J Comb Optim (2006) 11: 99–112 DOI 10.1007/s10878-006-5980-0 Minimum power assignment in wireless ad hoc networks with spanner property Yu Wang · Xiang-Yang Li ∗ Accepted: 25 October 2005 C  Springer Science + Business Media, Inc. 2006 Abstract Power assignment for wireless ad hoc networks is to assign a power for each wireless node such that the induced communication graph has some required properties. Re- cently research efforts have focused on finding the minimum power assignment to guarantee the connectivity or fault-tolerance of the network. In this paper, we study a new problem of finding the power assignment such that the induced communication graph is a spanner for the original communication graph when all nodes have the maximum power. Here, a spanner means that the length of the shortest path in the induced communication graph is at most a constant times of the length of the shortest path in the original communication graph. Polynomial time algorithm is given to minimize the maximum assigned power with spanner property. The algorithm also works for any other property that can be tested in polynomial time and is monotone. We then give a polynomial time approximation method to minimize the total transmission radius of all nodes. Finally, we propose two heuristics and conduct extensive simulations to study their performance when we aim to minimize the total assigned power of all nodes. Keywords Graph theory . Power assignment . Spanner . Wireless ad hoc networks 1. Introduction In this paper, we address the problem of finding minimum power assignment in wireless ad hoc networks such that the induced communication graph is a spanner of the communication graph when all nodes transmit at their maximum power. In a wireless network, each wireless ∗ The author is partially supported by NSF CCR-0311174. Y. Wang () Department of Computer Science, University of North Carolina at Charlotte, Charlotte, NC 28223, USA e-mail: ywang32@uncc.edu X.-Y. Li Department of Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA e-mail: xli@cs.iit.edu Springer