Yao spanners for wireless ad hoc networks

Wireless nodes are often powered by batteries and have limited memory resources. These characteristics make it critical to compute and maintain, at each node, only a subset of neighbors that the node communicates with. These subsets of neighbors define a topology and the problem of choosing “appropriate” subsets of neighbors is called the topology control problem. Highly relevant to the topology control problem is the class of Yao graphs, defined as follows. Let P be a set of points in the plane. The Yao graph Yk(P ), for k > 2, is defined as follows. At each point u ∈ P , any k equal-separated rays originated at u define k cones; in each non-empty cone Cu, pick the shortest edge {u, v} connecting u to its nearest neighbor v ∈ Cu (breaking ties arbitrarily), and add {u, v} to Yk. For a fixed value t ≥ 1, a graph G embedded in the plane is a length t-spanner if, for all pairs of vertices u, v ∈ P , the shortest path in G from u to v is no longer than t · |uv|; here |uv| denotes the Euclidean distance between u and v. It is a standing open question to decide whether the Yao structure Yk, for k ≤ 5, is a length t-spanner or not, for some constant t. We make progress towards resolving this question by showing that Y4 is a length spanner for sets of points in convex position. We prove that Y2 and Y3 are not length spanners. We show that a related structure, called Yao-Yao, is a length spanner for any set of points of bounded aspect ratio (defined as the ratio between the largest to the smallest interpoint distances).