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- Daniel A. Spielman, Shang-Hua Teng
- STOC
- 2004

We present algorithms for solving symmetric, diagonally-dominant linear systems to accuracy ε in time linear in their number of non-zeros and log (κ<inf>f</inf> (A) ε), where κ<inf>f</inf> (A) is the condition number of the matrix defining the linear system. Our algorithm applies the preconditioned Chebyshev iteration with… (More)

- Daniel A. Spielman, Shang-Hua Teng
- STOC
- 2001

We introduce the <italic>smoothed analysis of algorithms</italic>, which is a hybrid of the worst-case and average-case analysis of algorithms. Essentially, we study the performance of algorithms under small random perturbations of their inputs. We show that the shadow-vertex simplex algorithm has polynomial <italic>smoothed complexity</italic>.

- Xi Chen, Xiaotie Deng, Shang-Hua Teng
- J. ACM
- 2009

We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class <b>PPAD</b> (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991.
Our result, building upon the work of Daskalakis et al. [2006a] on the complexity of four-player Nash equilibria, settles a long standing… (More)

In this paper we discuss a new type of query in Spatial Databases, called the Trip Planning Query (TPQ). Given a set of points of interest in space, where each point belongs to a specific category, a starting point and a destination , TPQ retrieves the best trip that starts at , passes through at least one point from each category, and ends at . For… (More)

Generating local addresses and communication sets is an important issue in distributed-memory implementations of data-parallel languages such as High Performance Fortran. We show that for an array <italic>A</italic> affinely aligned to a <italic>template</italic> that is distributed across <italic>p</italic> processors with a <italic>cyclic(k)</italic>… (More)

- Arvind Sankar, Daniel A. Spielman, Shang-Hua Teng
- SIAM J. Matrix Analysis Applications
- 2006

Let A be any matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the smoothed precision needed by Gaussian elimination without pivoting.… (More)

- Daniel A. Spielman, Shang-Hua Teng
- SIAM J. Comput.
- 2013

We study the design of local algorithms for massive graphs. A local algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster—a subset of vertices whose internal connections are significantly richer than its external connections—… (More)

- Michael Elkin, Daniel A. Spielman, Shang-Hua Teng
- SIAM J. Comput.
- 2005

We show that every weighted connected graph <i>G</i> contains as a subgraph a spanning tree into which the edges of <i>G</i> can be embedded with average stretch <i>O</i> (log<sup>2</sup> <i>n</i> log log <i>n</i>). Moreover, we show that this tree can be constructed in time <i>O</i> (<i>m</i> log<sup>2</sup><i>n</i>) in general, and in time <i>O</i>… (More)

- Daniel A. Spielman, Shang-Hua Teng
- SIAM J. Comput.
- 2011

We introduce a new notion of graph sparsification based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original. We prove that… (More)

- Daniel A. Spielman, Shang-Hua Teng
- FOCS
- 1996

Spectral partitioning methods use the Fiedler vector—the eigenvector of the second-smallest eigenvalue of the Laplacian matrix—to find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning… (More)