ABS: A low-cost adaptive controller for prefetching in a banked shared last-level cache Hardware data prefetch is a very well known technique for hiding memory latencies. However, in a multicore system fitted with a shared Last-Level Cache (LLC), prefetch induced by a core consumes common resources such as shared cache space and main memory bandwidth. This may degrade the performance of other cores and even the overall system performance unless the prefetch aggressiveness of each core is controlled from a system standpoint. On the other hand, LLCs in commercial chip multiprocessors are more and more frequently organized in independent banks. In this contribution, we target for the first time prefetch in a banked LLC organization and propose ABS, a low-cost controller with a hill-climbing approach that runs stand-alone at each LLC bank without requiring inter-bank communication. Using multiprogrammed SPEC2K6 workloads, our analysis shows that the mechanism improves both user-oriented metrics (Harmonic Mean of Speedups by 27% and Fairness by 11%) and system-oriented metrics (Weighted Speedup increases 22% and Memory Bandwidth Consumption decreases 14%) over an eight-core baseline system that uses aggressive sequential prefetch with a fixed degree. Similar conclusions can be drawn by varying the number of cores or the LLC size, when running parallel applications, or when other prefetch engines are controlled. Hardware data prefetch is a very well known technique for hiding the long latencies involved in off-chip accesses. It tries to predict memory addresses in advance, requesting them to the next level, and loading the lines into the cache before the actual demands take place. Several commercial multicore processors implement some form of hardware data prefetch [Conway et al. 2010; Le et al. 2007] . These devices are fitted with a hierarchy of cache memories. Usually, the first levels are private and the Last Level Cache (LLC) is shared by all the cores in the system. In this work we consider this kind of system. Prefetches may be initiated in the first-level caches or directly from events occurring in the LLC, but in the end the prefetches reach the shared LLC and interfere with each other. That is, prefetches issued on behalf of one core may evict LLC lines previously allocated by other cores, either by a memory instruction or a prefetch request. In addition, the prefetch activity originated from a single core can reduce the overall available bandwidth, potentially increasing the latency seen by the demands or prefetches coming from the rest of cores.. Most prefetch proposals in multiprocessors deal with systems having only private caches [Cantin et al. 2006; Dahlgren et al. 1993; Koppelman 2000; Somogyi et al. 2009; Tcheun et al. 1997] , while prefetch for shared caches has received little attention [Ebrahimi et al. 2009] . Ebrahimi et al. [2009] tackle for the first time the problem of reducing the prefetch inter-core interference in a chip multiprocessor with a shared LLC. They propose the Hierarchical Prefetcher Aggressiveness Control (HPAC) mechanism, that monitors several global indexes (prefetch accuracy, inter-core pollution, and memory controller activities) to adjust the prefetch aggressiveness of each core. This assumes a centralized implementation of the LLC, internally organized in banks but with a single access port (see Figure 1(a) ). Thus, the global aggressiveness control and associated hardware structures are also centralized.. To the best of our knowledge, this is the first work where prefetch is studied in a multicore system fitted with a banked shared LLC, see Figure 1b . We assume an LLC organized in independent cache banks with an access port each, and an interconnection network attaching cores to cache banks (a crossbar is assumed, but other topologies can be considered) [Kongetira et al. 2005; Kottapalli and Baxter 2009] . Each bank is internally sub-banked in order to provide a higher throughput. We think this kind of LLC will become mainstream in the short term, because independent banks add layout flexibility and increase access bandwidth. Commercial processors from AMD, Sun, Intel, or IBM are using this design [Conway et al. 2010; Kongetira et al. 2005; Kottapalli and Baxter 2009; Le et al. 2007] .. In this scenario, we introduce the ABS controller, an Adaptive controller for prefetch in a Banked Shared LLC. ABS controllers are installed in all the LLC banks which are already fitted with a prefetch engine. Each ABS controller runs autonomously and gather local statistics to set the prefetch aggressiveness for each core in the bank it controls in order to maximize the overall system performance using a hill-climbing approach. Therefore, a given core is allowed to prefetch with different aggressiveness on different banks of the LLC.. Isolation between banks is a key factor of our proposal, meaning that both the ABS controller and the prefetcher in a bank are not influenced by their peers at other banks. Bank isolation achieves two essential benefits, namely (i) the prefetches generated from a given bank target itself, and it will always be possible to filter useless prefetches by looking up in the bank, thus saving memory bandwidth, and (ii) communicating prefetchers or ABS controllers among banks is not required, removing the need for a dedicated interconnection network or extra traffic in the existing one. As discussed in Section 5.2, bank isolation can be achieved by selecting a proper address interleaving among banks or by adjusting the prefetch distance.. Our results show that an eight-core system with ABS controllers running multiprogrammed SPEC2K6 workloads improves in both user-oriented metrics and systemoriented metrics over a baseline system with a fixed degree sequential prefetch. The results are consistent when varying the number of cores or LLC sizes. ABS control can be applied to other prefetch engines as long as they are able to operate at different aggressiveness levels. Specifically, we introduce ABS-controlled sequential streams. A comparison with HPAC-controlled sequential streams, such as that proposed by Ebrahimi et al. [2009] , shows higher performance at a very small fraction of the cost. Furthermore, when running multithreaded workloads from SPLASH-2 [Woo et al. 1995] and PARSEC [Bienia 2011 ], the ABS controllers also reduce the execution time and the consumed bandwidth over the baseline system. The paper is structured as follows. Section 2 describes the motivation behind the work. Section 3 gives some background and reviews related work. Section 4 introduces the ABS controller. Section 5 presents the prefetch framework. Section 6 shows the methodology followed. Section 7 shows the results when ABS controllers are evaluated in a variety of situations, and Section 8 concludes the paper. Figure 2 shows instructions per cycle (IPC) for eight SPEC2K6 applications running on a system with eight cores and a 4MB shared LLC. The simulation details are shown in Section 6. The figure shows four bars for each application. The first two bars represent programs running alone in the system, either without prefetch or with an aggressive (degree 16) sequential tagged prefetch (see Section 3). The last two bars represent the eight applications running together, either all without prefetch or all with the former aggressive prefetch turned on. When comparing the systems with prefetch (second and fourth bars), significant performance losses appear when resources are shared among cores. Note that prefetch involves virtually no performance loss in any application when running alone (first and second bars), while it causes losses in 5 out of 8 applications when running all together (third and fourth bars). Therefore, in order to boost the shared LLC performance by means of prefetch, a mechanism to control aggressiveness is called for. Such a mechanism should consider global metrics to realize when the prefetch activity of a core harms the overall system performance and it should be decreased in spite of the improvement achieved by that core. In this paper we focus on the shared, last level cache (LLC) of a multicore chip. We assume a LLC having a distributed implementation in multiple banks. In this scenario, we introduce the ABS controller (Adaptive prefetch control for a Banked Shared LLC), an adaptive mechanism to control the prefetch aggressiveness independently for each core in each LLC bank. The ABS controller implements a hill-climbing algorithm which runs stand-alone at each LLC bank, using only local information. Bank miss ratio and prefetch accuracy are sampled at fixed-length epochs and used as performance index.. For each bank at each epoch the aggressiveness of only one core is varied in order to establish a cause-effect relationship between the change in aggressiveness and the change in the performance index.. The ABS controllers are evaluated to adjust the aggressiveness level of variabledegree sequential tagged prefetchers. Our analysis using multiprogrammed SPEC2K6 workloads shows that the mechanism improves both user-oriented metrics (Harmonic Mean of Speedups by 27% and Fairness by 11%) and system-oriented metrics (Weighted Speedup increases by 22% and Memory Bandwidth Consumption decreases by 14%) over an eight-core baseline system that uses aggressive sequential prefetch with a fixed degree. Similar results have been obtained on a sixteen-core system, when doubling the LLC size or running parallel applications.. Besides, ABS is also compared to the mechanism for a centralized shared LLC proposed by Ebrahimi et al. [2009] but adapted for a banked LLC. For this comparison, variable-aggressiveness sequential stream prefetchers are used as prefetch engines. ABS performs better in all the performance indexes except in required bandwidth, which is somewhat greater. Summarizing, ABS controllers are able to control the aggressiveness of prefetch engines in a distributed fashion and with very low implementation costs. Their distributed nature assures scalability in the number of cores and cache banks for future multicore chips. In this section we evaluate the ABS controllers. Section 7.1 analyzes ABS controlling sequential tagged prefetch in the baseline system. Section 7.2 shows results for 16-core systems. In Section 7.3 we increase the LLC size. In Section 7.4 ABS controllers are compared with a previous proposal, and in Section 7.5 they are evaluated using parallel applications. Figure 6 shows the results of ABS controlling prefetch aggressiveness of the mix2 described in Section 2. The 8 programs run simultaneously and the bars corresponding to execution without prefetch (8apps no pref ) and with fixed-degree (16) aggressive prefetch (8apps pref ) are kept. A new bar corresponding to ABS prefetch (ABS) is added. As Figure 2 highlights, the benefit obtained by an application due to prefetch can decrease or even turn into losses if prefetch is simultaneously active in all cores. Therefore, our goal is to design a mechanism that dynamically controls the prefetch aggressiveness of each core in order to maximize system performance.. The impact on system performance can be assessed using global indexes such as aggregated IPC or shared LLC miss ratio. Both are obtained by adding quantities that are distributed in the cores or the cache banks, respectively. So, a centralized design of the prefetch aggressiveness control requires sending information from the places where events are counted to the centralized control point. Alternatively, we propose to place an ABS controller in each LLC bank. The controller uses bank-local information (i.e. bank miss ratio) to improve bank performance. Improving the performance of every bank will thus improve the system performance. Simics, a full-system execution-driven simulator, has been used to evaluate our proposal [Magnusson et al. 2002] . The Ruby plugin from the Multifacet GEMS toolset was used to model the memory hierarchy with a high degree of detail [Martin et al. 2005] , including coherence protocol, on-chip network, communication buffering, contention, etc. The prefetch system has been integrated into the coherence mechanism and a detailed DDR3 DRAM model has been added.. Multiprogrammed SPECCPU 2K6 workloads running on a Solaris 10 Operating System have been used. In order to discard the less demanding memory applications and locate the end of the initialization phase, all the SPARC binaries were run on a real machine until completion with the reference inputs and hardware counters were used. Eight applications were subsequently discarded and 21 selected, shown in the first column of Table I .. For an eight-core system, a set of 30 random mixes of 8 programs each, were taken from the previously selected 21 SPECCPU 2K6 programs (no effort has been made to distinguish between integer and floating point). In the next section, we usually show averages over this set of 30 mixes, but in order to gain a deeper insight into individual mix behaviors, sometimes a subset of 10 mixes is shown. This randomly chosen subset of mixes appears in Table I , along with the misses per kilo-instruction (MPKI) of each application in each mix when the application runs alone, that is, it runs using all the shared memory resources and with the remaining seven cores stopped. 4 Summarizing, we have created 30 checkpoints of 8 applications each. Each checkpoint guarantees that no application is in its initialization phase. The cycle-accurate simulation starts at these checkpoints, warming the memory hierarchy for 500 million cycles with the prefetch disabled. We then collect statistics for the next billion cycles.

