Finding the Consensus Shape for a Protein Family

We define and prove properties of the consensus shape for a protein family, a protein-like structure that provides a compact summary of the significant structural information for a protein family. If all members of the protein family exhibit a geometric relationship between corresponding -carbons, then that relationship is preserved in the consensus shape. In particular, distances and angles that are consistent across family members are preserved. For the consensus shape, the spacing between successive -carbons is variable, with small distances in regions where the members of the protein family exhibit significant variation and large distances (up to the standard spacing of about 3.8 Å) in regions where the family members agree. Despite this non-protein-like characteristic, the consensus shape preserves and highlights important structural information. We describe an iterative algorithm for computing the consensus shape and prove that the algorithm converges. As a by-product we produce a multiple structure alignment for the family members. We present the results of experiments in which we build consensus shapes for several known protein families.     

IsoMatch: Creating Informative Grid Layouts

Collections of objects such as images are often presented visually in a grid because it is a compact representation that lends itself well for search and exploration. Most grid layouts are sorted using very basic criteria, such as date or filename. In this work we present a method to arrange collections of objects respecting an arbitrary distance measure. Pairwise distances are preserved as much as possible, while still producing the specific target arrangement which may be a 2D grid, the surface of a sphere, a hierarchy, or any other shape. We show that our method can be used for infographics, collection exploration, summarization, data visualization, and even for solving problems such as where to seat family members at a wedding. We present a fast algorithm that can work on large collections and quantitatively evaluate how well distances are preserved.     

A k-server problem with parallel requests and unit distances

In this paper we consider k-server problems with parallel requests where several servers can also be located on one point. We will distinguish the surplus-situation where the request can be completely fulfilled by means of the k servers and the scarcity-situation where the request cannot be completely met. We use the method of the potential function by Bartal and Grove [2] in order to prove that a corresponding Harmonic algorithm is competitive for the more general k-server problem in the case of unit distances. For this purpose we partition the set of points in relation to the online and offline servers? positions and then use detailed considerations related to sets of certain partitions.     

Generalization of the incenter subdivision scheme

We introduce a new interpolatory subdivision scheme generalizing the incenter subdivision [8]. The proposed scheme is equipped with a shape controlling tension parameter, is Hermitian, and reproduces circles from non-uniform samples. We prove that for any value of the free parameter the limit curve is G¹ continuous. The scheme is shape preserving and avoids undesirable oscillations by producing curves with a finite number of inflection points at the regions where the control polygon suggests a change of convexity. Several examples are presented demonstrating the properties of the scheme.     

A new method for finding long consensus patterns in nucleic acid sequences.

We describe a fast computer algorithm for identifying consensus patterns in DNA sequences. The method requires no prior assumptions about the consensus pattern other than its length. In particular no previous knowledge of the frequency or spacing of consensus patterns is required. However, a priori information about the shape of the consensus pattern, or invariability of individual positions, or the overall conservation level, can be utilized to enhance the selectivity and sensitivity of search. As the number of all possible consensus words increases very rapidly with length, comprehensive searches have usually been restricted to a maximum of 10-12 nucleotides, even when large mainframes are used. Our algorithm enables searching for consensus patterns of this order on current mid-range and powerful microcomputers. Searches may be conducted on single, long sequences or a set of possibly aligned shorter sequences. We give examples of identified consensus patterns in both prokaryotic and eukaryotic DNA sequences, along with some typical program timings.     

The S-kernel: A measure of symmetry of objects

In this paper we introduce a new symmetry feature named “symmetry kernel” (SK) to support a measure of symmetry. Given any symmetry transform S, SK of a pattern P is the maximal included symmetric sub-set of P for all directions and shifts. We provide a first algorithm to exhibit this kernel where the centre of symmetry is assumed to be the centre of mass. Then we prove that, in any direction, the optimal axis corresponds to the maximal correlation of a pattern with its symmetric version. That leads to a second algorithm. The associated symmetry measure is a modified difference between the respective surfaces of a pattern and its kernel. A series of experiments supports the actual algorithm validation.     

