### Table 2: Dissemination Communication Times on a d-dimensional Torus

1997

"... In PAGE 8: ... The generalization to d dimensions and the analyses of similar algorithms for scattering are straightforward. The results are summarized in Table2 at the end of the paper. 5 All-to-All Communication Patterns Gossiping is the communication pattern in which each node broadcasts a message to all other nodes in the network.... In PAGE 9: ... Using q = k, the total cost of the all-ports 2-phase algorithm is: n 2q + q 2 + n 2 + q 2(q ? 1) + 3n 2 L (5) The number of rounds in Eq. 5 is larger than the number of rounds used by the one-port algorithm in [6] (see Table2 at the end of this paper). The increase occurs in the rst phase because all-ports gossiping can use at most q virtual cycles (every link is used in every round).... In PAGE 10: ... Combining these costs gives the following total cost to gossip in a d-dimensional torus using a hybrid algorithm in which the simultaneous gossiping operations in Phase 1 of each stage use the one-port cycle algorithm and Phase 2 of each stage uses the all-ports cycle algorithm. A similar analysis gives the total cost (shown in Table2 ) when all-ports cycle algorithms are used in both phases. d n 4(q ? 1) + q ? 1 + d n 2 + (2q ? 3)(q ? 1) + 3n 2 + n 2(q ? 1) ? q N ? 1 d(n ? 1) L : (7) In the multi-scattering pattern each node sends a personalized message to each other node.... In PAGE 11: ...d?1 = Nn . The numbers of rounds and the switching costs increase by a factor of d. The results are shown in Table 2. 6 Analysis of Results Table2 summarizes the upper bounds that we established in Sections 4 and 5. Our lower bounds are summarized in Table 1 in Section 3.... ..."

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### Table 2: A d-dimensional geometric file of N points.

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### Table 2: A d-dimensional geometric file of N points.

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

### Table 3: Communication Times on d-dimensional Tori

### Table 1: Probability P of a D-dimensional hypercube of side 2 centred in the mode of a D-dimensional normal distribution.

### Table 1. The number of vectors in a d-dimensional space with Hamming weight D.

1997

"... In PAGE 5: ... For simplicity we assume that all the (i) vectors have Hamming weight one. The number of vectors in a d-dimensional space with Hamming weight D is given by D X k=1 d k ! : Table1 shows the numerical values for several choices of D and d. Table 1.... ..."

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### Table 1: Theorem 1 The left null space of Gn is exactly d-dimensional.

1994

"... In PAGE 10: ... However, estimates for sn can still be obtained by solving ^sn = arg min sn2 d [^sn?N+m; ; ^sn?1; sn]Gn 2 F (9) This minimization is the basis of our recursive symbol estimation algorithm which is summarized in Table 1. A procedure for initializing the algorithm is not speci ed in Table1 . To be completely \blind quot; in the sense that both the channel and the symbol sequences are unknown, a block algorithm such... In PAGE 11: ... This delay is required for forming Gn = [Gn; ; Gn+Q?1]. Steps 1 and 3 in the algorithm (see Table1 ) call for shifts. This just amounts to throwing away old data and adding the new.... In PAGE 13: ...load has been reduced from O(Jd) to O(dJ). To exploit the source separation property in the RBSE algorithm in Table1 , simply replace the d-dimensional enumeration in Step 4 with d 1-dimensional searches in (10). Next we show that the enumeration in (10) is unnecessary in some cases.... ..."

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### Table 4. Results for a d-dimensional Torus T(h1,h2,...,hd)

2006

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