A formal approach to protocols and strategies for (legal) negotiation We propose a formal and executable framework for expressing protocols and strategies for automated (legal) negotiation. In this framework a party involved in a negotiation is represented through a software agent composed of four modules: (i) a communication module which manages the interaction with the other agents; (ii) a control module; (iii) a reasoning module specified as a defeasible theory; and (iv) a knowledge base which bridges the control and the reasoning modules, while keeping track of past decisions and interactions. The choice of defeasible logic is justified against a set of desirable criteria for negotiation automation languages. Moreover, the suitability of the framework is illustrated through two case studies. What is legal negotiation? With negotiation we mean a process involving at least two parties aimed at reaching an agreement that is acceptable by the parties involved. In this perspective a legal negotiation is either a negotiation about legal content or a negotiation where legal issues have to be taken into account.. Legal negotiations are by far more common than one might expect. For example any agreement can be conceived as a form of private contract, and contracts are subject to the current (relevant) legislation. Moreover, there are several kinds of negotiation and some of them are typical of legal areas (for example pleading, cf. [10] ) while others are more frequent in economics, commerce and related fields, but, in the end, even these cases have to be considered with a normative eye.. In addition to affecting its content, legal issues can also affect the negotiation process on itself. Indeed, any negotiation is guided £ This work was funded by the ARC Grant "Self-Describing Transactions Operating in a Large, Heterogeneous, Distributed Environment" involving QUT and GBST Pty Ltd.. 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. by a protocol, which describes the rules of the dispute, that is, how the parties exchange their offers, and how and when the negotiation can go on or terminate. Protocols can themselves be seen as agreements, that can take the form of a contract, and are thereby subject to law. Thus, for example, an auction is a one-to-many negotiation governed by the rules of the auction itself. In public auctions the rules (procedures) are established by law (cf., for example the articles 576 and 581 of the Italian Code of Civil Procedure), while private auctions are self-regulated: the auction house states the rules under which the auction will be conducted (yet, these rules are subject to the law).. It should be clear that one of the main normative aspects of negotiation concerns the definition of the protocols (rules) under which negotiations are carried out.. Negotiation protocols can be classified into one-to-one (e.g., bargaining, divorce), one-to-many (e.g., tenders, English auction), and many-to-many (double auctions) [35] . This classification can be further refined depending on whether the negotiation is carried out in a single step (i.e., sealed-bid auctions), or iteratively (open-cry auctions). Figure 1 provides a classification schema that captures these two dimensions. Single issue vs. multiple issues. Legal negotiations often involve several interdependent issues. Auctions involving multiple items [30] or multiple units of an item are another example of multiple-issue negotiations.. Direct vs. mediated. A mediator is a neutral party (e.g., a judge in a pleading), as opposed to a representative of one of the parties (e.g., a lawyer). Legal negotiations are often mediated.. ¯Anonymous vs. identity-open. Auction houses often carry out anonymous or at least semi-anonymous auctions.. This brief overview shows that protocols are manifold; it would thus be difficult, if not impossible, to capture all of them in a single universal framework. Still, we believe that a logic-based approach could cope easily with most of them. Moreover, this same kind of approach could be used to specify the behaviour of the parties involved in a negotiation, that is, their strategies. Interestingly, expressing both the protocols and the strategies in a logic-based framework, makes it possible to reason about their internal and their mutual consistencies.. With this motivation, this paper aims at assessing the feasibility and suitability of a logic-based approach to legal negotiation, and to develop a framework for specifying protocols and strategies in this setting.. The paper is organised as follows: in Section 2 we introduce a list of desiderata that an automated negotiation system should satisfy (Section 2.1) and we describe an agent based architecture for such systems (Section 2.2). In Section 2.3 we defend Defeasible Logic against the desiderata of Section 2.1, and then we introduce Defeasible Logic formally (Section 3). In Section 4 we develop two case studies in order to illustrate how the framework works. We conclude the paper with a discussion on related work (Section 5) and directions for future research (Section 6). In this paper we have proposed a qualitative logic-based approach to (automated) legal negotiation, and we have shown that it meets a set of desiderata that a negotiation framework should satisfy. In particular, we believe that the use of Defeasible Logic as the inference engine of the proposed architecture makes our approach "suitable" for legal negotiation in so far as it allows to capture effectively the defeasibility of law in a very close way.. It is worth noting that the present approach offers a great deal of flexibility when compared with other approaches, we have already argued that other approaches, and in particular those based on utility functions, fail to satisfy some of the criteria listed in section 2.1. Still, for some problems, very well understood and well behaving utility functions have been defined; in such cases it may be useful to combine the two approaches. Usually utility functions are applied to evaluate offers; thus, if one of the party is willing to concede on some issues, if the opponent party has conceded on other issues, then several utility functions can be defined with different weight for the conceded attributes. For example, let f 1 , f 2 and f 3 be utility functions defined over a set of attributes. We can supplement the given functions with a set of defeasible rules to choose the most appropriate utility function according to the solved issues, for example:. The meaning of the above rules is as follows: usually the offer is evaluated using f 1 , however if some concessions are given by the counter-party, utility functions f 2 or f 3 are used instead.. One of the advantages of this method is that it is possible to formulate trade-off strategies in an easy way, and utility functions can be simplified. Bellucci and Zeleznikow [6] propose a legal negotiation support system tailored for Australian family law. Such a system is based on bi-directional fuzzy cognitive map, and it relies essentially on numerical (fuzzy) values given to the negotiation issues by the parties involved. Then those values are used to determine the order of decomposition and allocation of the content of the dispute. Finally trade-off (fuzzy) relationships are used in resolving the conflicts. It is clear that such an approach does not meet some of the desiderata listed in Section 2.1: in general it is not easy to define the appropriate value functions and the corresponding trade-off relationship, and numerical functions are not easily explainable, thus this system does not seem a natural and intuitive approach to the problem. Secondly, it does not take into account legal issues nor deal with the defeasibility of normative reasoning.. The architecture of another legal negotiation support system is outlined in [22] . The main feature is a user interface which uses Toulmin's diagram to represent the status of the negotiation. Additionally it supports several defeasible reasoning mechanisms (prioritized logic programming), and an adviser. However, the reasoning modules are used only for negotiation contents and not to describe protocols, but, as we have seen, in a legal context, protocols for negotiation can be defeasible. Finally, the advisor module is based on utility functions, and we have argued that this approach lacks suitability and comprehensibility.. The Michigan AuctionBot [25, 35] is an auction management server supporting the creation, location and enactment of different kinds of auctions. Users can manually interact with the system through an HTML-based interface, or alternatively, they can develop their own arbitrarily complex bidding agents, and connect them to the auction manager through a TCP/IP-level API. This API is generic enough to deal with several kinds of auctions (e.g. English, Dutch, double, etc.) through a common set of primitives. The AuctionBot has recently been used to host an trading