More than fifty years ago, Bellman asked for the best escape path within a known forest but for an unknown starting position. This deterministic finite path is the shortest path that leads out of a given environment from any starting point. There are some worst case positions where the full path length is required. Up to now such a fixed ultimate optimal escape path for a known shape for any starting position is only known for some special convex shapes (i.e., circles, strips of a given width, fat convex bodies, some isosceles triangles). Therefore, we introduce a different, simple and intuitive escape path, the so-called certificate path. This escape path depends on the starting position s and takes the distances from s to the outer boundary of the environment into account. Due to the additional information, the certificate path always (for any position s) leaves the environment earlier than the ultimate escape path, in the above convex examples. Next we assume that fewer information is available. Neither the precise shape of the envir- onment, nor the location of the starting point is known. For a class of environments (convex shapes and shapes with kernel positions), we design an online strategy that always leaves the environment. We show that the path length for leaving the environment is always shorter than 3.318764 the length of the corresponding certificate path. We also give a lower bound of 3.313126, which shows that for the above class of environments the factor 3.318764 is (almost) tight.
We will see that our solution is a specific logarithmic spiral. In general, logarithmic spirals are natural candidates for optimal competitive search strategies, but in almost all cases the optimality remains a conjecture; see @cite_8 @cite_12 @cite_14 @cite_9 @cite_16 . In @cite_1 the optimality of spiral search was shown for searching a point in the plane with a radar. Many other conjectures are still open. For example Finch and Zhu @cite_9 considered the problem of searching for a line in the plane. .
@cite_14 @cite_8 @cite_9 @cite_1 @cite_16 @cite_12
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Searching for a point in the plane is a well-known search game problem introduced in the early eighties. The best known search strategy is given by a spiral and achieves a competitive ratio of 17.289 ... It was shown by Gal [14] that this strategy is the best strategy among all monotone and periodic strategies. Since then it was unknown whether the given strategy is optimal in general. This paper settles this old open fundamental search problem and shows that spiral search is indeed optimal. The given problem can be considered as the continuous version of the well-known m-ray search problem and also appears in several non-geometric applications and modifications. Therefore the optimality of spiral search is an important question considered by many researchers in the last decades. We answer the logarithmic spiral conjecture for the given problem. The lower bound construction might be helpful for similar settings, it also simplifies existing proofs on classical m-ray search. Logarithmic spirals are conjectured to be optimal escape paths from a half plane ocean. Assuming this, we find the rate of increase for both min-max and min-mean interpretations of optimal''. For the one-dimensional analog, which we call logarithmic coils, our min-mean solution differs from a widely-cited published account. Logarithmic spirals are ubiquitous in nature. This paper presents a novel mathematical definition of a 3D logarithmic spiral, which provides a proper description of objects found in nature. To motivate our work, we scanned spiral-shaped objects and studied their geometric properties. We consider the extent to which the existing 3D definitions capture these properties. We identify a property that is shared by the objects we investigated and is not satisfied by the existing 3D definitions. This leads us to present our definition in which both the radius of curvature and the radius of torsion change linearly along the curve. We prove that our spiral satisfies several desirable properties, including invariance to similarity transformations, smoothness, symmetry, extensibility, and roundness. Finally, we demonstrate the utility of our curves in the modeling of several animal structures. We consider the problem of searching for an object on a line at an unknown distance OPT from the original position of the searcher, in the presence of a cost of d for each time the searcher changes direction. This is a generalization of the well-studied linear-search problem. We describe a strategy that is guaranteed to find the object at a cost of at most 9 ċ OPT + 2d, which has the optimal competitive ratio 9 with respect to OPT plus the minimum corresponding additive term. Our argument for upper and lower bound uses an infinite linear program, which we solve by experimental solution of an infinite series of approximating finite linear programs, estimating the limits, and solving the resulting recurrences for an explicit proof of optimality. We feel that this technique is interesting in its own right and should help solve other searching problems. In particular, we consider the star search or cowpath problem with turn cost, where the hidden object is placed on one of m rays emanating from the original position of the searcher. For this problem we give a tight bound of (1 + 2mm (m - 1)m-1)OPT + m((m (m - 1))m-1 - 1)d. We also discuss tradeoffs between the corresponding coefficients and we consider randomized strategies on the line.