A spanner is a sparse subgraph that approximately preserves the pairwise distances of the original graph. It is well known that there is a smooth tradeoff between the sparsity of a spanner and the quality of its approximation, so long as distance error is measured multiplicatively . A central open question in the field is to prove or disprove whether such a tradeoff exists also in the regime of additive error. That is, is it true that for all ε > 0, there is a constant k ε such that every graph has a spanner on O ( n 1+ε ) edges that preserves its pairwise distances up to + k ε ? Previous lower bounds are consistent with a positive resolution to this question, while previous upper bounds exhibit the beginning of a tradeoff curve: All graphs have +2 spanners on O ( n 3/2 ) edges, +4 spanners on Õ ( n 7/5 ) edges, and +6 spanners on O ( n 4/3 ) edges. However, progress has mysteriously halted at the n 4/3 bound, and despite significant effort from the community, the question has remained open for all 0 < ε < 1/3. Our main result is a surprising negative resolution of the open question, even in a highly generalized setting. We show a new information theoretic incompressibility bound: There is no function that compresses graphs into O ( n 4/3 − ε ) bits so distance information can be recovered within + n o(1) error. As a special case of our theorem, we get a tight lower bound on the sparsity of additive spanners: the +6 spanner on O ( n 4/3 ) edges cannot be improved in the exponent, even if any subpolynomial amount of additive error is allowed. Our theorem implies new lower bounds for related objects as well; for example, the 20-year-old +4 emulator on O ( n 4/3 ) edges also cannot be improved in the exponent unless the error allowance is polynomial. Central to our construction is a new type of graph product, which we call the Obstacle Product . Intuitively, it takes two graphs G , H and produces a new graph G ⊗ H whose shortest paths structure looks locally like H but globally like G .

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A k-spanner of a graph G is a sparse subgraph H whose shortest path distances match those of G up to a multiplicative error k. In this paper we study spanners that are resistant to faults. A subgraph H ⊆ G is an f vertex fault tolerant (VFT) k-spanner if H \ F is a k-spanner of G \ F for any small set F of f vertices that might "fail." One of the main questions in the area is: what is the minimum size of an f fault tolerant k-spanner that holds for all n node graphs (as a function of f , k and n)? In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor k is fixed. Specifically, we prove that every (undirected, possibly weighted) n-node graph G has a (2k − 1)-spanner resilient to f vertex faults with O k (f 1−1/k n 1+1/k ) edges, and this is fully optimal (unless the famous Erdös Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating dist G\F (s, t) similarly can beat the space usage of our spanner in the worst case. To the best of our knowledge, this is the first instance in fault tolerant network design in which introducing fault tolerance to the structure increases the size of the (non-FT) structure by a sublinear factor in f . Another advantage of this result is that our spanners are constructed by a very natural and simple greedy algorithm, which is the obvious extension of the standard * Supported in part by NSF awards 1464239 and 1535887. Supported in part by NSF grants CCF-1417238, CCF-1528078 and CCF-1514339, and BSF grant BSF:2012338. greedy algorithm used to build spanners in the non-faulty setting.We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case k = 2 (and hence we close the EFT problem for 3-approximations), but it falls to Ω(f 1/2−1/(2k) · n 1+1/k ) for k ≥ 3. We leave it as an open problem to close this gap.

Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, sayÕ(n 1+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 ≥

We give a short and easy upper bound on the worst-case size of fault tolerant spanners, which improves on all prior work and is fully optimal at least in the setting of vertex faults. *