Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight.
Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm.
Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness.
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Epic Fail: Emulators Can Tolerate Polynomially Many Edge Faults for Free
A t-emulator of a graph G is a graph H that approximates its pairwise shortest path distances up to multiplicative t error. We study fault tolerant t-emulators, under the model recently introduced by Bodwin, Dinitz, and Nazari [ITCS 2022] for vertex failures. In this paper we consider the version for edge failures, and show that they exhibit surprisingly different behavior.
In particular, our main result is that, for (2k-1)-emulators with k odd, we can tolerate a polynomial number of edge faults for free. For example: for any n-node input graph, we construct a 5-emulator (k = 3) on O(n^{4/3}) edges that is robust to f = O(n^{2/9}) edge faults. It is well known that Ω(n^{4/3}) edges are necessary even if the 5-emulator does not need to tolerate any faults. Thus we pay no extra cost in the size to gain this fault tolerance. We leave open the precise range of free fault tolerance for odd k, and whether a similar phenomenon can be proved for even k.
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- Award ID(s):
- 1909111
- NSF-PAR ID:
- 10462948
- Date Published:
- Journal Name:
- Leibniz international proceedings in informatics
- Volume:
- 251
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 20:1 - 20:22
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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