1,448 research outputs found
Dynamic Ordered Sets with Exponential Search Trees
We introduce exponential search trees as a novel technique for converting
static polynomial space search structures for ordered sets into fully-dynamic
linear space data structures.
This leads to an optimal bound of O(sqrt(log n/loglog n)) for searching and
updating a dynamic set of n integer keys in linear space. Here searching an
integer y means finding the maximum key in the set which is smaller than or
equal to y. This problem is equivalent to the standard text book problem of
maintaining an ordered set (see, e.g., Cormen, Leiserson, Rivest, and Stein:
Introduction to Algorithms, 2nd ed., MIT Press, 2001).
The best previous deterministic linear space bound was O(log n/loglog n) due
Fredman and Willard from STOC 1990. No better deterministic search bound was
known using polynomial space.
We also get the following worst-case linear space trade-offs between the
number n, the word length w, and the maximal key U < 2^w: O(min{loglog n+log
n/log w, (loglog n)(loglog U)/(logloglog U)}). These trade-offs are, however,
not likely to be optimal.
Our results are generalized to finger searching and string searching,
providing optimal results for both in terms of n.Comment: Revision corrects some typoes and state things better for
applications in subsequent paper
Stratification Trees for Adaptive Randomization in Randomized Controlled Trials
This paper proposes an adaptive randomization procedure for two-stage
randomized controlled trials. The method uses data from a first-wave experiment
in order to determine how to stratify in a second wave of the experiment, where
the objective is to minimize the variance of an estimator for the average
treatment effect (ATE). We consider selection from a class of stratified
randomization procedures which we call stratification trees: these are
procedures whose strata can be represented as decision trees, with differing
treatment assignment probabilities across strata. By using the first wave to
estimate a stratification tree, we simultaneously select which covariates to
use for stratification, how to stratify over these covariates, as well as the
assignment probabilities within these strata. Our main result shows that using
this randomization procedure with an appropriate estimator results in an
asymptotic variance which is minimal in the class of stratification trees.
Moreover, the results we present are able to accommodate a large class of
assignment mechanisms within strata, including stratified block randomization.
In a simulation study, we find that our method, paired with an appropriate
cross-validation procedure ,can improve on ad-hoc choices of stratification. We
conclude by applying our method to the study in Karlan and Wood (2017), where
we estimate stratification trees using the first wave of their experiment
Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time
We give a nearly linear time randomized approximation scheme for the
Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an
undirected edge-weighted graph on edges and , the
algorithm outputs in time, with high probability, a
-approximation to the Held-Karp bound on the metric TSP instance
induced by the shortest path metric on . The algorithm can also be used to
output a corresponding solution to the Subtour Elimination LP. We substantially
improve upon the running time achieved previously
by Garg and Khandekar. The LP solution can be used to obtain a fast randomized
-approximation for metric TSP which improves
upon the running time of previous implementations of Christofides' algorithm
Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams
We consider preprocessing a set of points in convex position in the
plane into a data structure supporting queries of the following form: given a
point and a directed line in the plane, report the point of that
is farthest from (or, alternatively, nearest to) the point among all points
to the left of line . We present two data structures for this problem.
The first data structure uses space and preprocessing
time, and answers queries in time, for any . The second data structure uses space and
polynomial preprocessing time, and answers queries in time. These
are the first solutions to the problem with query time and
space.
The second data structure uses a new representation of nearest- and
farthest-point Voronoi diagrams of points in convex position. This
representation supports the insertion of new points in clockwise order using
only amortized pointer changes, in addition to -time
point-location queries, even though every such update may make
combinatorial changes to the Voronoi diagram. This data structure is the first
demonstration that deterministically and incrementally constructed Voronoi
diagrams can be maintained in amortized pointer changes per operation
while keeping -time point-location queries.Comment: 17 pages, 6 figures. Various small improvements. To appear in
Algorithmic
A tree-based kernel for graphs with continuous attributes
The availability of graph data with node attributes that can be either
discrete or real-valued is constantly increasing. While existing kernel methods
are effective techniques for dealing with graphs having discrete node labels,
their adaptation to non-discrete or continuous node attributes has been
limited, mainly for computational issues. Recently, a few kernels especially
tailored for this domain, and that trade predictive performance for
computational efficiency, have been proposed. In this paper, we propose a graph
kernel for complex and continuous nodes' attributes, whose features are tree
structures extracted from specific graph visits. The kernel manages to keep the
same complexity of state-of-the-art kernels while implicitly using a larger
feature space. We further present an approximated variant of the kernel which
reduces its complexity significantly. Experimental results obtained on six
real-world datasets show that the kernel is the best performing one on most of
them. Moreover, in most cases the approximated version reaches comparable
performances to current state-of-the-art kernels in terms of classification
accuracy while greatly shortening the running times.Comment: This work has been submitted to the IEEE Transactions on Neural
Networks and Learning Systems for possible publication. Copyright may be
transferred without notice, after which this version may no longer be
accessibl
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