Robot Localization -- Theory and Practice

We consider the first stage of the robot localization problem described as follows: a robot is at an unknown position in an indoor-environment and has to do a complete relocalization, that is, it has to enumerate all positions that it might be located at. This problem occurs if, for example, the robot "wakes up" after a breakdown (e.g., a power failure or maintenance works) and the possibility exists that it was moved meanwhile. An idealize

1. DIAMETER AND ECCENTRICITY LOWERING PROBLEMS

Given a tree T � (V, E) endowed with a length function l and a cost function c, the diameter lowering problem consists in finding the reals 0 ≤ x(e) ≤ l(e), e � E such that the tree obtained from T by decreasing the length of every edge e by x(e) units has a minimal diameter subject to the constraint ¥ e�Ec(e)x(e) ≤ B, where B is the available budget (analogously, one can minimize the cost of lowering subject to a diameter constraint). We present an O(�V � 2) algorithm for solving this problem by developing and using algorithms of similar complexity for related eccentricity lowering problems. © 2002 Wiley Periodicals, Inc