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Optimal Location of Transportation Devices
, 2007
"... We consider algorithms for finding the optimal location of a simple transportation device, that we call a moving walkway, consisting of a pair of points in the plane between which the travel speed is high. More specifically, one can travel from one endpoint of the walkway to the other at speed v> ..."
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We consider algorithms for finding the optimal location of a simple transportation device, that we call a moving walkway, consisting of a pair of points in the plane between which the travel speed is high. More specifically, one can travel from one endpoint of the walkway to the other at speed v> 1, but can only travel at unit speed between any other pair of points. The travel time between any two points in the plane is the minimum between the actual geometric distance, and the time needed to go from one point to the other using the walkway. A location for a walkway is said to be optimal with respect to a given finite set of points if it minimizes the maximum travel time between any two points of the set. We give a simple lineartime algorithm for finding an optimal location in the case where the points are on a line. We also give an Ω(n log n) lower bound for the problem of computing the travel time diameter of a set of n points on a line with respect to a given walkway. Then we describe an O(n log n) algorithm for locating a walkway with the additional restriction that the walkway must be horizontal. This algorithm is based on a recent generic method for solving quasiconvex programs with implicitly defined constraints. It is used in a (1+ε)approximation algorithm for optimal location of a walkway of arbitrary orientation.
Exact bounds for distributed graph colouring
, 2011
"... Abstract. We prove exact bounds on the time complexity of distributed graph colouring. If we are given a directed path that is properly coloured with n colours, by prior work it is known that we can find a proper 3colouring in 12 log ∗(n) ± O(1) communication rounds. We close the gap between upper ..."
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Cited by 2 (2 self)
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Abstract. We prove exact bounds on the time complexity of distributed graph colouring. If we are given a directed path that is properly coloured with n colours, by prior work it is known that we can find a proper 3colouring in 12 log ∗(n) ± O(1) communication rounds. We close the gap between upper and lower bounds: we show that for infinitely many n the time complexity is precisely 12 log ∗ n communication rounds. ar X iv
Moving walkways, escalators, and elevators
 CoRR
"... Abstract. We study a simple geometric model of transportation facility that consists of two points between which the travel speed is high. This elementary definition can model shuttle services, tunnels, bridges, teleportation devices, escalators or moving walkways. The travel time between a pair of ..."
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Abstract. We study a simple geometric model of transportation facility that consists of two points between which the travel speed is high. This elementary definition can model shuttle services, tunnels, bridges, teleportation devices, escalators or moving walkways. The travel time between a pair of points is defined as a time distance, in such a way that a customer uses the transportation facility only if it is helpful. We give algorithms for finding the optimal location of such a transportation facility, where optimality is defined with respect to the maximum travel time between two points in a given set. 1
and Problem Complexity—Nonnumerical Algorithms and Problems;
, 2005
"... The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in R d. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(nlog n)time algorithm for eval ..."
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The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in R d. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(nlog n)time algorithm for evaluating the dilation of a given star; a randomized O(n log n) expectedtime algorithm for finding an optimal center in R d; and for the case d = 2, a randomized O(n2 α(n) log 2 n) expectedtime algorithm for finding an optimal center among the input points.