agent competition [31] , as has another similar platform called FishMarket [7, 16] .. Unlike our work, neither FishMarket nor AuctionBot, address the issue of specifying bidding agent strategies.. More recently, yet another auction management server called eAuctionHouse has been released [30] . An interesting feature of eAuctionHouse is that it supports combinatorial auctions, that is, auctions in which a bidder may place bids on combinations of items, and may even issue simultaneous bids for several combinations, while ensuring that only one of his bids will win. eAuctionHouse also supports mobile agents that can issue bids on behalf of a user. However, the user is not allowed to specify his own bidding strategy: he has to choose between a set of predefined ones.. [17] classifies techniques for designing negotiation strategies into 3 categories: game-theoretic, heuristic and argumentationbased. The first approach, models a negotiation situation as a game, and attempts to find dominant strategies for a each participant by applying game theory techniques. For simple negotiation protocols, and under very specific assumptions about the rationality and attitude toward risk of each participant, this kind of analysis leads to simple and optimal strategies. However, very often these results do not generalize easily to more complex situations. In heuristicbased approaches on the other hand, a strategy consists of a family of tactics (i.e. a method for generating counter-offers), and a set of rules for selecting a particular tactic depending on the stage of the negotiation. The strategies for bargaining and bidding that we discussed in the previous section, can be seen as belonging to this family. Interestingly, our case studies show that defeasible logic is suitable to express both the tactics, and the rules for tactic selection. Finally, argumentation-based approaches extend heuristic ones, by introducing communication performatives such as threats (e.g., "this is my last offer"), rewards (e.g., "if you accept this offer, I will be more clever in our next negotiation"), etc. These performatives can be modeled in our approach through defeasible predicates shared by the participants of a negotiation. Concretely, in the bargaining case study of Section 4.1, in addition to having a predicate offer, the theory should also include predicates threat, reward, etc. as well as derivation rules for these predicates.. The use of logic-based formalisms with rule prioritisation in the context of automated negotiations has been studied in [9] and [29] .. Garcia et al. [9] suggests to use a "variant" of defeasible logic to express strategies for agents trading over stock markets. The "variant" of defeasible logic that the authors consider, does not support an explicit ranking of the rules within a theory, but rather derives this ranking through a specificity criterion over arguments. Roughly speaking, an argument is more specific than (and therefore can defeat) another argument, if it takes into account more information. In addition to the fact that this approach allows "tie-breaks" between defeasible arguments, it may sometimes lead to counterintuitive situations. Indeed, the semantics given by a user to the concept of "more information", may potentially not be in line with that given by the defeasible logic's inference method.. Reeves et al. [29] , use Corteous Logic Programs (CLP) [14] to express knowledge about user preferences, constraints, and negotiation structures. The authors do not address the issue of specifying negotiation strategies directly, but rather that of deriving the set of negotiation structures that have to be carried out in order to transform a contract template into an executable contract. Interestingly, Defeasible Logic Programming (DLP) is more expressive than CLP [5] , in the sense that it allows the user to express stratified theories. Hence, the question remains open whether it is possible to express useful contract templates in DLP that cannot be expressed in CLP.

O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Full-Information Lookups for Peer-to-Peer Overlays Abstract-Most peer-to-peer lookup schemes keep a small amount of routing state per node, typically logarithmic in the number of overlay nodes. This design assumes that routing information at each member node must be kept small so that the bookkeeping required to respond to system membership changes is also small, given that aggressive membership dynamics are expected. As a consequence, lookups have high latency as each lookup requires contacting several nodes in sequence. In this paper, we question these assumptions by presenting a peer-to-peer routing algorithm with small lookup paths. Our algorithm, called "OneHop," maintains full information about the system membership at each node, routing in a single hop whenever that information is up to date and in a small number of hops otherwise. We show how to disseminate information about membership changes quickly enough so that nodes maintain accurate complete membership information. We also present analytic bandwidth requirements for our scheme that demonstrate that it could be deployed in systems with hundreds of thousands of nodes and high churn. We validate our analytic model using a simulated environment and a real implementation. Our results confirm that OneHop is able to achieve high efficiency, usually reaching the correct node directly 99 percent of the time. S TRUCTURED peer-to-peer overlays like Chord [2] , CAN [3] , Pastry [4] , or Tapestry [5] provide a substrate for building large-scale distributed applications. These overlays allow applications to locate objects stored in the system in a limited number of overlay hops.. Peer-to-peer lookup algorithms strive to maintain a small amount of per-node routing state-typically Oðlog NÞ-because their designers expect that system membership changes frequently. This expectation has been confirmed for successfully deployed systems. A 2002 study of Gnutella [6] shows that the average session time was only 2.9 hours. This is equivalent to saying that in a system with 100,000 nodes, there are about 19 membership change events per second. More recent studies confirm that session times continue to be very short [7] , [8] .. Maintaining small tables helps keep the amount of bookkeeping required to deal with membership changes small. However, there is a price to pay: lookups have high latency since each lookup requires contacting several nodes in sequence.. This paper questions the need to keep routing state small. 1 We take the position that maintaining full routing state (i.e., a complete description of the system membership at each node) is viable even in a very large system, e.g., containing hundreds of thousands of nodes. We present techniques that show that in systems of this size, nodes can maintain membership information accurately, yet the communication costs are low. The results imply that a peer-to-peer system can route very efficiently even though the system is large and membership is changing rapidly.. We present a peer-to-peer lookup system called OneHop that maintains complete membership information at each node. We show analytic results that prove that the system meets our goals of reasonable accuracy and bandwidth usage. It is, of course, easy to achieve these goals for small systems. Our algorithm is designed to scale to large systems. Our analysis shows that we are able to use OneHop for systems of up to hundreds of thousands of nodes.. The paper presents simulation results that validate what our analytic models predict. These results show that performance does not degrade significantly as the system becomes larger nor in the presence of high churn.. We implemented the OneHop protocol as an additional protocol supported by the Chord framework for developing peer-to-peer applications [9] . We evaluated our implementation on a network that emulated wide-area internode latencies (using ModelNet [10] ), and our results corroborate the evaluation performed using simulations.. 