The Fast Marching Method [8] is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. In this paper we extend the Fast Marching Method to triangulated domains with the same computational complexity. As an application, we provide an optimal time algorithm for computing the geodesic distances and thereby extracting shortest paths on triangulated manifolds. 1 Introduction Sethian`s Fast Marching Method [8], is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M ) steps, where M is the total number of grid points in the domain. The technique hinges on producing numerically consistent approximations to the operators in the Eikonal equation that select the correct viscosity solution; this is done through the use of upwind nite dierence operators. The structure of this upwinding is then used to systematically construct the solution to the Eik...

The problems of finding a longest common subsequence of two sequences A and B and a shortest edit script for transforming A into B have long been known to be dual problems. In this paper, they are shown to be equivalent to finding a shortest/longest path in an edit graph. Using this perspective, a simple O(ND) time and space algorithm is developed where N is the sum of the lengths of A and B and D is the size of the minimum edit script for A and B. The algorithm performs well when differences are small (sequences are similar) and is consequently fast in typical applications. The algorithm is shown to have O(N +D expected-time performance under a basic stochastic model. A refinement of the algorithm requires only O(N) space, and the use of suffix trees leads to an O(NlgN +D ) time variation.

Fast Marching Methods are numerical schemes for computing solutions to the non-linear Eikonal equation and related static Hamilton-Jacobi equations. Based on entropy-satisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms. They are optimal in the sense that the computational complexity of the algorithms is O(N log N ), where N is the total number of points in the domain. The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shape-from-shading, photolithographic development, computing rst arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reection calculations. In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings, provide details of the computational schemes including higher order versions,...

This paper presents a comprehensive survey of existing techniques for interconnect optimization during the VLSI physical design process, with emphasis on recent studies on interconnect design and optimization for high-performance VLSI circuit design under the deep submicron fabrication technologies. First, we present a number of interconnect delay models and driver/gate delay models of various degrees of accuracy and efficiency which are most useful to guide the circuit design and interconnect optimization process. Then, we classify the existing work on optimization of VLSI interconnect into the following three categories and discuss the results in each category in detail: (i) topology optimization for highperformance interconnects, including the algorithms for total wire length minimization, critical path length minimization, and delay minimization; (ii) device and interconnect sizing, including techniques for efficient driver, gate, and transistor sizing, optimal wire sizing, and simultaneous topology construction, buffer insertion, buffer and wire sizing; (iii) highperfbrmance clock routing, including abstract clock net topology generation and embedding, planar clock routing, buffer and wire sizing for clock nets, non-tree clock routing, and clock schedule optimization. For each method, we discuss its effectiveness, its advantages and limitations, as well as its computational efficiency. We group the related techniques according to either their optimization techniques or optimization objectives so that the reader can easily compare the quality and efficiency of different solutions.

We present a new, interactive tool called Intelligent Scissors which we use for image seg-mentation. Fully automated segmentation is an unsolved problem, while manual tracing is inaccu-rate and laboriously unacceptable. However, Intelligent Scissors allow objects within digital images to be extracted quickly and accurately using simple gesture motions with a mouse. When the gestured mouse position comes in proximity to an object edge, a live-wire boundary “snaps” to, and wraps around the object of interest. Live-wire boundary detection formulates boundary detection as an optimal path search in a weighted graph. Optimal graph searching provides mathematically piece-wise optimal bound-aries while greatly reducing sensitivity to local noise or other intervening structures. Robustness is further enhanced with on-the-fly training which causes the boundary to adhere to the specific type of edge currently being followed, rather than simply the strongest edge in the neighborhood. Boundary cooling automatically freezes unchanging segments and automates input of additional seed points. Cooling also allows the user to be much more free with the gesture path, thereby increasing the efficiency and finesse with which boundaries can be extracted. (2) 1.

