BibTeX
@MISC{_csc2414-metric,
author = {},
title = {CSC2414- Metric Embeddings},
year = {}
}
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Abstract
Lecture 7: Lower bounds on the embeddability in ¡£ ¢ via expander graphs and some algorithmic connections to ¡¥¤ Notes taken by Periklis Papakonstantinou revised by Hamed Hatami Summary: In view of Bourgain’s upper bound a central question in finite metric embeddings concerns explicit constructions of families of metric spaces that incur distortion ¦¨§�©������� � when embedded into �� �. We will see that embedding constant degree expanders into �� � requires distortion ¦¨§�©������� �. On an independent direction we begin investigating algorithmic questions regarding the embeddability of metric spaces into �� � and �� �. We show that computing a minimum distortion embedding of a metric space into � � can be easily done in polynomial time. The corresponding question for � � appears to be a computationally hard task (unless ������ �). Even the weaker question of deciding whether a metric space embeds isometrically into � � is an �� �-complete problem. � � seems to intuitively maintain a strong connection with �� �-hard combinatorial optimization problems. We begin our investigation around algorithmic questions for � � by seeing how to represent an � � metric as a weighted sum of cut metrics. Furthermore, we introduce the sparsest cut problem which is of particular importance to the theory of approximation algorithms. 1 Terminology- notational conventions We use boldface to denote vectors. For �� � example corresponds to a vector whereas corresponds to a real number. We denote by �������������� � � the �-dimensional hamming cube; i.e. the hamming cube � � with points. Given a finite set of reals we denote by Avg ���������������������� � � � � � the arithmetic mean of �� � �. We consider graphs with vertex set ������������������������ � and denote the edges by �� � , where �������� � �� �. The term “expander graph ” corresponds to a regular, constant degree, and of constant expansion graph. For unweighted graphs the distance between two vertices is the length of the unweighted shortest path i.e. the minimum number of edges of a path connecting
Keyphrases
metric space csc2414 metric embeddings algorithmic question finite set central question algorithmic connection corresponding question example corresponds constant degree expanders path connecting unweighted shortest path minimum number terminology notational convention dimensional hamming cube real number independent direction unweighted graph term expander graph cut problem hamed hatami summary minimum distortion embedding hard combinatorial optimization problem expander graph cut metric constant expansion graph weighted sum strong connection hard task polynomial time approximation algorithm particular importance periklis papakonstantinou arithmetic mean constant degree complete problem
