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Let G = (V, E) be an undirected weighted graph with |V | = n and |E| = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k ) expected time, constructing a data structure of size O(kn 1+1/k ), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k −1. A 1963 girth conjecture of Erdős, implies that Ω(n 1+1/k ) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name "oracle". Previously, data structures that used only O(n 1+1/k ) space had a query time of Ω(n 1/k ).Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.

We describe several compact routing schemes for general weighted undirected networks. Our schemes are simple and easy to implement. The routing tables stored at the nodes of the network are all very small. The headers attached to the routed messages, including the name of the destination, are extremely short. The routing decision at each node takes constant time. Yet, the stretch of these routing schemes, i.e., the worst ratio between the cost of the path on which a packet is routed and the cost of the cheapest path from source to destination, is a small constant. Our schemes achieve a near-optimal tradeoff between the size of the routing tables used and the resulting stretch. More specifically, we obtain: 1. A routing scheme that uses onlyÕ(n 1=2 ) bits of memory at each node of an n-node network that has stretch 3. The space is optimal, up to logarithmic factors, in the sense that every routing scheme with stretch < 3 must use, on some networks, routing tables of total size (n 2 ), and every routing scheme with stretch < 5 must use, on some networks, routing tables of total size (n 3=2 ). The headers used are only (1 + o(1)) log 2 n-bit long and each routing decision takes constant time. A variant of this scheme with dlog 2 ne-bit headers makes routing decisions in O(log log n) time. 2. Also, for every integer k > 2, a general handshaking based routing scheme that usesÕ(n 1=k ) bits of memory at each node that has stretch 2k ; 1. A conjecture of Erdős from 1963, settled for k = 3 5, implies that the routing tables are of near-optimal size relative to the stretch. The handshaking is similar in spirit to a DNS lookup in TCP/IP. Headers are o(log 2 n) bits long and each routing decision takes constant time. Without handshaking, the stretch of the scheme increases to 4k ; 5.One ingredient used to obtain the routing schemes mentioned above, may be of independent practical and theoretical interest:Work supported in part by the Israel Science Foundation founded by The Israel Academy of Sciences and Humanities.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Copyright 2001 ACM 0-89791-88-6/97/05 ...$5.00. 3. A shortest path routing scheme for trees of arbitrary degree and diameter that assigns each vertex of an n-node tree a (1 + o(1)) log 2 n-bit label. Given the label of a source node and the label of a destination it is possible to compute, in constant time, the port number of the edge from the source that heads in the direction of the destination.The general scheme for k > 2 also uses a clustering technique introduced recently by the authors. The clusters obtained using this technique induce a sparse and low stretch tree cover of the network. This essentially ...

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We describe a novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small Sloan foundation.

We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs. Most of the bounds obtained depend solely on the number of edges in the graph in question, and not on the number of vertices. The bounds obtained improve upon various previously known results.

We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms.The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value inÕ(n 2+µ ) time, where µ satisfies the equation ω(1, µ, 1) = 1 + 2µ and ω(1, µ, 1) is the exponent of the multiplication of an n × n µ matrix by an n µ × n matrix. Currently, the best available bounds on ω(1, µ, 1), obtained by Coppersmith, imply that µ < 0.575. The running time of our algorithm is therefore O(n 2.575 ). Our algorithm improves on theÕ(n (3+ω)/2 ) time algorithm, where ω = ω(1, 1, 1) < 2.376 is the usual exponent of matrix multiplication, obtained by Alon, Galil and Margalit, whose running time is only known to be O(n 2.688 ).The second algorithm solves the APSP problem almost exactly for directed graphs with arbitrary nonnegative real weights. The algorithm runs inÕ((n ω /ǫ) log(W/ǫ)) time, where ǫ > 0 is an error parameter and W is the largest edge weight in the graph, after the edge weights are scaled so that the smallest non-zero edge weight in the graph is 1. It returns estimates of all the distances in the graph with a stretch of at most 1 + ǫ. Corresponding paths can also be found efficiently.