Hierarchical Spanners

Abstract

A hierarchical graph 𝒢 consists of a vertex set V(𝒢) and L pairwise disjoint edge sets E_1, … , E_L. Such a structure naturally defines a hierarchy of L unweighted graphs, where the 𝓁-th graph is G_𝓁 = (V, ⋃_{i=1}^𝓁 E_i).
In this paper, we initiate the study of hierarchical spanners, namely subgraphs of 𝒢 that approximately preserve distances among a given set of pairs of vertices in V(𝒢) at every level of the hierarchy. This notion generalizes classical spanners, and thus all known lower bounds extend to this setting; however, it is not clear whether the same size-stretch trade-offs can be achieved.
We investigate this question by devising both upper and lower bounds for hierarchical spanners under various types of stretch and pairs of vertices of interest whose approximate (or exact) distances are to be maintained. On the positive side, a trivial adaptation of the greedy construction yields (2k-1)-spanners of size O(n^{1+1/k}), matching the classical bounds. However, the non-hierarchical bounds do not extend to the hierarchical case when additive or nearly-additive spanners are considered. For instance, we prove that any β-additive single-pair hierarchical spanner must have size Ω (n √{n/(β+1)}) in the worst case. This bound is tight, as we provide a matching upper bound for every β ≥ 0, which in turn implies a O(n√n)-size single-pair hierarchical preserver.
Finally, we present additional positive results among which a 4-additive all-pairs hierarchical spanner of size Õ(n^{5/3}), an (essentially tight) single-source hierarchical (1+ε)-spanner of size Õ(n/ε), an all-pairs hierarchical spanner of size Õ(n√{n/(ε)}) achieving stretch (1+ε,2), for any constant value of ε > 0, and a subsetwise hierarchical preserver of size O(n √{n|S|}), where S ⊆ V(𝒢).