Joel Friedman

A deterministic view of random sampling and its use in geometry

The combination of divide-and-conquer and random sampling has proven very effective in the design of fast geometric algorithms. A flurry of efficient probabilistic algorithms have been recently discovered, based on this happy marriage. We show that all those algorithms can be derandomized with only polynomial overhead. In the process we establish results of independent interest concerning the covering of hypergraphs and we improve on various probabilistic bounds in geometric complexity. For example, givenn hyperplanes ind-space and any integerr large enough, we show how to compute, in polynomial time, a simplicial packing of sizeO(rd) which coversd-space, each of whose simplices intersectsO(n/r) hyperplanes.

On the second eigenvalue of random regular graphs

The following is an extended abstract for two papers, one written by Kahn and Szemeredi, the other written by Friedman, which have been combined at the request of the STOC committee. The introduction was written jointly, the second section by Kahn and Szemeredi, and the third by Friedman, Let G be a d-regular (i.e. each vertex has degree d) undirected graph on n nodes. It’s adjacency matrix is symmetric, and therefore has real eigenvalues Ar = d 2 x2 >_ *-. >_ X, with IX,] 5 d. Graphs for which X2 and

Expanding graphs contain all small trees

The assertion of the title is formulated and proved. The result is then used to construct graphs with a linear number of edges that, even after the deletion of almost all of their edges or almost all of their vertices, continue to contain all small trees.

Some geometric aspects of graphs and their eigenfunctions

We study three mathematical notions, that of nodal regions for eigenfunctions of the Laplacian, that of covering theory, and that of fiber products, in the context of graph theory and spectral theory for graphs. We formulate analogous notions and theorems for graphs and their eigenpairs. These techniques suggest new ways of studying problems related to spectral theory of graphs. We also perform some numerical experiments suggesting that the fiber product can yield graphs with small second eigenvalue.

Wide-sense nonblocking networks

A new method for constructing wide-sense nonblocking networks is presented. Application of this method yields (among other things) wide-sense nonblocking generalized connectors with n inputs and outputs and size

A Note on Matrix Rigidity

O( n\log n )

Computing Betti numbers via combinatorial Laplacians

, and with depth k and size

On the second eigenvalue of hypergraphs

O ( n^{1 + 1/k} ( \log n )^{1 - 1/k} )

Relative expanders or weakly relatively Ramanujan graphs

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On the Betti numbers of chessboard complexes

In this paper we give an explicit construction ofn×n matrices over finite fields which are somewhat rigid, in that if we change at mostk entries in each row, its rank remains at leastCn(logqk)/k, whereq is the size of the field andC is an absolute constant. Our matrices satisfy a somewhat stronger property, we will explain and call “strong rigidity”. We introduce and briefly discuss strong rigidity, because it is in a sense a simpler property and may be easier to use in giving explicit construction.