8 research outputs found
A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs
We consider the problem of computing all-pairs shortest paths in a directed
graph with real weights assigned to vertices.
For an 0-1 matrix let be the complete weighted graph
on the rows of where the weight of an edge between two rows is equal to
their Hamming distance. Let be the weight of a minimum weight spanning
tree of
We show that the all-pairs shortest path problem for a directed graph on
vertices with nonnegative real weights and adjacency matrix can be
solved by a combinatorial randomized algorithm in time
As a corollary, we conclude that the transitive closure of a directed graph
can be computed by a combinatorial randomized algorithm in the
aforementioned time.
We also conclude that the all-pairs shortest path problem for uniform disk
graphs, with nonnegative real vertex weights, induced by point sets of bounded
density within a unit square can be solved in time
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
Subquadratic High-Dimensional Hierarchical Clustering
International audienceWe consider the widely-used average-linkage, single-linkage, and Ward's methods for computing hierarchical clusterings of high-dimensional Euclidean inputs. It is easy to show that there is no efficient implementation of these algorithms in high dimensional Euclidean space since it implicitly requires to solve the closest pair problem, a notoriously difficult problem. However, how fast can these algorithms be implemented if we allow approxima-tion? More precisely: these algorithms successively merge the clusters that are at closest average (for average-linkage), minimum distance (for single-linkage), or inducing the least sum-of-square error (for Ward's). We ask whether one could obtain a significant running-time improvement if the algorithm can merge γ-approximate closest clusters (namely, clusters that are at distance (average, minimum , or sum-of-square error) at most γ times the distance of the closest clusters). We show that one can indeed take advantage of the relaxation and compute the approximate hierarchical clustering tree using r Opnq γ-approximate nearest neighbor queries. This leads to an algorithm running in time r Opndq`n 1`Op1{γq for d-dimensional Euclidean space. We then provide experiments showing that these algorithms perform as well as the non-approximate version for classic classification tasks while achieving a significant speed-up
Scribe: A Clustering Approach To Semantic Information Retrieval
Information retrieval is the process of fulfilling a user?s need for information by locating items in a data collection that are similar to a complex query that is often posed in natural language. Latent Semantic Indexing (LSI) was the predominant technique employed at the National Institute of Standards and Technology?s Text Retrieval Conference for many years until limitations of its scalability to large data sets were discovered. This thesis describes SCRIBE, a modification of LSI with improved scalability. SCRIBE clusters its semantic index into discrete volumes described by high-dimensional extensions to computer graphics data structures. SCRIBE?s clustering strategy limits the number of items that must be searched and provides for sub-linear time complexity in the number of documents. Experimental results with a large, natural language document collection demonstrate that SCRIBE achieves retrieval accuracy similar to LSI but requires 1/10 the time
Subquadratic Approximation Algorithms For Clustering Problems in High Dimensional Spaces
One of the central problems in information retrieval, data mining, computational biology, statistical analysis, computer vision, geographic analysis, pattern recognition, distributed protocols is the question of classification of data according to some clustering rule. Often the data is noisy and even approximate classification is of extreme importance. The difficulty of such classification stems from the fact that usually the data has many incomparable attributes, and often results in the question of clustering problems in high dimensional spaces. Since they require measuring distance between every pair of data points, standard algorithms for computing the exact clustering solutions use quadratic or "nearly quadratic" running time; i.e., O(dn 2\Gammaff(d) ) time where n is the number of data points, d is the dimen- Computer Science Department, University of Toronto. Part of this work was done while visiting Bell Communications Research. y Bell Communications Research, MCC-1C365..
Subquadratic Approximation Algorithms For Clustering Problems in High Dimensional Spaces
One of the central problems in information retrieval, data mining, computational biology, statistical analy-sis, computer vision, geographic analysis, pattern recognition, distributed protocols is the question of classification of data according to some clustering rule. Of-ten the data is noisy and even approximate classification is of extreme importance. The difficulty of such classi cation stems from the fact that usually the data has many incomparable attributes, and often results in the question of clustering problems in high dimensional spaces. Since they require measuring distance between every pair of data points, standard algorithms for computing the exact clustering solutions use quadratic or "nearly quadratic" running time; i.e., O(dn 2, (d) ) time where n is the number of data points, d is the dimen