A Bound on the Rounds to Reach Lattice Agreement

The lattice agreement decision problem is studied in the synchronous message passing model of distributed computation subject to crash failures Processors p p pn start with input values X X Xn respectively drawn from a lattice L the size of a maximal chain of elements of L that can be de ned start ing with fX X Xng as the joins of other elements is denoted joinheight L fX X Xng Each non faulty processor chooses a value greater than or equal to its original value and less than or equal to the join of the original values more over the chosen values must be pairwise comparable Thus lattice agreement is a weakening of traditional consensus Early stopping algorithms for the stronger consensus problem are known to re quire f rounds of communication for any execution in which f n processors crash We present an early stopping algorithm for lattice agreement whose perfor mance is superior to early stopping algorithms for consensus More speci cally each nonfaulty processor decides within minf joinheight L fX X Xng b p f cg rounds for any execution of the algorithm in which f n processors crash In particular this algorithm distinguishes itself from a comparable algorithm of Attiya et al that requires lgn communication rounds in every execution