a spatial data and sensor network application:

A Spatial Data and Sensor Network Application:

Each energized nano-sensor transmits a ping (location is triangulated from the ping). These locations are then translated to 3-dimensional coordinates at the display. The corresponding voxel on the display lights up. This is the expendable, one-time, cheap sensor version.

A more sophisticated CEASR device could sense and transmit the intensity levels, lighting up the display voxel with the same intensity.

Wherever threshold level is sensed (chem, bio, thermal...)a ping is registered in 1 compressed Ptree for that location.

Situation space

Nano-sensors droppedinto the Situation space

Soldier sees replica of sensedsituation prior to entering space

.:.:.:.:..::….:. : …:…:: ..:

. . :: :.:…: :..:..::. .:: ..:.::..

.:.:.:.:..::….:. : …:…:: ..:

. . :: :.:…: :..:..::. .:: ..:.::..

.:.:.:.:..::….:. : …:…:: ..:

. . :: :.:…: :..:..::. .:: ..:.::..

Using Alien Technology’s Fluidic Self-assembly (FSA) technology, clear layers are laminated into a cube, with a embedded nano-LED at each voxel.

==================================\ CARRIER /

CubE for Active Situation Replication (CEASR)

The Ptree is transmitted to the cube, where the pattern is reconstructed (uncompress Ptree, display on the cube).

Spatial Data

Pixel – a point in a spaceBand – feature attribute of the pixelsValue – usually one byte (0~255)Images have different numbers of bands

– TM4/5: 7 bands (B, G, R, NIR, MIR, TIR, MIR2)– TM7: 8 bands (B, G, R, NIR, MIR, TIR, MIR2, PC)– TIFF: 3 bands (B, G, R)– Ground data: individual bands (Yield, Moisture,

Nitrate level, Temperature, elevation…) These notes contain NDSU confidential &Proprietary material.Patents pending on Ptree technology

RSI dataset example

TIFF image Yield Map

RSI data can be viewed as collection of pixels. Each pixel has a value for each feature attribute

For example, the RSI dataset above has 320 rows and 320 columns of pixels (102,400 pixels) and 4 feature attributes (B,G,R,Y). The (B,G,R) feature bands are in the TIFF image and the Y feature is color coded in the Yield

Map.

Spatial Data Formats

BAND-1 254 127 (1111 1110) (0111 1111)

14 193 (0000 1110) (1100 0001)

BAND-237 240(0010 0101) (1111 0000)

200 19(1100 1000) (0001 0011)

BSQ format (2 files)

Band 1: 254 127 14 193 Band 2: 37 240 200 19

Existing formats

– BSQ (Band Sequential)

– BIL (Band Interleaved by Line)

– BIP (Band Interleaved by Pixel) New format

– bSQ (bit Sequential)

Spatial Data Formats (Cont.)

BAND-1 254 127 (1111 1110) (0111 1111)

14 193 (0000 1110) (1100 0001)

BAND-237 240(0010 0101) (1111 0000)

200 19(1100 1000) (0001 0011)


Band 1: 254 127 14 193 Band 2: 37 240 200 19

BIL format (1 file)

254 127 37 240 14 193 200 19

BAND-1 254 127 (1111 1110) (0111 1111)

14 193 (0000 1110) (1100 0001)

BAND-237 240(0010 0101) (1111 0000)

200 19(1100 1000) (0001 0011)


Band 1: 254 127 14 193 Band 2: 37 240 200 19

BIL format (1 file)

254 127 37 240 14 193 200 19

BIP format (1 file)

254 37 127 240 14 200 193 19

Spatial Data Formats (Cont.)

BAND-1 254 127 (1111 1110) (0111 1111)

14 193 (0000 1110) (1100 0001)

BAND-237 240(0010 0101) (1111 0000)

200 19(1100 1000) (0001 0011)


Band 1: 254 127 14 193 Band 2: 37 240 200 19

BIL format (1 file)

254 127 37 240 14 193 200 19

BIP format (1 file)

254 37 127 240 14 200 193 19

bSQ format (16 files)B11 B12 B13 B14 B15 B16 B17 B18 B21 B22 B23 B24 B25 B26 B27 B28 1 1 1 1 1 1 1 0 0 0 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1

Spatial Formats

Split each band into eight separate files, one for each bit position. Reasons of using bSQ format

– Different bits contribute to the value differently.

– bSQ format facilitates representation of a precision hierarchy (from 1 to 8 bit precision).

– bSQ format facilitates creation of an efficient data structure, the P-tree, algebra and cube.

BSQ and bSQ are “tabular” formats

– BSQ consist of a separate table for each feature band

– bSQ consist of a separate table for each bit of each band One can view it this way:

– The data set is initially 1 relation or table, R(K1,..,Kk, A1, …, An) where K1,..,Kk are structure attributes and Ai are feature attributes.

• Structure attributes of a 2-D image are X,Y coordinates of the pixels (rows).

• Feature attributes are the bands, B,G,R, NIR, …

• BSQ we separate each feature into a separate file and suppress the structure attributes altogether (assuming pixels are always arranged in raster order. (aka: Decomposition Storage Model (DSM), Copeland et al, SIGMOD85, 268-279.)

• bSQ, separate each bit of each feature into separate file (raster order assumption) (aka: Bit Transpose File (BTF) model, Wong et al, VLDB85, pp 448-457.)

An example of PC-tree

Peano or Z-ordering Pure (Pure-1/Pure-0) quadrant Root Count

Level Fan-out QID (Quadrant ID)

1 1 1 1 1 1 0 01 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1

55

16 8 15 16

3 0 4 1 4 4 3 4

1 1 1 0 0 0 1 0 1 1 0 1

16 16

55

0 4 4 4 4

158

1 1 1 0

3

0 0 1 0

1

1 1

3

0 1

Given a bSQ file, Bij, (shown in spatial positions also) we create its basic PC-tree, Pij as follows.

