arXiv subjects

Saeed Ilchi

Publications and source records attributed to Saeed Ilchi.

4 recordsLinked to original sources

arXiv2022

Deterministic Distributed Sparse and Ultra-Sparse Spanners and Connectivity Certificates

This paper presents efficient distributed algorithms for a number of fundamental problems in the area of graph sparsification: We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in $\textrm{polylog}(n)$ rounds in weighted graphs. Concretely, our algorithm outputs a spanning subgraph with only $n+o(n)$ edges in which the pairwise distances are stretched by a factor of at most $O(\log n \;\cdot\; 2^{O(\log^* n)})$. We provide a $\textrm{polylog}(n)$-round deterministic distributed algorithm that computes a spanner with stretch $(2k-1)$ and $O(nk + n^{1 + 1/k} \log k)$ edges in unweighted graphs and with $O(n^{1 + 1/k} k)$ edges in weighted graphs. We present the first $\textrm{polylog}(n)$-round randomized distributed algorithm that computes a sparse connectivity certificate. For an $n$-node graph $G$, a certificate for connectivity $k$ is a spanning subgraph $H$ that is $k$-edge-connected if and only if $G$ is $k$-edge-connected, and this subgraph $H$ is called sparse if it has $O(nk)$ edges. Our algorithm achieves a sparsity of $(1 + o(1))nk$ edges, which is within a $2(1 + o(1))$ factor of the best possible.

cs.DS↗

arXiv2022

Improved Distributed Network Decomposition, Hitting Sets, and Spanners, via Derandomization

This paper presents significantly improved deterministic algorithms for some of the key problems in the area of distributed graph algorithms, including network decomposition, hitting sets, and spanners. As the main ingredient in these results, we develop novel randomized distributed algorithms that we can analyze using only pairwise independence, and we can thus derandomize efficiently. As our most prominent end-result, we obtain a deterministic construction for $O(\log n)$-color $O(\log n \cdot \log\log\log n)$-strong diameter network decomposition in $\tilde{O}(\log^3 n)$ rounds. This is the first construction that achieves almost $\log n$ in both parameters, and it improves on a recent line of exciting progress on deterministic distributed network decompositions [Rozhoň, Ghaffari STOC'20; Ghaffari, Grunau, Rozhoň SODA'21; Chang, Ghaffari PODC'21; Elkin, Haeupler, Rozhoň, Grunau FOCS'22].

cs.DS↗

arXiv2022

Near-Optimal Distributed Dominating Set in Bounded Arboricity Graphs

We describe a simple deterministic $O( \varepsilon^{-1} \log Δ)$ round distributed algorithm for $(2α+1)(1 + \varepsilon)$ approximation of minimum weighted dominating set on graphs with arboricity at most $α$. Here $Δ$ denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation [Kuhn, Moscibroda, and Wattenhofer JACM'16]. Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized $O(α^2)$ approximation in $O(\log n)$ rounds [Lenzen and Wattenhofer DISC'10], a deterministic $O(α\log Δ)$ approximation in $O(\log Δ)$ rounds [Lenzen and Wattenhofer DISC'10], a deterministic $O(α)$ approximation in $O(\log^2 Δ)$ rounds [implicit in Bansal and Umboh IPL'17 and Kuhn, Moscibroda, and Wattenhofer SODA'06], and a randomized $O(α)$ approximation in $O(α\log n)$ rounds [Morgan, Solomon and Wein DISC'21]. We also provide a randomized $O(α\logΔ)$ round distributed algorithm that sharpens the approximation factor to $α(1+o(1))$. If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve $α- 1 - \varepsilon$ approximation [Bansal and Umboh IPL'17].

cs.DS↗

arXiv2020

On Statistical Learning of Simplices: Unmixing Problem Revisited

We study the sample complexity of learning a high-dimensional simplex from a set of points uniformly sampled from its interior. Learning of simplices is a long studied problem in computer science and has applications in computational biology and remote sensing, mostly under the name of `spectral unmixing'. We theoretically show that a sufficient sample complexity for reliable learning of a $K$-dimensional simplex up to a total-variation error of $ε$ is $O\left(\frac{K^2}ε\log\frac{K}ε\right)$, which yields a substantial improvement over existing bounds. Based on our new theoretical framework, we also propose a heuristic approach for the inference of simplices. Experimental results on synthetic and real-world datasets demonstrate a comparable performance for our method on noiseless samples, while we outperform the state-of-the-art in noisy cases.

cs.LG↗