An optimal-time construction of sparse Euclidean spanners with tiny diameter

In STOC'95 [5] Arya et al. showed that for any set of <i>n</i> points in R^<i>d</i>, a (1 + ε)-spanner with diameter at most 2 (respectively 3) and <i>O</i>(<i>n</i> log <i>n</i>) edges (resp., <i>O</i>(<i>n</i> log log <i>n</i>) edges) can be built in <i>O</i>(<i>n</i> log <i>n</i>) time. Moreover, Arya et al. [5] conjectured that one can build in <i>O</i>(<i>n</i> log <i>n</i>) time a (1 + ε)-spanner with diameter at most 4 and <i>O</i>(<i>n</i> log* <i>n</i>) edges. Since then, this conjecture became a central open problem in this area. Nevertheless, very little progress on this problem was reported up to this date. In particular, the previous state-of-the-art subquadratic-time construction of (1 + ε)-spanners with <i>o</i>(<i>n</i> log log <i>n</i>) edges due to Arya et al. [5] produces spanners with diameter 8. In addition, general tradeoffs between the diameter and number of edges were established [5, 26]. Specifically it was shown in [5, 26] that for any <i>k</i> ≥ 4, one can build in <i>O</i>(<i>n</i>(log <i>n</i>)2^<i>k</i><i>α</i>_<i>k</i>(<i>n</i>)) time a (1 + ε)-spanner with diameter at most 2<i>k</i> and <i>O</i>(<i>n</i>2^<i>k</i><i>α</i>_<i>k</i>(<i>n</i>)) edges. The function α_<i>k</i> is the inverse of a certain Ackermann-style function at the ⌊<i>k</i>/2⌋th level of the primitive recursive hierarchy, where α₀(<i>n</i>) = ⌈<i>n</i>/2⌉, α₁(<i>n</i>) = ⌈√<i>n</i>⌉, α₂(<i>n</i>) = ⌈log <i>n</i>⌉, α₃(<i>n</i>) = ⌈log log <i>n</i>⌉, α₄(<i>n</i>) = log* <i>n</i>, α₅(<i>n</i>) = ⌈1/2 log* <i>n</i>⌉,..., etc. It is also known [26] that if one allows quadratic time then these bounds can be improved. Specifically, for any <i>k</i> ≥ 4, a (1 + ε)-spanner with diameter at most <i>k</i> and <i>O</i>(<i>nkα</i>_<i>k</i>(<i>n</i>)) edges can be constructed in <i>O</i>(<i>n</i>²) time [26]. A major open question in this area is whether one can construct within time <i>O</i>(<i>n</i> log <i>n + nkα</i>_<i>k</i>(<i>n</i>)) a (1 + ε)-spanner with diameter at most <i>k</i> and <i>O</i>(<i>nkα</i>_<i>k</i>(<i>n</i>)) edges. This question in the particular case of <i>k</i> = 4 coincides with the aforementioned conjecture of Arya et al. [5]. In this paper we answer this long-standing question in the affirmative. Moreover, in fact, we provide a stronger result. Specifically we show that for any <i>k</i> ≥ 4, a (1 + ε)-spanner with diameter at most <i>k</i> and <i>O</i>(<i>nα</i>_<i>k</i>(<i>n</i>)) edges can be built in optimal time <i>O</i>(<i>n</i> log <i>n</i>). In particular, our tradeoff for <i>k</i> = 4 provides an <i>O</i>(<i>n</i> log <i>n</i>)-time construction of (1 + ε)-spanners with diameter at most 4 and <i>O</i>(<i>n</i> log* <i>n</i>) edges, thus settling the conjecture of Arya et al. [5]. The tradeoff between the diameter and number of edges of our spanner construction is tight up to constant factors in the entire range of parameters, even if one allows the spanner to use (arbitrarily many) Steiner points.