The dynamic shortest paths problem on planar graphs asks us to preprocess a planar graph G such that we may support insertions and deletions of edges in G as well as distance queries between any two nodes u, v subject to the constraint that the graph remains planar at all times. This problem has been extensively studied in both the theory and experimental communities over the past decades and gets solved millions of times every day by companies like Google, Microsoft, and Uber. The best known algorithm performs queries and updates inÕpn 2{3 q time, based on ideas of a seminal paper by Fakcharoenphol and Rao [FOCS'01]. A p1`εq-approximation algorithm of Abraham et al. [STOC'12] performs updates and queries inÕp ? nq time. An algorithm with a more practical Oppoly log nq runtime would be a major breakthrough. However, such runtimes are only known for a p1`εq-approximation in a model where only restricted weight updates are allowed due to Abraham et al. [SODA'16], or for easier problems like connectivity.In this paper, we follow a recent and very active line of work on showing lower bounds for polynomial time problems based on popular conjectures, obtaining the first such results for natural problems in planar graphs. Such results were previously out of reach due to the highly non-planar nature of known reductions and the impossibility of "planarizing gadgets". We introduce a new framework which is inspired by techniques from the literatures on distance labelling schemes and on parameterized complexity.Using our framework, we show that no algorithm for dynamic shortest paths or maximum weight bipartite matching in planar graphs can support both updates and queries in amortized Opn 1 2´ε q time, for ε ą 0, unless the classical all-pairs-shortest-paths problem can be solved in truly subcubic time, which is widely believed to be impossible. We extend these results to obtain strong lower bounds for other related problems as well as for possible trade-offs between query and update time. Interestingly, our lower bounds hold even in very restrictive models where only weight updates are allowed.

We show that there exists a graph G with Opnq nodes, such that any forest of n nodes is a node-induced subgraph of G. Furthermore, for constant arboricity k, the result implies the existence of a graph with Opn k q nodes that contains all n-node graphs of arboricity k as node-induced subgraphs, matching a Ωpn k q lower bound. The lower bound and previously best upper bounds were presented in Alstrup and Rauhe [FOCS'02]. Our upper bounds are obtained through a log 2 n`Op1q labeling scheme for adjacency queries in forests.We

For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an Opn 5{3 q-space distance oracle which answers exact distance queries in Oplog nq time for n-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space (i.e., space Opn 2´ q for some constant ą 0) either required query time polynomial in n or could only answer approximate distance queries.Furthermore, we show how to trade-off time and space: for any S ě n 3{2 , we show how to obtain an S-space distance oracle that answers queries in time Op n 5{2 S 3{2 log nq. This is a polynomial improvement over the previous planar distance oracles with opn 1{4 q query time.

In this paper we analyze a hash function for k-partitioning a set into bins, obtaining strong concentration bounds for standard algorithms combining statistics from each bin.This generic method was originally introduced by Flajolet and Martin [FOCS'83] in order to save a factor Ω(k) of time per element over k independent samples when estimating the number of distinct elements in a data stream. It was also used in the widely used HyperLogLog algorithm of Flajolet et al. [AOFA'97] and in large-scale machine learning by Li et al. [NIPS'12] for minwise estimation of set similarity.The main issue of k-partition, is that the contents of different bins may be highly correlated when using popular hash functions. This means that methods of analyzing the marginal distribution for a single bin do not apply. Here we show that a tabulation based hash function, mixed tabulation, does yield strong concentration bounds on the most popular applications of k-partitioning similar to those we would get using a truly random hash function. The analysis is very involved and implies several new results of independent interest for both simple and double tabulation, e.g. a simple and efficient construction for invertible bloom filters and uniform hashing on a given set.

We consider the Similarity Sketching problem: Given a universe rus " t0, . . . , u´1u we want a random function S mapping subsets A Ď rus into vectors SpAq of size t, such that similarity is preserved. More precisely: Given sets A, B Ď rus, define X i " rSpAqris " SpBqriss and X " ř iPrts X i . We want to have ErXs " t¨JpA, Bq, where JpA, Bq " |A X B|{|A Y B| and furthermore to have strong concentration guarantees (i.e. Chernoffstyle bounds) for X. This is a fundamental problem which has found numerous applications in data mining, large-scale classification, computer vision, similarity search, etc. via the classic MinHash algorithm. The vectors SpAq are also called sketches.The seminal tˆMinHash algorithm uses t random hash functions h 1 , . . . , h t , and stores pmin aPA h 1 pAq, . . . , min aPA h t pAqq as the sketch of A. The main drawback of MinHash is, however, its Opt¨|A|q running time, and finding a sketch with similar properties and faster running time has been the subject of several papers. Addressing this, Li et al. [NIPS'12] introduced one permutation hashing (OPH), which creates a sketch of size t in Opt`|A|q time, but with the drawback that possibly some of the t entries are "empty" when |A| " Optq. One could argue that sketching is not necessary in this case, however the desire in most applications is to have one sketching procedure that works for sets of all sizes. Therefore, filling out these empty entries is the subject of several follow-up papers initiated by Shrivastava and Li [ICML'14]. However, these "densification" schemes fail to provide good concentration bounds exactly in the case |A| " Optq, where they are needed.In this paper we present a new sketch which obtains essentially the best of both worlds. That is, a fast Opt log t`|A|q expected running time while getting the same strong concentration bounds as MinHash. Our new sketch can be seen as a mix between sampling with replacement and sampling without replacement. We demonstrate the power of our new sketch by considering popular applications in large-scale classification with linear SVM as introduced by Li et al. [NIPS'11] as well as approximate similarity search using the LSH framework of Indyk and Motwani [STOC'98]. In particular, for the j 1 , j 2 -approximate similarity search problem on a collection of n sets we obtain a data-structure with space usage Opn 1`ρ`ř APC |A|q and Opn ρ log n`|Q|q expected time for querying a set Q compared to a Opn ρ log n¨|Q|q expected query time of the classic result of Indyk and Motwani.