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Module Version: 126

# 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

```    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```

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