LAST but not Least: Online Spanners for Buy-at-Bulk
Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms
The online (uniform) buy-at-bulk network design problem asks us to design a network, where the edge-costs exhibit economy-of-scale. Previous approaches to this problem used treeembeddings, giving us randomized algorithms. Moreover, the optimal results with a logarithmic competitive ratio requires the metric on which the network is being built to be known up-front; the competitive ratios then depend on the size of this metric (which could be much larger than the number of terminals that arrive).
... inals that arrive). We consider the buy-at-bulk problem in the least restrictive model where the metric is not known in advance, but revealed in parts along with the demand points seeking connectivity arriving online. For the single sink buy-at-bulk problem, we give a deterministic online algorithm with competitive ratio that is logarithmic in k, the number of terminals that have arrived, matching the lower bound known even for the online Steiner tree problem. In the oblivious case when the buy-at-bulk function used to compute the edge-costs of the network is not known in advance (but is the same across all edges), we give a deterministic algorithm with competitive ratio polylogarithmic in k, the number of terminals. At the heart of our algorithms are optimal constructions for online Light Approximate Shortest-path Trees (LASTs) and spanners, and their variants. We give constructions that have optimal trade-offs in terms of cost and stretch. We also define and give constructions for a new notion of LASTs where the set of roots (in addition to the points) expands over time. We expect these techniques will find applications in other online network-design problems. The model of (uniform) buy-at-bulk network design captures economies-of-scale in routing problems. Given an undirected graph G = (V, E) with edge lengths d : E → R ≥0 -we can assume the lengths form a metric-the cost of sending x e flow over any edge e is d(e) · f (x e ) where f is some concave cost function. The total cost is the sum over all edges of the per-edge cost. Given some traffic matrix (a.k.a. demand), the goal is now to find a routing for the demand to minimize the total cost. This model is well studied both in the operations research and approximation algorithms communities, both in the offline and online settings. In the offline setting, an early result was an O(log k)-approximation due to Awerbuch and Azar, one of the first uses of tree embeddings in approximation algorithms [AA97]-here k is the number of demands. For the single-sink case, the first O(1)-approximation was given by [GMM09] . In fact, one can get a constant-factor even for the "oblivious" single-sink case where the demands are given, but the actual concave function f is known only after the network is built [GP12]. The problem is just as interesting in the online context: in the online single-sink problem, new demand locations (called terminals) are added over time, and these must be connected to the central root node as they arrive. This captures an increasing demand for telecommunication services as new customers arrive, and must be connected via access networks to a central core of nodes already provisioned for high bandwidth. The Awerbuch-Azar approach of embedding G into a tree metric T with O(log n) expected stretch (say using [FRT04]), and then routing on this tree, gives an O(log n)-competitive randomized algorithm even in the online case. But this requires that the metric is known in advance, and the dependence is on n, the number of nodes in the metric, and not on the number of terminals k! This may be undesirable in situations when n k; for example, when the terminals come from a Euclidean space R d for some large d. Moreover, we only get a randomized algorithm (competitive against oblivious adversaries). 1 In this paper, we study the Buy-at-Bulk problem in the online setting, in the least restrictive model where the metric is not known in advance, so the distance from some point to the previous points is revealed only when the point arrives. This forces us to focus on the problem structure, since we cannot rely on powerful general techniques like tree embeddings. Moreover, we aim for deterministic algorithms for the problem. Our first main result is an asymptotically optimal deterministic online algorithm for single-sink buy-at-bulk. Theorem 1.1 (Deterministic Buy-at-Bulk). There exists a deterministic O(log k)-competitive algorithm for online single-sink buy-at-bulk, where k is the number of terminals. Note that the guarantee is best possible, since it matches the lower bound [IW91] for the special case of a single cable type encoding the online Steiner tree problem. En route, we consider a generalization of the Light Approximate Shortest-path Trees (LASTs). Given a set of "sources" and a sink, a LAST is a tree of weight close to the minimum spanning tree (MST) on the sources and the sink, such that the tree distance from any source to the sink is close to the shortest-path distance. Khuller, Raghavachari, and Young [KRY95] defined and studied LASTs in the offline setting and showed that one can get constant stretch with cost constant times the MST. Ever since their work, LASTs have proved very versatile in network design applications. We 1 The tree embeddings of Bartal [Bar96] can indeed be done online with O(log k log ∆) expected stretch, where ∆ is the ratio of maximum to minimum distances in the metric. Essentially, this is because the probabilistic partitions used to construct the embedding can be computed online. This gives an O(log 2 k)-competitive randomized algorithm, alas sub-optimal by a logarithmic factor, and still randomized.