New Doubling Spanners: Better and Simpler

Abstract

In a seminal STOC 1995 paper, Arya et al. conjectured that spanners for lowdimensional Euclidean spaces with constant maximum degree, hop-diameter O(log n), and lightness O(log n) (i.e., weight O(log n) · w(MST)) can be constructed in O(n log n) time. This conjecture, which became a central open question in this area, was resolved in the affirmative by Elkin and Solomon in STOC 2013. In fact, Elkin and Solomon proved that the conjecture of Arya et al. holds even in doubling metrics. However, Elkin and Solomon's spanner construction is complicated. In this work we present a significantly simpler construction of spanners for doubling metrics with the same guarantees as above. Our construction is based on the basic net-tree spanner framework. However, by employing well-known properties of the net-tree spanner in conjunction with numerous new ideas, we managed to get significantly stronger results. First and foremost, our construction extends in a simple and natural way to provide k-fault tolerant spanners with maximum degree O(k 2 ), hopdiameter O(log n), and lightness O(k 2 log n). This is the first construction of fault-tolerant spanners (even for Euclidean metrics) that achieves good bounds (polylogarithmic in n and polynomial in k) on all the involved parameters simultaneously. Second, we show that the lightness bound of our construction can be improved to O(k 2 ) (with high probability), for random points in [0, 1] D , where 2 ≤ D = O(1). Introduction.An n-point metric space (X, d) can be represented by a complete weighted graph G = (X, E), where the weight w(e) of an edge e = (u, v) is given by d (u, v). A t-spanner of X is a weighted subgraph H = (X, E ) of G (where E ⊆ E has the same weights) that preserves all pairwise distances to within a factor of t, i.e., u, v) is the distance between u and v in H. The parameter t is called the stretch of the spanner H. A path between u and v in H with weight at most t · d(u, v) is called a t-spanner path.In this paper we focus on the regime of stretch t = 1 + for an arbitrarily small 0 < < 1 2 . In general, there are metric spaces (such as the one corresponding to uniformly weighted complete graph), where the only possible (1 + )-spanner is the complete graph. A special class of metric spaces, which has been subject to intensive research in the last decade, is the class of doubling metrics. The doubling dimension of a metric space (X, d), denoted by dim(X) (or dim when the context is clear), is the smallest value ρ such that every ball in X can be covered by 2 ρ balls of half the radius [3,11,22]. A metric space is called doubling if its doubling dimension is bounded by some constant. (We will sometimes disregard dependencies on and dim to avoid

Light spanners for Snowflake Metrics

“…In ICALP'13 Chan et al [14] showed that, in any doubling metric, one can build a (1 + ǫ)-spanner with constant degree and with logarithmic diameter and lightness, within O(n log n) time. A close examination of the construction of [14] shows that its lightness is dominated by the lightness of the standard net-tree spanner.…”

“…A close examination of the construction of [14] shows that its lightness is dominated by the lightness of the standard net-tree spanner. Plugging Theorem 4.7 in the construction of [14] gives rise to a logarithmic improvement in the lightness bound, for snowflake doubling metrics. This result is summarized in the following statement.…”

“…The construction of [14] was extended to the fault-tolerant (FT) setting in [14,33]. Plugging Theorem 4.7 in the FT constructions of [14,33] gives rise to a logarithmic improvement in the lightness bound, for snowflake doubling metrics.…”

“…Plugging Theorem 4.7 in the FT constructions of [14,33] gives rise to a logarithmic improvement in the lightness bound, for snowflake doubling metrics. This result is summarized in the following statement.…”

From hierarchical partitions to hierarchical covers

“…They showed that a k-FT (1 + ϵ)-spanner with degree O(k 2 ), diameter O(log n) and lightness O(k 2 log n), can be built in time O(n log n + k 2 n). Observe that the upper bounds of [15,41,16] are far from the known lower bounds: degree O(k 2 ) versus Ω(k), lightness O(k 2 log n) versus Ω(k 2 ), and runtime O(n log n + k 2 n) versus Ω(n log n + kn). (See Table 1 for a reference.)…”