Model

The Z-order floor

Z-order degrades sharply with d, and the damage is in the exponent.

L = U^{- 1/ d}, units touched \geq U^{1 - 1/ d}

$U$: Units; number of units
$d$: Dimensions; columns interleaved in the curve
$L$: Cell side; side of one cell, in coordinates scaled to 0–1

Trade-off: Generality vs. specialization

How it behaves

One input on the horizontal axis, the others held fixed.

d = 2d = 3d = 4

Data table

Fewest units a one-column query can touch, by table size
Units U	series	Floor on units touched
100	d = 2	10
115	d = 2	10.7
133	d = 2	11.5
154	d = 2	12.4
178	d = 2	13.3
205	d = 2	14.3
237	d = 2	15.4
274	d = 2	16.5
316	d = 2	17.8
365	d = 2	19.1
422	d = 2	20.5
487	d = 2	22.1
562	d = 2	23.7
649	d = 2	25.5
750	d = 2	27.4
866	d = 2	29.4
1K	d = 2	31.6
1.15K	d = 2	34
1.33K	d = 2	36.5
1.54K	d = 2	39.2
1.78K	d = 2	42.2
2.05K	d = 2	45.3
2.37K	d = 2	48.7
2.74K	d = 2	52.3
3.16K	d = 2	56.2
3.65K	d = 2	60.4
4.22K	d = 2	64.9
4.87K	d = 2	69.8
5.62K	d = 2	75
6.49K	d = 2	80.6
7.5K	d = 2	86.6
8.66K	d = 2	93.1
10K	d = 2	100
11.5K	d = 2	107
13.3K	d = 2	115
15.4K	d = 2	124
17.8K	d = 2	133
20.5K	d = 2	143
23.7K	d = 2	154
27.4K	d = 2	165
31.6K	d = 2	178
36.5K	d = 2	191
42.2K	d = 2	205
48.7K	d = 2	221
56.2K	d = 2	237
64.9K	d = 2	255
75K	d = 2	274
86.6K	d = 2	294
100K	d = 2	316
115K	d = 2	340
133K	d = 2	365
154K	d = 2	392
178K	d = 2	422
205K	d = 2	453
237K	d = 2	487
274K	d = 2	523
316K	d = 2	562
365K	d = 2	604
422K	d = 2	649
487K	d = 2	698
562K	d = 2	750
649K	d = 2	806
750K	d = 2	866
866K	d = 2	931
1M	d = 2	1K
1.15M	d = 2	1.07K
1.33M	d = 2	1.15K
1.54M	d = 2	1.24K
1.78M	d = 2	1.33K
2.05M	d = 2	1.43K
2.37M	d = 2	1.54K
2.74M	d = 2	1.65K
3.16M	d = 2	1.78K
3.65M	d = 2	1.91K
4.22M	d = 2	2.05K
4.87M	d = 2	2.21K
5.62M	d = 2	2.37K
6.49M	d = 2	2.55K
7.5M	d = 2	2.74K
8.66M	d = 2	2.94K
10M	d = 2	3.16K
100	d = 3	21.5
115	d = 3	23.7
133	d = 3	26.1
154	d = 3	28.7
178	d = 3	31.6
205	d = 3	34.8
237	d = 3	38.3
274	d = 3	42.2
316	d = 3	46.4
365	d = 3	51.1
422	d = 3	56.2
487	d = 3	61.9
562	d = 3	68.1
649	d = 3	75
750	d = 3	82.5
866	d = 3	90.9
1K	d = 3	100
1.15K	d = 3	110
1.33K	d = 3	121
1.54K	d = 3	133
1.78K	d = 3	147
2.05K	d = 3	162
2.37K	d = 3	178
2.74K	d = 3	196
3.16K	d = 3	215
3.65K	d = 3	237
4.22K	d = 3	261
4.87K	d = 3	287
5.62K	d = 3	316
6.49K	d = 3	348
7.5K	d = 3	383
8.66K	d = 3	422
10K	d = 3	464
11.5K	d = 3	511
13.3K	d = 3	562
15.4K	d = 3	619
17.8K	d = 3	681
20.5K	d = 3	750
23.7K	d = 3	825
27.4K	d = 3	909
31.6K	d = 3	1K
