Searching in The Plane

Ricardo A. Baeza-Yates, Joseph C. Culberson and Gregory J. E. Rawlins

Information and Computation, vol 106(2), 234-252, 1993

Abstract

In this paper we initiate a new area of study dealing with the best way to search a possibly unbounded region for an object. The model for our search algorithms is that we must pay costs proportional to the distance of the next probe position relative to our current position. This model is meant to give a realistic cost measure for a robot moving in the plane. We also examine the effect of decreasing the amount of a priori information given to search problems. Problems of this type are very simple analogues of non-trivial problems on searching an unbounded region, processing digitized images, and robot navigation. We show that for some simple search problems, the relative information of knowing the general direction of the goal is much higher than knowing the distance to the goal.

Here are some of the results presented in this paper that suggest that the relative information of knowing the general direction of a goal is much higher than knowing just the distance to the goal.

			Knowledge
Problem		Direction Distance	Nothing
Point on a line	   n	   3n		  9n
Point on m-rays	   n	  (2m-1)n	(1+2(m^m)/(m-1)^(m-1))n
Point in Lattice   n	    		in [2n^2+4n-1,2n^2+5n+2]
  " "  with Parity n	                <= 2n^2+4n+(n mod 2)
         The Advantage of Knowing Where Things Are

Ricardo Baeza-Yates rbaeza@dcc.uchile.cl

Gregory J. E. Rawlins rawlins@cs.indiana.edu

Joseph Culberson joe@cs.ualberta.ca