Let G = (V; E) be an undirected weighted graph with jV j = n and jEj = m. Let k 1 be an integer. We show that G = (V; E) can be preprocessed in O(kmn ) expected time, constructing a data structure of size O(kn ), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k \Gamma 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k \Gamma 1. We show that a 1963 girth conjecture of Erdos, implies ) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal.

We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors# /k) and # /k) for some constant c, where n and # denote the number of nodes and the largest degree in the graph. The number of rounds required in order to achieve a constant or even only a polylogarithmic approximation ratio is at log n/ log log n) and#1 #/ log log #). By a simple reduction, the latter lower bounds also hold for the construction of maximal matchings and maximal independent sets.

We consider the problem of labeling the nodes of an n-node graph G with short labels in such a way that the distance between any two nodes u; v of G can be approximated eciently (in constant time) by merely inspecting the labels of u and v, without using any other information. We develop such constant approximate distance labeling schemes for the classes of trees, bounded treewidth graphs, planar graphs, k-chordal graphs, and graphs with a dominating pair (including for instance interval, permutation, and AT-free graphs). We also show lower bounds, and prove that most of our schemes are optimal in length of labels generated and in the quality of the approximation, leaving some open problems.

We investigate the importance of space when solving problems based on graph distance in the streaming model. In this model, the input graph is presented as a stream of edges in an arbitrary order. The main computational restriction of the model is that we have limited space and therefore cannot store all the streamed data; we are forced to make space-efficient summaries of the data as we go along. For a graph of n vertices and m edges, we show that testing many graph properties, including connectivity (ergo any reasonable decision problem about distances) and bipartiteness, requires Ω(n) bits of space. Given this, we then investigate how the power of the model increases as we relax our space restriction. Our main result is an efficient randomized algorithm that constructs a (2t + 1)-spanner in one pass. With high probability, it uses O(t · n 1+1/t log 2 n) bits of space and processes each edge in the stream in O(t 2 · n 1/t log n) time. We find approximations to diameter and girth via the log n constructed spanner. For t = Ω (), the space log log n requirement of the algorithm is O(n·polylog n), and the per-edge processing time is O(polylog n). We also show a corresponding lower bound of t for the approximation ratio achievable when the space restriction is O(t · n1+1/t log 2 n). We then consider the scenario in which we are allowed multiple passes over the input stream. Here, we investigate whether allowing these extra passes will compensate for a given space restriction. We show that ∗This work was supported by the DoD University Research Initiative (URI) administered by the Office of Naval Research

The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined integer ¯ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value ¯ 0 (g) is attained by a graph whose girth is exactly g. The values of ¯ 0 (g) when 3 g 8 have been known for over thirty years. For these values of g each minimal graph is unique and, apart from the case g = 7, a simple lower bound is attained. This paper is mainly concerned with what happens when g 9, where the situation is quite different. Here it is known that the simple lower bound is attained if and only if g = 12. A number of techniques are described, with emphasis on the construction of families of graphs fG i g for which the number of vertices n i and the girth g i are such that n i 2 cg i for some finite constant c. The optimum value of c is known to lie between 0:5 and 0:75. At the end of the p...

The paper presents a deterministic distributed algorithm that, given k � 1, constructs in k rounds a (2k−1, 0)-spanner of O(kn 1+1/k)edgesforeveryn-node unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)-spanner with O(kn 1+1/k) edges.) Previous distributed solutions achieving such optimal stretch-size trade-off either make use of randomization providing performance guarantees in expectation only, or perform in log Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every ɛ>0, constructs a (1 + ɛ, 2)-spanner of O(ɛ −1 n 3/2)edgesin O(ɛ −1) rounds, without any prior knowledge on the graph. Our algorithms are complemented with lower bounds, which hold even under the assumption that n is known to the nodes. It is shown that any (randomized) distributed algorithm requires k rounds in expectation to compute a (2k − 1, 0)-spanner of o(n 1+1/(k−1))edgesfork ∈{2, 3, 5}. It is also shown that for every k>1, any (randomized) distributed algorithm that constructs a spanner with fewer than n 1+1/k+ɛ edges in at most n ɛ expected rounds must stretch some distances by an additive factor of n Ω(ɛ).Inotherwords, while additive stretched spanners with O(n 1+1/k) edges may exist, e.g., for k =2, 3, they cannot be computed distributively in a sub-polynomial number of rounds in expectation. Supported by the équipe-projet INRIA “DOLPHIN”. Supported by the ANR-project “ALADDIN”, and the

