We study algorithms for spectral graph sparsification. The input is a graph G with n vertices and m edges, and the output is a sparse graph G˜ that approximates G in an algebraic sense. Concretely, for all vectors x and any ϵ > 0, the graph G˜ satisfies (1-ϵ ) x T L G x ≤ x T L G˜ x ≤ (1+ϵ) x T L G x , where L G and G˜ are the Laplacians of G and G˜ respectively. The first contribution of this article applies to all existing sparsification algorithms that rely on solving solving linear systems on graph Laplacians. These algorithms are the fastest known to date. Specifically, we show that less precision is required in the solution of the linear systems, leading to speedups by an O (log n ) factor. We also present faster sparsification algorithms for slightly dense graphs: — An O ( m log n ) time algorithm that generates a sparsifier with O ( n log 3 n /ϵ 2 ) edges. — An O ( m log log n ) time algorithm for graphs with more than n log 5 n log log n edges. — An O ( m ) algorithm for graphs with more than n log 10 n edges. — An O ( m ) algorithm for unweighted graphs with more than n log 8 n edges. These bounds hold up to factors that are in O ( poly (log log n )) and are conjectured to be removable.

We present three spectral sparsification algorithms that, on input a graph G with n vertices and m edges, return a graph H with n vertices and O(n log n/epsilon^2) edges that provides a strong approximation of G. Namely, for all vectors x and any epsilon>0, we have (1-epsilon) x^T L_G x <= x^T L_H x <= (1+epsilon) x^T L_G x, where L_G and L_H are the Laplacians of the two graphs. The first algorithm is a simple modification of the fastest known algorithm and runs in tilde{O}(m log^2 n) time, an O(log n) factor faster than before. The second algorithm runs in tilde{O}(m log n) time and generates a sparsifier with tilde{O}(n log^3 n) edges. The third algorithm applies to graphs where m>n log^5 n and runs in tilde{O}(m log_{m/ n log^5 n} n time. In the range where m>n^{1+r} for some constant r this becomes softO(m). The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of dense SDD matrices.

Let G be a graph with n vertices and m edges. A sparsifier of G is a sparse graph on the same vertex set approximating $G$ in some natural way. It allows us to say useful things about $G$ while considering much fewer than m edges. The strongest commonly-used notion of sparsification is spectral sparsification; H is a spectral sparsifier of G if the quadratic forms induced by the Laplacians of G and H approximate one another well. This notion is strictly stronger than the earlier concept of combinatorial sparsification. In this paper, we consider a semi-streaming setting, where we have only ~O(n) storage space, and we thus cannot keep all of G. In this case, maintaining a sparsifier instead gives us a useful approximation to G, allowing us to answer certain questions about the original graph without storing all of it. In this paper, we introduce an algorithm for constructing a spectral sparsifier of G with O(n log n/epsilon^2) edges (where epsilon is a parameter measuring the quality of the sparsifier), taking ~O(m) time and requiring only one pass over G. In addition, our algorithm has the property that it maintains at all times a valid sparsifier for the subgraph of G that we have received. Our algorithm is natural and conceptually simple. As we read edges of $G,$ we add them to the sparsifier $H$. Whenever $H$ gets too big, we resparsify it in $tilde O(n)$ time. Adding edges to a graph changes the structure of its sparsifier's restriction to the already existing edges. It would thus seem that the above procedure would cause errors to compound each time that we resparsify, and that we should need to either retain significantly more information or reexamine previously discarded edges in order to construct the new sparsifier. However, we show how to use the information contained in $H$ to perform this resparsification using only the edges retained by earlier steps in nearly linear time.

It is a common belief that repetitive users can adapt to a spoken dialog system. In this paper we describe a dialog design that allows experienced users to make their interactions with the system more efficient and present experimental evaluation of such adaptation. In our study we focused on the application of dialog systems as a tool for real-time data collection for healthcare. Specifically, we implemented a dialog system, Pain Monitoring Voice Diary, for monitoring chronic pain patients and conducted a usability study involving 171 dialog sessions with 24 users. Breakdown of the data according to the level of user experience indicates that experienced users adapt and take advantage of dialog design to make their interaction more efficient.

This papers reports on the development of spoken dialog system as a tool for real-time data collection for healthcare, life and behavioral science. Specifically, we implemented a dialog system, Pain Monitoring Voice Diary, for monitoring chronic pain patients. We discuss the requirements and characteristics of this application, specifically, the needs for adaptive level of user support and high and controllable accuracy of data capture, and show their implications for the design of Pain Monitoring Voice Diary. In usability study involving 118 dialog sessions with 24 volunteers we measured 98 % data capture and 80 % dialog efficiency, as estimated by percentage of task-oriented prompts. 1.