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Feasible schedules for rotating transmissions Motivated by a scheduling problem that arises in the study of optical networks we prove
 

Summary: Feasible schedules for rotating transmissions
Noga Alon
Abstract
Motivated by a scheduling problem that arises in the study of optical networks we prove
the following result, which is a variation of a conjecture of Haxell, Wilfong and Winkler.
Let k, n be two integers, let wsj, 1 s n, 1 j k be non-negative reals satisfying
k
j=1 wsj < 1/n for every 1 s n and let dsj be arbitrary non-negative reals. Then there
are real numbers x1, x2, . . . , xn so that for every j, 1 j k, the n cyclic closed intervals
I
(j)
s = [xs + dsj, xs + dsj + wsj], (1 s n), where the endpoints are reduced modulo 1, are
pairwise disjoint on the unit circle.
The proof is based on some properties of multivariate polynomials and on the validity of
the Dyson Conjecture.
1 Introduction
Motivated by the study of information transmission in optical networks, the authors of [3] consid-
ered several variants of the following problem. Given n transmitters T1, T2 . . . , Tn and k receivers
R1, R2, . . . , Rk, our objective is to design a rotating schedule that will enable the transmitters to
transmit information to the receivers. We scale time so that the total length of the period in our

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics