Abstract
We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an \(\widetilde{\varOmega }(n)\) lower bound for computing the diameter in sparse networks, which was previously known only for dense networks. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define.
Moreover, our technique allows us to show \(\widetilde{\varOmega }(n)\) lower bounds for computing \((\frac{3}{2}-\varepsilon )\)-approximations of the diameter or the radius, and for computing a \((\frac{5}{3}-\varepsilon )\)-approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors, and for eccentricities the improvement is both in the lower bound and in the approximation factor.
Interestingly, our technique also allows showing an almost-linear lower bound for the verification of \((\alpha ,\beta )\)-spanners, for \(\alpha < \beta +1\).
A. Abboud—Supported by Virginia Vassilevska Williams’s NSF Grants CCF-1417238 and CCF-1514339, and BSF Grant BSF:2012338. K. Censor-Hillel and S.Khoury—Supported by ISF Individual Research Grant 1696/14.
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Notes
- 1.
The notations \(\widetilde{\varOmega }\) and \(\widetilde{O}\) hide factors that are polylogarithmic in n
- 2.
SETH is a pessimistic version of the \(\mathsf{P} \ne \mathsf{NP}\) conjecture, which essentially says that CNF-SAT cannot be solved in \((2-\varepsilon )^n\) time. More formally, SETH is the assumption that there is no \(\varepsilon >0\) such that for all \(k \ge 1\) we can solve k-SAT on n variables and m clauses in \((2-\varepsilon )^n \cdot poly(m)\) time.
- 3.
It would imply a new co-nondeterministic algorithm for SAT and refute the Nondeterministic-SETH, which is a strong version of \(\mathsf{NP} \ne \mathsf{CONP}\).
- 4.
A truly-subquadratic algorithm for computing the radius of a sparse graph refutes the Hitting Set Conjecture: there is no \(\varepsilon >0\) such that given two lists A, B of n subsets of a universe U of size \(poly\log {n}\) we can decide whether there is a set \(a \in A\) that intersects all sets \(b \in B\) in \(O(n^{2-\varepsilon })\) time.
- 5.
Note that for the sake of simplicity, some of the edges are omitted from Fig. 1.
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Acknowledgement
We thank Ami Paz for many discussions and helpful suggestions.
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Abboud, A., Censor-Hillel, K., Khoury, S. (2016). Near-Linear Lower Bounds for Distributed Distance Computations, Even in Sparse Networks. In: Gavoille, C., Ilcinkas, D. (eds) Distributed Computing. DISC 2016. Lecture Notes in Computer Science(), vol 9888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53426-7_3
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