20,095 research outputs found
A constant-time algorithm for middle levels Gray codes
For any integer a middle levels Gray code is a cyclic listing of
all -element and -element subsets of such that
any two consecutive subsets differ in adding or removing a single element. The
question whether such a Gray code exists for any has been the subject
of intensive research during the last 30 years, and has been answered
affirmatively only recently [T. M\"utze. Proof of the middle levels conjecture.
Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T.
M\"utze and J. Nummenpalo. An efficient algorithm for computing a middle levels
Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence
proof was turned into an algorithm that computes each new set in the Gray code
in time on average. In this work we present an algorithm for
computing a middle levels Gray code in optimal time and space: each new set is
generated in time on average, and the required space is
A Model for Optimal Human Navigation with Stochastic Effects
We present a method for optimal path planning of human walking paths in
mountainous terrain, using a control theoretic formulation and a
Hamilton-Jacobi-Bellman equation. Previous models for human navigation were
entirely deterministic, assuming perfect knowledge of the ambient elevation
data and human walking velocity as a function of local slope of the terrain.
Our model includes a stochastic component which can account for uncertainty in
the problem, and thus includes a Hamilton-Jacobi-Bellman equation with
viscosity. We discuss the model in the presence and absence of stochastic
effects, and suggest numerical methods for simulating the model. We discuss two
different notions of an optimal path when there is uncertainty in the problem.
Finally, we compare the optimal paths suggested by the model at different
levels of uncertainty, and observe that as the size of the uncertainty tends to
zero (and thus the viscosity in the equation tends to zero), the optimal path
tends toward the deterministic optimal path
An explicit universal cycle for the (n-1)-permutations of an n-set
We show how to construct an explicit Hamilton cycle in the directed Cayley
graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >...
k). The existence of such cycles was shown by Jackson (Discrete Mathematics,
149 (1996) 123-129) but the proof only shows that a certain directed graph is
Eulerian, and Knuth (Volume 4 Fascicle 2, Generating All Tuples and
Permutations (2005)) asks for an explicit construction. We show that a simple
recursion describes our Hamilton cycle and that the cycle can be generated by
an iterative algorithm that uses O(n) space. Moreover, the algorithm produces
each successive edge of the cycle in constant time; such algorithms are said to
be loopless
(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids
The Local Search problem, which finds a local minimum of a black-box function
on a given graph, is of both practical and theoretical importance to many areas
in computer science and natural sciences. In this paper, we show that for the
Boolean hypercube \B^n, the randomized query complexity of Local Search is
and the quantum query complexity is
. We also show that for the constant dimensional grid
, the randomized query complexity is for and the quantum query complexity is for . New
lower bounds for lower dimensional grids are also given. These improve the
previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04].
Finally we show for a new upper bound of on the quantum query complexity, which implies that Local Search on
grids exhibits different properties at low dimensions.Comment: 18 pages, 1 figure. v2: introduction rewritten, references added. v3:
a line for grant added. v4: upper bound section rewritte
Efficient computation of middle levels Gray codes
For any integer a middle levels Gray code is a cyclic listing of
all bitstrings of length that have either or entries equal to
1 such that any two consecutive bitstrings in the list differ in exactly one
bit. The question whether such a Gray code exists for every has been
the subject of intensive research during the last 30 years, and has been
answered affirmatively only recently [T. M\"utze. Proof of the middle levels
conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In this work we
provide the first efficient algorithm to compute a middle levels Gray code. For
a given bitstring, our algorithm computes the next bitstrings in the
Gray code in time , which is
on average per bitstring provided that
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