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Framework for security and privacy in automotive telematics Automotive telematics may be defined as the informationintensive applications that are being enabled for vehicles by a combination of telecommunications and computing technology. Telematics by its nature requires the capture of sensor data, storage and exchange of data to obtain remote services. In order for automotive telematics to grow to its full potential, telematics data must be protected. Data protection must include privacy and security for end-users, service providers and application providers. In this paper, we propose a new framework for data protection that is built on the foundation of privacy and security technologies. The privacy technology enables users and service providers to define flexible data model and policy models. The security technology provides traditional capabilities such as encryption, authentication, non-repudiation. In addition, it provides secure environments for protected execution, which is essential to limiting data access to specific purposes. Automotive telematics may be defined as the informationintensive applications that are being enabled for vehicles by a combination of telecommunications and computing technology. The automobile is, in effect, a computing platform to which mobile commerce services may be delivered. The services being delivered today on a regular basis and projected for the near future include navigation information, emergency roadside assistance, location-based services, delivery of digital information such as email, entertainment, diagnostics and prognostics, and pay-for-use rental and insurance. These applications are enabled by the collection and use of data which may include information on the location of a vehicle as a function of time, emergency situations including accidents and personal health emergencies, diagnostic data on the many systems within the vehicle, services and entertainment that are selected by the vehicle occupants, the demographics of the driver and passengers, and the behavior of the vehicle driver.. We can compare the growing automotive e-commerce telematics industry with that of the Web. The growth of e-commerce on the World Wide Web has been limited by the reluctance of consumers to release personal information. In "Building Consumer Trust in Online Environments" [1] the authors find that "Fully 94 percent of Web users have declined to provide personal information to Web sites at one time or another when asked and 40 percent who have provided demographic data have gone to the trouble of fabricating it". If potential automotive telematics users share the concerns of Web users, then a large segment of the potential telematics market, perhaps as much as fifty percent may be lost.. There is a significant potential for the misuse of collected data. End users or consumers may substitute false data or hack into invehicle applications. Telematics service providers and application providers may sell consumers' data to third parties without the permission of the consumers. Although, there are no current US regulations in place to "safeguard" the information collected, certain existing European regulations, and pending US and European statutes may soon impose strict controls on the collection, use, and storage of information about individuals. In general, telematics applications will be successful if providers know that the data that they receive is accurate and if end users know that their privacy is assured. Thus, data must be protected. Users must be assured that their privacy is respected and the security is in place to protect data from being divulged to unauthorized entities. Data protection consists of providing both privacy and security protection. Our goal is to achieve that protection while enabling the sharing of data.. Privacy protection today at a fundamental level requires a user to trust service providers to handle personal data according to stated terms. There is a certain degree of goodwill that is at stake to prevent a service provider from using the data in an inappropriate 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, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. manner. However, there are no safeguards in place to prevent inappropriate use of data; and no protection from insider abuse.. Likewise, there are no protections to assure a vehicle user that applications that are running in the car are secure. Kingpin and Mudge [2] [3]analyze the susceptibility of portable devices, primarily PDAs, to attack by malicious code. They make the point that you cannot have a secure application without a secure foundation. As with PDAs, it is key to the future of automotive telematics that end users, telematics and application service providers be assured of the security of their systems from end-toend. Security is a broad term encompassing many concepts and elements including confidentiality/secrecy (including privacy), integrity, and availability [15] . Security and privacy threats to systems similar to those used and being proposed for automotive telematics infrastructures have been studied for quite some time (e.g., see [16, 23] ). Here, our security focus will be on assuring the privacy and integrity of telematics information -user data, vehicle data, time and location information, and even executable software -that is generated or stored in, or transmitted to/from, the invehicle client platform during its life cycle.. In the following sections of this paper we provide a description of an automotive telematics application and a scenario, the challenges posed by automotive telematics data, and an overview of privacy technology used in the proposed framework. We then detail proposed data protection framework, and conclude the paper with a summary of our work. Figure 1 shows an overview of a typical automotive telematics application. Cars shown in the picture are equipped with a wireless communication device, variety of sensors, and a car computer that has a display, sufficient memory, storage, and processing to run complex embedded applications and middleware. The car computer interfaces to car bus and other car sensors, for example, Global Positioning System (GPS) sensor, and collects car engine performance data, safety information, and car location. In this paper, we have dealt with the protection of private data in the automotive telematics domain. As we have stated, our goal is to enable the controlled sharing of private data according to policies agreed to by the owner of the data. Further, we would like to assure services providers that the data is not tampered with at its point of origin or anywhere in the processing chain. We have outlined the various challenges to protecting automotive telematics data, and have presented a framework to address these challenges. Next, we intend to implement the proposed data protection framework within an end-to-end solution in order to enable real world applications.