1. This submission is based on a paper previously published in the First Symposium on Networked Systems Design and Implementation (NSDI) [1] . It extends the previous publication by presenting a complete implementation of the protocol and an evaluation of the protocol using that implementation. In the previous paper, we only had simulated the protocol. Other improvements include a revision of the analysis in Section 4 and an overall text revision.. The rest of the paper is organized as follows: Section 2 presents our system model. Section 3 describes the OneHop protocol. Section 4 shows how to parameterize the system and presents an analysis of the bandwidth requirements of this scheme. Section 5 presents an evaluation of OneHop using simulations. Section 6 presents the implementation of the protocol and an evaluation using that implementation. Section 7 discusses related work. We conclude with a discussion of what we have accomplished. These formulas allow us to compute values for k and u. For example, in a system of 10 5 nodes, we want roughly 500 slices, each containing five units. In a system of 10 6 nodes, we still have five units per slice, but now, there are 5,000 slices.. Given values for k and u, we can compute the unit size, and this in turn allows us to compute t small and t big . We find that we use the least bandwidth when. Thus, we choose 23 seconds for t big and 23 seconds for t small .. Given these values and the formulas in Table 1 , we can plot the bandwidth usage per slice leader in systems of various sizes. The results of this calculation are shown in Fig. 3 . Note that the load increases only linearly with the size of the system. The load is quite modest in a system with 10 5 nodes (17-Kbps upstream bandwidth), and therefore, even nodes behind cable modems can act as slice leaders in such a system. In a system with 10 6 nodes, the upstream bandwidth required at a slice leader is approximately 166 Kbps. Here, it would be more appropriate to limit slice leaders to being machines on reasonably provisioned local area networks. For larger networks, the bandwidth increases to a point where a slice leader would need to be a well-provisioned node. Fig. 4 shows the percentage overhead of this scheme in terms of aggregate bandwidth used in the system with respect to the hypothetical optimum scheme with zero overhead. In such a scheme, the cost is just the total bandwidth used in sending r events to every node in the system every second, i.e., r Á n Á m. Note that the overhead in our system comes from the per-message protocol overhead. The scheme itself does not propagate any redundant information. We note that the overhead is approximately 19 percent for a system containing 10 5 nodes and goes down to 4 percent for a system containing 10 6 nodes. This result is reasonable because messages get larger and the overhead becomes less significant as the system size increases. This paper questions the necessity of multihop lookups in peer-to-peer routing algorithms. We introduce a novel peerto-peer lookup algorithm that routes in one hop, unless the lookup fails and other routes need to be attempted. We designed our algorithm to provide a small fraction of lookup failures (e.g., 1 percent).. We present analytic results that show how we can parameterize the system to obtain reasonable bandwidth consumption, despite the fact that we are dealing with a highly dynamic membership. We present simulation results that support our analysis that the system delivers a large fraction of lookups within one hop. We also implemented this protocol and our experimental evaluation in an emulated wide area network to confirm the simulation results.. The OneHop protocol is now available as an additional routing choice for the Chord peer-to-peer DHT implementation, allowing applications that used this DHT to be deployed using one-hop lookups, without the need to modify their code. Fig. 12 . Query failure rate in a system with 600 nodes when 45 percent of the system nodes crash at t ¼ 3;600 seconds. Fig. 13 . Overall bandwidth usage in a system with 600 nodes when 45 percent of the system nodes crash at t ¼ 3;600 seconds. This section presents an analysis of how to parameterize the system to satisfy our goal of fast propagation. To achieve our desired success rate, we need to propagate information about events within some time period t tot ; we begin this section by showing how to compute this quantity. Yet we also require good performance, especially with respect to bandwidth utilization. Later in the section, we show how we satisfy this requirement by controlling the number of slices and units.. Our analysis considers only nonfailure situations. It does not take into account overheads of slice and unit leader failures because these events are rare. It also ignores message loss and delay since this simplifies the presentation, and the overhead introduced by message delays and retransmissions is small compared to other costs in the system.. Our analysis assumes that query targets are distributed uniformly throughout the ring. It is based on a worst-case pattern of events, queries, and notifications: we assume all events happen just after the last slice-leader notifications, and all queries happen immediately after that, so that none of the affected routing table entries has been corrected, and all queries targeted to those nodes (i.e., the nodes causing the events) fail. In a real deployment, queries would be interleaved with events and notifications, so fewer of them would fail.. This scenario is illustrated by the timeline in Fig. 2 . Here, t wait is the time slice leaders wait to communicate with their unit leaders, t small is the time it takes to propagate information throughout a unit, and t big is the time a slice leader waits between communications to some other slice leader. Within t wait þ t small (point 4), slices in which the events occurred all have correct entries for nodes affected by the respective events. t big units of time after the events (point 5), slice leaders notify other slice leaders. Within a further t wait þ t small (point 7), all nodes in the system receive notification about all events.. Thus, t tot ¼ t detect þ t wait þ t small þ t big . The quantity t detect represents the delay between the time an event occurs and the time when the leader of that slice first learns about it. In this section, we analyze the limitations of OneHop; in particular, we discuss its limits in terms of scalability and what can be done when these limits are reached.. The two main issues that hamper scalability are memory usage and network traffic. In terms of memory requirements, keeping complete membership information requires about 26 bytes per node (assuming that node IDs are based on the SHA-1 function that outputs 160 bits and assuming that network addresses consist of an IPv4 address and a port number). Therefore, assuming that 100 Mbytes is a reasonable amount of memory to be dedicated to this table, we can scale to four million nodes.. As far as bandwidth usage is concerned and assuming a worst-case scenario where slice leaders are behind a residential 100-Kbps uplink, our algorithm stops working at around 300,000 nodes, according to the previous analysis.. After this point, we probably want to move to a multihop overlay. The natural next step would be to use a two-hop algorithm such as the one we presented together with our original publication of OneHop [1] . Our analysis in that paper showed that it could scale to very large systems, even with tens of millions of nodes, which is the current scale for the largest peer-to-peer systems that are deployed.. Despite the scalability of a two-hop algorithm, we also analyzed the alternatives if we needed to support an even larger number of nodes. We considered the possibility of generalizing the two-hop protocol to obtain a k-hop system but decided to reject that solution because generalizing the two-hop system to support an arbitrary number of hops turned out to be complex, and so, it makes more sense to use a traditional overlay like Pastry [4] , which can already get close to achieving three or four hops (which would be the new hop counts a k-hop scheme would enable). This can be achieved both by caching and by appropriately choosing system parameters like the base used by the protocol. Rodrigues et al. [20] proposed a single-hop DHT, but they assumed much smaller peer dynamics, like those in a corporate environment, and therefore did not have to deal with the difficulties of rapidly handling a large number of membership changes with efficient bandwidth usage. Douceur et al. [21] present a system that routes in a constant number of hops, but that design assumes smaller peer dynamics, and searches can be lossy.. Kelips [22] uses ffiffiffi n p -sized tables per node and a gossip mechanism to propagate event notifications to provide constant-time lookups. Their lookups, however, are constant time only when the routing table entries are reasonably accurate. As seen before, these systems are highly dynamic, and the accuracy of the tables depends on how long it takes for the system to converge after an event. The expected convergence time for an event in Kelips is Oð ffiffiffi n p Â log 3 ðnÞÞ. While this will be tens of seconds for small systems of around 1,000 nodes, for systems having 10 5 to 10 6 nodes, it takes over an hour for an event to be propagated through the system. At this rate, a large fraction of the routing entries in each table is likely to be stale, and a correspondingly large fraction of queries would fail on their first attempt.. Mahajan et al. [23] also derive analytic models for the cost of maintaining reliability in the Pastry [4] peer-to-peer routing algorithm in a dynamic setting. This work differs substantially from ours in that the nature of the routing algorithms is