69,158 research outputs found
A Lower Bound on Opaque Sets
It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle
A lower bound on opaque sets
It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle. © Akitoshi Kawamura, Sonoko Moriyama, Yota Otachi, and János Pach
A lower bound on opaque sets
It is proved that the total length of any set of countably many rectifiable curves whose union meets all straight lines that intersect the unit square U is at least 2.00002. This is the first improvement on the lower bound of 2 known since 1964. A similar bound is proved for all convex sets U other than a triangle. (C) 2019 Published by Elsevier B.V
The opaque square
The problem of finding small sets that block every line passing through a
unit square was first considered by Mazurkiewicz in 1916. We call such a set
{\em opaque} or a {\em barrier} for the square. The shortest known barrier has
length . The current best lower
bound for the length of a (not necessarily connected) barrier is , as
established by Jones about 50 years ago. No better lower bound is known even if
the barrier is restricted to lie in the square or in its close vicinity. Under
a suitable locality assumption, we replace this lower bound by ,
which represents the first, albeit small, step in a long time toward finding
the length of the shortest barrier. A sharper bound is obtained for interior
barriers: the length of any interior barrier for the unit square is at least . Two of the key elements in our proofs are: (i) formulas established
by Sylvester for the measure of all lines that meet two disjoint planar convex
bodies, and (ii) a procedure for detecting lines that are witness to the
invalidity of a short bogus barrier for the square.Comment: 23 pages, 8 figure
Progressive Transactional Memory in Time and Space
Transactional memory (TM) allows concurrent processes to organize sequences
of operations on shared \emph{data items} into atomic transactions. A
transaction may commit, in which case it appears to have executed sequentially
or it may \emph{abort}, in which case no data item is updated.
The TM programming paradigm emerged as an alternative to conventional
fine-grained locking techniques, offering ease of programming and
compositionality. Though typically themselves implemented using locks, TMs hide
the inherent issues of lock-based synchronization behind a nice transactional
programming interface.
In this paper, we explore inherent time and space complexity of lock-based
TMs, with a focus of the most popular class of \emph{progressive} lock-based
TMs. We derive that a progressive TM might enforce a read-only transaction to
perform a quadratic (in the number of the data items it reads) number of steps
and access a linear number of distinct memory locations, closing the question
of inherent cost of \emph{read validation} in TMs. We then show that the total
number of \emph{remote memory references} (RMRs) that take place in an
execution of a progressive TM in which concurrent processes perform
transactions on a single data item might reach , which
appears to be the first RMR complexity lower bound for transactional memory.Comment: Model of Transactional Memory identical with arXiv:1407.6876,
arXiv:1502.0272
Inherent Limitations of Hybrid Transactional Memory
Several Hybrid Transactional Memory (HyTM) schemes have recently been
proposed to complement the fast, but best-effort, nature of Hardware
Transactional Memory (HTM) with a slow, reliable software backup. However, the
fundamental limitations of building a HyTM with nontrivial concurrency between
hardware and software transactions are still not well understood.
In this paper, we propose a general model for HyTM implementations, which
captures the ability of hardware transactions to buffer memory accesses, and
allows us to formally quantify and analyze the amount of overhead
(instrumentation) of a HyTM scheme. We prove the following: (1) it is
impossible to build a strictly serializable HyTM implementation that has both
uninstrumented reads and writes, even for weak progress guarantees, and (2)
under reasonable assumptions, in any opaque progressive HyTM, a hardware
transaction must incur instrumentation costs linear in the size of its data
set. We further provide two upper bound implementations whose instrumentation
costs are optimal with respect to their progress guarantees. In sum, this paper
captures for the first time an inherent trade-off between the degree of
concurrency a HyTM provides between hardware and software transactions, and the
amount of instrumentation overhead the implementation must incur
Competitive Imperfect Price Discrimination and Market Power
Two duopolists compete on price in the market for a homogeneous product. They can “profile” consumers, that is, identify their valuations with some probability. If both firms can profile consumers but with different abilities, then they achieve positive expected profits at equilibrium. This provides a rationale for firms to (partially and unequally) share data about consumers or for data brokers to sell different customer analytics to competing firms. Consumers prefer that both firms profile exactly the same set of consumers or that only one firm profiles consumers as this entails marginal cost pricing (so does a policy requiring list prices to be public). Otherwise, more protective privacy regulations have ambiguous effects on consumer surplus
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