23,011 research outputs found
Pivot Selection for Median String Problem
The Median String Problem is W[1]-Hard under the Levenshtein distance, thus,
approximation heuristics are used. Perturbation-based heuristics have been
proved to be very competitive as regards the ratio approximation
accuracy/convergence speed. However, the computational burden increase with the
size of the set. In this paper, we explore the idea of reducing the size of the
problem by selecting a subset of representative elements, i.e. pivots, that are
used to compute the approximate median instead of the whole set. We aim to
reduce the computation time through a reduction of the problem size while
achieving similar approximation accuracy. We explain how we find those pivots
and how to compute the median string from them. Results on commonly used test
data suggest that our approach can reduce the computational requirements
(measured in computed edit distances) by \% with approximation accuracy as
good as the state of the art heuristic.
This work has been supported in part by CONICYT-PCHA/Doctorado
Nacional/ through a Ph.D. Scholarship; Universidad Cat\'{o}lica
de la Sant\'{i}sima Concepci\'{o}n through the research project DIN-01/2016;
European Union's Horizon 2020 under the Marie Sk\l odowska-Curie grant
agreement ; Millennium Institute for Foundational Research on Data
(IMFD); FONDECYT-CONICYT grant number ; and for O. Pedreira, Xunta de
Galicia/FEDER-UE refs. CSI ED431G/01 and GRC: ED431C 2017/58
Leveraging graph dimensions in online graph search
Graphs have been widely used due to its expressive power to model complicated relationships. However, given a graph database DG = {g1; g2; ··· , gn}, it is challenging to process graph queries since a basic graph query usually involves costly graph operations such as maximum common subgraph and graph edit distance computation, which are NP-hard. In this paper, we study a novel DS-preserved mapping which maps graphs in a graph database DG onto a multidimensional space MG under a structural dimension Musing a mapping function φ(). The DS-preserved mapping preserves two things: distance and structure. By the distance-preserving, it means that any two graphs gi and gj in DG must map to two data objects φ(gi) and φ(gj) in MG, such that the distance, d(φ(gi); φ(gj), between φ(gi) and φ(gj) in MG approximates the graph dissimilarity δ(gi; gj) in DG. By the structure-preserving, it further means that for a given unseen query graph q, the distance between q and any graph gi in DG needs to be preserved such that δ(q; gi) ≈ d(φ(q); φ(gi)). We discuss the rationality of using graph dimension M for online graph processing, and show how to identify a small set of subgraphs to form M efficiently. We propose an iterative algorithm DSPM to compute the graph dimension, and discuss its optimization techniques. We also give an approximate algorithm DSPMap in order to handle a large graph database. We conduct extensive performance studies on both real and synthetic datasets to evaluate the top-k similarity query which is to find top-k similar graphs from DG for a query graph, and show the effectiveness and efficiency of our approaches. © 2014 VLDB
Hardness Amplification of Optimization Problems
In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products.
We say that an optimization problem ? is direct product feasible if it is possible to efficiently aggregate any k instances of ? and form one large instance of ? such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the k smaller instances. Given a direct product feasible optimization problem ?, our hardness amplification theorem may be informally stated as follows:
If there is a distribution D over instances of ? of size n such that every randomized algorithm running in time t(n) fails to solve ? on 1/?(n) fraction of inputs sampled from D, then, assuming some relationships on ?(n) and t(n), there is a distribution D\u27 over instances of ? of size O(n??(n)) such that every randomized algorithm running in time t(n)/poly(?(n)) fails to solve ? on 99/100 fraction of inputs sampled from D\u27.
As a consequence of the above theorem, we show hardness amplification of problems in various classes such as NP-hard problems like Max-Clique, Knapsack, and Max-SAT, problems in P such as Longest Common Subsequence, Edit Distance, Matrix Multiplication, and even problems in TFNP such as Factoring and computing Nash equilibrium
Approximating solution structure of the Weighted Sentence Alignment problem
We study the complexity of approximating solution structure of the bijective
weighted sentence alignment problem of DeNero and Klein (2008). In particular,
we consider the complexity of finding an alignment that has a significant
overlap with an optimal alignment. We discuss ways of representing the solution
for the general weighted sentence alignment as well as phrases-to-words
alignment problem, and show that computing a string which agrees with the
optimal sentence partition on more than half (plus an arbitrarily small
polynomial fraction) positions for the phrases-to-words alignment is NP-hard.
For the general weighted sentence alignment we obtain such bound from the
agreement on a little over 2/3 of the bits. Additionally, we generalize the
Hamming distance approximation of a solution structure to approximating it with
respect to the edit distance metric, obtaining similar lower bounds
Cell-Probe Bounds for Online Edit Distance and Other Pattern Matching Problems
We give cell-probe bounds for the computation of edit distance, Hamming
distance, convolution and longest common subsequence in a stream. In this
model, a fixed string of symbols is given and one -bit symbol
arrives at a time in a stream. After each symbol arrives, the distance between
the fixed string and a suffix of most recent symbols of the stream is reported.
The cell-probe model is perhaps the strongest model of computation for showing
data structure lower bounds, subsuming in particular the popular word-RAM
model.
* We first give an lower bound for
the time to give each output for both online Hamming distance and convolution,
where is the word size. This bound relies on a new encoding scheme and for
the first time holds even when is as small as a single bit.
* We then consider the online edit distance and longest common subsequence
problems in the bit-probe model () with a constant sized input alphabet.
We give a lower bound of which
applies for both problems. This second set of results relies both on our new
encoding scheme as well as a carefully constructed hard distribution.
* Finally, for the online edit distance problem we show that there is an
upper bound in the cell-probe model. This bound gives a
contrast to our new lower bound and also establishes an exponential gap between
the known cell-probe and RAM model complexities.Comment: 32 pages, 4 figure
- …