Orgad Keller

Range non-overlapping indexing and successive list indexing

We present two natural variants of the indexing problem: In the range non-overlapping indexing problem, we preprocess a given text to answer queries in which we are given a pattern, and wish to find a maximal-length sequence of occurrences of the pattern in the text, such that the occurrences do not overlap with one another. While efficiently solving this problem, our algorithm even enables us to efficiently perform so in substrings of the text, denoted by given start and end locations. The methods we supply thus generalize the string statistics problem [4,5], in which we are asked to report merely the number of non-overlapping occurrences in the entire text, by reporting the occurrences themselves, even only for substrings of the text. In the related successive list indexing problem, during query-time we are given a pattern and a list of locations in the preprocessed text. We then wish to find a list of occurrences of the pattern, such that the ith occurrence is the leftmost occurrence of the pattern which starts to the right of the ith location given by the input list. Both problems are solved by using tools from computational geometry, specifically a variation of the range searching for minimum problem of Lenhof and Smid [12], here considered over a grid, in what appears to be the first utilization of range searching for minimum in an indexing-related context.

On the longest common parameterized subsequence

The well-known problem of the longest common subsequence (LCS), of two strings of lengths n and m respectively, is O(nm)-time solvable and is a classical distance measure for strings. Another well-studied string comparison measure is that of parameterized matching, where two equal-length strings are a parameterized match if there exists a bijection on the alphabets such that one string matches the other under the bijection. All works associated with parameterized pattern matching present polynomial time algorithms. There have been several attempts to accommodate parameterized matching along with other distance measures, as these turn out to be natural problems, e.g., Hamming distance, and a bounded version of edit-distance. Several algorithms have been proposed for these problems. In this paper we consider the longest common parameterized subsequence problem which combines the LCS measure with parameterized matching. We prove that the problem is NP-hard, and then show a couple of approximation algorithms for the problem.

Generalized Substring Compression

In substring compression one is given a text to preprocess so that, upon request, a compressed substring is returned. Generalized substring compression is the same with the following twist. The queries contain an additional context substring (or a collection of context substrings) and the answers are the substring in compressed format, where the context substring is used to make the compression more efficient. We focus our attention on generalized substring compression and present the first non-trivial correct algorithm for this problem. In our algorithm we inherently propose a method for finding the bounded longest common prefix of substrings, which may be of independent interest. In addition, we propose an efficient algorithm for substring compression which makes use of range searching for minimum queries. We present several tradeoffs for both problems. For compressing the substring S [i . . j ] (possibly with the substring S [*** . . β ] as a context), best query times we achieve are O (C ) and

Finding the minimum-weight k -path

O\big(C\log\big(\frac{j-i}{C}\big)\big)

On the Longest Common Parameterized Subsequence

for substring compression query and generalized substring compression query, respectively, where C is the number of phrases encoded.

Generalized substring compression

Given a weighted n-vertex graph G with integer edge-weights taken from a range [−M,M], we show that the minimum-weight simple path visiting k vertices can be found in time

Approximate string matching with stuck address bits

\tilde{O}(2^k \mathrm{poly}(k) M n^\omega) = O^*(2^k M)

Approximating Bribery in Scoring Rules

. If the weights are reals in [1,M], we provide a (1+e)-approximation which has a running time of

New Approximations for Coalitional Manipulation in General Scoring Rules.

\tilde{O}(2^k \mathrm{poly}(k) n^\omega(\log\log M + 1/\varepsilon))

New Approximation for Borda Coalitional Manipulation

. For the more general problem of k-tree, in which we wish to find a minimum-weight copy of a k-node tree T in a given weighted graph G, under the same restrictions on edge weights respectively, we give an exact solution of running time