Shay Mozes

Shortest paths in directed planar graphs with negative lengths: A linear-space O ( n log 2 n )-time algorithm

We give an <i>O</i>(<i>n</i> log² <i>n</i>)-time, linear-space algorithm that, given a directed planar graph with positive and negative arc-lengths, and given a node <i>s</i>, finds the distances from <i>s</i> to all nodes.

An optimal decomposition algorithm for tree edit distance

The edit distance between two ordered rooted trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. In this paper, we present a worst-case O(n3)-time algorithm for this problem, improving the previous best O(n3 log n)-time algorithm [7]. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems, together with a deeper understanding of the previous algorithms for the problem. We prove the optimality of our algorithm among the family of decomposition strategy algorithms--which also includes the previous fastest algorithms--by tightening the known lower bound of Ω(n2 log2 n) [4] to O(n3), matching our algorithms running time. Furthermore, we obtain matching upper and lower bounds of Θ(nm2(1+log n/m)) when the two trees have sizes m and n where m < n.

Multiple-Source Multiple-Sink Maximum Flow in Directed Planar Graphs in Near-Linear Time

We give an O(n log3 n) algorithm that, given an n-node directed planar graph with arc capacities, a set of source nodes, and a set of sink nodes, finds a maximum flow from the sources to the sinks. Previously, the fastest algorithms known for this problem were those for general graphs.

Shortest paths in planar graphs with real lengths in O(n log 2 n/ log log n) time

Given an n-vertex planar directed graphwith real edge lengths and with no negative cycles, we show how to compute single-source shortest path distances in the graph in O(n log2 n/ log log n) time with O(n) space. This improves on a recent O(n log2 n) time bound by Klein et al.

Structured recursive separator decompositions for planar graphs in linear time

Given a triangulated planar graph G on n vertices and an integer rr--division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for each region, the total number of vertices on these faces is O(√ r). We provide an algorithm for computing r--divisions with few holes in linear time. In fact, our algorithm computes a structure, called decomposition tree, which represents a recursive decomposition of G that includes r--divisions for essentially all values of r. In particular, given an exponentially increasing sequence {vec r} = (r1,r2,...), our algorithm can produce a recursive {vec r}--division with few holes in linear time. r--divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our linear-time algorithm improves upon the decomposition algorithm used in the state-of-the-art algorithm for minimum st--cut (Italiano, Nussbaum, Sankowski, and Wulff-Nilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and bounded-genus graphs).

Speeding Up HMM Decoding and Training by Exploiting Sequence Repetitions

Abstract We present a method to speed up the dynamic program algorithms used for solving the HMM decoding and training problems for discrete time-independent HMMs. We discuss the application of our method to Viterbi’s decoding and training algorithms (IEEE Trans. Inform. Theory IT-13:260–269, 1967), as well as to the forward-backward and Baum-Welch (Inequalities 3:1–8, 1972) algorithms. Our approach is based on identifying repeated substrings in the observed input sequence. Initially, we show how to exploit repetitions of all sufficiently small substrings (this is similar to the Four Russians method). Then, we describe four algorithms based alternatively on run length encoding (RLE), Lempel-Ziv (LZ78) parsing, grammar-based compression (SLP), and byte pair encoding (BPE). Compared to Viterbi’s algorithm, we achieve speedups of Θ(log n) using the Four Russians method,

Deterministic dense coding with partially entangled states

\Omega(\frac{r}{\log r})

New construction for a QMA complete three-local Hamiltonian

using RLE,

Speeding up HMM decoding and training by exploiting sequence repetitions

\Omega(\frac{\log n}{k})

Short and Simple Cycle Separators in Planar Graphs

using LZ78,