1,308,067 research outputs found
Automata in SageMath---Combinatorics meet Theoretical Computer Science
The new finite state machine package in the mathematics software system
SageMath is presented and illustrated by many examples. Several combinatorial
problems, in particular digit problems, are introduced, modeled by automata and
transducers and solved using SageMath. In particular, we compute the asymptotic
Hamming weight of a non-adjacent-form-like digit expansion, which was not known
before
Unknown I.I.D. Prophets: Better Bounds, Streaming Algorithms, and a New Impossibility
A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who
sequentially observes random variables X1, . . . , Xn and selects one of them is at least an α fraction
of the maximum value in the sequence. We obtain three distinct improvements for a setting that
was first studied by Correa et al. (EC, 2019) and is particularly relevant to modern applications in
algorithmic pricing. In this setting, the random variables are i.i.d. from an unknown distribution and
the gambler has access to an additional βn samples for some β ≥ 0. We first give improved lower
bounds on α for a wide range of values of β; specifically, α ≥ (1 + β)/e when β ≤ 1/(e − 1), which is
tight, and α ≥ 0.648 when β = 1, which improves on a bound of around 0.635 due to Correa et al.
(SODA, 2020). Adding to their practical appeal, specifically in the context of algorithmic pricing,
we then show that the new bounds can be obtained even in a streaming model of computation
and thus in situations where the use of relevant data is complicated by the sheer amount of data
available. We finally establish that the upper bound of 1/e for the case without samples is robust
to additional information about the distribution, and applies also to sequences of i.i.d. random
variables whose distribution is itself drawn, according to a known distribution, from a finite set of
known candidate distributions. This implies a tight prophet inequality for exchangeable sequences
of random variables, answering a question of Hill and Kertz (Contemporary Mathematics, 1992),
but leaves open the possibility of better guarantees when the number of candidate distributions is
small, a setting we believe is of strong interest to applications
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