Annual IEEE Symposium on Foundations of Computer Science (FOCS) Amela Špica New Jersey, October 9-11, 2016 Field of theoretical computer science 307 submitted papers, 85 accepted

Linear hashing is awesome ℎ(𝑥) = ((𝑎𝑥 + 𝑏) 𝑚𝑜𝑑 𝑝) 𝑚𝑜𝑑 𝑚 𝑎,𝑏 𝜖 [𝑝] are chosen uniformly at random, p is prime (2-independent), p >> m Expected time for insertion is 𝑂(1) and expected length of longest chain is 𝑂(√𝑛) Can‘t improve expected query time Upper bound for expected length of longest chain is not known to be tight for h(x) Paper shows that it is 𝑛 1 3 +𝑜(1) – based on performance of 3-independent hash functions for which upper bound is proved to be 𝑜( 𝑛 1 3 ) Consequence of this proof: Letting 𝑋 = {𝑥1,… ,𝑥𝑛} be set of keys, h(x1) is a minimum hash value with probability 𝑂( 1 √𝑛 ) for min- wise hashing, in the paper this has been improved to 𝑛 (−1+𝑜(1)) For hash tables with linear probing, using a table of size n, and containing (1−𝜀)𝑛 keys worst case query time is 𝑂(√𝑛), this has been improved to 𝑛 𝑜(1)

Edit Distance: Sketching, Streaming and Document Exchange Edit distance – minimum number of insertions, deletions and substitutions to convert string s to string t Given threshold K - ed(s, t) ≤ K Three settings are considered Document exchange – Compute message msg based on string s (encoding) and recover msg with string t (decoding). Goal is to minimize message size and encoding/decoding time. In paper it is shown that message can be of size Ō(𝐾( 𝑙𝑜𝑔 2 𝐾 + 𝑙𝑜𝑔 𝑛)) bits, where Ō(𝑓) = 𝑓 𝑝𝑜𝑙𝑦 (𝑙𝑜𝑔 𝑓) and encoding/decoding can be done in 𝑂(𝑛 + 𝑝𝑜𝑙𝑦(𝐾)) Sketching – Compute sketches sk(s) and sk(t) (encoding) and send to referee to compute ed(s, t) and all edits using sk(s) and sk(t)(decoding). Goal is to minimize sketch size. In the paper, it is shown that sketches can be of size 𝑝𝑜𝑙𝑦(𝐾 𝑙𝑜𝑔 𝑛) bits Streaming - compute 𝑒𝑑(𝑠,𝑡) and all edits with limitations of scanning string s and then t, from left to right once, using a memory of small size. Goal is to minimize memory space. In this paper, it is shown that memory can be 𝑝𝑜𝑙𝑦(𝐾 𝑙𝑜𝑔 𝑛) bits of space.

Popular Conjectures as a Barrier for Dynamic Planar Graph Algorithms The dynamic shortest paths problem on planar graphs is to preprocess a planar graph G so that insertions and deletions of edges and distance queries between two nodes u, v in G, are supported Used in GPS navigation best known algorithm performs queries and updates in Ō( 𝑛 2 3 ) time Conjecture 1 (APSP Conjecture): There exists no algorithm for solving the all pairs shortest paths (APSP) problem in general weighted (static) graphs in time O( 𝑛 3 −ε) for any ε > 0 Using framework from authors of paper it has been shown that there is no algorithm which can support both updates and queries in O( 𝑛 1 2 −ε) for any ε > 0. It shows that if we want to have 𝑛 𝑜(1) for one operation, we likely need 𝑛 1 2 −𝑜(1) for the other.