Author: Sven Reichard

Date: 12:56:32 12/05/01

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I asked a similar question here before, however without any satisfactory result. The following is an argument why the Hamming distance can't be more than a crude heuristic, albeit possibly the best we have before. Sorry if it gets a bit mathematical. Assume we associate a bitstring to every piece-square combination. That is what's usually done in chess programs; some codes are added for the side to move, castling rights, e.p. squares, etc. We obtain the code of a position by xor-ing the codes of all the pieces contained in it. What we want to avoid is collisions at nodes close to the root. For nodes close to the leaves the cost of recomputing the score is smaller. Hence we want to avoid that x1^x2^...^xm = y1^y2^...^yn for codes xi, yi and small number m and n, and xi not equal to yj To translate that to a language that is more familiar - at least for people of a mathematical background - we consider the field F2 of two elements. The elements are 0 and 1, and we can add and multiply them as usual, with the additional rule that 1 + 1 = 0. This is really a field, just like the real or complex numbers, and we can do calculations as usual. Note that addition is just the exclusive or. Now the codes or bitstrings become vectors over the field F2, and the bitwise exclusive or becomes componentwise addition, i.e., usual addition of vectors. All these vectors form the vector space F2^k, where k is the length of the vectors. Typically, k = 64. So, what we want to avoid is an equation x1 + x2 + ... + xm = y1 + y2 + ... + yn or x1 + x2 + ... + xm + y1 + y2 + ... + yn = 0 since in F2, subtraction is the same as addition. Remembering some linear algebra, this just means that we want the set x1,...,xm,y1,...,yn to be linearly independent. This leads to the following criterion for picking a set of hashcodes: A set of vectors in F2^k is a good set of hash codes if each small subset of non-zero vectors is linearly independent. What is not clear here is the meaning of "small", but we want small to be as big as possible. In other words, we consider sets of size up to a certain size as small, and if we can make that size bigger, it is better, since this leads to unique codes deeper in the tree. However what is clear is that this quality criterion does not depend on the base of the vector space. I.e., if we have a good set and multiply each vector by an invertible matrix (in other words, if we rotate the vectors), the obtained set will be just as good, since the rotation does not change the linear independence. The Hamming distance, on the other hand, is highly dependent on the vector space base. Take for example the vectors (1,0) and (0,1) in F2^2; they have Hamming distance 2. If we multiply both of them by (1 1) (0 1) we get (1,1) and (0,1), which have Hamming distance 1. Actually we can change any distance to anything else (except for 0) by an appropriate matrix. Thus we try to approximate something that is independent from the base (the quality of our hash codes) by something that depends on it (the Hamming distance). Simple logic tells you that this approximation has to be real bad. An example where it doesn't work: It has been said that the Hamming distance shouldn't be to small or to big. So, vectors at a distance which is half the length should be ok, right? Let the length be 8 (I don't want to type too many 0's and 1's), and consider the vectors 11110000 11001100 00111100 They all have weight 4, their pairwise distance is 4, and yet they add up to 0. Just by looking at Hamming distances, you have no chance of detecting that. Summarizing I can say that I see no connection between the quality of hash codes and their Hamming distance. Using a good RNG like the one provided in GNU's stdlib will yield good hash codes ( you can actually prove that), and so I will take the codes as they are supplied by rand() or random() without messing with them and thereby most likely make them worse. Sorry for the length of this message and my rambling on. Thanks for your patience and any feedback. Actually I bet a case of beer that nobody reads up to this point (just kidding). Later, Sven.

- Re: About random numbers and hashing
**David Rasmussen***04:30:10 12/07/01*- Re: About random numbers and hashing
**Sune Fischer***05:45:47 12/07/01*- Re: About random numbers and hashing
**David Rasmussen***06:31:28 12/07/01*- Re: About random numbers and hashing
**Sune Fischer***07:06:59 12/07/01*- Re: About random numbers and hashing
**David Rasmussen***12:59:29 12/07/01*- Re: About random numbers and hashing
**Sune Fischer***14:34:37 12/07/01*- Re: About random numbers and hashing
**Sven Reichard***09:25:59 12/08/01*- Re: About random numbers and hashing
**David Rasmussen***03:57:28 12/09/01*- Re: About random numbers and hashing
**Sven Reichard***06:40:40 12/10/01* - Re: About random numbers and hashing
**Sune Fischer***04:31:48 12/09/01*- Re: About random numbers and hashing
**David Rasmussen***06:14:10 12/09/01*

- Re: About random numbers and hashing

- Re: About random numbers and hashing
- Re: About random numbers and hashing
**Ralf Elvsén***15:28:58 12/08/01*- Re: About random numbers and hashing
**Sune Fischer***15:50:45 12/08/01*

- Re: About random numbers and hashing

- Re: About random numbers and hashing

- Re: About random numbers and hashing

- Re: About random numbers and hashing
- Re: About random numbers and hashing
**Sune Fischer***07:29:19 12/07/01*

- Re: About random numbers and hashing

- Re: About random numbers and hashing

- Re: About random numbers and hashing

- Re: About random numbers and hashing
- Re: About random numbers and hashing
**Sven Reichard***11:35:57 12/06/01* - Re: About random numbers and hashing
**Ralf Elvsén***08:17:24 12/06/01* - Re: About random numbers and hashing
**Sune Fischer***17:47:56 12/05/01*- Re: About random numbers and hashing
**Sune Fischer***18:10:53 12/05/01*

- Re: About random numbers and hashing

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