1,265 research outputs found
Solving -SUM using few linear queries
The -SUM problem is given input real numbers to determine whether any
of them sum to zero. The problem is of tremendous importance in the
emerging field of complexity theory within , and it is in particular open
whether it admits an algorithm of complexity with . Inspired by an algorithm due to Meiser (1993), we show
that there exist linear decision trees and algebraic computation trees of depth
solving -SUM. Furthermore, we show that there exists a
randomized algorithm that runs in
time, and performs linear queries on the input. Thus, we show
that it is possible to have an algorithm with a runtime almost identical (up to
the ) to the best known algorithm but for the first time also with the
number of queries on the input a polynomial that is independent of . The
bound on the number of linear queries is also a tighter bound
than any known algorithm solving -SUM, even allowing unlimited total time
outside of the queries. By simultaneously achieving few queries to the input
without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we
deepen the understanding of this canonical problem which is a cornerstone of
complexity-within-.
We also consider a range of tradeoffs between the number of terms involved in
the queries and the depth of the decision tree. In particular, we prove that
there exist -linear decision trees of depth
Flag arrangements and triangulations of products of simplices
We investigate the line arrangement that results from intersecting d complete
flags in C^n. We give a combinatorial description of the matroid T_{n,d} that
keeps track of the linear dependence relations among these lines. We prove that
the bases of the matroid T_{n,3} characterize the triangles with holes which
can be tiled with unit rhombi. More generally, we provide evidence for a
conjectural connection between the matroid T_{n,d}, the triangulations of the
product of simplices Delta_{n-1} x \Delta_{d-1}, and the arrangements of d
tropical hyperplanes in tropical (n-1)-space. Our work provides a simple and
effective criterion to ensure the vanishing of many Schubert structure
constants in the flag manifold, and a new perspective on Billey and Vakil's
method for computing the non-vanishing ones.Comment: 39 pages, 12 figures, best viewed in colo
Regression Depth and Center Points
We show that, for any set of n points in d dimensions, there exists a
hyperplane with regression depth at least ceiling(n/(d+1)). as had been
conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n
hyperplanes in d dimensions there exists a point that cannot escape to infinity
without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our
approach to related questions on the existence of partitions of the data into
subsets such that a common plane has nonzero regression depth in each subset,
and to the computational complexity of regression depth problems.Comment: 14 pages, 3 figure
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