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 14The 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 4Level 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 - 1For 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
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.
Level Methods
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 6The block breakdowns are
+---------------+ ^
| \ ^ | | ^ / |
|\ \ 2 | | 3 / | = 2^k - 1
| \ \ | | / |
| 1\ \ | | / |
| v \ \+--+/ v
+----+
| |
+----+
| ^ /
| 0 /
| /
| /
+/
<----> = 2^k - 2Parts 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)^2Half 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] = 5OEIS
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 halfSEE 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/>.