We consider some network design problems with applications for wireless networks. The input for these problems is a metric space (
X,
d) and a finite subset
UX of terminals. In the
Steiner Tree with Minimum Number of Steiner Points (
STMSP) problem, the goal is to find a minimum size set
SX−
U of points so that the unit-disc graph of
S+
U is connected. Let Δ be the smallest integer so that for any finite
VX for which the unit-disc graph is connected, this graph contains a spanning tree with maximum degree Δ. The best known approximation ratio for
STMSP was Δ−1 [I.I. Măndoiu, A.Z. Zelikovsky, A note on the MST heuristic for bounded edge-length Steiner trees with minimum number of Steiner points, Information Processing Letters 75 (4) (2000) 165–167]. We improve this ratio to (Δ+1)/2+1+
ε.In the
Minimum Power Spanning Tree (
MPST) problem,
V=
X is finite, and the goal is to find a “range assignment”
on the nodes so that the edge set
contains a spanning tree, and ∑
vVp(
v) is minimized. We consider a particular case {0,1}-
MPST of
MPST when the distances are in {0,1}; here the goal is to find a minimum size set
SV of “active” nodes so that the graph (
V,
E0+
E1(
S)) is connected, where
, and
E1(
S) is the set the edges in
with both endpoints in
S. We will show that the (5/3+
ε)-approximation scheme for
MPST of [E. Althaus, G. Calinescu, I. Măndoiu, S. Prasad, N. Tchervenski, A. Zelikovsky, Power efficient range assignment for symmetric connectivity in static ad hoc wireless networks, Wireless Networks 12 (3) (2006) 287–299] achieves a ratio 3/2 for {0,1}-distances. This answers an open question posed in [E. Lloyd, R. Liu, S. Ravi, Approximating the minimum number of maximum power users in ad hoc networks, Mobile Networks and Applications 11 (2006) 129–142].
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