### Table 1: Conditions for which an object with consensus num-

"... In PAGE 3: ... This implementation employs objects with consensus number C, where C P.AsC varies, the quantum required for this consensus implementation to work correctly is as given in Table1 . In the last part of Sec.... In PAGE 5: ...ybrid schedulers. In Sec. 4.1, we presentamultiproces- sor consensus algorithm based on C-consensus primitives that is correct provided the quantum Q is as speci ed in Table1 .... In PAGE 6: ...4.1 Multiprocessor Consensus In this subsection, we establish the bounds on Q given in Table1 for hybrid-scheduled multiprocessor systems. Speci cally,we presentawait-free consensus algorithm for anynumber of processes executing on P processors, where the processes on each processor are scheduled us- ing a hybrid scheduler.... In PAGE 8: ...2 Lower-Bound Results In this subsection, weshow that if C = P, then our char- acterization of the quantum required for universalityis is asymptotically tight. As explained below, with only a few modi cations, the proof strategy given here can be used to show that, if C-consensus objects must be hard-wired, as in the algorithm of the previous subsec- tion, then all of the bounds on Q given in Table1 are asymptotically tight. Due to space limitations, we only sketch the main ideas of the C = P proof here.... In PAGE 10: ...rgument can be continued, i.e., A is not wait-free. If C-consensus objects must be hard-wired, then a vir- tually identical proof shows that all of the bounds on Q given in Table1 are asymptotically tight. In this case, as wemove from state to state, wemaintain the invari- ant that P ; Q + 1 processes exist that have executed at least Q ; 1 statements since last being preemptable, and of the remaining processes, one has executed at least Q ; 2 statements since last being preemptable, another has executed at least Q ; 3 statements since last being preemptable, and so on.... ..."

### Table 1.1: Upper and lower bounds on consensus global decision times in various models (t lt; n/2).

"... In PAGE 80: ...Table1 , which might be closed.... ..."

### Table 1. Tight performance ratios for given k.

"... In PAGE 115: ...swap push C8 BECZBV D1CPDC BG BF BG BF BK BJ C8 CZBV D1CPDC BE A0 BE D1B7BD BE A0 BE D1B7BD UB = BE A0 BE D1B7BD LB = BGD1 BFD1B7BD C9BECZBV D1CPDC BDB7 D4 BH BE BDB7 D4 BH BE D4 BDBJB7BD BG C9CZBV D1CPDC BDB7 D4 BGD1A0BF BE BDB7 D4 BGD1A0BF BE UB = BE A0 BE D1B7BD LB = BF BE A0 AF CABECZBV D1CPDC LB = D4 D1CPDC C7C8CC LB = D2 A0 BD undefined CACZBV D1CPDC LB = D4 D1CPDC C7C8CC LB = D4 D1CPDC A0BD C7C8CC undefined Table1 : performance guarantees: BV C4CB D1CPDC BPC7C8 CC same machine, then the swap neighborhood is empty; therefore, we define the swap neighborhood as one that consists of all possible jumps and all possible swaps. As can be seen in Table 1, the jump and swap neighborhoods have no constant performance guarantee for C9CZBV D1CPDC .... In PAGE 115: ...swap push C8 BECZBV D1CPDC BG BF BG BF BK BJ C8 CZBV D1CPDC BE A0 BE D1B7BD BE A0 BE D1B7BD UB = BE A0 BE D1B7BD LB = BGD1 BFD1B7BD C9BECZBV D1CPDC BDB7 D4 BH BE BDB7 D4 BH BE D4 BDBJB7BD BG C9CZBV D1CPDC BDB7 D4 BGD1A0BF BE BDB7 D4 BGD1A0BF BE UB = BE A0 BE D1B7BD LB = BF BE A0 AF CABECZBV D1CPDC LB = D4 D1CPDC C7C8CC LB = D2 A0 BD undefined CACZBV D1CPDC LB = D4 D1CPDC C7C8CC LB = D4 D1CPDC A0BD C7C8CC undefined Table 1: performance guarantees: BV C4CB D1CPDC BPC7C8 CC same machine, then the swap neighborhood is empty; therefore, we define the swap neighborhood as one that consists of all possible jumps and all possible swaps. As can be seen in Table1 , the jump and swap neighborhoods have no constant performance guarantee for C9CZBV D1CPDC . Therefore, we introduce a push neighborhood, for which any local optimum is at most a factor BE A0 BE D1B7BD of optimal for C9CZBV D1CPDC .... In PAGE 115: ... When pushing all jobs on the critical machines is unsuccessful, we are in a push optimal solution. In Table1 the performance guarantees for the various local optima and scheduling problems are given. UB = AQ denotes that AQ is a performance guarantee and LB = AQ denotes that the performance guarantee cannot be less than AQ; AQ denotes that UB = LB = AQ.... In PAGE 121: ...Empty Out-tree To approximate solution 0,079 0,005 Tolower bound 0,115 0,318 Table1 : Average relative errors of approximate solution of algorithm based on y jt -formulation to approximate solution of algorithm based on x jt -formulation and lower bound ( = 1). The graph of precedence constraints Empty Out-tree To approximate solution 0,048 0,001 Tolower bound 0,073 0,309 Table 2: Average relative errors of approximate solution of algorithm based on y jt -formulation to approximate solution of algorithm based on x jt -formulation and lower bound ( =1= p 2).... ..."

### Table 16: Breakdown of Consensus Cases and Singular Decisions

2000

### Table 1: Locally Computed Tightest Bounds

1999

"... In PAGE 11: ...M N O Q R S T U P 3 2 1 0 strata 4 Figure 3: Directed Tree (B; !) of computing tightest bounds at a node (the common nodes for Chaining and Fusion are lled black). Table1 shows the greatest lower and the least upper bounds that are computed at each node B of each stratum. More precisely, these bounds are 1 = inf Pr(BD)=Pr(B), 2 = sup Pr(BD)=Pr(B), 2 = sup Pr(BD)=Pr(B), and 2 = sup Pr(D)=Pr(B) subject to Pr j= KB and Pr(B) gt; 0.... In PAGE 11: ... More precisely, these bounds are 1 = inf Pr(BD)=Pr(B), 2 = sup Pr(BD)=Pr(B), 2 = sup Pr(BD)=Pr(B), and 2 = sup Pr(D)=Pr(B) subject to Pr j= KB and Pr(B) gt; 0. Table1 also shows the requested tight answer fx1=0:02; x2=0:17g, which is given by the tightest bounds 1 and 2 that are computed at the premise M.... ..."

Cited by 40

### Table 1: Tight bounds on closeness of synchronization.

2001

Cited by 11

### Table 1: Tightness of Heuristic Register Bound

"... In PAGE 16: ... Notice that underestimations are often xed by the sequencing algorithm. All the results presented in Table1 (a) are for small DDGs with a low MRR for which the ILP method can nd a solution.... In PAGE 17: ...E ectiveness of Heuristic Methods in Larger DDGs How good is our heuristic solution for the 181 DDGs for which the ILP approach failed to nd an optimal solution? The rst column of Table1 (b) reports the number of DDGs for which an instruction sequence was found that used HRB registers. The table also shows the number of DDGs for which the instruction sequence requires more registers than HRB, the average register increase and the maximum register increase for each method.... ..."