75 research outputs found

### A Simple proof of Johnson-Lindenstrauss extension theorem

Johnson and Lindenstrauss proved that any Lipschitz mapping from an $n$-point
subset of a metric space into Hilbert space can be extended to the whole space,
while increasing the Lipschitz constant by a factor of $O(\sqrt{\log n})$. We
present a simplification of their argument that avoids dimension reduction and
the Kirszbraun theorem.Comment: 3 pages. Incorporation of reviewers' suggestion

### Truly Online Paging with Locality of Reference

The competitive analysis fails to model locality of reference in the online
paging problem. To deal with it, Borodin et. al. introduced the access graph
model, which attempts to capture the locality of reference. However, the access
graph model has a number of troubling aspects. The access graph has to be known
in advance to the paging algorithm and the memory required to represent the
access graph itself may be very large.
In this paper we present truly online strongly competitive paging algorithms
in the access graph model that do not have any prior information on the access
sequence. We present both deterministic and randomized algorithms. The
algorithms need only O(k log n) bits of memory, where k is the number of page
slots available and n is the size of the virtual address space. I.e.,
asymptotically no more memory than needed to store the virtual address
translation table.
We also observe that our algorithms adapt themselves to temporal changes in
the locality of reference. We model temporal changes in the locality of
reference by extending the access graph model to the so called extended access
graph model, in which many vertices of the graph can correspond to the same
virtual page. We define a measure for the rate of change in the locality of
reference in G denoted by Delta(G). We then show our algorithms remain strongly
competitive as long as Delta(G) >= (1+ epsilon)k, and no truly online algorithm
can be strongly competitive on a class of extended access graphs that includes
all graphs G with Delta(G) >= k- o(k).Comment: 37 pages. Preliminary version appeared in FOCS '9

### Some applications of Ball's extension theorem

We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type 2 problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice {0,1,...,m}^n, equipped with the ā_p^n metric, in any 2-uniformly convex Banach space is of order min {n^(1/2 1/p),m^(1-2/p)}

### Multi-Embedding of Metric Spaces

Metric embedding has become a common technique in the design of algorithms.
Its applicability is often dependent on how high the embedding's distortion is.
For example, embedding finite metric space into trees may require linear
distortion as a function of its size. Using probabilistic metric embeddings,
the bound on the distortion reduces to logarithmic in the size.
We make a step in the direction of bypassing the lower bound on the
distortion in terms of the size of the metric. We define "multi-embeddings" of
metric spaces in which a point is mapped onto a set of points, while keeping
the target metric of polynomial size and preserving the distortion of paths.
The distortion obtained with such multi-embeddings into ultrametrics is at most
O(log Delta loglog Delta) where Delta is the aspect ratio of the metric. In
particular, for expander graphs, we are able to obtain constant distortion
embeddings into trees in contrast with the Omega(log n) lower bound for all
previous notions of embeddings.
We demonstrate the algorithmic application of the new embeddings for two
optimization problems: group Steiner tree and metrical task systems

### Scaled Enflo type is equivalent to Rademacher type

We introduce the notion of the scaled Enflo type of a metric space, and show that for Banach spaces, scaled Enflo type p is equivalent to Rademacher type p

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