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


Evaluating sentiment in financial news articles Can the choice of words and tone used by the authors of financial news articles correlate to measurable stock price movements? If so, can the magnitude of price movement be predicted using these same variables? We investigate these questions using the Arizona Financial Text (AZFinText) system, a financial news article prediction system, and pair it with a sentiment analysis tool. Through our analysis, we found that subjective news articles were easier to predict in price direction (59.0% versus 50.0% of chance alone) and using a simple trading engine, subjective articles garnered a 3.30% return. Looking further into the role of author tone in financial news articles, we found that articles with a negative sentiment were easiest to predict in price direction (50.9% versus 50.0% of chance alone) and a 3.04% trading return. Investigating negative sentiment further, we found that our system was able to predict price decreases in articles of a positive sentiment 53.5% of the time, and price increases in articles of a negative 2 sentiment 52.4% of the time. We believe that perhaps this result can be attributable to market traders behaving in a contrarian manner, e.g., see good news, sell; see bad news, buy. Predicting stock market behavior has always had a certain appeal to researchers. While numerous attempts have been made, the difficulty has always centered around the inability to model the behaviors of human traders. Worse yet, behavioral patterns are constantly changing, making accurate predictions quite difficult. To further add to this uncertainty, there are two entirely opposed philosophies of stock market research; fundamental and technical analysis techniques (Technical Analysis, 2005) . Fundamentalists seek to leverage a security's relative data, ratios and earnings, while technicians analyze charts and modeling techniques based on historical trading volume and pricing. The basic problem becomes does price history matter?. Within the past several years, the role of computers in stock prediction has increased dramatically. Many of these systems have simply followed the trend of automating existing fundamental and/or technical strategies with the goal of achieving better returns than human traders by removing the elements of emotion and bias from trading (Jelveh, 2006) . The downside of these types of systems is that they lack intuition and will continue to execute trades even after unfavorable news events, such as losing a costly court battle. In order to work effectively, these systems require news events to be translated into numeric data before appropriate decisions can be made. This problem introduces serious lag-time into decisions and in some cases human analysts must override trades.. Even with the addition of financial news in a quantitative trading system, other important features of the article may be missing. The addition of author sentiment in stock price prediction could improve accuracy (Das, 2010) and reduce risk (Lu and Chen, 2010) . By automatically recognizing not only the terms used in the financial news article, but also the emotional cues used by the author and reflected by their choice of words, we plan to implement a machine learning approach that can learn historical price movements using these features and build a price prediction model in which future news articles could be instantly evaluated.. Our motivation is to build and test such an artifact with an emphasis on the sentiment and author tone features incorporated within a financial news article. By evaluating the intricacies of article sentiment and author tone, we seek to discover the role author sentiment can play in price prediction.. This paper is arranged as follows. Section 2 provides an overview of literature concerning Stock Market prediction, textual representations and sentiment analysis techniques. Section 3 develops our research questions. Section 4 describes our approach to the problem and introduces the Arizona Financial Text (AZFinText) system. Section 5 gives an overview of our experimental design. Section 6 provides results of our experimental findings and a discourse of their impact on stock price prediction. Section 7 presents our conclusions and a brief discussion of future research directions. From our investigation we found several interesting results. The first of which was that AZFinText was best able to predict Subjective articles in Directional Accuracy (59.0% to 50.4%) and Trading Returns (3.30% to 2.41%), but not Closeness (0.103 versus 0.0516). We felt that the subjectivity of the articles may have had an impact on trading behavior. The second notable result was that AZFinText was best able to predict Negative Subjective articles in Directional Accuracy (50.9% to 50.4%) and Trading Returns (3.04% to 2.41%), but not Closeness (0.0576 versus 0.0516). We believe that these results are attributable to investors reacting more strongly to negative articles, which further adds to the ideas of Devitt and Ahmad (2007 We would also suggest several future directions for this area of research. The first of which is to investigate the role of verbs and adverbs as a textual representation method. Perhaps this representational scheme will lead to better predictivity than Proper Nouns alone. It would be beneficial in future studies to draw comparisons between various textual representations. The second future direction would be to investigate other machine learning techniques. While SVR has proven itself in the textual financial domain, perhaps other techniques could identify different types of hidden market patterns. The third future direction would be to explore the role of negation in the OpinionFinder tool and its impact on price direction. While OpinionFinder has satisfactory accuracy at 74%, an unaccounted for negation could introduce a small amount of noise into the results. It would be important to determine its overall impact before constructing a real-time trading system.