A note concerning the closest point pair algorithm

An algorithm, described by Sedgewick, finds the distance between the closest pair of n given points in a plane using a variant of mergesort. This takes O(nlogn) time. To prove this it is necessary to show that, in the merge phase of the algorithm, no more than a constant number of distances need to be checked for each point considered. Cormen, Leiserson and Rivest show that checking seven distances is sufficient while Sedgewick suggests that this should be four. This paper shows that checking three distances is sufficient and that a slight modification of the algorithm reduces the number to two.     

Efficient algorithms for consensus string problems minimizing both distance sum and radius

The consensus (string) problem is finding a representative string, called a consensus, of a given set S of strings. In this paper we deal with consensus problems considering both distance sum and radius, where the distance sum is the sum of (Hamming) distances from the strings in S to the consensus and the radius is the longest (Hamming) distance from the strings in S to the consensus. Although there have been results considering either distance sum or radius, there have been no results considering both, to the best of our knowledge.We present the first algorithms for two consensus problems considering both distance sum and radius for three strings: one problem is to find an optimal consensus minimizing both distance sum and radius. The other problem is to find a bounded consensus such that the distance sum is at most s and the radius is at most r for given constants s and r. Our algorithms are based on characterization of the lower bounds of distance sum and radius, and thus they solve the problems efficiently. Both algorithms run in linear time.     

A contour tracing algorithm that preserves common boundaries between regions

In most literatures, the boundary of a region R is defined as the set of pixels in R that have at least one neighbor outside R. Applying this definition to images with multi-labels, for example results of segmentation, one will end up with nonoverlapping boundaries between regions. This contradicts the physical boundary concept and makes shape analysis among regions difficult. This paper presents a new, efficient contour tracing algorithm based on the extended boundary concept, a common boundary representation which shifts the inter-pixel boundary by one-half pixel toward the right and lower directions. It is shown that, in addition to the common boundary representation, the new algorithm has the following advantages over the old approaches: (1) It provides more detailed topological information in each contour. (2) The implementation in a table lookup method is easier and more efficient. (3) The complexity can be reduced to half because only one of each pair of neighboring regions needs to be traced. (4) Parallel processing is feasible. (5) The shapes of regions are well preserved.     

On the Competitive Theory and Practice of Online List Accessing Algorithms

This paper concerns the online list accessing problem. In the first part of the paper we present two new families of list accessing algorithms. The first family is of optimal, 2-competitive, deterministic online algorithms. This family, called the MRI (MOVE-TO-RECENT-ITEM) family, includes as members the well-known MOVE-TO-FRONT (MTF) algorithm and the recent, more ``conservative'' algorithm TIMESTAMP due to Albers. So far MOVE-TO-FRONT and TIMESTAMP were the only algorithms known to be optimal in terms of their competitive ratio. This new family contains a sequence of algorithms { A(i) } <sub>i \geq 1</sub> where A(1) is equivalent to TIMESTAMP and the limit element A(∈fty) is \mtf. Further, in this class, for each i , the algorithm A(i) is more conservative than algorithm A(i+1) in the sense that it is more reluctant to move an accessed item to the front, thus giving a gradual transition from the conservative TIMESTAMP to the ``reckless'' MTF. The second new family, called the PRI (PASS-RECENT-ITEM) family, is also infinite and contains TIMESTAMP. We show that most algorithms in this family attain a competitive ratio of 3. In the second, experimental, part of the paper we report the results of an extensive empirical study of the performances of a large set of online list accessing algorithms (including members of our MRI and PRI families). The algorithms' access cost performances were tested with respect to a number of different request sequences. These include sequences of independent requests generated by probability distributions and sequences generated by Markov sources to examine the influence of locality. It turns out that the degree of locality has a considerable influence on the algorithms' absolute and relative costs, as well as on their rankings. In another experiment we tested the algorithms' performances as data compressors in two variants of the compression scheme of Bentley et al. In both experiments, members of the MRI and PRI families were found to be among the best performing algorithms.     