We explore problems related to computing graph distances in the data-stream model. The goal is to design algorithms that can process the edges of a graph in an arbitrary order given only a limited amount of working memory. We are motivated by both the practical challenge of processing massive graphs such as the web graph and the desire for a better theoretical understanding of the datastream model. In particular, we are interested in the trade-offs between model parameters such as perdata-item processing time, total space, and the number of passes that may be taken over the stream. These trade-offs are more apparent when considering graph problems than they were in previous streaming work that solved problems of a statistical nature. Our results include the following: (1) Spanner construction: There exists a single-pass, Õ(tn1+1/t)-space, Õ(t2n1/t)-time-per-edge algorithm that constructs a (2t + 1)-spanner. For t =Ω(logn/log log n), the algorithm satisfies the semistreaming space restriction of O(n polylog n) and has per-edge processing time O(polylog n). This resolves an open question from [J. Feigenbaum et al., Theoret. Comput. Sci., 348 (2005), pp. 207–216]. (2) Breadth-first-search (BFS) trees: For any even constant k, we show that any algorithm that computes the first k layers of a BFS tree from a prescribed node with probability at least 2/3 requires either greater than k/2 passes or ˜Ω(n1+1/k) space. Since constructing BFS trees is

An additive spanner of an unweighted undirected graph G with distortion d is a subgraph H such that for any two vertices u, v ∈ G, we have δH(u, v) ≤ δG(u, v) + d. For ln n every k = O (), we construct a graph G on n vertices ln ln n for which any additive spanner of G with distortion 2k − 1 has Ω ( 1 k n1+1/k) edges. This matches the lower bound previously known only to hold under a 1963 conjecture of Erdös. We generalize our lower bound in a number of ways. First, we consider graph emulators introduced by Dor, Halperin, and Zwick (FOCS, 1996), where an emulator of an unweighted undirected graph G with distortion d is like an additive spanner except H may be an arbitrary weighted graph such that δG(u, v) ≤ δH(u, v) ≤ δG(u, v) + d. We show a lower bound of Ω ( 1 k 2 n 1+1/k) edges for distortion-(2k − 1) emulators. These are the first non-trivial bounds for k> 3. Second, we parameterize our bounds in terms of the minimum degree of the graph. Namely, for minimum degree n 1/k+c for any c ≥ 0, we prove a bound of Ω ( 1 k n1+1/k−c(1+2/(k−1)) ) for addi-tive spanners and Ω ( 1 k 2 n 1+1/k−c(1+2/(k−1)) ) for emulators. For k = 2 these can be improved to Ω(n 3/2−c). This partially answers a question of Baswana et al (SODA, 2005) for additive spanners. Finally, we continue the study of pair-wise and source-wise distance preservers defined by Coppersmith and Elkin (SODA, 2005) by considering their approximate variants and their relaxation to emulators. We prove the first lower bounds for such graphs.

One of the fundamental trade-offs in compact routing schemes is between the space used to store the routing table on each node and the stretch factor of the routing scheme – the ratio between the cost of the route induced by the scheme and the cost of a minimum cost path between the same pair. Using a distributed Kolmogorov Complexity argument, we give a lower bound for the name-independent model that applies even to single-source schemes and does not require a girth conjecture. For any integer k ≥ 1 we prove that any routing scheme for networks with arbitrary weights and arbitrary node names (even a single-source routing scheme) with maximum stretch strictly less than 2k + 1 requires Ω((n log n) 1/k)-bit routing tables. We extend our results to lower bound the average-stretch, showing that for any integer k ≥ 1 any name-independent routing scheme with (n/(9k)) 1/k-bit routing tables has average-stretch of at least k/4 + 7/8. This result is in sharp contrast to recent results on the average-stretch of labeled routing schemes.

The main treasure that Paul Erdős has left us is his collection of problems, most of which are still open today. These problems are seeds that Paul sowed and watered by giving numerous talks at meetings big and small, near and far. In the past, his problems have spawned many areas in graph theory and beyond (e.g., in number theory, probability, geometry, algorithms and complexity theory). Solutions or partial solutions to Erdős problems usually lead to further questions, often in new directions. These problems provide inspiration and serve as a common focus for all graph theorists. Through the problems, the legacy of Paul Erdős continues (particularly if solving one of these problems results in creating three new problems, for example.) There is a huge literature of almost 1500 papers written by Erdős and his (more than 460) collaborators. Paul wrote many problem papers, some of which appeared in various (really hard-to-find) proceedings. Here is an attempt to collect and organize these problems in the area of graph theory. The list here is by no means complete or exhaustive. Our goal is to state the problems, locate the sources, and provide the references related to these problems. We will include the earliest and latest known references without covering the entire history of the problems because of space limitations. (The most up-to-date list of Erdős’ papers can be found in [65]; an electronic file is maintained by Jerry Grossman at