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 H-indexing per

Rolf Niedermeier, Klaus Reinhardt and Peter Sanders, "Towards Optimal Locality In Mesh Indexings", Discrete Applied Mathematics, volume 117, March 2002, pages 211-237. http://theinf1.informatik.uni-jena.de/publications/dam01a.pdf

It traverses an eighth 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 path 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.

It would be possible to take a level as N=0 to N=4^k-1 too, which would be a triangle against the Y axis. The 2*4^level - 1 is per the paper above.

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.

Level Methods

($n_lo, $n_hi) = $path->level_to_n_range($level)

Return (0, 2*4**$level - 1).

FORMULAS ^

Area

The area enclosed by curve in its triangular level k is

    A[k] = (2^k-1)^2
         = 0, 1, 9, 49, 225, 961, 3969, 16129, ...  (A060867)

For example level k=2 enclosed area marked by "@" signs,

      7 |   *---*---*---*---*---*---31
        |   |   | @ |   | @ |   | @ |
      6 |   *   *---*   *   *   *---*
        |   |           | @ |
      5 |   *   *---*   *   *
        |   |   | @ |   | @ |
      4 |   *---*   *   *---*         level k=2
        |   | @   @ |                 N=0 to N=31
      3 |   *-- *   *
        |   |   | @ |                 A[2] = 9
      2 |   *   *-- *
        |   |
      1 |   *
        |   |
    Y=0 |   0
        +------------------------------
           X=0  1   2   3   4   5   6

The block breakdowns are

    +---------------+     ^
    | \  ^ |  | ^  /      |
    |\ \ 2 |  | 3 /       | = 2^k - 1
    | \ \  |  |  /        |
    | 1\ \ |  | /         |
    | v \ \+--+/          v
    +----+
    |    |
    +----+
    | ^  /
    | 0 /
    |  /
    | /
    +/

    <---->  = 2^k - 2

Parts 0 and 3 are identical. Parts 1 and 2 are mirror images of 0 and 3 respectively. Parts 0 and 1 have an area in between 1 high and 2^k-2 wide (eg. 2^2-2=2 wide in the k=2 above). Parts 2 and 3 have an area in between 1 wide 2^k-1 high (eg. 2^2-1=3 high in the k=2 above). So the total area is

    A[k] = 4*A[k-1] + 2^k-2 + 2^k-1     starting A[0] = 0
         =    4^0     * (2*2^k - 3)
            + 4^1     * (2*2^(k-1) - 3)
            + 4^2     * (2*2^(k-2) - 3)
            + ...
            + 4^(k-1) * (2*2^1 - 3)
            + 4^k * A[0]
         = 2*2*(4^k - 2^k)/(4-2) - 3*(4^k - 1)/(4-1)
         = (2^k - 1)^2

Half Level Areas

Block 1 ends at the top-left corner and block 2 start there. The area before that midpoint enclosed to the Y axis can be calculated. Likewise the area after that midpoint to the top line. Both are two blocks, and with either 2^k-2 or 2^k-1 in between. They're therefore half the total area A[k], with the extra unit square going to the top AT[k].

    AY[k] = floor(A[k]/2)
          = 0, 0, 4, 24, 112, 480, 1984, 8064, 32512, ...  (A059153)

    AT[k] = ceil(A[k]/2)
          = 0, 1, 5, 25, 113, 481, 1985, 8065, 32513, ...  (A092440)
                                     15
                                      |
                                     14
                                      |
                                     13  10-- 9
                                      |   | @ |
                                     12--11   8
                                        @   @ |
                      3               3-- 4   7
                      |               |   | @ |
                      2               2   5-- 6
                      |               |
                      1               1
                      |               |
        0             0               0

    AY[0] = 0     AY[1] = 0       AY[2] = 4
       1       3-- 4   7       15--16  19--20  27--28  31
                   | @ |            | @ |   | @ |   | @ |
                   5-- 6           17--18  21  26  29--30
                                            | @ |
                                           22  25
                                            | @ |
                                           23--24

    AT[0] = 0   AT[1] = 1      AT[2] = 5

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

    A060867    area of level
    A059153    area of level first half
    A092440    area of level second half

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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