### 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 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.... ..."

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### 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.... ..."

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### 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 1: Some FBDD{sizes w.r.t. various tree{like FBDD{types compared with OBDD{sizes w.r.t. the ordering heuristic of [MWBS88] of some ISCAS85 and IS- CAS89 benchmarks. Abbreviations: f .. .. = fs..mtr..-f-fh; f apos; .. .. = fs..mtr..-f-SISh; d .. .. = ds..mtr..-f-fh; d apos; .. .. = ds..mtr..-f-SISh; W = wps-f-wpc; w = wps-wpc-f; s .. .. = ss..mtr..-f-fh; s apos; .. .. = ss..mtr..-f-SISh; t = ts..-f-fh; (:::) = Output ::: only.

"... In PAGE 8: ... Other abbreviations we currently use are fs ( at split), ds (depth split), wps (weight propagation split) and ts (tilt split) for split heuristics and wpc (weight propagation continue) and SISh (SIS-heuristic) for OBDD heuristics. The current list of improvements observed per circuit is given in Table1 . An OBDD for c2670 could not be computed due to restricted memory size.... ..."

### Table 2. Subclasses of Reduction and Scan

1997

"... In PAGE 6: ....3.3. Subclasses of Reduction and Scan. Functions in the classes Reduction and Scan have a local and a global in- put. Thus, the function g which determines the subclassifi- cation is exactly the binary function preS: g (a; b) = preS (a; b) Table2 lists different forms of function g with their cor- responding subclasses of Reduction and Scan. 3.... In PAGE 6: ...1. Reduction For all functions in the subclasses of Reduction which are listed in Table2 , we use a tree-like processor network with n processors which computes the result with time(n) = O(log n), costBrent(n) = O(n) and pipe(n) = O(1). The distributed input (as in Figure 1) determines the val- ues at the leaves; at each inner node, a function computes the value from the values received from the children; the result... ..."

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### 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.... ..."

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