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


Relative blocking in posets Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences.

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


Opportunity-Based Topology Control in Wireless Sensor Networks Abstract-Topology control is an effective method to improve the energy efficiency of wireless sensor networks (WSNs). Traditional approaches are based on the assumption that a pair of nodes is either "connected" or "disconnected." These approaches are called connectivity-based topology control. In real environments, however, there are many intermittently connected wireless links called lossy links. Taking a succeeded lossy link as an advantage, we are able to construct more energy-efficient topologies. Toward this end, we propose a novel opportunity-based topology control. We show that opportunity-based topology control is a problem of NP-hard. To address this problem in a practical way, we design a fully distributed algorithm called CONREAP based on reliability theory. We prove that CONREAP has a guaranteed performance. The worst running time is OðjEjÞ, where E is the link set of the original topology, and the space requirement for individual nodes is OðdÞ, where d is the node degree. To evaluate the performance of CONREAP, we design and implement a prototype system consisting of 50 Berkeley Mica2 motes. We also conducted comprehensive simulations. Experimental results show that compared with the connectivity-based topology control algorithms, CONREAP can improve the energy efficiency of a network up to six times.Index Terms-Topology control transitional region network reachability. T OPOLOGY control is an effective method to improve the energy efficiency of Wireless Sensor Networks (WSNs) [17] , [23] . In traditional methods, the network model is based on the assumption that a pair of nodes is either "connected" or "disconnected." When all nodes are connected to the network, the network is said to have the full connectivity. Traditional approaches aim to seek a topology of minimized energy cost and the full connectivity. We therefore call these approaches connectivity-based topology control. Connectivity-based topology control is a well-known NPC problem [12] and many heuristic algorithms have been proposed (e.g., [6] , [11] , [14] , [22] , [28] ).. In real application environments, however, this connectivity-based model is not practical due to the transitional region phenomenon. Beyond the "connected" region, there is a transitional region that allows wireless links to be intermittently connected. Such links are called lossy links [3] , [4] , [7] , [30] , [32] , [33] , [34] . Via lossy links, a pair of nodes are reachable but not always connected. As reported in the literature (e.g., [3] , [34] ), lossy links usually account a major part of a wireless network. In a specific setup, lossy links are more than 90 percent.. An interesting observation is that lossy links can be used to construct more energy-efficient topologies if used properly. Compared with these reliable links, lossy links allow a transmitter to reach more nodes when they succeed.. By taking the succeeded lossy links as an advantage, many transmissions can be saved and the cost is reduced.. The employment of lossy links in topology is not straightforward. When they are employed, the sink may not be able to reach all nodes for a given transmission. An important characteristic of WSNs is the large density and high data redundancy. For many applications, to require the sink reach all nodes for every transmission is not a necessity. As long as an expected percentage of the nodes can be reached, individual nodes are often less concern. This feature allows the lossy links to be employed during topology control by which some applications can trade certain part of the network for higher energy efficiency. We use networkwide broadcast to represent the routing protocol, which is the best that a routing protocol can do. No congestion or collision is considered. Under this transmission paradigm, we define the metric network reachability to quantify the percentage of the nodes that can be reached by the sink using a sink-initialized broadcast. Based on network reachability, we propose a novel opportunity-based topology control, targeting at more energy-efficient topologies.. An illustration is depicted in Fig. 1 . The x-axis is the energy cost and the y-axis is the network reachability. Connectivity-based topology control only uses reliable links. It seeks the optimal topology of the minimized cost and the full connectivity (point A in the figure). As this is an NPC problem, different heuristic algorithms were proposed to approach this point. Different from connectivity-based algorithms, opportunity-based topology control employs lossy links during topology control. It aims at a topology of higher energy efficiency with certain degraded network reachability (constrained by an application-dependent threshold th ). In the example of Fig. 1 , given the threshold th (0.8 in this case), point B is the target topology of opportunity-based topology control. Point C is of connectivity-based one. The energy gap between B and C is the main motivation of this work. In some extreme applications of th ¼ 1, opportunity-based topology control is equivalent to the traditional connectivity-based one.. The key challenge in opportunity-based topology control is the computation of the network reachability. As we will see later (Section 2.2), it is an NP-hard problem. We do not yet have efficient algorithms to compute the network reachability accurately, even given the global network knowledge. The key issue then becomes how to guarantee a derived network topology can satisfy the optimization constraint (i.e., the network reachability is no less than the threshold th ). To address the problem, we explore the reliability theory during our algorithm design.. The main contributions of this paper are as follows. First, we identify and highlight the opportunity of using lossy links during topology control in WSNs. We formulate the novel opportunity-based topology control by defining the metric "network reachability." To the best of our knowledge, we are the first one to attempt to employ the lossy links for energy-efficient topology control purpose. Second, we prove that opportunity-based topology control is a problem of NPhard. In order to address the problem in a practical manner, we design a fully distributed algorithm called CONREAP based on the reliability theory. We prove that the derived topology by CONREAP has a guaranteed network reachability, and the network energy cost is significantly reduced. The worst running time of CONREAP is OðjEjÞ, where E is the set of links in the original topology. The space requirement of individual nodes is OðdÞ, where d is the node degree. Third, in order to investigate the performance of CONREAP, we design and implement a prototype system consisting of 50 Berkeley Mica2 motes. We also conduct simulations to simulate a 200 node network. Our results show that CONREAP can improve energy efficiency of a network up to six times compared with the traditional connectivity-based algorithms.. The rest of the paper is organized as follows: In Section 2, we use an example to motivate our work. We then give an overview of opportunity-based topology control in Section 3. In Section 4, we present the proposed CONREAP algorithm and show some analytical results. Section 5 shows the implementation and simulation experiment results. We give an overview of the related work in Section 6. The last section concludes this paper and indicates possible future research directions. In this paper, we identify and highlight the opportunity of lossy links during energy-efficient topology control. To seize this opportunity, we propose a novel opportunity-based topology control. We focus on the reachability-preserving problem and show that this problem is NP-hard. In order to address this problem in a practical way, we propose CONREAP algorithm by exploring reliability theory. We prove that CONREAP has the guaranteed network reachability and the energy cost can be significantly reduced. The worst running time is OðjEjÞ and the space requirement is OðdÞ. Experimental results show that CONREAP is more appropriate for low-requirement, large transitional region environments. Compared with connectivity-based topology control, CONREAP can improve the energy efficiency up to six times.. Future work can be carried out along following directions. First, we study the reachability-preserving problem and leave other two problems open. Second, joint consideration with adjustable transmission power is also interesting. Third, we consider communications from sink to sensors. Optimizing topologies for other patterns are also important. Finally, some practical considerations are in concern. For example, we assume fixed-link reachability which is a function of time in practice. All these issues are calling for further investigations. . For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib. In this section, we use an example to motivate opportunitybased topology control. We first show some empirical studies to unclose the transitional region phenomenon. We then describe the network model that takes the transitional region phenomenon into consideration. Finally, we use an example to motivate the opportunity-based topology control. In this section, we give an overview of related works. As energy is the most concern in WSNs, many valuable efforts have been made in literature. One category is the traditional connectivity-based topology control [23] . While the network connectivity is preserved, various energy-saving schemes have been adopted. One such scheme is by reducing the transmission power of individual nodes. As this problem is NPC, a lot of heuristics have been proposed such as MST-based [19] , [22] and ConeBased algorithms (CBTC) [13] , [27] . Notice that [13] does not need location information such as [18] . In [14] , the authors studied k-connectivity issues. Interference-aware topology control (e.g., [25] ) focuses on interferences between consecutive hops which may result in a degradation of network throughput. All these works are based on the connectivity metric. Therefore, they are limited when leveraging the lossy links. Quality-based topology control (e.g., Xtc [29] ) attempts to preserve links of higher quality. It is, however, based on connectivity metric but uses lossy links as reliable links. It will suffer from extremely low network reachability. Very interesting, a recent work [15] also noticed the impact of lossy links to the topology control. The authors studied the network reachability (but a different term) using the probabilistic model instead of the traditional deterministic model. The results are, however, still preliminary as only a specific topology is studied. The result cannot be applied to a general network. Further more, no energy issue is studied.. Another energy-saving scheme is to reduce the transmitter set by seeking the MCDS of the network. It is also a known NPC problem [10] . Many heuristic algorithms were proposed such as [31] . We prove that reachability-preserving problem is NP-hard based on MCDS problem. However, MCDS is also a connectivity-based topology control algorithm, and thus, no lossy link advantage is taken.. Designers of routing algorithms also noticed the transitional region phenomenon in WSNs. In particular, ExOR [2] and MORE [5] are proposed to explore packet overhearing along lossy links in routing. These opportunistic routing schemes are, however, based on the network topology that provided by topology control algorithm. Without appropriate support from the beneath topology control, routing algorithms can hardly find those energyefficient routing schemes. For the example in Fig. 3 , no opportunistic routing can be carried out when the beneath topology is G 2 . In that sense, opportunity-based topology control aims at providing energy-efficient topology service for upper routing-layer protocols.

