Distributed algorithms for ultrasparse spanners and linear size skeletons
Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing
On the locality of distributed sparse spanner construction
Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Fault-tolerant spanners for general graphs
Proceedings of the forty-first annual ACM symposium on Theory of computing
ACM Transactions on Algorithms (TALG)
Local computation of nearly additive spanners
DISC'09 Proceedings of the 23rd international conference on Distributed computing
Additive spanners and (α, β)-spanners
ACM Transactions on Algorithms (TALG)
Additive spanners in nearly quadratic time
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners
ACM Transactions on Algorithms (TALG)
Fault Tolerant Spanners for General Graphs
SIAM Journal on Computing
Transitive-closure spanners: a survey
Property testing
Transitive-closure spanners: a survey
Property testing
Sparse spanners vs. compact routing
Proceedings of the twenty-third annual ACM symposium on Parallelism in algorithms and architectures
Multiplicative approximations of random walk transition probabilities
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
On approximate distance labels and routing schemes with affine stretch
DISC'11 Proceedings of the 25th international conference on Distributed computing
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Multipath spanners via fault-tolerant spanners
MedAlg'12 Proceedings of the First Mediterranean conference on Design and Analysis of Algorithms
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An additive spanner of an unweighted undirected graph G with distortion d is a subgraph H such that for any two vertices u, v \in G, we have \delta _H \left( {u,v} \right) = \delta _G \left( {u,v} \right) + d. For every k = O\left( {\frac{{\ln n}} {{\ln \ln n}}} \right), we construct a graph G on n vertices for which any additive spanner of G with distortion 2k - 1 has \\Omega \left( {\frac{1} {k}n^{1 + 1/k} } \right) edges. This matches the lower bound previously known only to hold under a 1963 conjecture of Erdos. We generalize our lower bound in a number of ways. First, we consider graph emulators introduced by Dor, Halperin, and Zwick (FOCS, 1996), where an emulator of an unweighted undirected graph G with distortion d is like an additive spanner except H may be an arbitrary weighted graph such that\delta _G \left( {u,v} \right)\leqslant\delta _H \left( {u,v} \right) \leqslant \delta _G \left( {u,v} \right) + d. We show a lower bound of \Omega \left( {\frac{1} {{k^2 }}n^{1 + 1/k} } \right) edges for distortion-(2k - 1) emulators. These are the first non-trivial bounds for k \ge 3. Second, we parameterize our bounds in terms of the minimum degree of the graph. Namely, for minimum degree n^{1/k+c} for any c \geqslant0, we prove a bound of \Omega \left( {\frac{1} {k}n^{1 + 1/k - c(1 + 2/(k - 1))} } \right) for additive spanners and \Omega \left( {\frac{1} {{k^2 }}n^{1 + 1/k - c(1 + 2/(k - 1))} } \right) for emulators. For k = 2 these can be improved to \Omega \left( {n^{3/2 - c} } \right). This partially answers a question of Baswana et al (SODA, 2005) for additive spanners. Finally, we continue the study of pair-wise and source-wise distance preservers defined by Coppersmith and Elkin (SODA, 2005) by considering their approximate variants and their relaxation to emulators. We prove the first lower bounds for such graphs.