quite different-Pastry uses only OðlogNÞ state but requires OðlogNÞ hops per lookup-and they focus their work on techniques to reduce their (already low) maintenance cost.. Liben-Nowell et al. [24] provide a lower bound on the cost of maintaining routing information in peer-to-peer networks that try to maintain a topological structure. We are designing a system that requires significantly larger bandwidth than in the lower bound because we aim to achieve a much lower lookup latency.. Accordion [25] proposes a system that adapts the size of its routing state in order to trade bandwidth for lookup efficiency. This work improves OneHop by allowing the routing substrate to adapt the number of hops it can deliver, according to the bandwidth the system can afford to spend.. Since we first published our OneHop protocol [1] , a number of other one-hop protocols have been proposed [26] , [27] , [28] . These protocols also achieve full-information lookups but improve on the original OneHop design by not requiring the network to maintain a fixed event dissemination hierarchy and avoiding an asymmetric bandwidth consumption across nodes. We have implemented the OneHop as an extension to the code developed for the Chord project [9] . This section describes the changes we made to the Chord code and some design details to make OneHop more practical.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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PSPACE Reasoning for Graded Modal Logics We present a PSPACE algorithm that decides satisfiability of the graded modal logic Gr(KR)-a natural extension of propositional modal logic KR by counting expressions-which plays an important role in the area of knowledge representation. The algorithm employs a tableaux approach and is the first known algorithm which meets the lower bound for the complexity of the problem. Thus, we exactly fix the complexity of the problem and refute a EXPTIME-hardness conjecture. We extend the results to the logic Gr(K R −1 ∩ ), which augments Gr(KR) with inverse relations and intersection of accessibility relations. This establishes a kind of "theoretical benchmark" that all algorithmic approaches can be measured against. Propositional modal logics have found applications in many areas of computer science. Especially in the area of knowledge representation, the description logic (DL) ALC, which is a syntactical variant of the propositional (multi-)modal logic K R [Sch91] , forms the basis of a large number of formalisms used to represent and reason about conceptual and taxonomical knowledge of the application domain. The graded modal logic Gr(K R ) extends K R by graded modalities [Fin72] , i.e., counting expressions which allow one to express statements of the form "there are at least (at most) n accessible worlds that satisfy . . . ". This is especially useful in knowledge representation because (a) humans tend to describe objects by the number of other objects they are related to (a stressed person is a person given at least three assignments that are urgent), and (b) qualifying number restrictions (the DL's analogue for graded modalities [HB91] ) are necessary for modeling semantic data models [CLN94] .. K R is decidable in PSPACE and can be embedded into a decidable fragment of predicate logic [AvBN98] . Hence, there are two general approaches for reasoning with K R : dedicated decision procedures [Lad77, SSS91, GS96] , and the translation into first order logic followed by the application of an existing first order theorem prover [OS97, Sch97] . To compete with the dedicated algorithms, the second approach has to yield a decision procedure and it has to be efficient, because the dedicated algorithms usually have optimal worst-case complexity. For K R , the first issue is solved and, regarding the complexity, experimental results show that the algorithm competes well with dedicated algorithms [HS97] . Since experimental result can only be partially satisfactory, a theoretical complexity result would be desirable, but there are no exact results on the complexity of the theorem prover approach.. The situation for Gr(K R ) is more complicated: Gr(K R ) is known to be decidable, but this result is rather recent [HB91] , and the known PSPACE upper complexity bound for Gr(K R ) is only valid if we assume unary coding of numbers in the input, which is an unnatural restriction. For binary coding no upper bound is known and the problem has been conjectured to be EXPTIME-hard [dHR95] . This coincides with the observation that a straightforward adaptation of the translation technique leads to an exponential blow-up in the size of the first order formula. This is because it is possible to store the number n in log k n-bits if numbers are represented in k-ary coding. In [OSH96] a translation technique that overcomes this problem is proposed, but a decision procedure for the target fragment of first order logic yet has to be developed.. In this work we show that reasoning for Gr(K R ) is not harder than reasoning for K R by presenting an algorithm that decides satisfiability in PSPACE, even if the numbers in the input are binary coded. It is based on the tableaux algorithms for K R and tries to prove the satisfiability of a given formula by explicitly constructing a model for it. When trying to generalise the tableaux algorithms for K R to deal with Gr(K R ), there are some difficulties: (1) the straightforward approach leads to an incorrect algorithm; (2) even if this pitfall is avoided, special care has to be taken in order to obtain a space-efficient solution. As an example for (1), we will show that the algorithm presented in [dHR95] to decide satisfiability of Gr(K R ) is incorrect. Nevertheless, this algorithm will be the basis of our further considerations. Problem (2) is due to the fact that tableaux algorithms try to prove the satisfiability of a formula by explicitly building a model for it. If the tested formula requires the existence of n accessible worlds, a tableaux algorithm will include them in the model it constructs, which leads to exponential space consumption, at least if the numbers in the input are not unarily coded or memory is not re-used. An example for a correct algorithm which suffers from this problem can be found in [HB91] and is briefly presented in this paper. Our algorithm overcomes this problem by organising the search for a model in a way that allows for the re-use of space for each successor, thus being capable of deciding satisfiability of Gr(K R ) in PSPACE.. Using an extension of these techniques we obtain a PSPACE algorithm for the logic Gr(K R −1 ∩ ), which extends Gr(K R ) by inverse relations and intersection of relations. This solves an open problem from [DLNN97] . This paper is an significantly extended and improved version of [Tob99] .. As usual, the modal operators R n and [R] n are dual: ♯R M (x, φ) > n means that in M more than n R-successors of x satisfy φ; ♯R M (x, ¬φ) ≤ n means that in M all but at most n R-successors satisfy φ.. In the following we will only consider formulae in negation normal form (NNF), a form in which negations have been pushed inwards and occur in front of propositional atoms only. We will denote the NNF of ¬φ by ∼φ. The NNF can always be generated in linear time and space by successively applying the following equivalences from left to right:. Before we present our algorithm for deciding satisfiability of Gr(K R ), for historic and didactic reasons, we present two other solutions: an incorrect one [dHR95] , and a solution that is less efficient [HB91] . From the fact that SAT(K R ) is PSPACE-complete [Lad77, HM92] , it immediately follows, that SAT(Gr(K R )) is PSPACE-hard. The algorithms we will consider decide the satisfiability of a given formula φ by trying to construct a model for φ. We have shown that by employing a space efficient tableaux algorithm satisfiability of the logic Gr(K R ) can be decided in PSPACE, which is an optimal result with respect to worstcase complexity. Moreover, we have extended the technique to the logic Gr(K R −1 ∩ ), which extends Gr(K R ) both by inverse relations and intersection of relations. This logic is a notational variant of the Description Logic ALCQIR, for which the complexity of concept satisfiability has also been open. This settles the complexity of the DL ALCN R for which the upper complexity bound with binary coding had also been an open problem [DLNN97] . allocate counters ♯ω S (x, ψ) for all ω ∈ Ω φ and ψ ∈ clos(φ). restart:

O O O O O O O O O O O O O O O O O B I O O O O O B I O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Multiple integral representations for some families of hypergeometric and other polynomials The main objective of this paper is to investigate several general families of hypergeometric and other polynomials and their associated multiple integral representations. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric and other polynomials.

