On-line single server dial-a-ride problems

In this paper results on the dial-a-ride problem with a single server are presented. Requests for rides consist of two points in a metric space, a source and a destination. A ride has to be made by the server from the source to the destination. The server travels at unit speed in the metric space and the objective is to minimize some function of the delivery times at the destinations. We study this problem in the natural on-line setting. Calls for rides come in while the server is travelling. This models e.g. the taxi problem, or, if the server has capacity more than 1 a minibus or courier service problem. For two versions of this problem, one in which the server has infinite capacity and the other in which the server has capacity 1, both having as objective minimization of the time the last destination is served, we will design algorithms that have competitive ratio's of 2. We also show that these are best possible, since no algorithm can have competitive ratio better than 2 for these problems. Then we study the on-line problem with objective minimization of the sum of completion times of the rides. We prove a lower bound on the competitive ratio of any algorithm of 1 + \sqrt{2} for a server with any capacity and of 3 for servers with capacity 1

A decomposition theory for vertex enumeration of convex polyhedra

In the last years the vertex enumeration problem of polyhedra has seen a revival in the study of metabolic networks, which increased the demand for efficient vertex enumeration algorit

Geologic framework, tectonic evolution, and displacement history of the Alexander Terrane

The Alexander terrane consists of upper Proterozoic(?)-Cambrian through Middle(?) Jurassic rocks that underlie much of southeastern (SE) Alaska and parts of eastern Alaska, western British Columbia, and southwestern Yukon Territory. A variety of geologic, paleomagnetic, and paleontologic evidence indicates that these rocks have been displaced considerable distances from their sites of origin and were not accreted to western North America until Late Cretaceous-early Tertiary time. Our geologic and U-Pb geochronologic studies in southern SE Alaska and the work of others to the north indicate that the terrane evolved through three distinct tectonic phases. During the initial phase, from late Proterozoic(?)-Cambrian through Early Devonian time, the terrane probably evolved along a convergent plate margin. Arc-type(?) volcanism and plutonism occurred during late Proterozoic(?)-Cambrian and Ordovician-Early Silurian time, with orogenic events during the Middle Cambrian-Early Ordovician (Wales orogeny) and the middle Silurian-earliest Devonian (Klakas orogeny). The second phase is marked by Middle Devonian through Lower Permian strata which accumulated in tectonically stable marine environments. Devonian and Lower Permian volcanic rocks and upper Pennsylvanian-Lower Permian syenitic to dioritic intrusive bodies occur locally but do not appear to represent major magmatic systems. The third phase is marked by Triassic volcanic and sedimentary rocks which are interpreted to have formed in a rift environment. Previous syntheses of the displacement history of the terrane emphasized apparent similarities with rocks in the Sierra-Klamath region and suggested that the Alexander terrane evolved in proximity to the California continental margin during Paleozoic time. Our studies indicate, however, that the geologic record of the Alexander terrane is quite different from that in the Sierra-Klamath region, and we conclude that the two regions were not closely associated during Paleozoic time. The available geologic, paleomagnetic, and paleontologic data are more consistent with a scenario involving (1) early Paleozoic origin and evolution of the Alexander terrane along the paleo-Pacific margin of Gondwana, (2) rifting from this margin during Devonian time, (3) late Paleozoic migration across the paleo-Pacific basin in low southerly paleolatitudes, (4) residence in proximity to the paleo-Pacific margin of South America during latest Paleozoic(?)-Triassic time, and (5) Late Permian(?)-Triassic rifting followed by northward displacement along the eastern margin of the Pacific basin