Measuring the similarity between 3D shapes is a fundamental problem, with applications in computer vision, molecular biology, computer graphics, and a variety of other fields. A challenging aspect of this problem is to find a suitable shape signature that can be constructed and compared quickly, while still discriminating between similar and dissimilar shapes. In this paper, we propose and analyze a method for computing shape signatures for arbitrary (possibly degenerate) 3D polygonal models. The key idea is to represent the signature of an object as a shape distribution sampled from a shape function measuring global geometric properties of an object. The primary motivation for this approach is to reduce the shape matching problem to the comparison of probability distributions, which is a simpler problem than the comparison of 3D surfaces by traditional shape matching methods that require pose registration, feature correspondence, or model fitting. We find that the dissimilarities be...

this paper, we propose and analyze a method for computing shape signatures for arbitrary (possibly degenerate) 3D polygonal models. The key idea is to represent the signature of an object as a shape distribution sampled from a shape function measuring global geometric properties of an object. The primary motivation for this approach is to reduce the shape matching problem to the comparison of probability distributions, which is simpler than traditional shape matching methods that require pose registration, feature correspondence, or model fitting

We present a new, practical algorithm to resolve the experimental data in restriction site analysis, which is a common technique for mapping DNA. Specifically, we assert that multiple digestions with a single restriction enzyme can provide sufficient information to identify the positions of the restriction sites with high probability. The motivation for the new approach comes from combinatorial results on the number of mutually homeometric sets in one dimension, where two sets of n points are homeometric if the multiset of n(n \Gamma 1)=2 distances they determine are the same. Since experimental data contains error, we propose algorithms for reconstructing sets from noisy interpoint distances, including the possibility of missing fragments. We analyze the performance of these algorithms under a reasonable probability distribution, establishing a relative error limit of r = \Theta(1=n 2 ) beyond which our technique becomes infeasible. Through simulations, we establish that our techni...

The Probed Partial Digestion mapping method partially digests a DNA strand with a restriction enzyme. A probe, which attaches to the DNA between two restriction enzyme cutting sites, is hybridized to the partially digested DNA, and the sizes of fragments to which the probe hybridizes are measured. The objective is to reconstruct the linear order of the restriction enzyme cutting sites from the multiset of measured lengths. In many cases, more than one underlying linear ordering is consistent with a multiset of measured lengths. This article shows that a multiset of N measured lengths can have as many as \Omega\Gamma N t ) solutions for any t ! i \Gamma1 (2) where i(t) is the Riemann Zeta Function and i \Gamma1 (2) ß 1:73. 1 Introduction The Probed Partial Digestion (or PPD) mapping scheme is used to generate physical maps of large DNA strands using restriction enzymes and probes. A DNA strand can be viewed as a finite sequence over the alphabet of four letters fA, C, G, Tg. A re...

Which point sets realize a given distance multiset? Interesting cases include the "turnpike problem" where the points lie on a line, the "beltway problem" where the points lie on a loop, and multidimensional versions. We are interested both in the algorithmic problem of determining such point sets for a given collection of distances and the combinatorial problem of finding bounds on the maximum number of different solutions. These problems have applications in genetics and crystallography.

In a previous paper we showed that, for any n ≥ m + 2, most sets of n points in R m are determined (up to rotations, reflections, translations and relabeling of the points) by the distribution of their pairwise distances. But there are some exceptional point configurations which are not reconstructible from the distribution of distances in the above sense. In this paper, we concentrate on the planar case m = 2 and present a reconstructibility test with running time O(n 11). The cases of orientation preserving rigid motions (rotations and translations) and scalings are also discussed.

In computational molecular biology, the aim of restriction mapping is to locate the restriction sites of a given enzyme on a DNA molecule. Double digest and partial digest are two well-studied techniques for restriction mapping. While double digest is NP-complete, there is no known polynomial algorithm for partial digest. Another disadvantage of the above techniques is that there can be multiple solutions for reconstruction. In this

The PARTIAL DIGEST problem well-known for its applications in computational biology and for the intriguingly open status of its computational complexity asks for the coordinates of n points on a line such that the pairwise distances of the points form a given multi-set of () distances. In an effort to model real-life data, we study the computational complexity of a minimization version of PARTIAL DIGEST, in which only a subset of all pairwise distances is given and the rest are lacking due to experimental errors. We show that this variation is NP-hard to solve exactly, thus making the existence of polynomial-time algorithms for this problem extremely unlikely. Our result answers an open question posed by Pevzner (2000). We then study a maximiza- tion version of PARTIAL DIGEST where a superset of all pairwise distances is given, with some additional distances due to inaccurate measurements. We show that this maximization version is NP-hard to approximate to within a factor of [D[ -c for any e 0, where [D[ is the number of input distances, which implies that polynomial-time algorithms cannot even guarantee to find a solution for the problem that comes close to the optimum. Our inapproximabilky result is tight up to low-order terms as we give a trivial approximation algorithm that achieves a matching approximation ratio. Our optimization variations model two different error types that occur in real-life data.

this paper we prove all cases except n = 2 (for which see Smith [37]) of the following proposition.

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