O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O


Discontinuous fluctuation distribution Abstract: This paper proposes an approach to the approximation of timedependent hyperbolic conservation laws which is both second order accurate in space and time (for any sufficiently smooth solution profile, even one containing turning points) and free of spurious oscillations for any time-step. The numerical algorithm is based on the concept of fluctuation distribution, applied on a space-time mesh of triangular prisms, for which second order accurate schemes already exist which are oscillation-free if the time-step satisfies a CFL-type constraint. This restriction is lifted here by combining the concept of a two-layer scheme with a representation of the solution which is allowed to be discontinuous in time. Numerical results are presented in two space dimensions, using unstructured meshes of space-time triangular prisms, for the scalar advection equation, Burgers' equation and the Euler equations of gasdynamics. Discontinuous residual distribution : A route to unconditional positivity and high order accuracyRésumé : This paper proposes an approach to the approximation of timedependent hyperbolic conservation laws which is both second order accurate in space and time (for any sufficiently smooth solution profile, even one containing turning points) and free of spurious oscillations for any time-step. The numerical algorithm is based on the concept of fluctuation distribution, applied on a space-time mesh of triangular prisms, for which second order accurate schemes already exist which are oscillation-free if the time-step satisfies a CFL-type constraint. This restriction is lifted here by combining the concept of a two-layer scheme with a representation of the solution which is allowed to be discontinuous in time. Numerical results are presented in two space dimensions, using unstructured meshes of space-time triangular prisms, for the scalar advection equation, Burgers' equation and the Euler equations of gasdynamics. The fluctuation distribution framework was developed as an alternative to the finite volume approach for approximating hyperbolic systems of conservation laws which would allow a natural representation of genuinely multidimensional flow features. The resulting algorithms are closely related to conforming finite elements, but their structure makes it far simpler to construct nonlinear approximation schemes, and therefore to avoid unphysical oscillations in the numerical solution. Recent reviews of the state-of-the-art and descriptions of these schemes can be found in [1, 6, 12] .. The original formulation, which has been successfully used to simulate steady flow around complete aircraft configurations [8] , led naturally to schemes which can achieve second order accuracy in smooth flows without introducing spurious numerical oscillations in the vicinity of discontinuities. Those schemes, however, reduce to first order accuracy away from the steady state, so much recent research has aimed to impose the same combination of properties on time-dependent simulations, leading to the space-time formulation [5] described briefly in Section 2.. All of the aforementioned schemes are based on the assumption that the solution varies continuously across the computational domain, but it is straightforward to generalise the framework to allow discontinuities across the faces of the mesh [3, 7] . This additional flexibility provides a number of advantages, e.g. h-and p-adaptivity are more easily applied, solution discontinuities can be captured exactly when aligned with the mesh, and characteristic boundary conditions can be implemented in a fully conservative manner. However, for the purposes of this work, the most important property is that allowing the solution to be discontinuous in time means that the distribution scheme used in the space-time cells no longer has to be upwind in time. This removes the CFLlike positivity constraint on the time-step so that a scheme can be constructed which is both second order accurate in space and time and unconditionally positive. This scheme will be summarised in Section 3 and numerical results will be presented in Section 4. A new formulation has been presented for fluctuation distribution schemes which based on a solution representation which is discontinuous in time. This formulation allows arbitrarily large time-step to be taken without introducing unphysical oscillations in to the numerical solution. It has also been shown that for scalar linear problems with smooth initial conditions the scheme is second order accurate.. The penalty incurred with respect to a standard implicit discretization is the doubling of the number of unknowns/nonlinear equations to be solved at each time step. Work remains in assessing the efficiency of the approach with respect to the accuracy versus computational time.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A nonparametric hierarchical bayesian framework for information filtering Information filtering has made considerable progress in recent years. The predominant approaches are content-based methods and collaborative methods. Researchers have largely concentrated on either of the two approaches since a principled unifying framework is still lacking. This paper suggests that both approaches can be combined under a hierarchical Bayesian framework. Individual content-based user profiles are generated and collaboration between various user models is achieved via a common learned prior distribution. However, it turns out that a parametric distribution (e.g. Gaussian) is too restrictive to describe such a common learned prior distribution. We thus introduce a nonparametric common prior, which is a sample generated from a Dirichlet process which assumes the role of a hyper prior. We describe effective means to learn this nonparametric distribution, and apply it to learn users' information needs. The resultant algorithm is simple and understandable, and offers a principled solution to combine content-based filtering and collaborative filtering. Within our framework, we are now able to interpret various existing techniques from a unifying point of view. Finally we demonstrate the empirical success of the proposed information filtering methods. Information filtering denotes a family of techniques that help users to find the right information items while filtering out undesired ones. In a wide range of applications, such as spam email filtering, news filtering, and recommender systems for products (e.g. books), information filtering is playing an increasingly important role. Content-based filtering (CBF) and collaborative filtering (CF) represent the two major information filtering technologies. CBF [19, 2, 10, 13] has its root in the concept of relevance feedback in the information retrieval literature (e.g. Rocchio's algorithm [19] ). CBF explores the similarity of contents between information items (e.g. articles, images, music), to infer which of the yet unseen items might be of interest for the active user, based on some annotated examples previously given by the user. In contrast, collaborative filtering methods [18, 20, 4] typically accumulate a database of item ratings-explicitly or implicitly-cast by a large set of users. The prediction of ratings for the active user is solely based on the ratings provided by all other users, under the assumption that likeminded users are sharing similar information needs. The method does not rely on a description of item content.. One major difficulty in designing CBF systems lies in extracting content features that are sufficiently indicative. There is often a large gap between low-level content features (visual, auditory, or others) and high-level user interests (like or dislike a painting or a CD). In some other circumstances, the features are not available at all. Fortunately, the information on personal preferences and interests are all carried in (explicit or implicit) user ratings. Thus CF systems can make use of these high level features rather easily, by combining the ratings of other like-minded users.. Pure CF only relies solely on user preferences, without incorporating the actual content of items. CF often suffers from the extreme sparsity of available data, in the sense that users typically rate only very few items, thus making it difficult to compare the interests of two users. Furthermore, pure CF can not handle items for which no user has previously given a rating. Such cases are easily handled in CBF systems, which can make predictions based on the content of the new item.. Naturally, previous researchers have worked on compensating the drawbacks of each particular approach. Many approaches have focused on hybrid filtering to unify both approaches [12, 7, 2, 3, 16] . However, due to the lack of a unifying framework for information filtering, existing solutions were developed mostly in heuristic or ad-hoc ways. The chal-lenge is therefore to find a principled approach to combine CBF and CF. A recent publication [21] made first attempts to unify CF and CBF under a hierarchical Bayesian framework. The work is extended in this