We study how to improve the throughput of high-bandwidth traffic such as large file transfers in a network where resources are fairly shared among connections. While it is possible to devise priority or reservation-based schemes that give high-bandwidth traffic preferential treatment at the expense of other connections, we focus on the use of routing algorithms that improve resource allocation while maintaining max-min fair share semantics. In our approach, routing is closely coupled with congestion control in the sense that congestion information, such as the rates allocated to existing connections, is used by the routing algorithm. To reduce the amount of routing information that must be distributed, an abstraction of the congestion information is introduced. Using an extensive set of simulation, we identify a link-cost or cost metric for "shortest-path" routing that performs uniformly better than the minimal-hop routing and shortest-widest path routing algorithms. To further improve throughput without reducing the fair share of single-path connections, we propose a novel prioritized multi-path routing algorithm in which low priority paths share the bandwidth left unused by higher priority paths. This leads to a conservative extension of max-min fairness called prioritized multi-level max-min fairness. Simulation results confirm the advantages of our multi-path routing algorithm.

Efficient algorithms are known for computing a minimum spann.ing tree, or a shortest path. tree (with a fixed vertex as the root). The weight of a shortest path tree can be much more than the weight of a minimum spa,nning tree. Conversely, the distance bet,ween the root, and any vertex in a minimum spanning tree may be much more than the distance bet#ween the two vertices in the graph. Consider the problem of balancing between the two kinds of trees: Does every graph contain a tree that is “light ” (at most a constant times heavier than the minimum spanning t,ree), such that the distance from the root to any vertex in t,he tree is no more than a constant times the true distance? This paper answers the question in the affirmative. It is shown that there is a continuous tradeoff between the two parameters. For every y> 0, there is a tree in the graph whose total weight is at most 1 + $? times the weight of a minimum spanning tree, such that the di&nce in the tree between the root, and any vertex is at, most 1 + &y times the true distance. Efficient sequential and parallel algorithms achieving these factors are provided. The algorithms are shown to be optimal in two ways. First, it is shown that no algorithm can achieve better factors in all graphs, because there a.re graphs that do not have better trees. Second, it is shown that even on a per-graph basis, finding trees that achieve better factors is NP-hard.

We describe techniques for coordinating the actions of large numbers of small-scale robots to achieve useful large-scale results in surveillance, reconnaissance, hazard detection, and path finding. We exploit the biologically inspired notion of a "virtual pheromone," implemented using simple transceivers mounted atop each robot. Unlike the chemical markers used by insect colonies for communication and coordination, our virtual pheromones are symbolic messages tied to the robots themselves rather than to fixed locations in the environment. This enables our robot collective to become a distributed computing mesh embedded within the environment, while simultaneously acting as a physical embodiment of the user interface. This leads to notions of world-embedded computation and world-embedded displays that provide different ways to think about robot colonies and the types of distributed computations that such colonies might perform.

Networking unattended sensors is expected to have a significant impact on the efficiency of many military and civil applications. Sensors in such systems are typically disposable and expected to last until their energy drains. Therefore, energy is a very scarce resource for such sensor systems and has to be managed wisely in order to extend the life of the sensors for the duration of a particular mission. In this paper, we present a novel approach for energyaware management of sensor networks that maximizes the lifetime of the sensors while maintaining desired quality of service attributes related to sensed data delivery. The approach is to dynamically set routes and arbitrate medium access to minimize energy consumption and maximize sensor life. We give a brief overview of the energyaware routing and describe a Time-Division-Multiple-Access (TDMA)-based Medium Access Control (MAC) protocol. We discuss algorithms for assigning time slots for the communicating sensor nodes. The approach is evaluated through simulation. Simulation results have confirmed the effectiveness of our new approach. 1.

We present critical-sink routing tree (CSRT) constructions which exploit available critical-path information to yield high-performance routing trees. Our CS-Steiner and "Global Slack Removal" algorithms together modify traditional Steiner tree constructions to optimize signal delay at identified critical sinks. We further propose an iterative Elmore routing tree (ERT) construction which optimizes Elmore delay directly, as opposed to heuristically abstracting linear or Elmore delay as in previous approaches. Extensive timing simulations on industry IC and MCM interconnect parameters show that our methods yield trees that significantly improve (by averages of up to 67%) over minimum Steiner routings in terms of delays to identified critical sinks. ERTs also serve as generic high-performance routing trees when no critical sink is specified: for 8-sink nets in standard IC (MCM) technology, we improve average sink delay by 19% (62%) and maximum sink delay by 22% (52%) over the mini...