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


Power grid correction using sensitivity analysis under an RC model Verifying an RC model of the power grid requires one to check if the steady state voltage drops on all the nodes of the grid do not exceed a certain threshold. We propose an approach to correct the grid, in case some voltage drops violate the threshold condition, by making minor changes to the original design. Previous work has been done in [1] on the DC model of the grid and this paper deals with the transient model. Rather than directly reducing the steady state voltage drops below the threshold we work on reducing the first time step voltage drops. The method uses current constraints proposed in [2] to find the first time step voltage drop whose distance to the corresponding threshold is the largest. It then tries to estimate it as a function of the metal widths on the grid. A non-linear optimization problem is then formulated and the required metal line width changes that reduce the first time step voltage drops by a sufficient amount are then determined. The reduction of the first time step voltage drop by that amount will make the steady state voltage drops of all the nodes less than the threshold. With technology scaling, the supply and threshold voltages are decreasing. Hence, voltage drops on the power grid become more significant and can result in longer circuit delays, leading to soft errors. Checking the integrity of the voltage on the power grid has become crucial in reliable chip design.. Power grid verification via traditional circuit simulation, requires full knowledge of the current waveforms drawn by every circuit block attached to the grid. Once these waveforms are known, the grid is simulated and the voltage drop at every node is determined. This voltage drop is then compared to a threshold to check if any node on the grid is unsafe. Verifying the grid using this approach requires the simulation of a comprehensive set of currents and full knowledge of the circuit. The latter is problematic in case one would like to do the verification early in * This work was supported by Advanced Micro Devices (AMD) 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. the design flow, before all the circuit details are available, which is typically the case.. A new approach was proposed in [2] to deal with these problems. It is based on the current constraints concept to capture the uncertainty about the circuit details and circuit behavior. These current constraints can be obtained from simulations of the circuit or from the knowledge of the overall power dissipation of the circuit blocks. They are a set of upper bounds on the currents that would be drawn by the underlying circuit. A linear program (LP) is then formulated using these constraints, to check if the voltage drop at any of the nodes exceeds a certain threshold under all possible current waveforms that satisfy the constraints. In case none of the nodes violates its voltage drop requirement, the grid is said to be robust. Unlike the simulation based approach, the current constraints approach allows the verification to be done early in the design process when grid modifications can be most easily incorporated.. Once the grid verification is completed, we may have some nodes violating the voltage threshold. It then becomes useful to be able to make some minor changes on the grid so that it becomes safe without having to redo all the design from scratch. A method has been presented in [1] to correct an R model of the grid. This work presents an efficient method to correct an RC model of the grid. Rather than working on reducing the steady state voltage drops below the threshold in a direct way, we work with the first time step voltage drops. This is possible and in fact much easier because of the relation between the upper bound of the steady state voltage drop and the first time step voltage drop. Reducing the first time step voltage drop also reduces the steady state voltage drop, as we will see in this paper. The correction of the grid will be achieved by doing minor changes to the widths of some metal branches.. The method presented here uses linear programming concepts to express the first time step voltage drops as a function of the width parameters. It then uses a nonlinear optimization method to find the width changes that cause a sufficient reduction to the first time step voltage drop. The steady state voltage drops are consequently reduced. This paper is organized as follows. In sections 2 and 3, the grid model and the constraint-based approach are explained. The concept of transient robustness is then defined in section 4. In section 5, the basic linear programming terminology that will be used to formulate the proposed method is introduced. The formulation of our problem is then presented in section 6, and the correction approach is given in section 7. Finally, in sections 9 and 10 the experimental results and some concluding remarks are given. With the rising demand of low voltage designs, supply voltage integrity verification has become a crucial step in reliable highspeed chip design. In this paper, we propose a fast and easy approach to correct the grid with minimal changes, when some nodes do not satisfy the threshold. We extend the work done in [1] to the case of an RC model of the grid. Instead of reducing the steady state voltage drops directly, we work on reducing the first time step voltage drops, which will have an impact at infinity. We use linear programming concepts and the current constraints-based verification approach and we formulate a nonlinear optimization problem to find the required width changes needed. Our method showed good results: A grid of 70421 nodes was corrected in a total time of 6.8 hours and the changes needed in the widths of metal branches on some levels of the grid were less than 16%.

