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For matrices with all nonnegative entries, the Perron-Frobenius theorem guarantees the existence of an eigenvector with all nonnegative components. We show that the existence of such an eigenvector is also guaranteed for a very different class of matrices, namely real symmetric matrices with exactly two eigenvalues. We also prove a partial converse, that among real symmetric matrices with any more than two eigenvalues there exist some having no nonnegative eigenvector.A nonnegative vector is one whose components are all nonnegative. This concept has no place in pure linear algebra, as it is highly basis dependent. However, nonnegative vectors (and their cousins, positive vectors) sometimes crop up and prove useful in applications. For example, one of the consequences of the Perron-Frobenius theorem is that a matrix with nonnegative entries has a nonnegative (or even positive, under appropriate hypotheses) eigenvector, which fact is of great consequence for, e.g., ranking pages in search engine results [2].In this note, we prove that the existence of a nonnegative eigenvector is also guaranteed for a very different class of matrices, namely real symmetric matrices having only two distinct eigenvalues. Recall that a symmetric matrix has a set of orthogonal eigenvectors that span the ambient space. This is the only fact about symmetric matrices that we will need.Let M ∈ R n×n be our matrix of interest. Since we suppose M has only two eigenvalues, it has two eigenspaces V and W which are orthogonal and satisfy V + W = R n . Hence W = V ⊥ (with respect to the standard inner product on R n ) and vice versa. Thus the existence of a nonnegative eigenvector of M is an immediate corollary of the following proposition.Proposition. For any subspace V ⊆ R n , either V contains a nonzero, nonnegative vector or V ⊥ does.Some commentary before commencing with the proof: Although this is ostensibly a result about linear algebra, we have noted already that the notion of nonnegativity is inherently not a purely linear algebraic property. Hence it should not be surprising that the proof should require other ideas. It turns out that convexity is the key here. Proof of Proposition. Define setsR n ≥0 is the set of all nonnegative vectors, and proving the proposition amounts to showing that V or V ⊥ intersects R n ≥0 in a nonzero vector. Because V and R n ≥0 are both January 2014] NOTES 1

Large search engines process thousands of queries per second over billions of documents, making query processing a major performance bottleneck. An important class of optimization techniques called early termination achieves faster query processing by avoiding the scoring of documents that are unlikely to be in the top results. We study new algorithms for early termination that outperform previous methods. In particular, we focus on safe techniques for disjunctive queries, which return the same result as an exhaustive evaluation over the disjunction of the query terms. The current state-of-the-art methods for this case, the WAND algorithm by Broder et al. [11] and the approach of Strohman and Croft [30], achieve great benefits but still leave a large performance gap between disjunctive and (even non-early terminated) conjunctive queries.We propose a new set of algorithms by introducing a simple augmented inverted index structure called a block-max index. Essentially, this is a structure that stores the maximum impact score for each block of a compressed inverted list in uncompressed form, thus enabling us to skip large parts of the lists. We show how to integrate this structure into the WAND approach, leading to considerable performance gains. We then describe extensions to a layered index organization, and to indexes with reassigned document IDs, that achieve additional gains that narrow the gap between disjunctive and conjunctive top-k query processing.

The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem.We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in O(log n log ∆) rounds with high probability, where n is the number of nodes, ∆ is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within O(log ∆) of the optimal in expectation and within O(log n) of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements.