969 research outputs found
On Formal Specification of Maple Programs
This paper is an example-based demonstration of our initial results on the
formal specification of programs written in the computer algebra language
MiniMaple (a substantial subset of Maple with slight extensions). The main goal
of this work is to define a verification framework for MiniMaple. Formal
specification of MiniMaple programs is rather complex task as it supports
non-standard types of objects, e.g. symbols and unevaluated expressions, and
additional functions and predicates, e.g. runtime type tests etc. We have used
the specification language to specify various computer algebra concepts
respective objects of the Maple package DifferenceDifferential developed at our
institute
Quality Measures of Parameter Tuning for Aggregated Multi-Objective Temporal Planning
Parameter tuning is recognized today as a crucial ingredient when tackling an
optimization problem. Several meta-optimization methods have been proposed to
find the best parameter set for a given optimization algorithm and (set of)
problem instances. When the objective of the optimization is some scalar
quality of the solution given by the target algorithm, this quality is also
used as the basis for the quality of parameter sets. But in the case of
multi-objective optimization by aggregation, the set of solutions is given by
several single-objective runs with different weights on the objectives, and it
turns out that the hypervolume of the final population of each single-objective
run might be a better indicator of the global performance of the aggregation
method than the best fitness in its population. This paper discusses this issue
on a case study in multi-objective temporal planning using the evolutionary
planner DaE-YAHSP and the meta-optimizer ParamILS. The results clearly show how
ParamILS makes a difference between both approaches, and demonstrate that
indeed, in this context, using the hypervolume indicator as ParamILS target is
the best choice. Other issues pertaining to parameter tuning in the proposed
context are also discussed.Comment: arXiv admin note: substantial text overlap with arXiv:1305.116
Clustered Integer 3SUM via Additive Combinatorics
We present a collection of new results on problems related to 3SUM,
including:
1. The first truly subquadratic algorithm for
1a. computing the (min,+) convolution for monotone increasing
sequences with integer values bounded by ,
1b. solving 3SUM for monotone sets in 2D with integer coordinates
bounded by , and
1c. preprocessing a binary string for histogram indexing (also
called jumbled indexing).
The running time is:
with
randomization, or deterministically. This greatly improves the
previous time bound obtained from Williams'
recent result on all-pairs shortest paths [STOC'14], and answers an open
question raised by several researchers studying the histogram indexing problem.
2. The first algorithm for histogram indexing for any constant alphabet size
that achieves truly subquadratic preprocessing time and truly sublinear query
time.
3. A truly subquadratic algorithm for integer 3SUM in the case when the given
set can be partitioned into clusters each covered by an interval
of length , for any constant .
4. An algorithm to preprocess any set of integers so that subsequently
3SUM on any given subset can be solved in
time.
All these results are obtained by a surprising new technique, based on the
Balog--Szemer\'edi--Gowers Theorem from additive combinatorics
Threesomes, Degenerates, and Love Triangles
The 3SUM problem is to decide, given a set of real numbers, whether any
three sum to zero. It is widely conjectured that a trivial -time
algorithm is optimal and over the years the consequences of this conjecture
have been revealed. This 3SUM conjecture implies lower bounds on
numerous problems in computational geometry and a variant of the conjecture
implies strong lower bounds on triangle enumeration, dynamic graph algorithms,
and string matching data structures.
In this paper we refute the 3SUM conjecture. We prove that the decision tree
complexity of 3SUM is and give two subquadratic 3SUM
algorithms, a deterministic one running in
time and a randomized one running in time with
high probability. Our results lead directly to improved bounds for -variate
linear degeneracy testing for all odd . The problem is to decide, given
a linear function and a set , whether . We show the
decision tree complexity of this problem is .
Finally, we give a subcubic algorithm for a generalization of the
-product over real-valued matrices and apply it to the problem of
finding zero-weight triangles in weighted graphs. We give a
depth- decision tree for this problem, as well as an
algorithm running in time
Subquadratic harmonic functions on Calabi-Yau manifolds with Euclidean volume growth
We prove that on a complete Calabi-Yau manifold with Euclidean volume
growth, a harmonic function with subquadratic polynomial growth is the real
part of a holomorphic function. This generalizes a result of Conlon-Hein. We
prove this result by proving a Liouville type theorem for harmonic -forms,
which follows from a new local estimate of the differential. We also give
another proof based on the construction of harmonic functions with polynomial
growth in Ding, and the algebraicity of tangent cones in Liu-Sz\'ekelyhidi.Comment: 30 pages. Comments are welcom
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