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


Adaptive medial-axis approximation for sphere-tree construction Hierarchical object representations play an important role in performing efficient collision handling. Many different geometric primitives have been used to construct these representations, which allow areas of interaction to be localized quickly. For timecritical algorithms, there are distinct advantages to using hierarchies of spheres, known as sphere-trees, for object representation. This article presents a novel algorithm for the construction of sphere-trees. The algorithm presented approximates objects, both convex and non-convex, with a higher degree of fit than existing algorithms. In the lower levels of the representations, there is almost an order of magnitude decrease in the number of spheres required to represent the objects to a given accuracy. Collision handling is a major bottleneck in any interactive simulation. Ensuring that objects interact in the correct manner is very computationally intensive and much research has addressed the issues involved with trying to reduce the computational requirements. Researchers often utilize hybrid collision detection algorithms to tackle the problem in various phases. The initial phase of such an algorithm, the broad phase, aims to efficiently cull out pairs of objects that cannot possibly be interacting. A number of different techniques have been used to achieve this coarse grain detection. These include Sweep & Prune [Cohen et al. 1995; Ponamgi et al. 1997] , global bounding volume tables [Palmer and Grimsdale 1995] and overlap tables [Wilson et al. 1998 ].. Having determined which objects are potentially interacting, the hybrid algorithm uses a finer grained algorithm to narrow in on the regions of the objects that are in contact. This narrow phase processing typically traverses hierarchical representations of the objects, such as that shown in Figure 1 , to hone in on the regions of interest. This reduces the amount of work that is required to perform the "exact" collision detection using such algorithms as those described by Palmer and Grimsdale [1995] and Moore and Wilhelms [1988] .. This work was supported by The Higher Education Authority and Enterprise Ireland. Authors' address: Image Synthesis Group, Department of Computer Science, Trinity College, Dublin, Ireland; email: gareth bradshaw@yahoo.com.uk; Carol.OSullivan@cs.tcd.ie. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 1515 Broadway, New York, NY 10036 USA, fax: +1 (212) 869-0481, or permissions@acm.org. c 2004 ACM 0730-0301/04/0100-0001 $5.00 Many different geometric primitives have been used for constructing the "Bounding Volume Hierarchies" (BVH) used to perform narrow phase processing. These include: Spheres [Quinlan 1994; Palmer and Grimsdale 1995; Hubbard 1995a Hubbard , 1995b Hubbard , 1995c Hubbard , 1996a O'Sullivan and Dingliana 1999] , Axis Aligned Bounding Boxes (AABBs) [van den Bergen 1997] , Oriented Bounding Boxes (OBBs) [Gottschalk et al. 1996; ], Discrete Oriented Polytopes (k-DOPs) [Klosowski et al. 1998 ], Quantized Orientation Slabs with Primary Orientations (QuOSPOs) [He 1999 ], Spherical Shells ] and Sphere Swept Volumes (SSVs) [Larsen et al. 1999] .. There is often a trade-off between the complexity of the bounding volume primitives and the tightness of fit that they can achieve. Simpler primitives, such as spheres and AABBs, are quite inexpensive to test for intersections. However, as they provide relatively poor approximations, large numbers are often required to approximate the objects effectively. More complex bounding volume primitives require more expensive intersection tests but, as they often provide tighter approximations, fewer primitives (and hence intersection tests) are required. The following equation has been used by Gottschalk et al. [1996] , van den Bergen [1997] and Klosowski et al. [1998] to evaluate various types of bounding volume hierarchies:. where :. T is the total cost of collision detection between two objects, N u is the number of primitives updated during the traversal, C u is the cost of updating a primitive's position/orientation, N v is the number of overlap tests that are performed, C v is the cost of an overlap test between a pair of primitives.. Simple primitives, such as spheres, exhibit low values for C u and C v . As spheres are rotationally invariant, a node can be updated simply by translating its center point to reflect the change in the object's position and orientation. Testing spheres for overlap is also computationally inexpensive, requiring only a simple distance test. However, as these primitives often do not fit objects well, N v (and hence N u ) can be quite large. More complex primitives, such as OBBs and SSVs, represent the objects more closely and hence N v and N u are lower, but the overlap tests and update costs are generally higher. The table below compares spheres and OBBs (using the Separating Axis Test [Gottschalk et al. 1996] ) with respect to Eq. (1):. Primitive C u C v Sphere 21 = 1*t 10 OBB 75 = 3*r + 1*t 80-200. Although spheres often do not provide particularly tight bounding volumes, there are a number of distinct advantages to their use. Not only are they computationally inexpensive to use at run-time, but each level of the sphere-tree can be used to generate an approximate response [O'Sullivan and Dingliana 1999; Dingliana and O'Sullivan 2000] . This is particularly advantageous when using the interruptible collision detection algorithm. This algorithm, introduced by Hubbard, uses a time-critical traversal of the sphere-trees that is terminated when the allotted time-slice has expired, allowing consistent frame-rate to be maintained [Hubbard 1995a [Hubbard , 1995b [Hubbard , 1995c [Hubbard , 1996a . As the collisions may never be resolved down to the surface of the objects, the spheres are used to approximate the collisions.. This article presents a novel sphere-tree construction algorithm, based on Hubbard's original medial axis method. The algorithm maintains a medial axis approximation that can be refined as the sphere-tree construction progresses. This ensures that there will always be enough information for the construction of a tight fitting approximation. The rest of the article is organized as follows: Section 2 discusses the requirements of a good bounding volume hierarchy and gives a brief overview of some of the existing algorithms for the construction of sphere-trees. Section 3 overviews the process of spheretree construction within our frame-work and discusses how object subdivision is managed. Section 4 presents the adaptive medial axis approximation algorithm, which extends the existing algorithm to allow the approximation to be updated as required. Section 5 presents a number of new algorithms for the generation of sphere sets from the medial axis approximation. Section 6 evaluates the new algorithms for the approximation of objects, the construction of sphere-trees and their use within interactive simulations. Section 7 presents information about how we construct our sphere-trees for use in real-time interactive simulations. Finally, Section 8 presents conclusions and mentions some future work. This article has presented some novel work in the areas of medial axis approximation and spheretree construction. The adaptive medial axis approximation algorithm addresses a number of problems that existed with previous algorithms. The algorithm guarantees to produce approximations that cover the object completely, that is, do not contain gaps, without adversely affecting the fit to the original object. This results in sphere-trees that always represent a complete approximation of the object.. The adaptive nature of the medial axis approximation algorithm allows the approximation to be refined during the sphere-tree construction process. Combining this with our improved sphere generation algorithms, which generate subsections of the sphere-tree from the medial axis approximation, provides significant improvements in the resulting sphere-trees. The new sphere-trees typically exhibit about 1 2 the error of those generated with the best existing algorithm. This represents almost an order of magnitude decrease in the number of spheres required to approximate an object to a desired level. The tighter fitting sphere-trees have been proven to provide a significant reduction in the number of false positives generated by the collision detection algorithm and reduce the errors present in the approximated collisions.. There are a number of interesting areas of research that can build upon this work. A number of the algorithms presented may be applicable to areas other than collision detection. The adaptive sampling algorithm, combined with the expand algorithm, provides a method for approximating rigid objects with very low variances. It is often very desirable to be able to approximate objects with a high level of consistency. It would also be interesting to explore a combined collision detection strategy that uses different representations for different classes of objects, allowing static objects, such as buildings, to be modelled more efficiently. Another area of interest would be to use the same spheretrees for collision handling and rendering, using an algorithm such as that presented by Levoy [2000, 2001] . Many researchers have used bounding volume hierarchies, ranging in complexity, as a means of performing efficient collision detection. Swept sphere volumes, used by Larsen et al. [1999] , provide a number of different types of bounding volume, which can be chosen to best approximate individual regions of the objects. However, all their primitives are based on spheres. Using a wider range of bounding volumes could provide a more efficient means of approximating the objects.

