Difference between revisions of "Binary traversal"

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Both the decimal and the binary numbers really have an infinite number of leading zeros before them, but we can still maintain a unique sequence of digits to represent every integer even if we omit all leading zeros as we have in this table. The point is that this rule means that every single number except for 0 always ends with a non-zero digit (numbers are read from right to left). While this may not be of much interest for most number systems, you will notice that with binary it means that every number except zero ends with a 1 - and if we forget about the two first items in the list, then we could drop the last bit altogether. As can be seen from the ''Nodal Path'' column in the list, that's exactly what we do in path traversal. This can't usually work with normal numeric data types because ''0'' and ''00'' evaluate to the same, but a ''sequence'' of binary digits is more like a string comparison of ''"0"'' and ''"00"''.
 
Both the decimal and the binary numbers really have an infinite number of leading zeros before them, but we can still maintain a unique sequence of digits to represent every integer even if we omit all leading zeros as we have in this table. The point is that this rule means that every single number except for 0 always ends with a non-zero digit (numbers are read from right to left). While this may not be of much interest for most number systems, you will notice that with binary it means that every number except zero ends with a 1 - and if we forget about the two first items in the list, then we could drop the last bit altogether. As can be seen from the ''Nodal Path'' column in the list, that's exactly what we do in path traversal. This can't usually work with normal numeric data types because ''0'' and ''00'' evaluate to the same, but a ''sequence'' of binary digits is more like a string comparison of ''"0"'' and ''"00"''.
  
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So that's how the binary conversion from a decimal index goes, but there's one more inefficiency to remove; every item in list-space is composed of three list-item-keys (which are each integer indexes of other items) and each index points to the middle of the three. So the index of the root item is 1, the next item is 4, then 7, 10, 13... so as you can see, a division by three can be done to ensure that we're not missing out any binary paths. Actually we subtract one, then divide by three giving us a proper zero-based integer range of binary paths.
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So that's how the binary conversion from a decimal index goes, but in the actual code, a multiplication by the size of three [[list item]]s is also needed, and an addition of one so that the indexes 1 and 2 map to the single-bit sequences of 0 and 1. Traversing the index of zero ([[root]]) means to return the subject unchanged with no traversal occuring (i.e. a zero-length traversal).
  
 
;See also
 
;See also

Revision as of 23:35, 8 November 2006

The array elements of a Nodal Space are all identified by an integer index which starts at 0 for the first element created, and one for the next and so on. All integers can be represented as a unique binary sequence as in the decimal and binary columns of the following list of example index numbers:

Example integers and their equivalent binary paths
Decimal   Binary   Nodal Path
0    0    -
1    1    -
2    10    0
3    11    1
4    100    0-0
⋮    ⋮    ⋮
100    1100100    0-0-1-0-0-1
⋮    ⋮    ⋮
1000    1111101000    0-0-0-1-0-1-1-1-1
⋮    ⋮    ⋮
1023    1111111111    1-1-1-1-1-1-1-1-1
1024    10000000000    0-0-0-0-0-0-0-0-0-0
1025    10000000001    1-0-0-0-0-0-0-0-0-0

Both the decimal and the binary numbers really have an infinite number of leading zeros before them, but we can still maintain a unique sequence of digits to represent every integer even if we omit all leading zeros as we have in this table. The point is that this rule means that every single number except for 0 always ends with a non-zero digit (numbers are read from right to left). While this may not be of much interest for most number systems, you will notice that with binary it means that every number except zero ends with a 1 - and if we forget about the two first items in the list, then we could drop the last bit altogether. As can be seen from the Nodal Path column in the list, that's exactly what we do in path traversal. This can't usually work with normal numeric data types because 0 and 00 evaluate to the same, but a sequence of binary digits is more like a string comparison of "0" and "00".

So that's how the binary conversion from a decimal index goes, but in the actual code, a multiplication by the size of three list items is also needed, and an addition of one so that the indexes 1 and 2 map to the single-bit sequences of 0 and 1. Traversing the index of zero (root) means to return the subject unchanged with no traversal occuring (i.e. a zero-length traversal).

See also
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