a graduate student at Cornell University and was supported by the Packard Fellowship of Jon Kleinberg. Preliminary version of this paper has appeared in 24th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), 2005.     

Adjacency queries in dynamic sparse graphs

We deal with the problem of maintaining a dynamic graph so that queries of the form “is there an edge between u and v?” are processed fast. We consider graphs of bounded arboricity, i.e., graphs with no dense subgraphs, like, for example, planar graphs. Brodal and Fagerberg [G.S. Brodal, R. Fagerberg, Dynamic representations of sparse graphs, in: Proc. 6th Internat. Workshop on Algorithms and Data Structures (WADS'99), in: Lecture Notes in Comput. Sci., vol. 1663, Springer, Berlin, 1999, pp. 342-351] described a very simple linear-size data structure which processes queries in constant worst-case time and performs insertions and deletions in O(1) and O(logn) amortized time, respectively. We show a complementary result that their data structure can be used to get O(logn) worst-case time for query, O(1) amortized time for insertions and O(1) worst-case time for deletions. Moreover, our analysis shows that by combining the data structure of Brodal and Fagerberg with efficient dictionaries one gets O(logloglogn) worst-case time bound for queries and deletions and O(logloglogn) amortized time for insertions, with size of the data structure still linear. This last result holds even for graphs of arboricity bounded by O(logkn), for some constant k.     

Distributed algorithms for finding and maintaining a k-tree core in a dynamic network

A k-core C_k of a tree T is subtree with exactly k leaves for k?n_l, where n_l the number of leaves in T, and minimizes the sum of the distances of all nodes from C_k. In this paper first we propose a distributed algorithm for constructing a rooted spanning tree of a dynamic graph such that root of the tree is located near the center of the graph. Then we provide a distributed algorithm for finding k-core of that spanning tree. The spanning tree is constructed in two stages. In the first stage, a forest of trees is generated. In the next stage these trees are connected to form a single rooted tree. An interesting aspect of the first stage of proposed spanning algorithm is that it implicitly constructs the (convex) hull of those nodes which are not already included in the spanning forest. The process is repeated till all non root nodes of the graph have chosen a unique parent. We implemented the algorithms for finding spanning tree and its k-core. A core can be quite useful for routing messages in a dynamic network consisting of a set of mobile devices.     

Dynamic detection of subgraphs in computer networks

Most distributed algorithms for computer networks are designed to work with arbitrary graph structures. Most networks, however, can usually be decomposed into subgraphs with a specific structure. Detecting and exploiting these subgraphs can considerably reduce the storage and communication cost of the algorithm. In this paper we propose a distributed algorithm for detecting and exploiting tree subgraphs. In a network with fixed topology, the algorithm is optimal in terms of communication complexity. The algorithm also dynamically adapts to changes in network topology caused by link failure and recovery. The dynamic operation of the algorithm is incremental as only nodes that may be affected by the change reinitiate the algorithm. Another important property of our algorithm is that it requires no node identities or sequence numbers. We examine how this idea can be extended to other subgraph structures.     

Static Frequency Assignment in Cellular Networks

A cellular network is generally modeled as a subgraph of the triangular lattice. In the static frequency assignment problem, each vertex of the graph is a base station in the network, and has associated with it an integer weight that represents the number of calls that must be served at the vertex by assigning distinct frequencies per call. The edges of the graph model interference constraints for frequencies assigned to neighboring stations. The static frequency assignment problem can be abstracted as a graph multicoloring problem. We describe an efficient algorithm to multicolor optimally any weighted even or odd length cycle representing a cellular network. This result is further extended to any outerplanar graph. For the problem of multicoloring an arbitrary connected subgraph of the triangular lattice, we demonstrate an approximation algorithm which guarantees that no more than 4/3 times the minimum number of required colors are used. Further, we show that this algorithm can be implemented in a distributed manner, where each station needs to have knowledge only of the weights at a small neighborhood. Received May 13, 1997; revised August 24, 1998.     

A network flow based approach for network selection in dynamic spectrum access networks

In this paper, we present a network flow based approach for dynamic network and channel selection for secondary users in dynamic spectrum access networks. Most approaches in the current literature on dynamic spectrum access networks do not consider dynamic network and channel selection. We present a network flow framework for network selection. We show that our approach can enable re-assignment of networks to secondary users and also re-assignment of channels to secondary users within the same network. The assignments and re-assignments take into account, the interference caused to primary users, the price each secondary user is willing to pay and the quality of service (QoS) obtained by each secondary user. We obtain a bound for the maximum number of re-assignments.     

On Certificates and Lookahead in Dynamic Graph Problems

Recent work in dynamic graph algorithms has led to efficient algorithms for dynamic undirected graph problems such as connectivity. However, no efficient deterministic algorithms are known for the dynamic versions of fundamental directed graph problems like strong connectivity and transitive closure, as well as some undirected graph problems such as maximum matchings and cuts. We provide some explanation for this lack of success by presenting quadratic lower bounds on the certificate complexity of the seemingly difficult problems, in contrast to the known linear certificate complexity for the problems which have efficient dynamic algorithms. In many applications of dynamic (di)graph problems, a certain form of lookahead is available. Specifically, we consider the problems of assembly planning in robotics and the maintenance of relations in databases. These give rise to dynamic strong connectivity and dynamic transitive closure problems, respectively. We explain why it is reasonable, and indeed natural and desirable, to assume that lookahead is available in these two applications. Exploiting lookahead to circumvent their inherent complexity, we obtain efficient dynamic algorithms for strong connectivity and transitive closure. Received August 11, 1995; revised August 23, 1996.     

On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage

In this note we consider the following problem. Suppose a set of sensors is jointly trying to estimate a process. One sensor takes a measurement at every time step and the measurements are then exchanged among all the sensors. What is the sensor schedule that results in the minimum error covariance? We describe a stochastic sensor selection strategy that is easy to implement and is computationally tractable. The problem described above comes up in many domains out of which we discuss two. In the sensor selection problem, there are multiple sensors that cannot operate simultaneously (e.g., sonars in the same frequency band). Thus measurements need to be scheduled. In the sensor coverage problem, a geographical area needs to be covered by mobile sensors each with limited range. Thus from every position, the sensors obtain a different view-point of the area and the sensors need to optimize their trajectories. The algorithm is applied to these problems and illustrated through simple examples.     

Memoryless search algorithms in a network with faulty advice

In this paper, we present a randomized algorithm for a mobile agent to search for an item stored at a node t of a network, without prior knowledge of its exact location. Each node of the network has a database that will answer queries of the form “how do I find t?” by responding with the first edge on a shortest path to t. It may happen that some nodes, called liars, give bad advice. We investigate a simple memoryless algorithm which follows the advice with some fixed probability q>1/2 and otherwise chooses a random edge. If the degree of each node and number of liars k are bounded, we show that the expected number of edges traversed by the agent before finding t is bounded from above by O(d+r^k), where d is the distance between the initial and target nodes and . We also show that this expected number of steps can be significantly improved for particular topologies such as the complete graph and the torus.