paper and the theoretical basis is further developed and clarified. This paper describes a nonparametric hierarchical Bayesian framework to information filtering. In the framework, each user is modelled by a parametric content-based profile model, whose parameters θ are generated from a common prior distribution p(θ), which is shared by all the users. Then the model allows different models to inherit knowledge from each other. We adopt a nonparametric form for the common prior, which is generated from a Dirichlet process (i.e. the hyper prior). We have shown that this nonparametric Bayesian treatment offers great advantages, including (1) flexible modelling ability and (2) meaningful insights into the problem. The hyper prior reflects the prior knowledge that users are related. We derive effective ways to learn the model from data. The resultant formulism is generally applicable to a wide range of probabilistic profiling models. The connections of this work to other related approaches are also clarified. Many independently proposed methods can now be interpreted from a unified view. Our empirical study demonstrates the excellent performance of our approaches.. Nonparametric Bayesian modelling has a long tradition in statistics and was recently introduced into the machine learning community. Its applications to information retrieval, like text modelling, document clustering, and user grouping, is still at the earliest stage. One recent related approach is the hierarchical topic model proposed by Blei et al. [5] , which applies Dirichlet process using MCMC to cluster documents by using MCMC. Hierarchical modelling using Dirichlet process with MCMC can discover the clustering structure of data and naturally finds the number of clusters [8, 17] , while optimizing the number of clusters in most other clustering methods is a technically difficult problem. Our current work aimed at realizing efficient learning and prediction based on an EM algorithm. However, as part of future work, it would also be interesting to implement the full MCMC solution to discover groups of related users.

O O O O B O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O B O O B O O B O O O O O O O O O O O O B O O O O B O O O O O O O B O O B O O O O O O O B O O O O O O O B O O O O O O O O O O O O O B O B O O O O O O O O B O O O O O B O B O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O B O O O O O B O O B O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B


Pad assignment for die-stacking System-in-Package design Wire bonding is the most popular method to connect signals between dies in System-in-Package (SiP) design nowadays. Pad assignment, which assigns inter-die signals to die pads so as to facilitate wire bonding, is an important physical design problem for SiP design because the quality of a pad assignment solution affects both the cost and performance of a SiP design. In this paper, we study a pad assignment problem, which prohibits the generation of illegal crossings and aims to minimize the total signal wirelength, for die-stacking SiP design. We first consider a variety of special cases and present a minimum-cost maximum-flow based approach to optimally solve them in polynomial time. We then describe an approach, which uses a modified left edge algorithm and an integer linear programming technique, to solve the general case. Encouraging experimental results are shown to support our approaches. Comparing System-in-Package (SiP) with System-on-Chip (SoC), SiP is a more economical option than SoC in many consumer electronic products because of the high process complexity and cost associated with SoC. On the other hand, compared with traditional system integration where multiple dies with separate packaging are mounted on a PCB, SiP * This work was partially supported by National Science Council of Taiwan under Grants NSC 98-2220-E-007-011, NSC 98-2220-E-007-012, and NSC 98-2220-E-007-013.. has the advantages of smaller size, lower cost, higher performance, lower power, and shorter time to market. So, today SiP is already widely used in consumer electronics such as cell phones. Currently, SiP design is mostly done by ad hoc methods and the quality of a design is heavily dependent on the designers' expertise. Tool support specific to SiP design is still inadequate [5, 6, 9] .. Wire bonding is the most popular method to connect signals between different dies in SiP nowadays. As shown in Figure 1 , a die-stacking SiP design using wire bonding has the following properties: (1) Dies with different sizes are stacked and pad signals are connected by bonding wires. (2) Die pads can only be located on die boundaries. If a direct inter-die connection cannot be made for a signal between two dies, then we need to route the signal from the two dies to the package substrate by two bonding wires and use the substrate to complete its routing which will not only affect performance but is much more costly (because many SiPs use gold for bonding wires). It is desirable to maximize the number of direct inter-die connections and minimize the total bonding wirelength both for cost and performance consideration. In this paper, we have presented novel pad assignment approaches for SiP design. The experimental results have been provided to demonstrate the efficiency and effectiveness of our approaches.

B I O O B O B I O O O O O O O O O O O B O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O


On the frontiers of polynomial computations in tropical geometry We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We characterize the borderline between tractable and hard computations by proving N P-hardness and #P-hardness results under various strong restrictions of the input data, as well as providing polynomial time algorithms for various other restrictions. Geometry over the tropical semiring (R, ⊕, ⊙) := (R, min, +) has received much attention in the last years (see the surveys [15, 20, 25] and the references therein) with applications in counting curves [14] , studying phylogenetic trees [16] , and the analysis of amoebas of algebraic varieties [15] . From the viewpoint of polynomial equations, the modern birth of tropical geometry originates in the book [26] which pinpoints the central role of tropical geometry as a link between algebraic geometry, symbolic computation, and discrete geometry, thus providing computationally-accessible methods for studying algebraic-geometric problems. Indeed, one of the early roots of the developments in tropical geometry can be seen in the polyhedral homotopy methods by Huber and Sturmfels [9] , providing a state-of-the-art technique for numerically solving systems of polynomial equations based on a deformation to a (discrete) tropical problem.. Some major algorithmic results in tropical geometry are based on Gröbner basis computations and thus may become intractable already for small dimensions [3] . On the positive side, there also exist some algorithmic problems (such as computing the tropical determinant) which can be efficiently solved using techniques from linear programming, polyhedral computation and combinatorial optimization (see, e.g., [10, 20] ). For many tropical problems, their computational complexity has not been clarified yet.. In this paper, we make a first step towards systematically studying the frontiers of polynomial time computations in tropical geometry. For this, we consider three natural algorithmic problems concerning the intersection of tropical hypersurfaces, so-called tropical prevarieties. The algorithmic problems are to decide whether this intersection Key words and phrases. Tropical geometry, tropical varieties, tropical prevarieties, computational complexity, NP-hard, polynomial time algorithms.. Part of this work was done while the author was a Feodor Lynen fellow of the Alexander von Humboldt Foundation at Yale University. P is nonempty (Tropical Intersection), whether P is a tropical variety (Tropical Consistency), and whether P is connected (Tropical Connectivity).. Our results refer to the the standard Turing machine model, and we mainly aim at characterizing the borderline between tractable (in the sense of polynomial time solvable) and hard (in the sense of N P-hard) computations. Our main results can roughly be stated as follows. If the number of hypersurfaces is part of the input then the three problems become N P-hard or co-N P-hard, and this hardness persists even under various restrictions to the input data. As a particular example, already for quadratic input polynomials it is co-N P-hard to decide whether a tropical prevariety is a tropical variety. Hence, efficient algorithms cannot be expected in this setting. We contrast these hardness results by polynomial time algorithms for a fixed number of tropical hypersurfaces. For a precise statement of the results see Theorems 3.1-3.3.. The paper is structured as follows. In Section 2 we introduce the relevant notation from tropical geometry and computational complexity. In Section 3 we formally state our main results, and Section 4 contains the proofs of these theorems. We close the paper with a short discussion of related computational aspects on amoebas.