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


A demo for affinity based content delivery architectures Streaming video data over best effort networks is a challenging task concerning the quality of the received content. The quality decreases with the number of frames that are corrupted, lost or received after the playback time. The main reasons for lost, delayed or corrupted frames are overloaded streaming servers and crowded network paths. In order to deal with overloaded streaming servers and crowded network paths, we present the prototype of an innovative architecture called Proxy-to-Proxy (X2X). The three main components in the architecture are proxies (provided by end users), videos (replicated from original servers) and endclients (requesting shared content). The behavior of the three components is defined by the so-called affinity model which consists of three levels. The first level covers the content replication from original to surrogate servers (also called proxies), the second level covers the collaboration between the proxies, and the third level covers the delivery from the proxies to the end-clients. Proxies, videos and end-clients have a certain affinity to each other. Only by changing the input parameters to the affinity functions, the overlay network behaves either like a classical 1) Content Delivery Network, 2) a Peer-to-Peer System or 3) a Proxy-to-Proxy overlay network -which is a combination and generalization of the two former ones. Delivering videos in good quality over best effort networks can still be regarded as an unsolved problem. If a video is streamed from an arbitrary server to an arbitrary client, the perceived quality is typically a question of luck. The classical server/client approach suffers from poor scalability and a single point of failure. In case of best effort based video delivery, an additional difficulty is the "long distance 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. problem". Over long distances (high number of traversed network elements) packet delays accumulate and result in an unacceptable poor quality, even in case of lightly loaded servers. An alternative to the classical client/server approach is provided by improved video distribution methods, such as Content Delivery Networks (CDNs) and Peer-to-Peer systems (P2P). The basic principle of CDNs is transparent redirection of client requests from the origin server to a cluster of surrogates. Replication (from the servers to the cluster) is performed on demand, whenever the content is not cached. The main drawback is that CDNs have no notion of a quality of a video and of a streaming service. They try to find the best connected cluster, but will not check, whether this is good enough for video streaming. As opposed to Content Delivery Networks (which are typically closed systems, with little public information), P2P file sharing has been investigated fairly well in the recent years [4] . A major drawback is the fragility of P2P systems, as clients, leaving the system, is treated as a normal case. If the time for streaming of a video and that of an on-line period of client machines are in the same order of magnitude then hiccups become "normal" -which is hardly acceptable.

