### Table 1: Summary of the Communication costs and lower bounds

### Table 1: Lower Bounds on Communication Time on d-dimensional Tori

### Table 1: Summary of the communication complexity for one-shot and continuous threshold monitoring of different frequency moments. The randomized bounds are expected communication bounds for randomized algorithms with failure probability lt; 1=2.

"... In PAGE 3: ... The algorithm is a sophisticated variation of GLOBAL above, with multiple rounds, us- ing different sketch summaries at multiple levels of accuracy. Table1 summarizes our results. For comparison, we also include the one-shot costs: observe that for F0 and F1, the cost of continuous monitoring is no higher than the one- shot computation and close to the lower bounds; only for F2 is there a clear gap to address.... ..."

### Table 1. Lower bounds for the different components of communication cost.

2006

"... In PAGE 3: ... When the amount of data is small, the cost of initiating messages, AB, tends to dominate, and algorithms should strive to reduce this cost. In other words, the lower bound on the latency in Table1 be- comes the limiting factor. When the amount of data is large, the cost per item sent and/or computed, AC and/or AD, becomes the lim- iting factors.... In PAGE 3: ... When the amount of data is large, the cost per item sent and/or computed, AC and/or AD, becomes the lim- iting factors. In this case the lower bound due to bandwidth and/or computation in Table1 is the limiting factor. There are several classes of algorithms for collective communi- cation, but we concentrate on these two cases: short-vector algo- rithms and long-vector algorithms.... ..."

Cited by 3

### Table 1: Lower bounds for the different components of communication cost. Pay particular attention to the conditions for the lower bound, given in the text.

1999

Cited by 1

### Table 2: Lower bounds of termination delay (gossiping) in communication steps

1996

"... In PAGE 12: ... Therefore, we can use the gossiping results for edge-colored graphs to compare with the above lower bounds. In the following, if the gossiping time in some colored graph is optimal with respect to the corresponding lower bound in Table2 , then the termination delay due to our algorithm using the same coloring scheme is optimal or near optimal. Precisely, if we let g to be the gossiping time, then the termination delay is less than g + 2 communication steps.... In PAGE 14: ... By comparing Theorem 5.1 and Table2 , we nd that our termination detection algorithm is optimal for the chain and the even ring (d ? g is a constant 2 to 4 time steps).... In PAGE 15: ... Moreover, the edge-coloring technique used in the algorithm allows e cient communication while avoiding the possibility of message collisions and congestions. From Table2 , we note that the numbers of time steps for the optimal cases under the 1-port model is actually equal to the respective absolute lower bounds for information dissemination in these structures regardless of the communication model . Therefore, if we use the all-port communication model, our algorithm performs as well as the broadcast-based algorithm or any other algorithm in these structures.... ..."

Cited by 4

### Table 2: Lower bounds of termination delay (gossiping) in communication steps

1996

"... In PAGE 11: ... Therefore, we can use the gossiping results for edge-colored graphs to compare with the above lower bounds. In the following, if the gossiping time in some colored graph is optimal with respect to the corresponding lower bound in Table2 , then the termination delay due to our algorithm using the same coloring scheme is optimal or near optimal. Precisely, if we let g to be the gossiping time, then the termination delay is less than g + 2 communication steps.... In PAGE 13: ... By comparing Theorem 5.1 and Table2 , we nd that our termination detection algorithm is optimal for the chain and the even ring (d ? g is a constant 2 to 4 time steps).... In PAGE 14: ... Moreover, the edge-coloring technique used in the algorithm allows e cient communication while avoiding the possibility of message collisions and congestions. From Table2 , we note that the numbers of time steps for the optimal cases under the 1-port model is actually equal to the respective absolute lower bounds for information dissemination in these structures regardless of the communication model. Therefore, if we use the all-port communication model, our algorithm performs as well as the broadcast-based algorithm or any other algorithm in these structures.... ..."

Cited by 4