Provided by: libmath-planepath-perl_125-1_all bug

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, 2015, 2016, 2017 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/>.