Abstract—For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertex-cover number (equiv-alently, the size of the maximum matching in C is bounded

Abstract. Vassilevska and Williams (STOC 2009) showed how to count simple paths on k vertices and matchings on k/2 edges in an n-vertex graph in time nk/2+O(1). In the same year, two different algorithms with the same runtime were given by Koutis and Williams (ICALP 2009), and Björklund et al

Color Coding is an algorithmic technique for deciding efficiently if a given input graph contains a path of a given length (or another small subgraph of constant tree-width). Applications of the method in computational biology motivate the study of similar algorithms for counting the number

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:

Abstract not found

Parents, who directed me to follow the scientific path and where fully supportive at all times. Thanks to all the members of LSV – it was a pleasure working with You. I would like to thank the awesome people whom I have been working with in the same office on the 4th floor. Without all the nice riddles

of the thesis can be summarized as follows. • We prove that for all Coxeter graphs constructed from an n-path of unlabelled edges by adding a new labelled edge and a new vertex (sometimes two new edges and vertices), there is a permutational representation of the corresponding group. Group elements correspond

and whose edges link such strips to their ‘good continuations’. The point cloud data are reduced to counts measuring membership in strips. Each anisotropy graph is reduced to a subgraph consisting of strips with ‘significant’ counts. The algorithm rejects H0 whenever some such subgraph contains a path

, and we can say the Campbell bandwidth is the minimum average bandwidth for en-coding the process across all possible distortion levels. IX. CONCLUSION We have presented two new derivations of the coefficient rate in-troduced by Campbell. One derivation solidifies its interpretation as a coefficient rate, and shows that the spectral entropy of a random process is proportional to the logarithm of the equivalent bandwidth of the smallest frequency band that contains most of the energy. The second derivation implies that the number of samples of a particular component should be proportional to the variance of that component. We discussed the implications of the latter result for realization-adap-tive source coding and provided a connection with the familiar reverse water-filling result from rate distortion theory. From the coefficient rate, we defined a quantity called the Campbell bandwidth of a random process, and we contrasted Fourier bandwidth, Shannon bandwidth, and Campbell bandwidth. ACKNOWLEDGMENT The authors are indebted to the referees for their constructive com-ments and insights.

2010 to my wife, Joyce, and my family...- Résumé-