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.
For circles and fat convex bodies, it was shown that the diameter is the ultimate optimal escape path; see Finch and Wetzel @cite_2 . For the infinite strip of width @math , the ultimate optimal escape path is due to Zalgaller @cite_24 @cite_0 . For the simple equilateral triangle of side length @math , the zig-zag path of Besicovitch @cite_22 of length @math is optimal; see also @cite_19 . Furthermore, in 1961 Gluss @cite_5 introduced the problem of searching for a circle @math of given radius @math and given distance @math away from the start @math . Two different cases can be considered, either @math is inside @math or not. Interestingly, in the latter case and for @math a certificate path with length @math and an arc of length @math is the best one can do.
@cite_22 @cite_24 @cite_19 @cite_0 @cite_2 @cite_5
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From a random point O in an infinite strip of width 1, we move in a randomly chosen direction along a curve Г. What shape of Г gives the minimum value to the expectation of the length of the path that reaches the boundary of the strip? After certain arguments suggesting that the desired curve belongs to one of four classes, it is proved that the best curve in those classes consists of four parts: an interval OA of length a, its smooth continuation, an arc AB of radius 1 and small length φ, an interval BD that is smooth continuation of the arc, and an interval DF (with a corner at the point D). If O is the origin and OA is the x axis of a coordinate system, then the coordinates of the above-mentioned points are as follows: A(a, 0), B(a + sin φ, 1 − cos φ), F(a, 1), and @math For the best curve, we have: a ≈ 0.814 and φ ≈ 0.032. Related questions are discussed. Bibliography: 9 titles. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation. Abstract The problem of finding the maximum diameter of n equal mutually disjoint circles inside a unit square is addressed in this paper. Exact solutions exist for only n = 1, …, 9,10,16,25,36 while for other n only conjectural solutions have been reported. In this work a max-min optimization approach is introduced which matches the best reported solutions in the literature for all n ⩽ 30, yields a better configuration for n = 15, and provides new results for n = 28 and 29. The problem of determining shortest paths through a weighted planar polygonal subdivision with n vertices is considered. Distances are measured according to a weighted Euclidean metric: The length of a path is defined to be the weighted sum of (Euclidean) lengths of the subpaths within each region. An algorithm that constructs a (restricted) “shortest path map” with respect to a given source point is presented. The output is a partitioning of each edge of the subdivion into intervals of e-optimality, allowing an e-optimal path to be traced from the source to any query point along any edge. The algorithm runs in worst-case time O ( ES ) and requires O ( E ) space, where E is the number of “events” in our algorithm and S is the time it takes to run a numerical search procedure. In the worst case, E is bounded above by O ( n 4 ) (and we give an O( n 4 ) lower bound), but it is likeky that E will be much smaller in practice. We also show that S is bounded by O ( n 4 L ), where L is the precision of the problem instance (including the number of bits in the user-specified tolerance e). Again, the value of S should be smaller in practice. The algorithm applies the “continuous Dijkstra” paradigm and exploits the fact that shortest paths obey Snell's Law of Refraction at region boundaries, a local optimaly property of shortest paths that is well known from the analogous optics model. The algorithm generalizes to the multi-source case to compute Voronoi diagrams.