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


Performance analysis of multimedia based web traffic with QoS constraints During the recent years, there has been a tremendous growth in the development and deployment of multimedia based networked applications such as video streaming, IP telephony, interactive games, among others. These applications, in contrast to elastic applications such as email and data sharing, are delay and delay jitter sensitive but can tolerate certain level of packet loss. A vital element of end-to-end delay and delay jitter is the random queueing delays in network switches and routers. Analysis of robust mechanisms for buffer management at network routers needs to be carried out in order to reduce end-to-end delay for traffic generated by multimedia applications. In this context, a threshold based buffer management scheme for accommodating multiple class multimedia traffic in network routers has been analysed. This technique effectively controls the allocation of buffer to various traffic classes according to their delay constraints. The forms of the joint state probabilities, as well as basic performance measures such as blocking probabilities are analytically established at equilibrium. Typical numerical experiments are included to illustrate the credibility of the proposed mechanism in the context of different quality of service (QoS) grades for various network traffic classes. This model, therefore, can be used as a powerful tool to provide a required grade of service to a particular class of multimedia based web traffic in any heterogeneous network. During the recent years real-time applications such as video streaming, interactive games and voice over IP have become increasingly popular among computer users. These applications are generally delay sensitive and need preferential treatment in order to satisfy a desired level of Quality of Service (QoS) constraints. Many enterprises demand applications development using software which integrates the support for real-time applications with the support for conventional computations. Such demands have posed various challenges to network community for robust design of the communication infrastructure. In that, significant developments have been made to design networks with the ability to guarantee the QoS for the real time data [1] .. Traffic generated by the multimedia applications is generally very sensitive to the transmission. End-to-end delay and delay jitter are usually introduced due to random queueing in the network routers. Traditionally finite capacity queues with tail drop (TD) mechanisms have been employed in the network routers. Such queues temporarily accommodate the arriving packets when the server is busy. The arriving packets are dropped when the queue reaches its maximum capacity. Although this technique is simple, it suffers various problems, e.g., lock out, global synchronisation and full queue [2] . The main problem among these is the full queue which causes longer delays and makes this technique an inappropriate choice for time sensitive applications.. To support these multimedia applications along with traditional non-real-time services such as data transfer and emails, traditional queue management schemes need replacing with sophisticated and effective mechanisms. The use of thresholds for controlling congestion in communication buffers is well known and is used, for example, active queue management (AQM) in current Internet routers. The basic principle is a simple one: if the mean queue length exceeds a pre-determined threshold, the arriving packets are dropped or marked with a specific probability. Then the position of the threshold and the value of the drop probability define a specific trade-off between packet delay and packet loss which can be adjusted to suit a particular type of service and its quality of service (QoS) requirements. This technique maintains a small size steady state queue, thus results in reduced packet loss, decreased end-to-end delay and the avoidance of lock out behaviour thus using the network resources more efficiently.. A number of studies have been reported in the literature to implement AQM. These include random early detection (RED) [3] , random early marking (REM) [4, 5] , a virtual queue based scheme where the virtual queue is adaptive [6] [7] [8] and a proportional integral controller mechanism [9] , among others. Of the above schemes to implement AQM, the RED mechanism is the one recommended by The Internet Society in [2] : Quote: Unless a developer has reasons to provide another equivalent mechanism we recommend that RED be used. This mechanism has the potential to overcome some of the problems discovered in drop tail mechanisms which are specific to the Internet traffic, such as synchronisation of TCP flows and correlation of the drop events (multiple packets dropped in sequence) within a TCP flow and it is therefore this mechanism that we will focus on in this paper.. The main aim of this paper is to formulate such a model with multiple queue thresholds and examine the queueing behaviour for multimedia type traffic under first come first served (FCFS) service discipline. To facilitate this aim, we study a stable GE/GE/1/N censored queue with a single server, finite capacity and multiple classes under a complex but effective buffer management scheme. The external bursty traffic and service time have been modelled using the generalised exponential (GE) distribution. The analysis has been carried out using the principle of maximum entropy (ME).. The remaining paper is organised as follows: Section 2 presents related work. Some preliminaries are outlined in Section 3. The ME solution for a stable GE/GE/1/N censored queue with buffer thresholds and FCFS scheduling discipline is characterised in Section 4. Numerical results, involving GE interarrival and service time distributions, are included in Section 5. Section 6 finally concludes the paper. This paper presents analysis of a threshold based queue for multimedia based networked applications in order to reduce end-to-end delay for traffic generated by these applications. In this context, product-form approximation, based on the principle of ME, for a stable GE/GE/1/N queue with FCFS scheduling discipline under PBS buffer management scheme has been proposed as a useful performance evaluation tool. This scheme effectively controls the allocation of buffer to various traffic classes according to their delay constraints. Closed form analytical expressions for state and blocking probabilities have been derived. The proposed model has been implemented using the GE-type external traffic represent the bursty nature of the multimedia traffic. Typical numerical examples have been included to show the impact of buffer threshold settings on various performance metrics for delay tolerant and delay sensitive video streams. Traffic congestion in the Internet routers occurs when the aggregate demand exceeds the available capacity of resources. Performance modelling techniques help to design effective mechanisms capable of providing interactive services (such as voice, data and video) by developing performance models. The rapid development of communication networks and technologies in recent years has imposed great challenges for network support due to introduction of various multimedia type applications. As most current networking protocols and congestion control techniques were designed mainly for delay tolerant or elastic services, many buffer management schemes that make them efficient for these services are no longer true for new delay sensitive applications and cause severe performance degradation problems.. Feedback is a traditional technique for indicating the status of congestion within the network. Helali et al. [10] presented a multi-profile communication environment to ensure end-to-end QoS management. It supports dynamic assignment of application requirements to the network resources. The main contribution lies in the prediction of network congestion using feedback control algorithm to avoid overloading with streaming multimedia traffic.. Issues related to QoS and reliability design of packet networks have been addressed by mapping the end-user performance constraints into transport-layer performance constraints and then into network-layer performance constraints [11] . A collection of heuristic algorithms have also been presented and their performance has been validated against simulation. However, their analysis completely ignores employment of buffer management policies in the network routers.. Manvi and Venkataram [12] proposed an agent based scheme for adaptive bandwidth allocation when congestion occurs. This scheme functions at the network nodes and adaptively finds an alternative path for every congested or failed link and reallocates the bandwidth for the affected multimedia. The analysis of the proposed model is based on the simple assumption of modelling of constant bit rate (CBR) and variable bit rate (VBR) traffic sources using uniform distribution.. Kusmeirek and Du have simulated the problem of monitoring the available network resources by modelling the network behaviour and identifying the factors of vital importance [13] . Their approach relies on simple early congestion notification (ECN)-based mechanism to obtain a feedback from the network and end-point observations to determine available bandwidth and adjust the transmission rate using 3-rate adaptation mechanism. Their study considers only CBR-based traffic sources.. Most of the current studies are either based simple traffic assumptions or do not consider any buffer management for effectively dealing with the delay and delay jitter caused by the network queueing. This paper analyses a threshold based partial buffer sharing scheme which gives preferential treatment to delay sensitive traffic over the delay tolerant traffic in order to reduce the queueing delay. It models the external bursty traffic using GE distribution to closely match the real video streams.

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 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 Stochastic Model for the Interconnection Topology of Digital Circuits Abstract-Rent's rule has been successfully applied to a priori estimation of wire length distributions. However, this approach is very restrictive: the circuits are assumed to be homogeneous. In this brief, recursive clustering is described as a more advanced model for the partitioning behavior of digital circuits. It is applied to predict the variance of the terminal count distribution. First, the impact of the block degree distribution is analyzed with a simple model. A more refined model incorporates the effect of stochastic self-similarity. Finally, the model is further extended to describe the effects of heterogeneity. This model is a promising candidate for more accurate a priori estimation tools. T HE COMPLEXITY of the interconnection topology of a digital circuit is well captured by Rent's rule [1] and the corresponding Rent exponent. This empirical law states that, when a circuit is partitioned into a set of more or less equally sized modules subject to a certain terminal minimization objective, there exists a relationship between the average terminal count T and the module size B:. The parameters k and p are known as the Rent coefficient and the Rent exponent, respectively. Rent's rule has been applied numerous times, especially in the field of a priori wire length estimation [2] , [3] , [4] . All these techniques use Rent's rule as a simple model for the partitioning behavior of logic circuits. The circuits are assumed to be hierarchically homogeneous (absence of Rent region II [1] ) and spatially homogeneous in a very strict sense: for any partition all modules are assumed to have an identical number of terminals. Rent's rule is used in a deterministic sense, which is clearly a very restrictive interpretation.. In reality the terminal counts for a set of modules in a partition show a certain spread. Fig. 1 shows the partitioning behavior and Rent fit for ISPD98 [5] circuit ibm02. The partitions were obtained by recursively bipartitioning the circuit with hMetis [6] . Rent's rule describes the sample mean of these terminal counts, which is only a first-order statistic. In this brief we apply recursive clustering as a more realistic model to describe the partitioning behavior of circuits. This model essentially captures the stochastic properties of self-similarity in circuit graphs. Therefore, it allows the estimation of higher-order statistics such as the sample variance of the terminal count distribution. Rent's rule states a relationship between the average terminal count and the module size of a partitioned circuit. For wire length estimation this rule is interpreted in a very strict sense: the circuits are assumed to be hierarchically and spatially homogeneous. As a first-order estimate, all modules are assumed to have the exact number of terminals as predicted by Rent's rule. This deterministic interpretation of Rent's rule is a very restrictive approximation.. Recursive clustering can be used to model the partitioning behavior of digital circuits. Because the stochastic properties of self-similarity in digital circuits are adequately captured, recursive clustering appears to be a valuable candidate for the next generation of a priori estimation tools. We applied recursive clustering to model the terminal count distribution. First, a simple model was derived that only takes the block degree distribution into account. Then, the model was refined to incorporate the effects of stochastic self-similarity. Finally, the model was further extended to describe the effects of heterogeneity more accurately. The model was validated by predicting the variance of the terminal count distribution for all ISPD98 benchmark circuits. Some deviations were observed due to hierarchical heterogeneity.

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