655,028 research outputs found
The -Server Problem on Bounded Depth Trees
We study the -server problem in the resource augmentation setting i.e.,
when the performance of the online algorithm with servers is compared to
the offline optimal solution with servers. The problem is very
poorly understood beyond uniform metrics. For this special case, the classic
-server algorithms are roughly -competitive when
, for any . Surprisingly however, no
-competitive algorithm is known even for HSTs of depth 2 and even when
is arbitrarily large.
We obtain several new results for the problem. First we show that the known
-server algorithms do not work even on very simple metrics. In particular,
the Double Coverage algorithm has competitive ratio irrespective of
the value of , even for depth-2 HSTs. Similarly the Work Function Algorithm,
that is believed to be optimal for all metric spaces when , has
competitive ratio on depth-3 HSTs even if . Our main result
is a new algorithm that is -competitive for constant depth trees,
whenever for any . Finally, we give a general
lower bound that any deterministic online algorithm has competitive ratio at
least 2.4 even for depth-2 HSTs and when is arbitrarily large. This gives
a surprising qualitative separation between uniform metrics and depth-2 HSTs
for the -server problem, and gives the strongest known lower bound for
the problem on general metrics.Comment: Appeared in SODA 201
On a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems
In 1996, H. Volkmer observed that the inequality
is
satisfied with some positive constant for a certain class of functions
on if the eigenfunctions of the problem form a Riesz basis of the Hilbert space
. Here the weight is assumed to satisfy
a.e. on .
We present two criteria in terms of Weyl-Titchmarsh -functions for the
Volkmer inequality to be valid. Using these results we show that this
inequality is valid if the operator associated with the spectral problem
satisfies the linear resolvent growth condition. In particular, we show that
the Riesz basis property of eigenfunctions is equivalent to the linear
resolvent growth if is odd.Comment: 26 page
Directed Ramsey number for trees
In this paper, we study Ramsey-type problems for directed graphs. We first
consider the -colour oriented Ramsey number of , denoted by
, which is the least for which every
-edge-coloured tournament on vertices contains a monochromatic copy of
. We prove that for any oriented
tree . This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In
general, it is tight up to a constant factor.
We also consider the -colour directed Ramsey number
of , which is defined as above, but, instead
of colouring tournaments, we colour the complete directed graph of order .
Here we show that for any
oriented tree , which is again tight up to a constant factor, and it
generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined
the -colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure
On the linear fractional self-attracting diffusion
In this paper, we introduce the linear fractional self-attracting diffusion
driven by a fractional Brownian motion with Hurst index 1/2<H<1, which is
analogous to the linear self-attracting diffusion. For 1-dimensional process we
study its convergence and the corresponding weighted local time. For
2-dimensional process, as a related problem, we show that the renormalized
self-intersection local time exists in L^2 if .Comment: 14 Pages. To appear in Journal of Theoretical Probabilit
Optimal Binary Search Trees with Near Minimal Height
Suppose we have n keys, n access probabilities for the keys, and n+1 access
probabilities for the gaps between the keys. Let h_min(n) be the minimal height
of a binary search tree for n keys. We consider the problem to construct an
optimal binary search tree with near minimal height, i.e.\ with height h <=
h_min(n) + Delta for some fixed Delta. It is shown, that for any fixed Delta
optimal binary search trees with near minimal height can be constructed in time
O(n^2). This is as fast as in the unrestricted case.
So far, the best known algorithms for the construction of height-restricted
optimal binary search trees have running time O(L n^2), whereby L is the
maximal permitted height. Compared to these algorithms our algorithm is at
least faster by a factor of log n, because L is lower bounded by log n
- …