### Table 5.1: Composite elds GF ((2n)m) up to GF (232), their generating polynomials and the space complexities for parallel multipliers

in i Preface

### Table 1: Composite elds GF ((2n)m) up to nm = 32, primitive eld polynomials, and the space complexities and theoretical delays of parallel multipliers

1996

"... In PAGE 11: ...Table 1: Composite elds GF ((2n)m) up to nm = 32, primitive eld polynomials, and the space complexities and theoretical delays of parallel multipliers 5 Results Table1 gives insight in the complexities and architectures of parallel multipliers in composite elds GF (2k) k = 2; 4; : : : ; 32. For each eld an optimized eld polynomial P (x) and a multiplier with a minimum complexity is given.... ..."

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### Table 1 gives an overview of the state of a airs w. r. t. eight complexity classes named - - for 2 fI; IOg, 2 fFPT; Pg, and 2 fT; Sg. Component indi- cates whether the complexity is related to input (I) or input and output OBDDs (IO); separates xed-parameter (FPT) from polynomial (P) complexities; separates time (T) from space (S) complexity. The classes are related as follows:

2006

"... In PAGE 5: ...MaxFlow no y no no ? yes no y no no ? yes APSP no y no no ? yes no y no no ? yes PW-APSP no y yes y no ? yes y no y ? no ? yes SSSP no y no no ? yes no y no no ? yes PW-SSSP no y ? no ? yes no y ? no ? yes Reachability no no no ? yes no no no ? yes TransClos no no no ? yes no no no ? yes MST no no ? yes no no ? yes s{t-Conn. no y - yes - no y - yes - Connected no y - yes - no y - yes - Bipartite no y - yes - no y - yes - Acyclic no y - yes - no y - yes - Euler Cycle no y - yes - no y - yes - Table1 . The complexity of graph problems on OBDD-represented inputs, unless P=PSPACE.... ..."

### Table 1: The refinement operator for ordering the space of polynomial equations.

2004

"... In PAGE 4: ... The refinement operator increases the complexity of the equation by 1, either by adding a new linear term or by adding a variable to an existing term. First, we can add an arbitrary linear (first degree) term (that is a single variable from V \{vd}) to the current equation as presented in the first (upper) part of Table1 . Special care is taken that the newly introduced term is different from all the terms in the current equation.... In PAGE 5: ...135 Figure 1: The search space of polynomial equations over the set of variables V = {x, y, z}, where z is the dependent variable, as ordered by the refinement operator from Table1 . Note that for simplicity, real-valued constants are omitted from the equations.... In PAGE 6: ... The value of the constant parameter const is fitted against the training data D using linear regression. In each search iteration, the refinements of the equations in the current beam are generated (using the refinement operator from Table1 ) and collected in Qr (line 5). In case when redundant equations are generated due to the sub-optimality of the refinement operator, the duplicate equations are filtered out from the set Qr (each refined equation structure is compared to the equations from the current version of Qr: if it is already included in Qr, we skip it and proceed with the next refinement).... ..."

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### Table 4: Characteristics of polynomial systems in benchmark: dimension n, total degree D, best known B ezout bound B and mixed volume MV . The last two columns contain the total number of nite isolated solutions, respectively in complex and in real space.

1998

"... In PAGE 12: ... 7.1 An overview on the benchmark In Table4 we give an overview of our benchmark. To save space, we omit the algebraic description of the problems (available from the web site at [22]) and provide only the refer-... In PAGE 13: ... The last two columns contain the total number of nite isolated solutions, respectively in complex and in real space. The root counts D, B and MV listed in Table4 give an idea about the intrinsic cost of the respective homotopy continuation methods based on these root counts. The gap between the actual number of nite solutions illustrates their (sometimes poor) performance.... ..."

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### Table 1. The re nement operator for ordering the space of polynomial equations.

2004

"... In PAGE 3: ... The re nement operator increases the complexity of the equation by 1, either by adding a new linear term or by adding a variable to an existing term. First, we can add an arbitrary linear ( rst degree) term (that is a single variable from V n fvdg) to the current equation as presented in the rst (upper) part of Table1 . Special care is taken that the newly introduced term is di erent from all the terms in the current equation.... In PAGE 5: ... The value of the constant parameter const is tted against the training data D using linear regression. In each search iteration, the re nements of the equations in the current beam are generated (using the re nement operator from Table1 ) and collected in Qr (line 5). In case when redundant equations are generated due to the sub- optimality of the re nement operator, the duplicate equations are ltered out from the set Qr.... ..."

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### Table 1: Characteristic (columns - CP) and minimal (rows - MP) polynomial types corre- sponding to the Segre types of Rab in 5-D Lorentzian spaces. polynomial, while the associated characteristic polynomials are, respectively, of types f32g and f41g. We also remark that the Segre types [2(111)] and [(21)(11)] have the same type for both polynomials, namely f32g and k21k.

"... In PAGE 7: ... We also remark that the Segre types [2(111)] and [(21)(11)] have the same type for both polynomials, namely f32g and k21k. Table1 collects together the characteristic (columns - CP) and minimal polynomial (rows - MP) types corresponding to the possible Segre types of a symmetric two-tensor in 5-D Lorentzian spaces. It should be noticed that the characteristic polynomial for the complex Segre types [z z 111], [z z 1(11)] and [z z (111)] have been denoted, respectively, by fz z111g, f21z zg and f3z zg.... ..."

### Table 1: Asymptotic complexity for matrix construction method time space

1997

"... In PAGE 1: ... Yet, other polynomial multiplication methods, such as Karatsuba apos;s, may o er simpler though asymptotically slower alternatives; the latter may be advantageous in cer- tain circumstances, as discussed in section 8. Table1 com- pares the existing and the achieved complexities, in terms of matrix row and column dimension, respectively denoted a and c and the number of variables n, as explained in sec- tion 6. Note that a gt; c and typically a; c n.... ..."

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### Table 1. Asymptotic complexity for matrix construction method time space

2002

"... In PAGE 2: ... Yet, for smaller input sizes, other polynomial multiplication methods, such as Karatsuba apos;s, may o er simpler though asymptotically slower alternatives. Table1 compares the existing and the achieved complexities, in terms of row and column dimension, respectively denoted a and c, and the number of variables n, as explained in section 6. Note that a gt; c and typically c n.... ..."

Cited by 7

### Table 1: Space Complexity

2001

"... In PAGE 4: ... This will become clear in Section 3 where algorith- mic routing is explained in detail. Table1 summarizes the space complexity analysis. Computational complexity for flat routing, which adopts the Dijkstra algorithm, is C7B4C6BFB5.... ..."

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