Kevin Ryde > Math-PlanePath > Math::PlanePath::HIndexing

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NAME ^

Math::PlanePath::HIndexing -- self-similar right-triangle traversal

SYNOPSIS ^

 use Math::PlanePath::HIndexing;
 my $path = Math::PlanePath::HIndexing->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION ^

This is an infinite integer version of the H-indexing by Rolf Niedermeier, Klaus Reinhardt and Peter Sanders.

"Towards Optimal Locality In Mesh Indexings", Discrete Applied Mathematics, volume 117, March 2002. http://theinf1.informatik.uni-jena.de/publications/dam01a.pdf

It traverses an octant of the plane by self-similar right triangles. Notice the "H" shapes that arise from the backtracking, for example N=8 to N=23, and repeating above it.

        |                                                           |
     15 |  63--64  67--68  75--76  79--80 111-112 115-116 123-124 127
        |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
     14 |  62  65--66  69  74  77--78  81 110 113-114 117 122 125-126
        |   |           |   |           |   |           |   |
     13 |  61  58--57  70  73  86--85  82 109 106-105 118 121
        |   |   |   |   |   |   |   |   |   |   |   |   |   |
     12 |  60--59  56  71--72  87  84--83 108-107 104 119-120
        |           |           |                   |
     11 |  51--52  55  40--39  88  91--92  99-100 103
        |   |   |   |   |   |   |   |   |   |   |   |
     10 |  50  53--54  41  38  89--90  93  98 101-102
        |   |           |   |           |   |
      9 |  49  46--45  42  37  34--33  94  97
        |   |   |   |   |   |   |   |   |   |
      8 |  48--47  44--43  36--35  32  95--96
        |                           |
      7 |  15--16  19--20  27--28  31
        |   |   |   |   |   |   |   |
      6 |  14  17--18  21  26  29--30
        |   |           |   |
      5 |  13  10-- 9  22  25
        |   |   |   |   |   |
      4 |  12--11   8  23--24
        |           |
      3 |   3-- 4   7
        |   |   |   |
      2 |   2   5-- 6
        |   |
      1 |   1
        |   |
    Y=0 |   0
        +-------------------------------------------------------------
           X=0  1   2   3   4   5   6   7   8   9  10  11  12  13  14

The tiling is essentially the same as the Sierpinski curve (see Math::PlanePath::SierpinskiCurve). The following is with two points per triangle. Or equally well it could be thought of with those triangles further divided to have one point each, a little skewed.

    +---------+---------+--------+--------/
    |  \      |      /  | \      |       /
    | 15 \  16| 19  /20 |27\  28 |31    /
    |  |  \  ||  | /  | | | \  | | |  /
    | 14   \17| 18/  21 |26  \29 |30 /
    |       \ | /       |     \  |  /
    +---------+---------+---------/
    |       / |  \      |       /
    | 13  /10 | 9 \  22 | 25   /
    |  | /  | | |  \  | |  |  /
    | 12/  11 | 8   \23 | 24/
    |  /      |      \  |  /
    +-------------------/
    |  \      |       /
    | 3 \   4 | 7    /
    | |  \  | | |  /
    | 2   \ 5 | 6 /
    |       \ |  /
    +----------/
    |         /
    | 1     /
    | |   /
    | 0  /
    |  /
    +/

The correspondence to the SierpinskiCurve is as follows. The 4-point verticals like N=0 to N=3 are a Sierpinski horizontal, and the 4-point "U" parts like N=4 to N=7 are a Sierpinski vertical. In both cases there's an X,Y transpose and bit of stretching.

    3                                       7
    |                                      /
    2         1--2             5--6       6
    |  <=>   /    \            |  |  <=>  |
    1       0      3           4  7       5
    |                                      \
    0                                       4

Level Ranges

Counting the initial N=0 to N=7 section as level 1, the X,Y ranges for a given level is

    Nlevel = 2*4^level - 1
    Xmax = 2*2^level - 2
    Ymax = 2*2^level - 1

For example level=3 is N through to Nlevel=2*4^3-1=127 and X,Y ranging up to Xmax=2*2^3-2=14 and Xmax=2*2^3-1=15.

On even Y rows, the N on the X=Y diagonal is found by duplicating each bit in Y except the low zero (which is unchanged). For example Y=10 decimal is 1010 binary, duplicate to binary 1100110 is N=102.

FUNCTIONS ^

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::HIndexing->new ()

Create and return a new path object.

($x,$y) = $path->n_to_xy ($n)

Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list.

OEIS ^

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

http://oeis.org/A097110 (etc)

    A097110    Y at N=2^k, being successively 2^j-1, 2^j

SEE ALSO ^

Math::PlanePath, Math::PlanePath::SierpinskiCurve

HOME PAGE ^

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE ^

Copyright 2011, 2012, 2013, 2014 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.

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