The algorithm uses a directed tree T(u) to replace the directed star for
each node u. Therefore, if nodes u and v are connected by a path in ??’??’?†’ Y Gk,
they are also connected by a path in ??’??’?†’ Y G??—k. It is already known that ??’??’?†’ Y Gk is
strongly connected if UDG(V) is connected, so is ??’??’?†’ Y G??—k. In [24], the authors
also proved that the power stretch factor of ??’??’?†’ Y G??—k(V ) is at most ( 1
1??’(2 sin ??
k )?? )2
and the maximum degree of ??’??’?†’ Y G??—k(V ) is at most (k + 1)2 ??’ 1.
Sparsified Yao Graph
Another Yao-based algorithm is proposed by Li et al [35] that constructs a
sparser and bounded-degree topology. The basic idea is to apply reverse Yao
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Yu Wang
structure on ??’??’?†’ Y Gk to bound the in-degree. Node u chooses a node v from each
cone, if there is any, so the directed link ??’?†’vu has the smallest length among all
directed links ??’?†’ wu in ??’??’?†’ Y Gk in that cone. The union of all chosen directed links
is the final network topology, denoted by ??’??’?†’ Y Y k(V ). Notice that in [38,39], they
reinvestigate ??’??’?†’ Y Y k(V ) structure, and call it a Sparsified Yao graph or Yao Yao
graph.
In [35], the authors proved ??’??’?†’ Y Y k(V ) is strongly connected if UDG(V ) is
connected and k > 6. It was proved in [40] that ??’??’?†’ Y Y k(V ) is a spanner in
civilized unit disk graphs (also called ?»-precision unit disk graphs [41]). Here
a unit disk graph is a civilized graph if the distance between any two nodes
in this graph is larger than a positive constant ?».
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