Bounding the expected number of rectilinear full Steiner trees

TY - JOUR

T1 - Bounding the expected number of rectilinear full Steiner trees

AU - Wulff-Nilsen, Christian

PY - 2010/8

Y1 - 2010/8

N2 - Given a finite set Z of n points, called terminals, in Rd , the Rectilinear Steiner Tree Problem asks for a tree of minimal L1-length spanning Z . An optimal solution has a unique decomposition into full Steiner trees (FSTs). By using geometric properties and combinatorial arguments, we bound the expected number of FSTs satisfying simple necessary conditions for being part of an optimal solution. More specifically, we show that the expected number of FSTs spanning exactly K terminals and satisfying the empty lune property, a weak version of the bottleneck property, and the so-called empty hyperbox property is O(n(log log n)2(d-1)) for K = 3 and O(n(log log n)d-1 logK-2 n) for K > 3, assuming terminals are randomly distributed in a hypercube with aniform distribution. In the plane, we improve an earlier bound by showing that the expected number of FSTs with the Hwang form spanning exactly K terminals and satisfying the empty lune property and the so-called disjoint lunes property is O(nπK ).

AB - Given a finite set Z of n points, called terminals, in Rd , the Rectilinear Steiner Tree Problem asks for a tree of minimal L1-length spanning Z . An optimal solution has a unique decomposition into full Steiner trees (FSTs). By using geometric properties and combinatorial arguments, we bound the expected number of FSTs satisfying simple necessary conditions for being part of an optimal solution. More specifically, we show that the expected number of FSTs spanning exactly K terminals and satisfying the empty lune property, a weak version of the bottleneck property, and the so-called empty hyperbox property is O(n(log log n)2(d-1)) for K = 3 and O(n(log log n)d-1 logK-2 n) for K > 3, assuming terminals are randomly distributed in a hypercube with aniform distribution. In the plane, we improve an earlier bound by showing that the expected number of FSTs with the Hwang form spanning exactly K terminals and satisfying the empty lune property and the so-called disjoint lunes property is O(nπK ).

U2 - 10.1002/net.20342

DO - 10.1002/net.20342

M3 - Journal article

SN - 0028-3045

VL - 56

SP - 1

EP - 10

JO - Networks

JF - Networks

IS - 1

ER -