On the coordinator's rule for Fast Paxos

Fast Paxos is an algorithm for consensus that works by a succession of rounds, where each round tries to decide a value v that is consistent with all past rounds. Rounds are started by a coordinator process and consistency is guaranteed by the rule used by this process for the selection of v and by the properties of process sets called quorums. We show a simplified version of this rule for the specific case where the quorums are defined by the cardinality of these process sets. This rule is of special interest for implementors of the algorithm.     

Average Case Analysis of Moore’s State Minimization Algorithm

We prove that the average complexity of Moore’s state minimization algorithm is $\mathcal {O}(kn\log n)$ , where n is the number of states of the input and k the size of the alphabet. This result holds for a whole family of probabilistic models on automata, including the uniform distribution over deterministic and accessible automata, as well as uniform distributions over classical subclasses, such as complete automata, acyclic automata, automata where each state is final with probability γ∈(0,1), and many other variations.     

Mesh Segmentation via Spectral Embedding and Contour Analysis

We propose a mesh segmentation algorithm via recursive bisection where at each step, a sub-mesh embedded in 3D is first spectrally projected into the plane and then a contour is extracted from the planar embedding. We rely on two operators to compute the projection: the well-known graph Laplacian and a geometric operator designed to emphasize concavity. The two embeddings reveal distinctive shape semantics of the 3D model and complement each other in capturing the structural or geometrical aspect of a segmentation. Transforming the shape analysis problem to the 2D domain also facilitates our segmentability analysis and sampling tasks. We propose a novel measure of the segmentability of a shape, which is used as the stopping criterionfor our segmentation. The measure is derived from simple area- and perimeter-based convexity measures. We achieve invariance to shape bending through multi-dimensional scaling (MDS) based on the notion of inner distance. We also utilize inner distances to develop a novel sampling scheme to extract two samples along a contour which correspond to two vertices residing on different parts of the sub-mesh. The two samples are used to derive a spectral linear ordering of the mesh faces. We obtain a final cut via a linear search over the face sequence based on part salience, where a choice of weights for different factors of part salience is guided by the result from segmentability analysis.     

A bivalency proof of the lower bound for uniform consensus

Bivalency argument is a widely-used technique that employs forward induction to show impossibility results and lower bounds related to consensus. However, for a synchronous distributed system of n processes with up to t potential and f actual crash failures, applying bivalency argument to prove the lower bound for reaching uniform consensus is still an open problem. In this paper, we address this problem by presenting a bivalency proof that the lower bound for reaching uniform consensus is (f+2)-rounds where 0?f?t−2.     

Universal Timestamp-Scheduling for real-time networks

Consider a network of computers interconnected by point-to-point communication channels. For each flow of packets through the network, the network reserves a fraction of the packet rate of each channel along the path of the flow. We define a family of scheduling protocols, called Universal Timestamp-Scheduling, to forward packets in this network, such that all members of the protocol family provide the same upper bound on packet delay as the well-known packet delay of Virtual Clock scheduling. The protocol family is called universal because it encompasses a wide variety of protocols. To show this, we prove that many scheduling protocols in the literature are members of the protocol family, and thus provide the above guarantee. In addition, we show that the protocols in the literature have only considered one side of the spectrum of possible scheduling protocols, and that there is another side of the spectrum that deserves attention and remains to be investigated.     

