Jesper Jansson

Compressed dynamic tries with applications to LZ-compression in sublinear time and space

The dynamic trie is a fundamental data structure which finds applications in many areas. This paper proposes a compressed version of the dynamic trie data structure. Our data-structure is not only space efficient, it also allows pattern searching in o(|P|) time and leaf insertion/ deletion in o(log n) time, where |P| is the length of the pattern and n is the size of the trie. To demonstrate the usefulness of the new data structure, we apply it to the LZ-compression problem. For a string S of length s over an alphabet A of size σ, the previously best known algorithms for computing the Ziv-Lempel encoding (lz78) of S either run in: (1) O(s) time and O(s log s) bits working space; or (2) O(sσ) time and O(sHk + s log σ/ logσ s) bits working space, where Hk is the k- order entropy of the text. No previous algorithm runs in sublinear time. Our new data structure implies a LZ-compression algorithm which runs in sublinear time and uses optimal working space. More precisely, the LZ-compression algorithm uses O(s(log σ +log logσ s)/ logσ s) bits working space and runs in O(s(log log s)2/(logσ s log log log s)) worst-case time, which is sublinear when σ = 2o(log slog log log s/(log log s)2).

Approximation Algorithms for the Graph Orientation Minimizing the Maximum Weighted Outdegree

Given an undirected graph G= (V,E) and a weight function w: Ei¾?i¾?+, we consider the problem of orienting all edges in Eso that the maximum weighted outdegree among all vertices is minimized. In this paper (1) we prove that the problem is strongly NP-hard if all edge weights belong to the set {1,k}, where kis any integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1 + 1/k) unless P=NP; (2) we present a polynomial time algorithm that approximates the general version of the problem within a factor of (2 i¾? 1/k), where kis the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1,k} within a factor of 3/2 for k= 2 (note that this matches the inapproximability bound above), and (2 i¾? 2/(k+ 1)) for any ki¾? 3, respectively, in polynomial time.

GRAPH ORIENTATION TO MAXIMIZE THE MINIMUM WEIGHTED OUTDEGREE

We study a new a variant of the graph orientation problem called MAXMINO where the input is an undirected, edge-weighted graph and the objective is to assign a direction to each edge so that the minimum weighted outdegree (taken over all vertices in the resulting directed graph) is maximized. All edge weights are assumed to be positive integers. This problem is closely related to the job scheduling on parallel machines, called the machine covering problem, where its goal is to assign jobs to parallel machines such that each machine is covered as much as possible. First, we prove that MAXMINO is strongly NP-hard and cannot be approximated within a ratio of 2 = ∈ for constant ∈ ≫ 0 in polynomial time unless P=NP, even if all edge weights belong to {2}, every vertex has degree at most three, and the input graph is bipartite or planar. Next, we show how to solve MAXMINO exactly in polynomial time for the special case in which all edge weights are equal to 1. This technique gives us a simple polynomial-time wmax/wmin-approximation algorithm for MAXMINO where wmax and wmin denote the maximum and minimum weights among all the input edges. Furthermore. we also observe that this approach yields an exact algorithm for the general case of MAXMINO whose running time is polynomial whenever the number of edges having weight larger than wmin is at most logarithmic in the number of vertices. Finally, we, show that MAXMINO is solvable in polynomial time if the input is a cactus graph.

Polynomial-Time Algorithms for the Ordered Maximum Agreement Subtree Problem

AbstractFor a set of rooted, unordered, distinctly leaf-labeled trees, the NP-hard maximum agreement subtree problem (MAST) asks for a tree contained (up to isomorphism or homeomorphism) in all of the input trees with as many labeled leaves as possible. We study the ordered variants of MAST where the trees are uniformly or non-uniformly ordered. We provide the first known polynomial-time algorithms for the uniformly and non-uniformly ordered homeomorphic variants as well as the uniformly and non-uniformly ordered isomorphic variants of MAST.nOur algorithms run in time

Algorithms for finding a most similar subforest

O(kn^3)

Ultra-succinct representation of ordered trees

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Compressed random access memory

O(n^3 min {kn,, n+log^{k-1}n})

On the complexity of computing evolutionary trees

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Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree

O(kn^3)

On the approximability of maximum and minimum edge clique partition problems

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