### Table 1: Lower bounds for randomized algorithms.

"... In PAGE 8: ... For n = 15 and s = 1 + i=100 for i = 1; : : : ; 99, we have calculated the optimal values of this linear program by using CPLEX. We display the actual lower bounds in Table1 . The value c(i) is a lower bound for any randomized online algorithm at speed s = 1 + i=100.... ..."

### Table 5. Evaluation of the MARS Algorithm based on 12 nouns, 1 verb, 1 adjective in LDOCE. Word Pos #Senses #Done #Correct Prec

### Table 3: Comparison of the Lower Bounds on the Competitiveness

"... In PAGE 6: ... Whereas Gormley presents an adversary strategy and a formal algorithm to determine the lower bounds of any online request answer games that be represented as linear programs. Table3 presents the comparison of the results of the lower bound analysis. Table 3: Comparison of the Lower Bounds on the Competitiveness ... ..."

### Table 4 Lower bound L, upper bound U, approximation A, and tree size T for random permutations on n elements.

"... In PAGE 25: ... In the third set of experiments we studied the quality of the approximation algo- rithm and the upper bound algorithm on random permutations. Results are given in Table4 , where dev denotes the standard deviation. Note that the maximum di erence between the upper and lower bound, which is what limits the range of optimal solu- tion, was at most 2 reversals for n up to 30, while the average di erence between the bounds was around 2.... ..."

### Table 1: Hardness versus randomness trade-o s for AM. If the hardness condition on the left-hand side of Table 1 holds for in nitely many input lengths, then the corresponding derandomization on the right-hand side works for in nitely many input lengths. Similar weak interpretations hold for all our results. As a corollary to the weak version of Table 1, we obtain that every language in AM, and graph nonisomorphism in particular, has subexponential size proofs for in nitely many input sizes unless the polynomial-time hierarchy collapses.Using other hardness measures and various oracles B, we get derandomization results for other complexity classes. We summarize the situation in Table 2. lower bound on: derandomizes:

1998

"... In PAGE 2: ... We show that the existence of an exponential-time decidable language with high worst-case nonuniform complexity when the circuits have access to an oracle for satis ability, implies nontrivial derandomizations of AM-games. The trade-o s are presented in Table1 . We use CB to denote circuit complexity given access to oracle B, and similarly ~ CB given only parallel access to the oracle.... In PAGE 2: ... See Section 2 for precise de nitions. The parameter s in Table1 can be any space constructible function, the interesting range lying between logarithmic and subpolynomial, e.... In PAGE 8: ...4 and 3.13 to the nondeterministic setting, and thus relax the hardness assumptions in Table1 from circuit complexity for parallel access to SAT to nondeterministic circuit complexity .... ..."

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### Table 6.1: A compilation of known lower and upper bounds for k-sets and j-facets. Currently best bounds are marked with *.

2000

### Table 6.2: A compilation of known lower and upper bounds for ( k)-sets and ( j)-facets. Currently best bounds are marked with *.

2000