### Table 6.12: Fraction of the rules which are necessary to replace at rules as IP ! PHI.N PHI.P PHI.VL PHI.N PHI.VR PM BREAK in such a manner that trees like in Figure 6.6 can be generated. The table shows only one sixth of the IP-rules those which have a PHI.N as their left-most child node. The same collection of rules exists for the other 5 PHI-categories PHI.A PHI.E PHI.VL PHI.VR, and PHI.P.

### Table 1: Upper bounds for OBDDs for functions representable by tree-like circuits.

1996

"... In PAGE 12: ... Proof: We have size(f; f) = apos;(size(f); size(f; f)) apos;(size(f)= size(f; f); 1)?1 apos;(1; 2) n apos;(0:5; 1)?1 = 2 n : The claim follows with the result of Theorem 4. 2 In Table1 we present concrete values of our upper bounds.... ..."

Cited by 2

### Table 1: Upper bounds for OBDDs for functions representable by tree-like circuits.

1996

"... In PAGE 12: ... Proof: We have size(f; f) = apos;(size(f); size(f; f)) apos;(size(f)= size(f; f); 1)?1 apos;(1; 2) n apos;(0:5; 1)?1 = 2 n : The claim follows with the result of Theorem 4. 2 In Table1 we present concrete values of our upper bounds.... ..."

Cited by 2

### Table 1 describes the equivalence between di erent types of tree-like circuits of polynomial degree and programs of polynomial length, and gives the relation with some important complexity classes.

"... In PAGE 8: ... Table1 : Relationship between di erent complexity classes, polynomial-length pro- grams and polynomial-degree tree-like circuits... In PAGE 9: ...be tempted to conjecture that the only inequality in Table1 can be replaced by an equality. However, things are not so clear, as the following discussion shows.... ..."

### Table 2. Duration and magnitude estimates of 15 dry and 15 wet spells.

"... In PAGE 6: ... It is character- ized by a slight dry period from through 1902, a wet spell through a drought from 1911-1913, the major drought of the 1930 apos;s, another downward trend in growth in the 1950 apos;s and yet another in 1988 which corresponds with the occurrence of several major forest fires in the Black Hills. Table2 shows the rank, estimated magnitude, and duration of the major dry and wet spells as suggested by the examination of this tree ring chronolo- gy. The magnitude of the droughts and wet spells are estimated by summing the consecutive negative and consecutive positive standardized index values, respectively, in the years shown.... ..."

### Table 1: Summary of results. The data structures are one-dimensional tree-like dictionaries and the update bounds are position given. Oa2a4a3 a5 , Oa2a6a3a7a5 and a8 Oa2a6a3a7a5 denote worst case, average and high probability bounds respectively, n is the number of servers, and d is the distance between guessed and actual position.

2000

"... In PAGE 2: ... Its logarithmic height, kept with highly local criteria without dependence on data distribution, makes this effort quite appealing. Table1 summarizes our results. In section 2 we briefly describe Skip Lists and an alternative way of viewing them.... ..."

Cited by 1

### Table 3: Reduction of computational effort of the proposed approach over the standard multiscale algorithm (measured via raw FLOP count). As expected, the benefit increases for more difficult problems: the reduction factor increases for larger trees and more finely sampled domains. The results are based on the tree-like prior of Figure 9.

"... In PAGE 14: ... Our research goal is the development of statistical methods for very large problems, so in this section we focus on computational issues. Table3 shows the improvement in speed of our proposed approach over the standard singly-rooted mul- tiscale algorithm[7, 22], when applied to the Markov random field problem of Figure 9. The extra states in- troduced by our multiply-rooted approach cannot be justified for extremely small or poorly-sampled problems (upper left of Table 3), however as the problem size and sampling density increase (lower right) the decompo- sition offered by the multiply-rooted approach becomes more competitive.... In PAGE 14: ... Table 3 shows the improvement in speed of our proposed approach over the standard singly-rooted mul- tiscale algorithm[7, 22], when applied to the Markov random field problem of Figure 9. The extra states in- troduced by our multiply-rooted approach cannot be justified for extremely small or poorly-sampled problems (upper left of Table3 ), however as the problem size and sampling density increase (lower right) the decompo- sition offered by the multiply-rooted approach becomes more competitive. For large, densely sampled trees, computational improvements in excess of a factor of twenty were observed.... ..."

### Table 3: Results using the Sequential ILP Approach At this stage the problem owner felt that the solutions obtained with the simple sequential approach were capable of improvement and, in particular, was con dent that the catfood problem was capable of solution with only two templates. As an attempt to test this assertion we used a form of the restricted global model in which the template population was generated by multiple runs of the sequential approach. At each stage, instead of using only the optimal template provided by the previous stage to derive the residual demand, we use each of the best (up to 10) templates found during the branch and bound search as a starting point. Each of these generates several more templates so that a population of templates is generated in a tree-like fashion. Given this population of templates we attempt to nd a solution using a 6

1998

Cited by 7