We present a collection of new techniques for designing and analyzing efficient external-memory algorithms for graph problems and illustrate how these techniques can be applied to a wide variety of specific problems. Our results include: ffl Proximate-neighboring. We present a simple method for deriving external-memory lower bounds via reductions from a problem we call the "proximate neighbors" problem. We use this technique to derive non-trivial lower bounds for such problems as list ranking, expression tree evaluation, and connected components. ffl PRAM simulation. We give methods for efficiently simulating PRAM computations in external memory, even for some cases in which the PRAM algorithm is not work-optimal. We apply this to derive a number of optimal (and simple) external-memory graph algorithms. ffl Time-forward processing. We present a general technique for evaluating circuits (or "circuit-like" computations) in external memory. We also use this in a deterministic list rank...

We present a randomized linear-time algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered linear-time algorithm for verifying a minimum spanning tree. Our computational model is a unit-cost random-access machine with the restriction that the only operations allowed on edge weights are binary comparisons.

We solve a shortest path problem that is motivated by recent interest in pricing networks or other computational resources. Informally, how much is an edge in a network worth to a user who wants to send data between two nodes along a shortest path? If the network is a decentralized entity, such as the Internet, in which multiple self-interested agents own different parts of the network, then auctionbased pricing seems appropriate. A celebrated result from auction theory shows that the use of Vickrey pricing motivates the owners of the network resources to bid truthfully. In Vickrey's scheme, each agent is compensated in proportion to the marginal utility he brings to the auction. In the context of shortest path routing, an edge's utility is the value by which it lowers the length of the shortest path---the difference between the shortest path lengths with and without the edge. Our problem is to compute these marginal values for all the edges of the network efficiently. The na ve method requires solving the single-source shortest path problem up to n times, for an n-node network. We show that the Vickrey prices for all the edges can be computed in the same asymptotic time complexity as one single-source shortest path problem. This solves an open problem posed by Nisan and Ronen [12]. 1.

We consider rule sets for internet packet routing and filtering, where each rule consists of a range of source addresses, a range of destination addresses, a priority, and an action. A given packet should be handled by the action from the maximum priority rule that matches its source and destination. We describe new data structures for quickly finding the rule matching an incoming packet, in near-linear space, and a new algorithm for determining whether a rule set contains any conflicts, in time O(n 3/2 ). 1 Introduction The working of the current Internet and its posited evolution depend on efficient packet filtering mechanisms: databases of rules, maintained at various parts of the network, which use patterns to filter out sets of IP packets and specify actions to be performed on those sets. Typical filter patterns are based on packet header information such as the source or destination IP addresses. The actions to be performed depend on where the packet filtering is performed i...

Following in the spirit of data structure and algorithm correctness checking, authenticated data structures provide cryptographic proofs that their answers are as accurate as the author intended, even if the data structure is being maintained by a remote host. We present techniques for authenticating data structures that represent graphs and collection of geometric objects. We use a model where a data structure maintained by a trusted source is mirrored at distributed directories, with the directories answering queries made by users. When a user queries a directory, it receives a cryptographic proof in addition to the answer, where the proof contains statements signed by the source. The user verifies the proof trusting only the statements signed by the source. We show how to efficiently authenticate data structures for fundamental problems on networks, such as path and connectivity queries, and on geometric objects, such as intersection and containment queries.

We present two new data structure tools---disjoint set union with bottom-up linking, and pointer-based radix sort---and combine them with bottom-level microtrees to devise the first linear-time pointer-machine algorithms for off-line least common ancestors, minimum spanning tree (MST) verification, randomized MST construction, and computing dominators in a flowgraph.

Eliciting truthful responses from self-interested agents is an important problem in game theory and microeconomics, and it is studied under mechanism design or implementation theory. Truthful mechanisms have received considerable interest within computer science recently for designing protocols for Internet-based applications, which typically involve cooperation of multiple selfinterested agents. A cornerstone of the mechanism design field is the Vickrey mechanism, or more generally the class of Vickrey-Clarke-Groves mechanisms. These mechanisms are known to be incentive-compatible, meaning that rational agents maximize their utility by truthfully revealing their preferences. In the Vickrey-Clarke-Groves (VCG) mechanism, each agent receives a "payment" for his participation, and this payment is proportional to the added "value" he brings to the system. Implementing the VCG mechanism often requires solving a (non-trivial) optimization problem n + 1 times, once with all agents, and once corresponding to each agent's deletion to determine his incremental value. An important algorithmic challenge is to reduce this computational overhead.

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We describe a randomized CRCW PRAM algorithm that finds a minimum spanning forest of an n-vertex graph in O(log n) time and linear work. This shaves a factor of 2 log n off the best previous running time for a linear-work algorithm. The novelty in our approach is to divide the computation into two phases, the first of which finds only a partial solution. This idea has been used previously in parallel connected components algorithms. 1 Introduction We describe the first work-optimal minimum spanning forest (MSF) algorithm that runs in O(log n) time. The algorithm uses a random-sampling technique previously used by Karger, Klein, and Tarjan in a sequential linear-time algorithm and by Cole, Klein, and Tarjan in a parallel algorithm. These previous algorithms have the following form. Choose a random subset of edges, and recursively calculate the MSF of the sample graph, the graph consisting of the chosen edges. Use the recursively calculated minimum spanning forest to identify edges ...

Lower and upper bounds for finding a minimum spanning tree (MST) in a weighted undirected graph on the BSP model are presented. We provide the first non-trivial lower bounds on the communication volume required to solve the MST problem. Let p denote the number of processors, n the number of nodes of the input graph, and m the number of edges of the input graph. We show that in the worst case, a total of \Omega\Gamma \Delta min(m; pn)) bits need to be communicated in order to solve the MST problem, where is the number of bits required to represent a single edge weight. This implies that if each message communicates at most bits, any BSP algorithm for finding an MST requires communication time \Omega\Gamma g \Delta min(m=p; n)), where g is the gap parameter of the BSP model. In addition, we present two algorithms with communication requirements that match our lower bound in different situations. Both algorithms perform linear work for appropriate values of n, m and p, and use a numbe...