Exercise 3.6.4 : Generalize Exercise 3.6.3 to give, as a function of r and b, the point of maximum slope and the value of that slope, for families of functions defined from the minhash functions by: A b-way OR construction followed by an r-way AND construction.

Exercise 3.6.3 : Any function of probability p, such as that of Fig. 3.11, has a slope given by the derivative of the function. The maximum slope is where that derivative is a maximum. Find the value of p that gives a maximum slope for the S-curves given by Fig. 3.11 and Fig. 3.12. What are the values of these maximum slopes?

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(1-(1-P)) 0.0140 0.1215 0.3334 0.5740 0.7725 0.9015 0.9680 0.9936 0.6 0.7 0.8 Figure 3.12: Effect of the 4-way OR-construction followed by the 4-way AND construction 1 p 0.2 0.3 0.4 0.5 0.6 -(1 - p+)4 0.0064 0.0320 0.0985 0.2275 0.4260 0.6666 0.8785 0.9860 0.7 0.8 0.9 Figure 3.11: Effect of the 4-way AND-construction followed by the 4-way OR- construction (1-(1-P)) 0.0140 0.1215 0.3334 0.5740 0.7725 0.9015 0.9680 0.9936 0.6 0.7 0.8 Figure 3.12: Effect of the 4-way OR-construction followed by the 4-way AND construction 1 p 0.2 0.3 0.4 0.5 0.6 -(1 - p+)4 0.0064 0.0320 0.0985 0.2275 0.4260 0.6666 0.8785 0.9860 0.7 0.8 0.9 Figure 3.11: Effect of the 4-way AND-construction followed by the 4-way OR- construction