Enoch Peserico

Enoch Peserico

In an array of N elements, M positions and M elements are "marked". We show how to permute the elements in the array so that all marked elements end in marked positions, in time O(N) (in the standard word-RAM model), deterministically, and obliviously - i.e. with a sequence of memory accesses that depends only on N and not on which elements or positions are marked. As a corollary, we answer affirmatively to an open question about the existence of a deterministic oblivious algorithm with O(N) running time for tight compaction (move the M marked elements to the first M positions of the array), a building block for several cryptographic constructions. Our O(N) result improves the running-time upper bounds for deterministic tight compaction, for randomized tight compaction, and for the simpler problem of randomized loose compaction (move the M marked elements to the first O(M) positions) - until now respectively O(N lg N), O(N lg lg N), and O(N lg*N).

Marco Bressan, Enoch Peserico, Luca Pretto

We study the complexity of local graph centrality estimation, with the goal of approximating the centrality score of a given target node while exploring only a sublinear number of nodes/arcs of the graph and performing a sublinear number of elementary operations. We develop a technique, that we apply to the PageRank and Heat Kernel centralities, for building a low-variance score estimator through a local exploration of the graph. We obtain an algorithm that, given any node in any graph of $m$ arcs, with probability $(1-δ)$ computes a multiplicative $(1\pmε)$-approximation of its score by examining only $\tilde{O}(\min(m^{2/3} Δ^{1/3} d^{-2/3},\, m^{4/5} d^{-3/5}))$ nodes/arcs, where $Δ$ and $d$ are respectively the maximum and average outdegree of the graph (omitting for readability $\operatorname{poly}(ε^{-1})$ and $\operatorname{polylog}(δ^{-1})$ factors). A similar bound holds for computational complexity. We also prove a lower bound of $Ω(\min(m^{1/2} Δ^{1/2} d^{-1/2}, \, m^{2/3} d^{-1/3}))$ for both query complexity and computational complexity. Moreover, our technique yields a $\tilde{O}(n^{2/3})$ query complexity algorithm for the graph access model of [Brautbar et al., 2010], widely used in social network mining; we show this algorithm is optimal up to a sublogarithmic factor. These are the first algorithms yielding worst-case sublinear bounds for general directed graphs and any choice of the target node.