Intuition. Try to use a cheap lower bounding calculation as often as possible. Only do the expensive, full calculations when it is absolutely necessary. Lower Bound of Xxx et. al. C A D B LB_Kim Xxx, S, Park, S, & Chu, W. An index-based approach for similarity search supporting time warping in large sequence databases. ICDE 01, pp 607-614 The squared difference between the two sequence’s first (A), last (D), minimum
Intuition. In the presence of a trusted hardware, any party receiving a valid message from the sender can attest to this message using the Append() call to their trusted hardware in a specified log and sequence number. Sending this attestation back to the sender guarantees both provability as well as termination against a corruption threshold of < 1/2. For provability, if delivery for every honest party requires attesting at a specific position in a log as a proof, then receiving n +1 attestations from a set of parties P is a sufficient proof to state that parties in P cannot deliver a different message. For termination, if the sender is honest and eventually the sender’s messages arrives at honest parties, then all honest parties will attest to this message in the correct log and sequence number and deliver m; the attestations sent back to the sender will allow it to deliver m with a proof σout consisting of all the attestations it received. However, the proof σout is not O(1) words. Observe that the attestations from the small trusted hardware from a linear number of parties provide us with O(n) signatures. Thus, the challenge is to ensure that the proof σout remains O(1) without relying on the hardware to generate threshold signatures. Our key idea is to verify the existence of n + 1 attestations by spreading this work evenly among the parties. We aim for two seemly opposing goals: On the one hand, each party needs to check just a constant number of attestations to be locally satisfied. On the other hand, if a majority of parties say they are locally satisfied then, even if t of them are lying then it is still the case that there were n + 1 attestations. We obtain this through the magic of expander graphs. Every party communicates their attestation with their neighbors in the network. The expansion properties of the graph ensure that receiving correct information from a small fraction of honest parties, specifically sn, suffices to learn the state about a majority of the parties in the network. In particular, on receiving some vote messages (containing a threshold signature share) from n + 1 parties (see lines 4–7 in Algorithm 2), out of which at least sn parties are honest, the sender can learn that a majority of parties (not necessarily honest) have attested to a message m in the log and sequence number corresponding to the instance, and thus cannot deliver a different message with a valid attestation. To ensure that the proof σout is O(1) words, the sender can ...
Intuition. We design three executions 𝑒0, 𝑒1, 𝑒2 so that and not too much to continue to ensure liveness as long as
Intuition. We design three executions 𝑒0, 𝑒1, 𝑒2 so that information to ensure accountability in case of disagreement
Intuition. The ABA algorithm’s accuracy improves with increasing accuracy probability of non-faulty processes and decreasing proportion of faulty processes. Theorem (Probabilistic Accuracy) The ABA algorithm guarantees accuracy with probability greater than 1 ( − e−δ (1−δ)(1−δ) )µ, where µ = p(1/2 + d ) and δ = (2d − r )/(2d + 1) if for each iteration: Accuracy probability of non-faulty processes > (1/2 + d ) and, r = Σweight of faulty processes (q) < 2d Σweight of non-faulty processes (p)
Intuition. The ABA algorithm is almost always accurate if there is at least one process that is almost always accurate [Kal 07].
Intuition. The ABA algorithm is almost always accurate if there is at least one process that is almost always accurate [Kal 07]. Theorem (At-Least-One Accuracy) The ABA algorithm is inaccurate at most 2(1 + ǫ)b + (2/ǫ) log n times if at least one process is inaccurate at most b times. Introduction Contributions ABA Problem ABA Algorithm ABA Accuracy Guarantees Updating weights (always or on inaccuracy) Optimal policies to update weights Introduction Contributions ABA Problem ABA Algorithm ABA Accuracy Guarantees
