### Table 3. Lower and upper bounds on the bisection width of De Bruijn graphs. Exact bounds on the bisection width of De Bruijn graphs are not known. As can be seen from Table 3, the gap between upper and lower bounds is still very large. The lower bound uses the technique of embedding a complete graph that was already described. Because the edge congestion is not uniform, this bound is weak. Some upper bound can be found using the complex plane embedding of the De Bruijn graph that de nes an O(N= log N) bisection [26]. Optimizing the embedding in terms of edge congestion can improve the lower bound. The column LoB in Table 3 gives the lower bound obtained with standard shortest path routing (note that an embedding of a complete graph de nes a routing scheme). LoB(SA) is an improved bound derived from optimized shortest path routing using simulated annealing [10] and UpB is an upper bound found with the bisection heuris- tic from [11]. As can be seen, there is still a large gap between the lower and upper bounds. This results mainly from the weakness of the lower bound. The upper bounds are not very

1994

Cited by 11

### Table 2: Comparison between lower bounds on shortest paths through nodes in closed list

2004

"... In PAGE 22: ... Given that this is merely a lower bound, PHA* is expected to perform better in practice. Table2 provides a comparison between PHA* and the MST lower bound on the shortest path as described above. The average cost entered for each graph size was computed over 250 randomly generated instances.... ..."

Cited by 8

### Table 2. Typical functional forms for the lower bounds in Table 1

2004

"... In PAGE 11: ...Table2... In PAGE 11: ...Table 2. Typical functional forms for the lower bounds in Table 1 It can be easily shown (using MATHEMATICATM for instance) that if we substitute the lower bounds in Table2 for T and S in the corresponding time-space tradeoff expressions given in Table 1, they vanish asymptotically as n ! 1, thereby verifying the results in Table 2. Upper bounds from known efficient algorithms.... ..."

### Table 1: Lower bounds

1997

"... In PAGE 4: ... a b c d e f g h k j i C D E F G H I J K L M N O P Q R S B A l m n Figure 5: Sokoban problem 4 only one way to get to the left of the stone { by backing it out and then back into the room. Table1 shows the e ectiveness of our lower bound estimate. The table shows the lower bound achieved by minimum matching, inclusion of the linear con icts enhancement, inclusion of the backout enhancement, and the combination of all three features.... ..."

Cited by 6

### Table 1: Shortest Path Problems Classification

2005

Cited by 3

### Table 2. Shortest Path Problem Connection matrix

2007

Cited by 1

### Table 9 Shortest Path Problem Connection matrix

### Table 1: Results with the shortest paths, and 2- and 3- multi- path routes together with the respective single-path routes.

"... In PAGE 5: ...maximumfluxof0:3763 , i.e., the flux corresponding to the outer radial-ring paths at the boundary. The numerical results are given in Table1 , where rows indicated with 1) and 2) correspond to the optimal weights for randomized path selection with the given two and three path sets, respectively, and column multi-path con- tains the corresponding maximum scalar fluxes. However, according to Proposition 1, multi-path routes mp1 and mp2 cannot be an optimal solution to the load balancing problem, and, in particular, the corresponding single-path routes, denoted by sp1 and sp2, obtained using (14) yield a lower maximum scalar packet flux.... In PAGE 5: ... However, according to Proposition 1, multi-path routes mp1 and mp2 cannot be an optimal solution to the load balancing problem, and, in particular, the corresponding single-path routes, denoted by sp1 and sp2, obtained using (14) yield a lower maximum scalar packet flux. This maximum scalar flux can be computed numer- ically and the corresponding results are given in column single- path in Table1 . We note that in both cases combining the multi- path traffic flows to single-path improves the situation considerably, as expected.... In PAGE 6: ...4 0.5 sp1 sp3 mod (r) r modified sp1 sp3 circular (2 path sets) (3 path sets) Figure 5: Resulting scalar flux as a function of distance r from the origin for modified circular paths (see [12]), and the optimal single-path routes sp1 and sp3 (rows 1) and 3) in Table1 ). Three dimensional plots illustrate the same situation.... ..."

### Table 1: Lower bounds Type of Protocol Time Bits Time Bits

"... In PAGE 4: ... They have considered three types of protocols: arbitrary, balanced and oblivious and have shown lower bounds on time-message trade-o , as well as an upper bound of O(N log2 N) for the oblivious case. The lower bound results from [5] are summarized in Table1 . The gaps between the O(N log2 N) upper bound and the lower bounds of (N loglogN) and (N logN) for arbitrary and balanced protocols, respectively, have been stated as open problems.... In PAGE 5: ... The results are summarized in Table 2. (Here, as well as in Table1 the results are obtained for a xed parameter t related to the time complexity.) The rest of this paper is organized as follows.... ..."

### Table 1: The lower bound example.

"... In PAGE 3: ... We have numer- ically calculated the length of this curve (as a discrete path) for different values of n. The numerical results, presented in Table1 , shows that for large values of n, Figure 4: A lower bound example and an interesting... ..."