This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of labeling the nodes of any n-node graph is such a way that given the labels of two nodes u and v, one can decide whether u and v are k-vertex connected in G, i.e., whether there exist k vertex disjoint paths connecting u and v. The paper establishes an upper bound of k 2 log n on the number of bits used in a label. The best previous upper bound for the label size of such labeling scheme is 2 k log n.

We study the verification problem in distributed networks, stated as follows. Let H be a subgraph of a network G where each vertex of G knows which edges incident on it are in H. We would like to verify whether H has some properties, e.g., if it is a tree or if it is connected (every node knows in the end of the process whether H has the specified property or not). We would like to perform this verification in a decentralized fashion via a distributed algorithm. The time complexity of verification is measured as the number of rounds of distributed communication. In this paper we initiate a systematic study of distributed verification, and give almost tight lower bounds on the running time of distributed verification algorithms for many A full version of this paper is available as [5] at

Awerbuch, Afek, Plotkin, and Saks identified an important fundamental problem inherent to distributed networks, which they called the Resource Controller problem. Consider, first, the problem in which one node (called the ‘root’) is required to estimate the number of events that occurred all over the network. The counting can be viewed as a useful variant of the heavily studied and used task of topology update (that deals with collecting all remote information). The Resource Controller problem generalizes the counting problem further: such remote events are considered as requests, and the counting node, i.e., the ‘root’, also issues permits for the requests. That way, the number of request granted can be controlled (bounded). In the paper by Awerbuch et al., it was assumed that the network is spanned by a tree that may only grow, and only by allowing leaves to join the tree (after receiving a permit). In contrast, the Resource Controller presented here can operate under a more general dynamic model. Specifically, the dynamic model considered in this paper allows both controlled insertions and deletions of leaves as well as controlled insertions and deletions of internal nodes. Despite the more dynamic network model we allow, the message complexity of our controller is always at most the message complexity of the

An ancestry labeling scheme labels the nodes of any tree in such a way that ancestry queries between any two nodes can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry scheme is by its label size, that is the maximum number of bits stored in a label, taken over all n-node trees. The design of ancestry labeling schemes finds applications in XML search engines. In these contexts, even small improvements in the label size are important. As a result, following the proposal of a simple interval based ancestry scheme with label size 2 log n bits (Kannan et al., STOC 88), a considerable amount of work was devoted to improve the bound on the label size. The current state of the art upper bound is log n + O ( √ log n) bits (Abiteboul et al., SICOMP 06) which is still far from the known log n + Ω(log log n) lower bound (Alstrup et al., SODA 03). Motivated by the fact that typical XML trees have extremely small depth, this paper parameterizes the quality measure of an ancestry scheme not only by the number of nodes in the given tree but also by its depth. Our main result is the construction of an ancestry scheme that labels n-node trees of depth d with labels of size log n + 2 log d + O(1). In addition to our main result, we prove a result that may be of independent interest concerning the existence of a small universal graph for the family of trees with bounded depth. 1

Abstract. Let f be a function on pairs of vertices. An f-labeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and every two vertices u, v ∈ G, f(u, v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in F. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes. Let F(n) be a family of connected graphs of size at most n and let C(F, n) denote the collection of graphs of size at most n, such that each graph in C(F, n) is composed of a disjoint union of some graphs in F(n). We first investigate methods for translating f-labeling schemes for F(n) to f-labeling schemes for C(F, n). In particular, we show that in many cases, given an f-labeling scheme of size g(n) for a graph family F(n), one can construct an f-labeling scheme of size g(n) + log log n + O(1) for C(F, n). We also show that in several cases, the above mentioned extra additive term of log log n + O(1) is necessary. In addition, we show that the family of n-node graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size log n + O(1). This illustrates a nontrivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5 log n + O(1) and a lower bound of 4/3 log n − O(1) for the size of any such labeling scheme.

This paper describes the invited talk given at the 8th International

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In this paper, we solve the ancestry problem, which was introduced more than twenty years ago by Kannan et al. [STOC ’88], and is among the most well-studied problems in the field of informative labeling schemes. Specifically, we construct an ancestry labeling scheme for n-node trees with label size log 2 n + O(log log n) bits, thus matching the log 2 n + Ω(log log n) bits lower bound given by Alstrup et al. [SODA ’03]. Besides its optimal label size, our scheme assigns the labels in linear time, and guarantees that any ancestry query can be answered in constant time. In addition to its potential impact in terms of improving the performances of XML search engines, our ancestry scheme is also useful in the context of partially ordered sets. Specifically, for any fixed integer k, our scheme enables the construction of a universal poset of size O(n k log 4k n) for the family of n-element posets with tree-dimension at most k. This bound is almost tight thanks to a lower bound of n k−o(1) due to Alon and Scheinerman [Order ’88].

This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yes-instance there must exist a proof that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on locally checkable proofs that can be verified with a constanttime distributed algorithm. For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a 2-colouring of the graph, which only takes 1 bit per node. However, it is more difficult to prove that a graph is not bipartite—it turns out that any locally checkable proof requires Ω(log n) bits per node. In this work we classify graph problems according to their local proof complexity, i.e., how many bits per node are needed in a locally checkable proof. We establish tight or near-tight results for classical graph properties such as the chromatic number. We show that the proof complexities form a natural hierarchy of complexity classes: for many classical graph problems, the proof complexity is either 0, Θ(1), Θ(log n), or poly(n) bits per node. Among the most difficult graph properties are symmetric graphs, which require Ω(n 2) bits per node, and non-3-colourable graphs, which require Ω(n 2 / log n) bits per node—any pure graph property admits a trivial proof of size O(n 2).

Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and ( ∆ + 1)-coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the O(∆2)-coloring algorithm by Linial [28]. Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, they assume that all nodes know good estimations of one or more global parameters of the network, e.g., the maximum degree ∆ or the number of nodes n. This paper provides a rather general method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all of the state of the art non-uniform algorithms regarding MIS and Maximal Matching, as well as to many results concerning the coloring problem. (In particular, it applies to all aforementioned algorithms.) To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.