Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric tspanner 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 tspanner G_i for S_i for all i, while minimizing the number of edges, and their total weight.
Under the L₂norm 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^{1d}log ε^{1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1d})⋅ 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 instanceoptimal 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 L₁norm.
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 = (2k1)(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 tradeoff 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.
more »
« less
Stochastic Online Metric Matching
We study the minimumcost metric perfect matching problem under online i.i.d arrivals. We are given a fixed metric with a server at each of the points, and then requests arrive online, each drawn independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost of the matching. Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a tight O(log n)competitiveness for the randomorder arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of O(log n) has long been conjectured and remains a tantalizing open question. In this paper, we show that the i.i.d model admits substantially better algorithms: our main result is an O((log log log n)^2)competitive algorithm in this model, implying a strict separation between the i.i.d model and the adversarial and random order models. Along the way we give a 9competitive algorithm for the line and tree metrics  the first O(1)competitive algorithm for any nontrivial arrival model for these muchstudied metrics.
more »
« less
 NSFPAR ID:
 10121532
 Date Published:
 Journal Name:
 International Colloquium on Automata, Languages, and Programming
 Page Range / eLocation ID:
 67:1  67:14
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric tspanner 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 tspanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂norm 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^{1d}log ε^{1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1d})⋅ 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 instanceoptimal 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 L₁norm. 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 = (2k1)(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 tradeoff 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.more » « less

Megow, Nicole ; Smith, Adam (Ed.)In this paper, we study the weighted kserver problem on the uniform metric in both the offline and online settings. We start with the offline setting. In contrast to the (unweighted) kserver problem which has a polynomialtime solution using mincost flows, there are strong computational lower bounds for the weighted kserver problem, even on the uniform metric. Specifically, we show that assuming the unique games conjecture, there are no polynomialtime algorithms with a subpolynomial approximation factor, even if we use cresource augmentation for c < 2. Furthermore, if we consider the natural LP relaxation of the problem, then obtaining a bounded integrality gap requires us to use at least 𝓁 resource augmentation, where 𝓁 is the number of distinct server weights. We complement these results by obtaining a constantapproximation algorithm via LP rounding, with a resource augmentation of (2+ε)𝓁 for any constant ε > 0. In the online setting, an exp(k) lower bound is known for the competitive ratio of any randomized algorithm for the weighted kserver problem on the uniform metric. In contrast, we show that 2𝓁resource augmentation can bring the competitive ratio down by an exponential factor to only O(𝓁² log 𝓁). Our online algorithm uses the twostage approach of first obtaining a fractional solution using the online primaldual framework, and then rounding it online.more » « less

he noisy broadcast model was first studied by [Gallager, 1988] where an ncharacter input is distributed among n processors, so that each processor receives one input bit. Computation proceeds in rounds, where in each round each processor broadcasts a single character, and each reception is corrupted independently at random with some probability p. [Gallager, 1988] gave an algorithm for all processors to learn the input in O(log log n) rounds with high probability. Later, a matching lower bound of Omega(log log n) was given by [Goyal et al., 2008]. We study a relaxed version of this model where each reception is erased and replaced with a `?' independently with probability p, so the processors have knowledge of whether a bit has been corrupted. In this relaxed model, we break past the lower bound of [Goyal et al., 2008] and obtain an O(log^* n)round algorithm for all processors to learn the input with high probability. We also show an O(1)round algorithm for the same problem when the alphabet size is Omega(poly(n)).more » « less

Abernethy, Jacob ; Agarwal, Agarwal (Ed.)Motivated by problems in controlled experiments, we study the discrepancy of random matrices with continuous entries where the number of columns $n$ is much larger than the number of rows $m$. Our first result shows that if $\omega(1) = m = o(n)$, a matrix with i.i.d. standard Gaussian entries has discrepancy $\Theta(\sqrt{n} \, 2^{n/m})$ with high probability. This provides sharp guarantees for Gaussian discrepancy in a regime that had not been considered before in the existing literature. Our results also apply to a more general family of random matrices with continuous i.i.d. entries, assuming that $m = O(n/\log{n})$. The proof is nonconstructive and is an application of the second moment method. Our second result is algorithmic and applies to random matrices whose entries are i.i.d. and have a Lipschitz density. We present a randomized polynomialtime algorithm that achieves discrepancy $e^{\Omega(\log^2(n)/m)}$ with high probability, provided that $m = O(\sqrt{\log{n}})$. In the onedimensional case, this matches the best known algorithmic guarantees due to Karmarkar–Karp. For higher dimensions $2 \leq m = O(\sqrt{\log{n}})$, this establishes the first efficient algorithm achieving discrepancy smaller than $O( \sqrt{m} )$.more » « less