Distributed Strong Diameter Network Decomposition

For a pair of positive parameters $D,\chi$, a partition ${\cal P}$ of the vertex set $V$ of an $n$-vertex graph $G = (V,E)$ into disjoint clusters of diameter at most $D$ each is called a $(D,\chi)$ network decomposition, if the supergraph ${\cal G}({\cal P})$, obtained by contracting each of the clusters of ${\cal P}$, can be properly $\chi$-colored. The decomposition ${\cal P}$ is said to be strong (resp., weak) if each of the clusters has strong (resp., weak) diameter at most $D$, i.e., if for every cluster $C \in {\cal P}$ and every two vertices $u,v \in C$, the distance between them in the induced graph $G(C)$ of $C$ (resp., in $G$) is at most $D$. Network decomposition is a powerful construct, very useful in distributed computing and beyond. It was shown by Awerbuch \etal \cite{AGLP89} and Panconesi and Srinivasan \cite{PS92}, that strong $(2^{O(\sqrt{\log n})},2^{O(\sqrt{\log n})})$ network decompositions can be computed in $2^{O(\sqrt{\log n})}$ distributed time. Linial and Saks \cite{LS93} devised an ingenious randomized algorithm that constructs {\em weak} $(O(\log n),O(\log n))$ network decompositions in $O(\log^2 n)$ time. It was however open till now if {\em strong} network decompositions with both parameters $2^{o(\sqrt{\log n})}$ can be constructed in distributed $2^{o(\sqrt{\log n})}$ time. In this paper we answer this long-standing open question in the affirmative, and show that strong $(O(\log n),O(\log n))$ network decompositions can be computed in $O(\log^2 n)$ time. We also present a tradeoff between parameters of our network decomposition. Our work is inspired by and relies on the "shifted shortest path approach", due to Blelloch \etal \cite{BGKMPT11}, and Miller \etal \cite{MPX13}. These authors developed this approach for PRAM algorithms for padded partitions. We adapt their approach to network decompositions in the distributed model of computation

A simple linear-time algorithm for finding path-decompositions of small width

We described a simple algorithm running in linear time for each fixed constant $k$, that either establishes that the pathwidth of a graph $G$ is greater than $k$, or finds a path-decomposition of $G$ of width at most $O(2^{k})$. This provides a simple proof of the result by Bodlaender that many families of graphs of bounded pathwidth can be recognized in linear time.Comment: 9 page

The geometry of quantum learning

Concept learning provides a natural framework in which to place the problems solved by the quantum algorithms of Bernstein-Vazirani and Grover. By combining the tools used in these algorithms--quantum fast transforms and amplitude amplification--with a novel (in this context) tool--a solution method for geometrical optimization problems--we derive a general technique for quantum concept learning. We name this technique "Amplified Impatient Learning" and apply it to construct quantum algorithms solving two new problems: BATTLESHIP and MAJORITY, more efficiently than is possible classically.Comment: 20 pages, plain TeX with amssym.tex, related work at http://www.math.uga.edu/~hunziker/ and http://math.ucsd.edu/~dmeyer

Approximating kk-Median via Pseudo-Approximation

We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of $1+\sqrt{3}+\epsilon$, improving upon the decade-old ratio of $3+\epsilon$. Our approach is based on two components, each of which, we believe, is of independent interest. First, we show that in order to give an $\alpha$-approximation algorithm for $k$-median, it is sufficient to give a \emph{pseudo-approximation algorithm} that finds an $\alpha$-approximate solution by opening $k+O(1)$ facilities. This is a rather surprising result as there exist instances for which opening $k+1$ facilities may lead to a significant smaller cost than if only $k$ facilities were opened. Second, we give such a pseudo-approximation algorithm with $\alpha= 1+\sqrt{3}+\epsilon$. Prior to our work, it was not even known whether opening $k + o(k)$ facilities would help improve the approximation ratio.Comment: 18 page

Improved Distance Queries and Cycle Counting by Frobenius Normal Form

Consider an unweighted, directed graph G with the diameter D. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time O-tilde(n^omega). The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the following problems efficiently. * All Nodes Shortest Cycles - for every node return the length of the shortest cycle containing it. We give an O-tilde(n^omega) algorithm that improves the previous O-tilde(n^((omega + 3)/2)) algorithm for unweighted digraphs. * We show how to compute all D sets of vertices lying on cycles of length c in {1, ..., D} in randomized time O-tilde(n^omega). It improves upon an algorithm by Cygan where algorithm that computes a single set is presented. * We present a functional improvement of distance queries for directed, unweighted graphs. * All Pairs All Walks - we show almost optimal O-tilde(n^3) time algorithm for all walks counting problem. We improve upon the naive O(D n^omega) time algorithm