Modeling and Exploring Co‐variations in the Geometry and Configuration of Man‐made 3D Shape Families

We introduce co‐variation analysis as a tool for modeling the way part geometries and configurations co‐vary across a family of man‐made 3D shapes. While man‐made 3D objects exhibit large geometric and structural variations, the geometry, structure, and configuration of their individual components usually do not vary independently from each other but in a correlated fashion. The size of the body of an airplane, for example, constrains the range of deformations its wings can undergo to ensure that the entire object remains a functionally‐valid airplane. These co‐variation constraints, which are often non‐linear, can be either physical, and thus they can be explicitly enumerated, or implicit to the design and style of the shape family. In this article, we propose a data‐driven approach, which takes pre‐segmented 3D shapes with known component‐wise correspondences and learns how various geometric and structural properties of their components co‐vary across the set. We demonstrate, using a variety of 3D shape families, the utility of the proposed co‐variation analysis in various applications including 3D shape repositories exploration and shape editing where the propagation of deformations is guided by the co‐variation analysis. We also show that the framework can be used for context‐guided orientation of objects in 3D scenes.     

Probably Approximately Symmetric: Fast Rigid Symmetry Detection With Global Guarantees

We present a fast algorithm for global rigid symmetry detection with approximation guarantees. The algorithm is guaranteed to find the best approximate symmetry of a given shape, to within a user‐specified threshold, with very high probability. Our method uses a carefully designed sampling of the transformation space, where each transformation is efficiently evaluated using a sublinear algorithm. We prove that the density of the sampling depends on the total variation of the shape, allowing us to derive formal bounds on the algorithm's complexity and approximation quality. We further investigate different volumetric shape representations (in the form of truncated distance transforms), and in such a way control the total variation of the shape and hence the sampling density and the runtime of the algorithm. A comprehensive set of experiments assesses the proposed method, including an evaluation on the eight categories of the COSEG data set. This is the first large‐scale evaluation of any symmetry detection technique that we are aware of.     

Scale Normalization for Isometric Shape Matching

We address the scale problem inherent to isometric shape correspondence in a combinatorial matching framework. We consider a particular setting of the general correspondence problem where one of the two shapes to be matched is an isometric (or nearly isometric) part of the other up to an arbitrary scale. We resolve the scale ambiguity by finding a coarse matching between shape extremities based on a novel scale‐invariant isometric distortion measure. The proposed algorithm also supports (partial) dense matching, that alleviates the symmetric flip problem due to initial coarse sampling. We test the performance of our matching algorithm on several shape datasets in comparison to state of the art. Our method proves useful, not only for partial matching, but also for complete matching of semantically similar hybrid shape pairs whose maximum geodesic distances may not be compatible, a case that would fail most of the conventional isometric shape matchers.     

Distributed discrete-time coordinated tracking with a time-varying reference state and limited communication

This paper studies a distributed discrete-time coordinated tracking problem where a team of vehicles communicating with their local neighbors at discrete-time instants tracks a time-varying reference state available to only a subset of the team members. We propose a PD-like discrete-time consensus algorithm to address the problem under a fixed communication graph. We then study the condition on the communication graph, the sampling period, and the control gain to ensure stability and give the quantitative bound of the tracking errors. It is shown that the ultimate bound of the tracking errors is proportional to the sampling period. The benefit of the proposed PD-like discrete-time consensus algorithm is also demonstrated through comparison with an existing P-like discrete-time consensus algorithm. Simulation results are presented as a proof of concept.     

Fault-tolerant flocking for a group of autonomous mobile robots

Consider a system composed of mobile robots that move on the plane, each of which independently executing its own instance of an algorithm. Given a desired geometric pattern, the flocking problem consists in ensuring that the robots form this pattern and maintain it while moving together on the plane. In this paper, we explore flocking in the presence of faulty robots, where the desired pattern is a regular polygon. We propose a distributed fault tolerant flocking algorithm assuming a semi-synchronous model with a k-bounded scheduler, in the sense that no robot is activated no more than k times between any two consecutive activations of any other robot.The algorithm is composed of three parts: failure detector, ranking assignment, and flocking algorithm. The role of the rank assignment is to provide a persistent and unique ranking for the robots. The failure detector identifies the set of currently correct robots in the system. Finally, the flocking algorithm handles the movement and reconfiguration of the flock, while maintaining the desired shape. The difficulty of the problem comes from the combination of the three parts, together with the necessity to prevent collisions and allow the rotation of the flock. We formally prove the correctness of our proposed solution.