FractionalGeometry-Chap2

If this gives a string not already in l then loop to

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Unformatted text preview: APTER 2. ITERATED FUNCTION SYSTEMS 21 Figure 2.31: Address 21 for Prob. 2.4.10. length 10. If, say, the probability of applying transformation T1 is 1/10, then about 1010 applications of the random algorithm would be needed to visit the pixel with address 110 . This is a long time to wait. Can we do better? A de Bruijn sequence B (k, N ) of order N over an alphabet of size k is sequence, of length k N , in which every sequence of length N over the alphabet occurs exactly once. Given an IFS {T1 , . . . , Tk }, applying the transformations in the order imposed by a de Bruijn sequence B (k, N ) visits every adddress length N region in the attractor of the IFS. For the N = 10 and k = 4 example mentioned above, a de Bruijn sequence has length 410 = 10488576, so much shorter than 1010 . That there are any de Bruijn sequences at all is not so obvious, though in 1894 Camille Flye Sainte-Marie established the existence of binary de Bruijn sequences. Generalizations to arbitrary (finite, of course) alphabets were done by Nicolaas...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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