Oren Weimann

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.

Consequences of Faster Alignment of Sequences

The Local Alignment problem is a classical problem with applications in biology. Given two input strings and a scoring function on pairs of letters, one is asked to find the substrings of the two input strings that are most similar under the scoring function. The best algorithms for Local Alignment run in time that is roughly quadratic in the string length. It is a big open problem whether substantially subquadratic algorithms exist. In this paper we show that for all e > 0, an O(n 2 − e ) time algorithm for Local Alignment on strings of length n would imply breakthroughs on three longstanding open problems: it would imply that for some δ > 0, 3SUM on n numbers is in O(n 2 − δ ) time, CNF-SAT on n variables is in O((2 − δ) n ) time, and Max Weight 4-Clique is in O(n 4 − δ ) time. Our result for CNF-SAT also applies to the easier problem of finding the longest common substring of binary strings with don’t cares. We also give strong conditional lower bounds for the more general Multiple Local Alignment problem on k strings, under both k-wise and SP scoring, and for other string similarity problems such as Global Alignment with gap penalties and normalized Longest Common Subsequence.

On Cartesian Trees and Range Minimum Queries

We present new results on Cartesian trees with applications in range minimum queries and bottleneck edge queries. We introduce a cache-oblivious Cartesian tree for solving the range minimum query problem, a Cartesian tree for the bottleneck edge query problem on trees and undirected graphs, and a proof that no Cartesian tree exists for the two-dimensional version of the range minimum query problem.

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,

Gene Proximity Analysis across Whole Genomes via PQ Trees1

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

A Unified Algorithm for Accelerating Edit-Distance Computation via Text-Compression

using RLE,

Using PQ trees for comparative genomics

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

The Stackelberg Minimum Spanning Tree Game

using LZ78,

Near linear time construction of an approximate index for all maximum consecutive sub-sums of a sequence

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