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Distance X Branching {m'Xm} Branching Ratio Local from Factor Factor of Max Start Cubes to Classes 0 1 1 0 1 12 12.000 1 1.000 12.000 0 2 114 9.500 5 5.000 22.800 0 3 1,068 9.368 25 5.000 42.720 0 4 10,011 9.374 219 8.760 45.712 0 5 93,840 9.374 1,978 9.032 47.442 0 6 878,880 9.366 18,395 9.300 47.778 0 7 8,221,632 9.355 171,529 9.325 47.931 0 8 76,843,595 9.347 1,601,725 9.338 47.976 0 9 717,789,576 9.341 14,956,266 9.338 47.993 0 10 6,701,836,858 9.337 139,629,194 9.336 47.997 11 62,549,615,248 9.333 1,303,138,445 9.333 47.9992

This chart includes a column for local maxima, which my charts

usually do not. With all the data kept in files instead of memory,

it is not a very natural calculation to determine which positions

are local maxima. With the data in memory, for any position X

you would calculate the 12 neighbors Xq, and immediately determine

which of the 12 neighbors were one move closer to Start. It is

easy to identify local maxima in this situation. With the data

written to files, the neighbors Xq are sorted before determining

which are closer to Start, and there is no (easy) way to relate

a given Xq back to its original X.

However, let me describe the sorting/merging process in a little

more detail. There is a file containing all cubes X such that

|X|=n. The neighbors Xq are written to a file. The file

is sorted, with duplicates deleted. (Actually, the problem is so

large that there are *many* files containing the neighbors Xq.

Each file is sorted, and then the results are merged). Finally, the

resulting file is matched against another file containing all

cubes Y such that |Y|=(n-1). Any matches are deleted, and whatever

is left over becomes the file containing all cubes Z such that

|Z|=(n+1). The difference between the number of matches deleted

and the number of cubes in the n-1 file is the number of local

maxima of length n-1. (Remember that all the X's and Y's and Z's

and Xq's are representative elements of M-conjugacy classes.)

The last time through this process, I generated neighbors of level

10 to create level 11, sorted and deleted duplicates, and matched

against level 9 deleting matches. Hence, the last level for which

I have local maxima information is level 9.

There are not any local maxima through level 9. I am not really

expecting any until Pons Asinorum at level 12. However, it would

be nice to verify that Pons Asinorum is the shortest local maximum.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU