By Burkhard Monien, Karsten Tiemann (auth.), Andrzej Pelc (eds.)
Read Online or Download Distributed Computing: 21st International Symposium, DISC 2007, Lemesos, Cyprus, September 24-26, 2007. Proceedings PDF
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Extra info for Distributed Computing: 21st International Symposium, DISC 2007, Lemesos, Cyprus, September 24-26, 2007. Proceedings
That would reduce the overall storage overhead to O(log n) in Theorem 1, but for simplicity we will just assume that copies of d are stored. Notice that as long as the set of DoS-attacked nodes is static, our storage strategy could be transformed into an eﬃcient insert protocol together with techniques in  to avoid congestion problems. However, for a dynamically changing set of DoS-attacked nodes this is tricky since some non-DoS-attacked nodes are now missing information that is necessary for our lookup protocol to work correctly.
Theorem 1 shows that the process converges before O(n log n) interactions with high probability. Suppose the process converges at some time τ = O(n log n). Then √ | = O( n log n) throughout this interval by Azuma’s inequality we have that |uτ − u0√ with high probability. So if u0 = u0 = ω( n log n), it follows that u0 ≥ u0 ≥ 0 throughout the execution, and in particular that the process does not terminate before convergence and that u is non-negative at convergence. But this excludes the y = n case, so the process converges to the initial majority value.
So we must also accept a small probability that we reach the blank configuration quickly, and the assurance that we reach it with probability 1 after a very long time. However, we can show that with high probability neither outcome occurs within a polynomial number of steps. Our technique is to adjust the potential functions used by the non-Byzantine process to account for Byzantine transitions. We then use these adjusted potential functions to show that (a) strong pressure exists to keep the process out of the high-b corner and in the high-x and high-y corners, and (b) the number of interactions (including Byzantine interactions) to reach the x or y corner is still small.