Math::PlanePath::HIndexing -- self-similar right-triangle traversal
use Math::PlanePath::HIndexing; my $path = Math::PlanePath::HIndexing->new; my ($x, $y) = $path->n_to_xy (123);
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
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.
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 $n < 0
then the return is an empty list.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, 2*4**$level - 1)
.
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
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
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
Math::PlanePath, Math::PlanePath::SierpinskiCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015 Kevin Ryde
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