11,845 research outputs found
A Static Optimality Transformation with Applications to Planar Point Location
Over the last decade, there have been several data structures that, given a
planar subdivision and a probability distribution over the plane, provide a way
for answering point location queries that is fine-tuned for the distribution.
All these methods suffer from the requirement that the query distribution must
be known in advance.
We present a new data structure for point location queries in planar
triangulations. Our structure is asymptotically as fast as the optimal
structures, but it requires no prior information about the queries. This is a
2D analogue of the jump from Knuth's optimum binary search trees (discovered in
1971) to the splay trees of Sleator and Tarjan in 1985. While the former need
to know the query distribution, the latter are statically optimal. This means
that we can adapt to the query sequence and achieve the same asymptotic
performance as an optimum static structure, without needing any additional
information.Comment: 13 pages, 1 figure, a preliminary version appeared at SoCG 201
Query Learning with Exponential Query Costs
In query learning, the goal is to identify an unknown object while minimizing
the number of "yes" or "no" questions (queries) posed about that object. A
well-studied algorithm for query learning is known as generalized binary search
(GBS). We show that GBS is a greedy algorithm to optimize the expected number
of queries needed to identify the unknown object. We also generalize GBS in two
ways. First, we consider the case where the cost of querying grows
exponentially in the number of queries and the goal is to minimize the expected
exponential cost. Then, we consider the case where the objects are partitioned
into groups, and the objective is to identify only the group to which the
object belongs. We derive algorithms to address these issues in a common,
information-theoretic framework. In particular, we present an exact formula for
the objective function in each case involving Shannon or Renyi entropy, and
develop a greedy algorithm for minimizing it. Our algorithms are demonstrated
on two applications of query learning, active learning and emergency response.Comment: 15 page
Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games
We study quantum algorithms on search trees of unknown structure, in a model
where the tree can be discovered by local exploration. That is, we are given
the root of the tree and access to a black box which, given a vertex ,
outputs the children of .
We construct a quantum algorithm which, given such access to a search tree of
depth at most , estimates the size of the tree within a factor of in steps. More generally, the same algorithm can
be used to estimate size of directed acyclic graphs (DAGs) in a similar model.
We then show two applications of this result:
a) We show how to transform a classical backtracking search algorithm which
examines nodes of a search tree into an time
quantum algorithm, improving over an earlier quantum backtracking algorithm of
Montanaro (arXiv:1509.02374).
b) We give a quantum algorithm for evaluating AND-OR formulas in a model
where the formula can be discovered by local exploration (modeling position
trees in 2-player games). We show that, in this setting, formulas of size
and depth can be evaluated in quantum time . Thus,
the quantum speedup is essentially the same as in the case when the formula is
known in advance.Comment: Fixed some typo
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Parallel data compression
Data compression schemes remove data redundancy in communicated and stored data and increase the effective capacities of communication and storage devices. Parallel algorithms and implementations for textual data compression are surveyed. Related concepts from parallel computation and information theory are briefly discussed. Static and dynamic methods for codeword construction and transmission on various models of parallel computation are described. Included are parallel methods which boost system speed by coding data concurrently, and approaches which employ multiple compression techniques to improve compression ratios. Theoretical and empirical comparisons are reported and areas for future research are suggested
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