Given an undirected graph @math with @math edges, @math vertices, and non-negative edge weights, and given an integer @math , we show that for some universal constant @math , a @math -approximate distance oracle for @math of size @math can be constructed in @math time and can answer queries in @math time. We also give an oracle which is faster for smaller @math . Our results break the quadratic preprocessing time bound of Baswana and Kavitha for all @math and improve the @math time bound of Thorup and Zwick except for very sparse graphs and small @math . When @math and @math , our oracle is optimal w.r.t. both stretch, size, preprocessing time, and query time, assuming a widely believed girth conjecture by Erd o s.
A problem related to distance oracles is that of finding spanners. We have already mentioned the linear-time algorithm of Baswana and Sen @cite_0 to find a spanner of stretch @math . There has also been interest in so-called @math -spanners, where @math and @math are real numbers. Such a spanner @math of a graph @math ensures that for all vertices @math and @math , @math . In other words, @math allows an stretch in addition to a multiplicative stretch. Thorup and Zwick @cite_3 showed the existence of @math -spanners of size @math for any constant @math , where @math for some constant @math . A @math -spanner of size @math was presented by Dor, Halperin, and Zwick @cite_9 . The size was later improved slightly by Elkin and Peleg to @math @cite_10 . Baswana, Kavitha, Mehlhorn, and Pettie @cite_14 gave a spanner of size @math which has additive stretch @math and no multiplicative stretch. This is currently the smallest known spanner with constant additive stretch and no multiplicative stretch.
@cite_14 @cite_9 @cite_3 @cite_0 @cite_10
2508647357 2809995963 2951859740 1564010364
Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say O(n1+Δ) bits. There is an inherent tradeoff between the sparsity parameter Δ and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when Δ ≥ 1 3 there are schemes with constant additive stretch (distance d is stretched to at most f(d) = d + O(1)), and recent results of Abboud and Bodwin show that when Δ In this paper we show that the lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that characterize the asymptotic behavior of the optimal stretch function f for sparsity parameter Δ ∈ (0,1 3). Specifically, for any integer k ≥ 2, any compression scheme with size [EQUATION] has a sublinear additive stretch function f: f(d) = d + Ω(d1−1 k). This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's (1 + ϵ, β)-spanners have an essentially optimal tradeoff between Δ, ϵ, and β, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of (1 + ϵ, O(k ϵ)k−1)-spanners with size [EQUATION], where hk Our lower bound technique exhibits several interesting degrees of freedom in the framework of Abboud and Bodwin. By carefully exploiting these freedoms, we are able to obtain lower bounds for several related combinatorial objects. We get lower bounds on the size of (β, ϵ)-hopsets, matching Elkin and Neiman's construction (2016), and lower bounds on shortcutting sets for digraphs that preserve the transitive closure. Our lower bound simplifies Hesse's (2003) refutation of Thorup's conjecture (1992), which stated that adding a linear number of shortcuts suffices to reduce the diameter to polylogarithmic. Finally, we show matching upper and lower bounds for graph compression schemes that work for graph metrics with girth at least 2γ + 1. One consequence is that 's (2010) additive O(γ)-spanners with size [EQUATION] cannot be improved in the exponent. An (α,β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v, H) ≤ α·dist(u, v, G) + β for every pair (u, v) ∈ V ×V, where dist(u,v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1,0)-spanner of size O(n 1 + 1 k ). This size-stretch bound is essentially optimal by the girth conjecture. Yet, it is important to note that any argument based on the girth only applies to adjacent vertices. It is therefore intriguing to ask if one can “bypass” the conjecture by settling for a multiplicative stretch of 2k − 1 only for neighboring vertex pairs, while maintaining a strictly better multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of k-hybrid spanners, in which non neighboring vertex pairs enjoy a multiplicative k stretch and the neighboring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted n-vertex graph G, there is a (polynomially constructible) k-hybrid spanner with O(k 2 ·n 1 + 1 k ) edges. This should be compared against the current best (α,β) spanner construction of [5] that obtains (k,k − 1) stretch with O(k ·n 1 + 1 k ) edges. An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs S ×V for a given subset of vertices S ⊆ V referred to here as sources. Spanners in which the distances in S ×V are bounded are referred to as sourcewise spanners. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more). Given an undirected @math -node unweighted graph @math , a spanner with stretch function @math is a subgraph @math such that, if two nodes are at distance @math in @math , then they are at distance at most @math in @math . Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the @math - @math distance only for pairs @math in a given set @math . Such @math -spanners were studied before [Coppersmith,Elkin'05] only in the special case that @math is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same @math ) and of the best known spanners (with the same @math ). In more detail, for arbitrary @math , we show that there exists a @math -spanner of size @math with @math . Alternatively, for any @math , there exists a @math -spanner of size @math with @math . We also consider the relevant special case that there is a critical set of nodes @math , and we wish to approximate either the distances within nodes in @math or from nodes in @math to any other node. We show that there exists an @math -spanner of size @math with @math , and an @math -spanner of size @math with @math . All the mentioned pairwise spanners can be constructed in polynomial time. Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn @math ) expected time and construct a data structure (a (2k–1)-approximate distance oracle) of size O(kn @math ) capable of returning in O(k) time an approximation @math of the distance δ(u,v) from u to v in G that satisfies @math , for any two vertices u,v∈ V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn @math log n). We present here a deterministic O(kmn @math ) time algorithm for constructing oracles of size O(kn @math ). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor. Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn @math ) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.