There has recently been a notable interest in the organization of routing information to enable fast lookup of IP addresses. The interest is primarily motivated by the goal of building multigigabit routers for the Internet, without having to rely on multilayer switching techniques. We address this problem by using an LC-trie, a trie structure with combined path and level compression. This data structure enables us to build efficient, compact, and easily searchable implementations of an IP-routing table. The structure can store both unicast and multicast addresses with the same average search times. The search depth increases as 2(log log n) (log log n) (log log n) with the number of entries in the table for a large class of distributions, and it is independent of the length of the addresses. A node in the trie can be coded with four bytes. Only the size of the base vector, which contains the search strings, grows linearly with the length of the addresses when extended from 4 to 16 bytes, as mandated by the shift from IP version 4 to IP version 6. We present the basic structure as well as an adaptive version that roughly doubles the number of lookups/s. More general classifications of packets that are needed for link sharing, qualityof-service provisioning, and multicast and multipath routing are also discussed. Our experimental results compare favorably with those reported previously in the research literature.
We show that a unit-cost RAM with a word length of w bits can sort n integers in the range 0 } } } 2 w &1 in O(n log log n) time for arbitrary w log n, a significant improvement over the bound of O(n -log n) achieved by the fusion trees of Fredman and Willard. Provided that w (log n) 2+= for some fixed =>0, the sorting can even be accomplished in linear expected time with a randomized algorithm. Both of our algorithms parallelize without loss on a unit-cost PRAM with a word length of w bits. The first one yields an algorithm that uses O(log n) time and O(n log log n) operations on a deterministic CRCW PRAM. The second one yields an algorithm that uses O(log n) expected time and O(n) expected operations on a randomized EREW PRAM, provided that w (log n) 2+= for some fixed =>0. Our deterministic and randomized sequential and parallel algorithms generalize to the lexicographic sorting of multiple-precision integers represented in several words. ]
We consider the three-dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress-free boundary condition. Building on our recently developed theory for difference methods for second order hyperbolic systems [H.-O. Kreiss, N. A. Petersson, J. Yström, SIAM J. Numer. Anal., 40 (2002), pp. 1940-1967, we develop an explicit, second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities. The spatial discretization is self-adjoint, and the stability is obtained through an energy estimate. Seismic events are often modeled using singular source terms, and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source. Several numerical examples are given.
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