O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I B I B I B I B B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


On the set multi-cover problem in geometric settings We consider the set multi-cover problem in geometric settings. Given a set of points P and a collection of geometric shapes (or sets) F, we wish to find a minimum cardinality subset of F such that each point p ∈ P is covered by (contained in) at least d(p) sets. Here d(p) is an integer demand (requirement) for p. When the demands d(p) = 1 for all p, this is the standard set cover problem. The set cover problem in geometric settings admits an approximation ratio that is better than that for the general version. In this paper, we show that similar improvements can be obtained for the multi-cover problem as well. In particular, we obtain an O(log opt) approximation for set systems of bounded VCdimension, and an O(1) approximation for covering points by half-spaces in three dimensions and for some other classes of shapes. The set cover problem is the following. Given a universe U of n elements and a collection of sets F = {S1, . . . , Sm} where each Si is a subset of U, find a minimum cardinality sub-collection C ⊆ F such that C covers U; in other words, the union of the sets in C is U. In the weighted version each set Si has a non-negative weight wi and the goal is to find a minimum weight cover C. In this paper, we are primarily interested in a generalization of the set cover problem; namely, the set multi-cover problem. In this version, each element e ∈ U has an integer demand or requirement d(e) and a multi-cover is a sub-collection C ⊆ F such that for each e ∈ U there are d(e) distinct sets in C that contain e. 1 The set cover problem and its variants arise directly and indirectly in a wide variety of settings and have numerous applications. Often F is available only in an implicit fashion and could be exponential in the size of U or even infinite (for example F could be the set of all disks in the plane). The set cover problem is NP-Hard and consequently approximation algorithms for it have received considerable attention. A simple greedy algorithm, that iteratively adds a set from F that covers the most uncovered elements, is known to give a (1 + ln n) approximation, where n = |U|. (In the weighted case, the algorithm picks the set with minimum average cost for the uncovered elements.) Similar bounds can also be achieved via rounding a linear programming relaxation. The advantage of the greedy algorithm is that it is also applicable in settings where F is given implicitly, but there exists a polynomial time oracle to (approximately) implement the greedy step in each iteration. It is also known that unless P = NP there is no o(log n) approximation for the set cover problem [18] . Moreover, unless NP ⊂ DTIME(n O(log log n) ) there is no (1 − o(1)) ln n approximation [13] . Thus the approximability of the general set cover problem is essentially resolved if P = NP. However, there are many set systems of interest for which the hardness of approximation result does not apply. There has been considerable effort to understand the approximability of set cover in restricted settings. Set systems that arise in geometric settings are the focus of this paper. Previous work has shown that the set cover problem admits improved approximation ratios in various geometric cases.. In the geometric setting, we use (P, F) to describe a set system (also referred to as a range space) where P is a set of points and F is a collection of sets (also called objects or ranges). We are typically interested in the case where F is a set of "well-behaved shapes". Examples of such shapes include disks, pseudo-disks, and convex polygons. The goal is to cover a given finite set of points P in IR d by a collection 1 A related and somewhat easier variant allows a set to be picked multiple times. In this paper, unless explicitly stated, we use multi-cover for the version where only one copy of a set is allowed to be picked.. of objects from F. At a higher level of abstraction, one can consider set systems of small (or constant) VC dimension. In addition to the inherent theoretical interest in geometric set systems, there is also motivation from applications in wireless and sensor networks. In these applications the coverage of a wireless or sensor transmitter can be reasonably approximated as a disk-like region in the plane. The problem of locating transmitters to optimize various metrics of coverage and connectivity is a well-studied topic; see [22] for a survey. Brönnimann and Goodrich [3] , extending the work of Clarkson [7] , used the reweighting technique to give an O(log opt) approximation for the set cover problem when the VC dimension of the set system is bounded 2 . Here opt is the size of an optimum solution. Known hardness results [18] preclude such an approximation ratio for the general set cover problem. The reweighting technique and its application to set cover [7, 3] show that the approximation ratio for set cover can be related to bounds on ε-nets for set systems. Using this observation, [3] showed an improved O(1) approximation ratio for the set cover problem in some cases, including the problem of covering points by disks in the plane. Long [17] made an explicit connection between the integrality gap of the natural LP relaxation for the set cover problem and bounds on the ε-nets for the set system (see also [11] ). This allows opt in the approximation ratio to be replaced by f, where f is the value of an optimum solution to the LP relaxation (i.e., the optimal fractional solution). Clarkson and Varadarajan [9] developed a framework to obtain useful bounds on the ε-net size via bounds on the union complexity of a set of geometric shapes. Using this framework they gave improved approximations for various set systems/shapes. Recently, Aronov, Ezra and Sharir [1] , and Varadarajan [23] slightly improve the bounds of Clarkson and Varadarajan [9] .. The geometric set cover problem induced by covering points by disks in the plane is strongly NP-Hard [12] ; existing results yield a constant factor approximation although a PTAS for this problem has so far not been ruled out. Some other geometric coverage problems are known to be APX-hard [14] ; that is, there is a constant c > 1 such that unless P = NP, there is no c approximation for them.. Our results: In this paper, we consider the multi-cover problem in the geometric setting. In addition to the set system (P, F), each point p ∈ P has an integer demand d(p). Now a cover needs to include, for each point p, d(p) sets that contain p. For general set systems, the greedy algorithm and other methods such as randomized rounding, which work for the set cover problem, can be adapted to the multi-cover problem, yielding a (1 + ln n) approximation [24] . In contrast, the ε-net based approach for geometric set cover does not generalize to the multi-cover setting in a straight-forward fashion. Nevertheless, we are able to use related ideas, in a somewhat more sophisticated way, to obtain approximation ratios for the geometric set multi-cover problem that essentially match the ratios known for the corresponding set cover problem. In particular, we obtain the following bounds. In all the bounds, f ≤ opt is the value of an optimum (fractional) solution to the natural LP relaxation, and opt is the value of an optimum (integral) solution.. • O(log f) approximation for set multi-cover of set systems of bounded VC dimension.. • O(1) approximation for (multi) covering points in IR 3 by halfspaces. This immediately leads to a similar result for multi-cover of disks by points in the plane.. • O(log log log f) approximation for covering points by fat triangles (or