### 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.... ..."

Cited by 2

### TABLE IV Antenna types and the corresponding minimum-energy broadcast heuristic algorithms. p is the number of antennas per node and n is the number of nodes in the network.

2006

Cited by 1

### TABLE I A TAXONOMY OF MINIMUM-ENERGY BROADCAST HEURISTIC ALGORITHMS

### Table 1: Classification of the costs of implementing minimum energy routing protocol features

2002

"... In PAGE 8: ... AF Radio hardware modification cost: This reflects the cost of the changes that have to be made in the wireless Ethernet card hardware in order to support the feature. Table1 shows the classification of the costs for each of the features. The link energy cost computation fea- ture needs changes only in the routing header to in- clude the transmit power information of the packet so that the energy cost of the link can be calculated at the receiver node.... ..."

Cited by 40

### 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.... ..."

Cited by 5

### 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.... ..."

Cited by 1

### Table 3. The correlation coefficients between principal axes of embedding in D-dimensional space

"... In PAGE 7: ... We apply this to n = 2. In Table3 , nDX denotes the X-th principal axis for the n-dimensional embedding. As can be seen from the table, we may conclude that 2D is enough to capture the main features in the data.... ..."

### Table 1. The Isomap algorithm takes as input the distances dX(i,j) between all pairs i,j from N data points in the high-dimensional input space X, measured either in the standard Euclidean metric (as in Fig. 1A) or in some domain-speci c metric (as in Fig. 1B). The algorithm outputs coordinate vectors yi in a d-dimensional Euclidean space Y that (according to Eq. 1) best represent the intrinsic geometry of the data. The only free parameter (e or K) appears in Step 1.

"... In PAGE 2: ... These approxima- tions are computed efficiently by finding shortest paths in a graph with edges connect- ing neighboring data points. The complete isometric feature mapping, or Isomap, algorithm has three steps, which are detailed in Table1 . The first step deter- mines which points are neighbors on the manifold M, based on the distances dX(i,j) between pairs of points i,j in the input space X.... ..."

### Table 1. The Isomap algorithm takes as input the distances dX(i,j) between all pairs i,j from N data points in the high-dimensional input space X, measured either in the standard Euclidean metric (as in Fig. 1A) or in some domain-speci c metric (as in Fig. 1B). The algorithm outputs coordinate vectors yi in a d-dimensional Euclidean space Y that (according to Eq. 1) best represent the intrinsic geometry of the data. The only free parameter (e or K) appears in Step 1.

"... In PAGE 2: ... These approxima- tions are computed efficiently by finding shortest paths in a graph with edges connect- ing neighboring data points. The complete isometric feature mapping, or Isomap, algorithm has three steps, which are detailed in Table1 . The first step deter- mines which points are neighbors on the manifold M, based on the distances dX(i,j) between pairs of points i,j in the input space X.... ..."