226 research outputs found
Finite Volume Spaces and Sparsification
We introduce and study finite -volumes - the high dimensional
generalization of finite metric spaces. Having developed a suitable
combinatorial machinery, we define -volumes and show that they contain
Euclidean volumes and hypertree volumes. We show that they can approximate any
-volume with multiplicative distortion. On the other hand, contrary
to Bourgain's theorem for , there exists a -volume that on vertices
that cannot be approximated by any -volume with distortion smaller than
.
We further address the problem of -dimension reduction in the context
of volumes, and show that this phenomenon does occur, although not to
the same striking degree as it does for Euclidean metrics and volumes. In
particular, we show that any metric on points can be -approximated by a sum of cut metrics, improving
over the best previously known bound of due to Schechtman.
In order to deal with dimension reduction, we extend the techniques and ideas
introduced by Karger and Bencz{\'u}r, and Spielman et al.~in the context of
graph Sparsification, and develop general methods with a wide range of
applications.Comment: previous revision was the wrong file: the new revision: changed
(extended considerably) the treatment of finite volumes (see revised
abstract). Inserted new applications for the sparsification technique
Ascending auctions and Walrasian equilibrium
We present a family of submodular valuation classes that generalizes gross
substitute. We show that Walrasian equilibrium always exist for one class in
this family, and there is a natural ascending auction which finds it. We prove
some new structural properties on gross-substitute auctions which, in turn,
show that the known ascending auctions for this class (Gul-Stacchetti and
Ausbel) are, in fact, identical. We generalize these two auctions, and provide
a simple proof that they terminate in a Walrasian equilibrium
Extremal problems on shadows and hypercuts in simplicial complexes
Let be an -vertex forest. We say that an edge is in the
shadow of if contains a cycle. It is easy to see that if
is "almost a tree", that is, it has edges, then at least
edges are in its shadow and this is tight.
Equivalently, the largest number of edges an -vertex cut can have is
. These notions have natural analogs in higher
-dimensional simplicial complexes, graphs being the case . The results
in dimension turn out to be remarkably different from the case in graphs.
In particular the corresponding bounds depend on the underlying field of
coefficients. We find the (tight) analogous theorems for . We construct
-dimensional "-almost-hypertrees" (defined below) with an empty
shadow. We also show that the shadow of an "-almost-hypertree"
cannot be empty, and its least possible density is . In
addition we construct very large hyperforests with a shadow that is empty over
every field.
For even, we construct -dimensional -almost-hypertree whose shadow has density .
Finally, we mention several intriguing open questions
New Sublinear Algorithms and Lower Bounds for LIS Estimation
Estimating the length of the longest increasing subsequence (LIS) in an array
is a problem of fundamental importance. Despite the significance of the LIS
estimation problem and the amount of attention it has received, there are
important aspects of the problem that are not yet fully understood. There are
no better lower bounds for LIS estimation than the obvious bounds implied by
testing monotonicity (for adaptive or nonadaptive algorithms). In this paper,
we give the first nontrivial lower bound on the complexity of LIS estimation,
and also provide novel algorithms that complement our lower bound.
Specifically, for every constant , every nonadaptive
algorithm that outputs an estimate of the length of the LIS in an array of
length to within an additive error of has to make
queries. Next, we design nonadaptive LIS
estimation algorithms whose complexity decreases as the the number of distinct
values, , in the array decreases. We first present a simple algorithm that
makes queries and approximates the LIS length with an
additive error bounded by . We then use it to construct a
nonadaptive algorithm with query complexity that, for an array with LIS length at least , outputs a multiplicative -approximation to the LIS length.
Finally, we describe a nonadaptive erasure-resilient tester for sortedness,
with query complexity . Our result implies that nonadaptive tolerant
testing is strictly harder than nonadaptive erasure-resilient testing for the
natural property of monotonicity.Comment: 32 pages, 3 figure
Online embedding of metrics
We study deterministic online embeddings of metrics spaces into normed spaces
and into trees against an adaptive adversary. Main results include a polynomial
lower bound on the (multiplicative) distortion of embedding into Euclidean
spaces, a tight exponential upper bound on embedding into the line, and a
-distortion embedding in of a suitably high
dimension.Comment: 15 pages, no figure
Online Embedding of Metrics
We study deterministic online embeddings of metric spaces into normed spaces of various dimensions and into trees. We establish some upper and lower bounds on the distortion of such embedding, and pose some challenging open questions
Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs
In this paper, we present a simple factor 6 algorithm for approximating the
optimal multiplicative distortion of embedding a graph metric into a tree
metric (thus improving and simplifying the factor 100 and 27 algorithms of
B\v{a}doiu, Indyk, and Sidiropoulos (2007) and B\v{a}doiu, Demaine, Hajiaghayi,
Sidiropoulos, and Zadimoghaddam (2008)). We also present a constant factor
algorithm for approximating the optimal distortion of embedding a graph metric
into an outerplanar metric. For this, we introduce a general notion of metric
relaxed minor and show that if G contains an alpha-metric relaxed H-minor, then
the distortion of any embedding of G into any metric induced by a H-minor free
graph is at meast alpha. Then, for H=K_{2,3}, we present an algorithm which
either finds an alpha-relaxed minor, or produces an O(alpha)-embedding into an
outerplanar metric.Comment: 27 pages, 4 figires, extended abstract to appear in the proceedings
of APPROX-RANDOM 201
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