36.5K	d = 3	1.1K
42.2K	d = 3	1.21K
48.7K	d = 3	1.33K
56.2K	d = 3	1.47K
64.9K	d = 3	1.62K
75K	d = 3	1.78K
86.6K	d = 3	1.96K
100K	d = 3	2.15K
115K	d = 3	2.37K
133K	d = 3	2.61K
154K	d = 3	2.87K
178K	d = 3	3.16K
205K	d = 3	3.48K
237K	d = 3	3.83K
274K	d = 3	4.22K
316K	d = 3	4.64K
365K	d = 3	5.11K
422K	d = 3	5.62K
487K	d = 3	6.19K
562K	d = 3	6.81K
649K	d = 3	7.5K
750K	d = 3	8.25K
866K	d = 3	9.09K
1M	d = 3	10K
1.15M	d = 3	11K
1.33M	d = 3	12.1K
1.54M	d = 3	13.3K
1.78M	d = 3	14.7K
2.05M	d = 3	16.2K
2.37M	d = 3	17.8K
2.74M	d = 3	19.6K
3.16M	d = 3	21.5K
3.65M	d = 3	23.7K
4.22M	d = 3	26.1K
4.87M	d = 3	28.7K
5.62M	d = 3	31.6K
6.49M	d = 3	34.8K
7.5M	d = 3	38.3K
8.66M	d = 3	42.2K
10M	d = 3	46.4K
100	d = 4	31.6
115	d = 4	35.2
133	d = 4	39.2
154	d = 4	43.7
178	d = 4	48.7
205	d = 4	54.2
237	d = 4	60.4
274	d = 4	67.3
316	d = 4	75
365	d = 4	83.5
422	d = 4	93.1
487	d = 4	104
562	d = 4	115
649	d = 4	129
750	d = 4	143
866	d = 4	160
1K	d = 4	178
1.15K	d = 4	198
1.33K	d = 4	221
1.54K	d = 4	246
1.78K	d = 4	274
2.05K	d = 4	305
2.37K	d = 4	340
2.74K	d = 4	379
3.16K	d = 4	422
3.65K	d = 4	470
4.22K	d = 4	523
4.87K	d = 4	583
5.62K	d = 4	649
6.49K	d = 4	723
7.5K	d = 4	806
8.66K	d = 4	898
10K	d = 4	1K
11.5K	d = 4	1.11K
13.3K	d = 4	1.24K
15.4K	d = 4	1.38K
17.8K	d = 4	1.54K
20.5K	d = 4	1.72K
23.7K	d = 4	1.91K
27.4K	d = 4	2.13K
31.6K	d = 4	2.37K
36.5K	d = 4	2.64K
42.2K	d = 4	2.94K
48.7K	d = 4	3.28K
56.2K	d = 4	3.65K
64.9K	d = 4	4.07K
75K	d = 4	4.53K
86.6K	d = 4	5.05K
100K	d = 4	5.62K
115K	d = 4	6.26K
133K	d = 4	6.98K
154K	d = 4	7.77K
178K	d = 4	8.66K
205K	d = 4	9.65K
237K	d = 4	10.7K
274K	d = 4	12K
316K	d = 4	13.3K
365K	d = 4	14.9K
422K	d = 4	16.5K
487K	d = 4	18.4K
562K	d = 4	20.5K
649K	d = 4	22.9K
750K	d = 4	25.5K
866K	d = 4	28.4K
1M	d = 4	31.6K
1.15M	d = 4	35.2K
1.33M	d = 4	39.2K
1.54M	d = 4	43.7K
1.78M	d = 4	48.7K
2.05M	d = 4	54.2K
2.37M	d = 4	60.4K
2.74M	d = 4	67.3K
3.16M	d = 4	75K
3.65M	d = 4	83.5K
4.22M	d = 4	93.1K
4.87M	d = 4	104K
5.62M	d = 4	115K
6.49M	d = 4	129K
7.5M	d = 4	143K
8.66M	d = 4	160K
10M	d = 4	178K

Worked examples, recomputed

Every number the mechanisms quote for this relationship is recomputed from it, so none of them disagrees with it.

Where it appears

Each mechanism states it in its own names. The general symbol each name stands for is shown with it.

Storage: Physical Organization
What should sit next to what, given that I must decide at write time and find out at read time?
$L = U^{- 1/ d}, U^{1 - 1/ d}$
Read it in place →