Online Euclidean Spanners
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In this paper, we study the online Euclidean spanners problem for points in ℝ^d. Suppose we are given a sequence of n points (s_1,s_2,…, s_n) in ℝ^d, where point s_i is presented in step i for i=1,…, n. The objective of an online algorithm is to maintain a geometric t-spanner on S_i={s_1,…, s_i} for each step i. First, we establish a lower bound of Ω(ε^-1log n / logε^-1) for the competitive ratio of any online (1+ε)-spanner algorithm, for a sequence of n points in 1-dimension. We show that this bound is tight, and there is an online algorithm that can maintain a (1+ε)-spanner with competitive ratio O(ε^-1log n / logε^-1). Next, we design online algorithms for sequences of points in ℝ^d, for any constant d≥ 2, under the L_2 norm. We show that previously known incremental algorithms achieve a competitive ratio O(ε^-(d+1)log n). However, if the algorithm is allowed to use additional points (Steiner points), then it is possible to substantially improve the competitive ratio in terms of ε. We describe an online Steiner (1+ε)-spanner algorithm with competitive ratio O(ε^(1-d)/2log n). As a counterpart, we show that the dependence on n cannot be eliminated in dimensions d ≥ 2. In particular, we prove that any online spanner algorithm for a sequence of n points in ℝ^d under the L_2 norm has competitive ratio Ω(f(n)), where lim_n→∞f(n)=∞. Finally, we provide improved lower bounds under the L_1 norm: Ω(ε^-2/logε^-1) in the plane and Ω(ε^-d) in ℝ^d for d≥ 3.
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