Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [Shiri Chechik and Christian Wulff-Nilsen, 2018] improved the state-of-the-art for light spanners by constructing a (2k-1)(1+epsilon)-spanner with O(n^(1+1/k)) edges and O_epsilon(n^(1/k)) lightness. Soon after, Filtser and Solomon [Arnold Filtser and Shay Solomon, 2016] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of O(mn^(1+1/k)) (which is faster than [Shiri Chechik and Christian Wulff-Nilsen, 2018]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Omega_epsilon(kn^(1/k)), even when randomization is used. The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an O_epsilon(n^(2+1/k+epsilon')) time spanner construction which achieves the state-of-the-art bounds. Our second result is an O_epsilon(m + n log n) time construction of a spanner with (2k-1)(1+epsilon) stretch, O(log k * n^(1+1/k) edges and O_epsilon(log k * n^(1/k)) lightness. This is an exponential improvement in the dependence on k compared to the previous result with such running time. Finally, for the important special case where k=log n, for every constant epsilon>0, we provide an O(m+n^(1+epsilon)) time construction that produces an O(log n)-spanner with O(n) edges and O(1) lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any k = omega(1). To achieve our constructions, we show a novel deterministic incremental approximate distance oracle. Our new oracle is crucial in our construction, as known randomized dynamic oracles require the assumption of a non-adaptive adversary. This is a strong assumption, which has seen recent attention in prolific venues. Our new oracle allows the order of the edge insertions to not be fixed in advance, which is critical as our spanner algorithm chooses which edges to insert based on the answers to distance queries. We believe our new oracle is of independent interest.

A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U. We give upper and lower bounds for the size of induced universal graphs for the family of graphs with n vertices of maximum degree D. Our new bounds improve several previous results except for the special cases where D is either near-constant or almost n/2. For constant even D Butler [Graphs and Combinatorics 2009] has shown O(n^(D/2)) and recently Alon and Nenadov [SODA 2017] showed the same bound for constant odd D. For constant D Butler also gave a matching lower bound. For generals graphs, which corresponds to D = n, Alon [Geometric and Functional Analysis, to appear] proved the existence of an induced universal graph with (1+o(1)) \cdot 2^((n-1)/2) vertices, leading to a smaller constant than in the previously best known bound of 16 * 2^(n/2) by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015]. In this paper we give the following lower and upper bound of binom(floor(n/2))(floor(D/2)) * n^(-O(1)) and binom(floor(n/2))(floor(D/2)) * 2^(O(sqrt(D log D) * log(n/D))), respectively, where the upper bound is the main contribution. The proof that it is an induced universal graph relies on a randomized argument. We also give a deterministic upper bound of O(n^k / (k-1)!). These upper bounds are the best known when D <= n/2 - tilde-Omega(n^(3/4)) and either D is even and D = omega(1) or D is odd and D = omega(log n/log log n). In this range we improve asymptotically on the previous best known results by Butler [Graphs and Combinatorics 2009], Esperet, Arnaud and Ochem [IPL 2008], Adjiashvili and Rotbart [ICALP 2014], Alon and Nenadov [SODA 2017], and Alon [Geometric and Functional Analysis, to appear].

Finding cycles in graphs is a fundamental problem in algorithmic graph theory. In this paper, we consider the problem of finding and reporting a cycle of length 2k in an undirected graph G with n nodes and m edges for constant k≥ 2. A classic result by Bondy and Simonovits [J. Combinatorial Theory, 1974] implies that if m ≥ 100k n1+1/k, then G contains a 2k-cycle, further implying that one needs to consider only graphs with m = O(n1+1/k).

Graph reconstruction algorithms seek to learn a hidden graph by repeatedly querying a black-box oracle for information about the graph structure. Perhaps the most well studied and applied version of the problem uses a distance oracle, which can report the shortest path distance between any pair of nodes. We introduce and study the betweenness oracle, where bet(a, m, z) is true iff m lies on a shortest path between a and z. This oracle is strictly weaker than a distance oracle, in the sense that a betweenness query can be simulated by a constant number of distance queries, but not vice versa. Despite this, we are able to develop betweenness reconstruction algorithms that match the current state of the art for distance reconstruction, and even improve it for certain types of graphs. We obtain the following algorithms: (1) Reconstruction of general graphs in O(n^2) queries, (2) Reconstruction of degree-bounded graphs in ~O(n^{3/2}) queries, (3) Reconstruction of geodetic degree-bounded graphs in ~O(n) queries In addition to being a fundamental graph theoretic problem with some natural applications, our new results shed light on some avenues for progress in the distance reconstruction problem.

Min-wise hashing is an important method for estimating the size of the intersection of sets, based on a succinct summary (a "min-hash") of each set. One application is estimation of the number of data points that satisfy the conjunction of m >= 2 simple predicates, where a min-hash is available for the set of points satisfying each predicate. This has application in query optimization and for approximate computation of COUNT aggregates.