By Claude Brezinski
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ACM Transactions on Algorithms 2, 557–577 (2006) 13. : A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Structures and Algorithms 30, 532– 563 (2007) 14. : More algorithms for all-pairs shortest paths in weighted graphs. In: Proceedings of 39th Annual ACM Symposium on Theory of Computing, pp. 590– 598 (2007) 15. : Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing 28, 210–236 (1998) 16. : All-pairs small stretch paths.
5 Complexity of Exact Rounding No complexity bounds can be deduced using only Precondition A. To deduce bounds, we need to invoke the ε-discreteness conditions of Precondition B (even when our algorithms are based on METHOD A). If Δε (f (x), G) holds, the complexity of rounding f (x) in G is basically the time to compute f (x) ± ε/2. When f is an elementary function, Brent  tells us that the running time is O(M (log(1/ε)) log(1/ε)) where M (n) is the time to multiply two n-bit numbers. , it is only applicable when x lies in a bounded range .
Until the interval [y(n) ± 2−n ] = [y(n) − 2−n , y(n) + 2−n ] contains no grid points, [y(n) ± 2−n ] ∩ G = ∅. (3) We then output y(n) G . Correctness: By our assumption about the eﬀective grid G, we can check (3) since this amounts to checking that | y(n) G − y(n)| > 2−n and | y(n) G − y(n)| > 2−n . Furthermore, (3) ensures that y G = y(n) G . Finally, observe that (3) will eventually hold since y ∈ G. ¶3. METHOD B. In contrast to the precondition A, the ε-discreteness property Δε (y, G) does not exclude the possibility that y ∈ G.
Algorithmes d'acceleration de la convergence: etude numerique by Claude Brezinski