other fat convex polygonal shapes of constant descriptive complexity) in the plane.. The second and third results follow from a general framework for a class of "well-behaved" shapes based on the union complexity of the shapes. This is inspired by a similar framework from [9, 1] . Our work differs from previous work for set cover in geometric settings in two ways. First, we use the LP relaxation in an explicit fashion in several ways, demonstrating its effectiveness. Second, our work points out the usefulness of shallow cuttings for the multi-cover problem. We hope that these directions will be further developed in the future. We presented improved approximation algorithms for set multi-cover in geometric settings. Our key insight was to produce a "small" instance of the problem by clustering the given instance. This in turn was done by using a variant of shallow cuttings. We believe that this approach might be useful for other problems in geometric settings.. An interesting open problem, is to obtain improved algorithms for the set cover and the set multi-cover problems in geometric settings when the sets/shapes have costs associated with them and the goal is to find a cover of lowest cost. Can the results from [7, 3, 9] and this paper be extended to this more general setting? APPENDIX A. A SHALLOW CUTTING LEMMA FOR "NICE" SHAPES. In this section, we prove Theorem 5.1, a variant of the shallow cutting lemma of Matoušek in a slightly different setting. We include the details for the sake of completeness, which are not hard in light of Matoušek's work [19] . Our description is somewhat informal, for simplicity. The family of shapes that we consider needs to satisfy the assumptions outlined in Section 5.. Building (1/r)-cuttings: When computing cuttings, one first picks a random sample R of size r of the objects of F, and computes the decomposition A || (R) of the arrangement of the random sample. For a cell in this decomposition, let cl( ) be the list of shapes of F whose boundaries intersect the interior of . If |cl( )| ≤ n/r then it is acceptable, and we add it to the resulting cutting.. Otherwise, we need to do a local patching up, by partitioning each such cell further. Specifically, let t = cl( ) /(n/r) be the excess of . We take a random sample R of size O(t log(t )) from cl( ). With constant probability, this is a 1/t -net of cl( ) (for ranges formed by our decomposition). We verify that it is such a net, and if not, we resample, and repeatedly do so until we obtain a 1/t -net. To do the verification, we build the arrangement of R inside , and compute its decomposition, and check that all the cells in this decomposition intersect at most n/r boundaries of the shapes of F. Let dcmp( ) denote this decomposition of

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Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers We study arithmetical and combinatorial properties of β-integers for β being the root of the equation x 2 = mx − n, m, n ∈ N, m ≥ n + 2 ≥ 3. We determine with the accuracy of ±1 the maximal number of β-fractional positions, which may arise as a result of addition of two β-integers. For the infinite word u β coding distances between consecutive β-integers, we determine precisely also the balance. The word u β is the fixed point of the morphism A → A m−1 B and B → A m−n−1 B. In the case n = 1 the corresponding infinite word u β is sturmian and therefore 1-balanced. On the simplest non-sturmian example with n ≥ 2, we illustrate how closely the balance and arithmetical properties of β-integers are related. In this paper, we focus on study of arithmetical and combinatorial properties of β-integers for β being a quadratic algebraic integer with positive norm. The notion of β-integer is related to greedy algorithm searching for the expansion of a real number x in base β > 1; this algorithm has been introduced in [23] by Rényi. The real number x is called β-integer if its β-expansion has the form ± n k=0 x k β k , i.e. if all of its coefficients at powers β −k vanish for k > 0. The set of β-integers (denoted by Z β ) equals in case of β ∈ N to the set of integers Z. If β is not an integer, the set Z β has much more interesting properties: Z β . From results [26, 22] , it follows that Z β has this property if and only if the Rényi expansion of unity in base β is eventually periodic. Such numbers β are called Parry numbers. It can be easily shown that every Parry number β is an algebraic integer, i.e. it is a root of a monic polynomial having integer coefficients. The task to describe which algebraic integers are Parry numbers has not been solved yet. It is known that each Pisot number is as well Parry. Let us remind that an algebraic integer β is a Pisot number if all of its conjugates have modulus less than 1. In case of β being a Pisot number, β-integers form the Meyer set, i.e. it holds Z β − Z β ⊂ Z β + F for a finite set F ⊂ R. Thus, the notion of Meyer set generalizes the notion of lattice, which is crucial for description of crystals. As we have already mentioned, in case of β being a Parry number, the set Z β disposes with a finite number of distances between neighbors. If we associate to different gaps different letters, it is possible to encode the set Z β as an infinite word u β over a finite alphabet. Combinatorial properties of words u β have been studied in several papers: [12, 13] is devoted to description of factor complexity of u β , palindromes of u β are described in papers [3, 6] . So far the least studied problem is the balance of u β , i.e. the maximal difference in numbers of different letters in factors of the same length. Balance is clearly known for Z β which corresponds to sturmian words, i.e. for β being a quadratic unit. In [27] , the balance property for u β , where β is the larger root of the quadratic polynomial x 2 − mx − n, m, n ∈ N, m ≥ n ≥ 1, has been studied. For other types of irrationalities, the balance property has not been described yet.. The sets of ordinary integers and β-integers are very different also from the arithmetical point of view. Z β is not closed under addition and multiplication for any β ∈ N. Sum of two β-integers may even not have a finite β-expansion. So far unsolved and likely very difficult is the question of characterization of those β for which this pathological situation does not appear. Mathematically expressed it means to describe β for which the set F in(β), i.e. the set of numbers with finite β-expansion, is a subring of R. Frougny and Solomyak have shown in [14] that the necessary condition for this so-called finiteness property is that β is a Pisot number. Some sufficient conditions can be found in [2, 14, 16] . If sum or product of two β-integers has a finite β-expansion, there arises a question how long is the β-fractional part of the sum or product. This problem has been investigated in [4, 7, 10, 15, 19] .. Here, the main attention is devoted to investigation of arithmetics of β-integers for a non-simple quadratic Parry number β, i.e. for β being the root of the equation x 2 = mx − n, m, n ∈ N, m ≥ n + 2 ≥ 3. We determine with the accuracy of ±1 the maximal number of β-fractional positions L ⊕ (β), which may arise as a result of addition of two β-integers. So we improve considerably the estimate from the paper [15] . We determine accurately also the balance of u β . On this easiest non-sturmian example, we illustrate how closely the arithmetical and combinatorial properties of Z β are related. Particulary, we show the relation between L ⊕ (β) and the balance property. Our method might be applied also for determination of the balance property for words coding β-integers with irrationalities of a higher degree.

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