Abstract
We present an $$O\left( n\log ^2{n} +(1/\varepsilon )^5 n\log {n}\right) $$ -time algorithm that computes a $$(1+\varepsilon )$$ -approximation of the diameter of a non-negatively-weighted, undirected planar graph of n vertices. This is an improvement over the algorithm of Weimann and Yuster (ACM Trans Algorithms 12(1):12, 2016) of $$O\left( (1/\varepsilon )^4 n\log ^4{n}+2^{O(1/\varepsilon )}n\right) $$ time in two regards. First we eliminate the exponential dependency on $$1/\varepsilon $$ by adapting and specializing Cabello’s recent Voronoi-diagram-based technique (Cabello, in: Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2017) for approximation purposes. Second we shave off two logarithmic factors by choosing a better sequence of error parameters in the recursion. Moreover, using similar techniques we obtain a variant of Gu and Xu’s $$(1+\varepsilon )$$ -approximate distance oracle (Gu and Xu, in: Proceedings of the 26th International Symposium on Algorithms and Computation (ISAAC), 2015) with polynomial dependency on $$1/\varepsilon $$ in the preprocessing time and space and $$O\left( \log (1/\varepsilon )\right) $$ query time.
Highlights
In this paper we study the problem of computing the diameter of a weighted, undirected planar graph of n vertices with non-negative edge lengths1, defined as the longest shortest path distance between two vertices of the graph
Poly-logarithmic speedups were given by Chan [5] in 2006 and by Wulff-Nilsen [22] in 2010; the algorithm of the former works for the unweighted case and requires O n2 log log n/ log n time; the algorithm of the latter requires the same amount of time for the unweighted case and O n2(log log n)4/ log n time for the weighted
Gawrychowski et al [8] recently improved Cabello’s algorithm [4] for computing the exact diameter in planar graphs; their algorithm is deterministic instead of randomized and requires O(n5/3) time instead of O(n11/6). It is worth investigating whether the techniques therein could be used to make our approximation algorithm deterministic and perhaps shave off some 1/ε factors
Summary
In this paper we study the problem of computing the diameter of a weighted, undirected planar graph of n vertices with non-negative edge lengths, defined as the longest shortest path distance between two vertices of the graph. Given a weighted, undirected planar graph of n vertices with nonnegative edge lengths, we can compute a (1 + ε)-approximation of its diameter in expected O n log n log n + (1/ε) time Another important problem in planar graphs is the construction of efficient (1 + ε)approximate distance oracles, i.e., data structures that in a query for a pair of vertices u, v of a planar graph G, return a value dsuch that dG(u, v) ≤ d ≤ (1+ε)dG(u, v), where dG(u, v) is the shortest path distance from u to v in G. Gu and Xu [9] combined the ideas of those results with the techniques of the diameter algorithm of Weimann and Yuster [21] to obtain the first distance oracle with constant query time (independent of both n and ε); it requires O n log n (1/ε) log n + 2O(1/ε) preprocessing time and O n log n (1/ε) log n + 2O(1/ε) space. We assume that all the planar graphs under discussion have a fixed, combinatorial embedding and are triangulated
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