THE ROUND COMPLEXITY OF l - S 0 LVABLE TASKS

A distributed task T is I-solvable if there exists a protocol that solves it in the presence of (at most) one crash failure. A precise characterization of the I-solvable tasks was given in [BMZ]. In this paper we detennine the number of rounds of communication that are required, in the worst case, by a protocol which I-solves a given I-solvable task T for n processors. We define the radius R(T) of T, and show that if R (T) is finite, then this number is 8(logn R (T) ); more precisely, we give a lower bound of logn(n_l)R (T), and an upper bound of 2+rlog(n-l)R (T~. The upper bound implies, for example, that each of the following tasks: renaming, order preserving renaming ([ABDKPRJ) and binary monotone consensus [BMZ] can be solved in the presence of one fault in 3 rounds of communications. All previ­ ous protocols that I-solved these tasks required Q(n) rounds. The result is also generalized to tasks whose radii are not bounded, e.g., the approximate consensus and its variants [DLPSW , BMZ].