7,087 research outputs found
Online Carpooling Using Expander Decompositions
We consider the online carpooling problem: given n vertices, a sequence of edges arrive over time. When an edge e_t = (u_t, v_t) arrives at time step t, the algorithm must orient the edge either as v_t ? u_t or u_t ? v_t, with the objective of minimizing the maximum discrepancy of any vertex, i.e., the absolute difference between its in-degree and out-degree. Edges correspond to pairs of persons wanting to ride together, and orienting denotes designating the driver. The discrepancy objective then corresponds to every person driving close to their fair share of rides they participate in.
In this paper, we design efficient algorithms which can maintain polylog(n,T) maximum discrepancy (w.h.p) over any sequence of T arrivals, when the arriving edges are sampled independently and uniformly from any given graph G. This provides the first polylogarithmic bounds for the online (stochastic) carpooling problem. Prior to this work, the best known bounds were O(?{n log n})-discrepancy for any adversarial sequence of arrivals, or O(log log n)-discrepancy bounds for the stochastic arrivals when G is the complete graph.
The technical crux of our paper is in showing that the simple greedy algorithm, which has provably good discrepancy bounds when the arriving edges are drawn uniformly at random from the complete graph, also has polylog discrepancy when G is an expander graph. We then combine this with known expander-decomposition results to design our overall algorithm
The Iray Light Transport Simulation and Rendering System
While ray tracing has become increasingly common and path tracing is well
understood by now, a major challenge lies in crafting an easy-to-use and
efficient system implementing these technologies. Following a purely
physically-based paradigm while still allowing for artistic workflows, the Iray
light transport simulation and rendering system allows for rendering complex
scenes by the push of a button and thus makes accurate light transport
simulation widely available. In this document we discuss the challenges and
implementation choices that follow from our primary design decisions,
demonstrating that such a rendering system can be made a practical, scalable,
and efficient real-world application that has been adopted by various companies
across many fields and is in use by many industry professionals today
Online Discrepancy Minimization for Stochastic Arrivals
In the stochastic online vector balancing problem, vectors
chosen independently from an arbitrary distribution in
arrive one-by-one and must be immediately given a sign.
The goal is to keep the norm of the discrepancy vector, i.e., the signed
prefix-sum, as small as possible for a given target norm.
We consider some of the most well-known problems in discrepancy theory in the
above online stochastic setting, and give algorithms that match the known
offline bounds up to factors. This substantially
generalizes and improves upon the previous results of Bansal, Jiang, Singla,
and Sinha (STOC' 20). In particular, for the Koml\'{o}s problem where
for each , our algorithm achieves
discrepancy with high probability, improving upon the previous
bound. For Tusn\'{a}dy's problem of minimizing the
discrepancy of axis-aligned boxes, we obtain an bound for
arbitrary distribution over points. Previous techniques only worked for product
distributions and gave a weaker bound. We also consider the
Banaszczyk setting, where given a symmetric convex body with Gaussian
measure at least , our algorithm achieves discrepancy with
respect to the norm given by for input distributions with sub-exponential
tails.
Our key idea is to introduce a potential that also enforces constraints on
how the discrepancy vector evolves, allowing us to maintain certain
anti-concentration properties. For the Banaszczyk setting, we further enhance
this potential by combining it with ideas from generic chaining. Finally, we
also extend these results to the setting of online multi-color discrepancy
A Gaussian Fixed Point Random Walk
In this note, we design a discrete random walk on the real line which takes
steps (and one with steps in ) where at least
of the signs are in expectation, and which has as a
stationary distribution. As an immediate corollary, we obtain an online version
of Banaszczyk's discrepancy result for partial colorings and
signings. Additionally, we recover linear time algorithms for logarithmic
bounds for the Koml\'{o}s conjecture in an oblivious online setting.Comment: 8 page
Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing
A well-known result of Banaszczyk in discrepancy theory concerns the prefix
discrepancy problem (also known as the signed series problem): given a sequence
of unit vectors in , find signs for each of them such
that the signed sum vector along any prefix has a small -norm?
This problem is central to proving upper bounds for the Steinitz problem, and
the popular Koml\'os problem is a special case where one is only concerned with
the final signed sum vector instead of all prefixes. Banaszczyk gave an
bound for the prefix discrepancy problem. We
investigate the tightness of Banaszczyk's bound and consider natural
generalizations of prefix discrepancy:
We first consider a smoothed analysis setting, where a small amount of
additive noise perturbs the input vectors. We show an exponential improvement
in compared to Banaszczyk's bound. Using a primal-dual approach and a
careful chaining argument, we show that one can achieve a bound of
with high probability in the smoothed setting.
Moreover, this smoothed analysis bound is the best possible without further
improvement on Banaszczyk's bound in the worst case.
We also introduce a generalization of the prefix discrepancy problem where
the discrepancy constraints correspond to paths on a DAG on vertices. We
show that an analog of Banaszczyk's bound continues
to hold in this setting for adversarially given unit vectors and that the
factor is unavoidable for DAGs. We also show that the
dependence on cannot be improved significantly in the smoothed case for
DAGs.
We conclude by exploring a more general notion of vector balancing, which we
call combinatorial vector balancing. We obtain near-optimal bounds in this
setting, up to poly-logarithmic factors.Comment: 22 pages. Appear in ITCS 202
Internal states of model isotropic granular packings. III. Elastic properties
In this third and final paper of a series, elastic properties of numerically
simulated isotropic packings of spherical beads assembled by different
procedures and subjected to a varying confining pressure P are investigated. In
addition P, which determines the stiffness of contacts by Hertz's law, elastic
moduli are chiefly sensitive to the coordination number, the possible values of
which are not necessarily correlated with the density. Comparisons of numerical
and experimental results for glass beads in the 10kPa-10MPa range reveal
similar differences between dry samples compacted by vibrations and lubricated
packings. The greater stiffness of the latter, in spite of their lower density,
can hence be attributed to a larger coordination number. Voigt and Reuss bounds
bracket bulk modulus B accurately, but simple estimation schemes fail for shear
modulus G, especially in poorly coordinated configurations under low P.
Tenuous, fragile networks respond differently to changes in load direction, as
compared to load intensity. The shear modulus, in poorly coordinated packings,
tends to vary proportionally to the degree of force indeterminacy per unit
volume. The elastic range extends to small strain intervals, in agreement with
experimental observations. The origins of nonelastic response are discussed. We
conclude that elastic moduli provide access to mechanically important
information about coordination numbers, which escape direct measurement
techniques, and indicate further perspectives.Comment: Published in Physical Review E 25 page
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