1111110011111000111111001111111011111111111111111111111101111111

55

16 8 15 16

3 0 4 1 4 4 3 4

1 1 1 0 0 0 1 0 1 1 0 1

Our example of PC-tree (again)

Peano or Z-ordering Pure (Pure-1/Pure-0) quadrant Root Count

Level Fan-out QID (Quadrant ID)

1 1 1 1 1 1 0 01 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1

0 1 2 3

111

( 7, 1 ) ( 111, 001 )

2

3

2 . 2 . 3

001

Level-0

Level-3

Level-2

Level-1

10.10.11

P-tree variation – PM-tree

Peano Mask tree (PM-tree) uses mask instead of count. 1 denotes pure-1, 0 denotes pure-0 and m denotes mixed. It provides an efficient way for ANDing. Most compact form (all lossless)

– Predicate Tree (1 iff predicate is true for quadrant)• E.g., Pure1-Tree (predicate: quad is all 1’s)

1 1 1 1 1 1 0 01 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1

m

1 m m 1

m 0 1 m 1 1 m 1

1 1 1 0 0 0 1 0 1 1 0 1

PM-tree1: m ______/ / \ \______ / / \ \ / / \ \ 1 m m 1 / / \ \ / / \ \ m 0 1 m 1 1 m 1 //|\ //|\ //|\ 1110 0010 1101

PM-tree2: m ______/ / \ \______ / / \ \ / / \ \ 1 0 m 0 / / \ \ 1 1 1 m //|\ 0100

AND Result: m ________ / / \ \___ / ____ / \ \ / / \ \ 1 0 m 0 / | \ \ 1 1 m m //|\ //|\ 1101 0100

0 100 101 102 12 132 20 21 220 221 223 23 3 & 0 20 21 22 231 RESULT0 0 0 20 20 20 21 21 21 220 221 223 22 220 221 223 23 231 231

Depth-first Pure 1 path code

Ptree Algebra And Or Complement Other

Ptree: 55 ____________/ / \ \___________ / ___ / \___ \ / / \ \ 16 ____8__ _15__ 16 / / | \ / | \ \ 3 0 4 1 4 4 3 4 //|\ //|\ //|\ 1110 0010 1101

Complement: 9 ____________/ / \ \___________ / ___ / \___ \ / / \ \ 0 ____8__ __1__ 0 / / | \ / | \ \ 1 4 0 3 0 0 1 0 //|\ //|\ //|\ 0001 1101 0010

Basic, Value and Tuple Ptrees

Tuple Ptrees (predicate: quad is purely target tuple) e.g., P(1, 2, 3) = P(001, 010, 111) = P1, 001 AND P2, 010 AND P3, 111

AND

Value Ptrees (predicate: quad is purely target value in target attribute) e.g., P1, 5 = P1, 101 = P11 AND P12’ AND P13

AND

Target Attribute Target Value

Basic Ptrees (a Pure1-Trees predicate-tree for target bit of target attribute)e.g., P11, P12, …, P18, P21, …, P28, …, P71, …, P78

Target Attribute Target Bit Position

Cube Ptrees (predicate: quad is purely in target cube (product of intervals)

e.g., P([13],, [0.2]) = (P1,1 OR P1,2 OR P1,3) AND (P3,0 OR P3,1 OR P3,2)

AND/OR

Creating Peano-Count-trees (PC-trees) from Spatial Relations

Take any spatial relation, R(K1,..,Kk, A1, A2, …, An) (Ki=structure, Ai=feature attributes).

•Eg, Structure attributes of a 2-D image = X-Y coords, feature attribs = bands (e.g., B,G,R)

•We create BSQ files from it by projection, Bi = R[Ai].

•We create bSQ files from each of these BSQ files, Bi1, Bi2 , …, Bin

•We create a Peano Tree, Pij, from each bSQ file, Bij

Peano trees (P-trees):P-tree represents bSQ, BSQ, relational data in a recursive quadrant-by-quadrant,

lossless, compressed, datamining-ready format.

P-trees come in many forms

Count-trees (PC-trees);

Predicate-trees (P1, P0, PN1, PNZ, value-P-trees, tuple-P-trees, cube-P-trees)

Other forms: Predicate Ptrees (1 if condition is true thruout the quadrant, else 0) (P1 and P0 are lossless)

Pure1Tree (P1T) .---- 0 ----. / / \ \1 0 0 1 // \ \ // \ \ 0 0 1 0 11 0 1 //|\ //|\ //|\1110 0010 1101

PCT: .--- 55 ---. / / \ \16 8 15 16 // \ \ // \\ 3 0 4 1 44 3 4 //|\ //|\ //|\ 1110 0010 1101

1 1 1 1 1 1 0 01 1 1 1 1 0 0 01 1 1 1 1 1 0 01 1 1 1 1 1 1 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 1 1 1 1 1 1 1

Pure0Tree (P0T) .---- 0 ----. / / \ \0 0 0 0 // \ \ // \ \ 0 1 0 0 00 0 0 //|\ //|\ //|\0001 1101 0010

NotPure0(NP0T) .---- 1 ----. / / \ \1 1 1 1 // \ \ // \ \ 1 0 1 1 11 1 1 //|\ //|\ //|\1110 0010 1101

NotPure1(NP1T) .---- 1 ----. / / \ \0 1 1 0 // \ \ // \\ 1 1 0 1 00 10 //|\ //|\ //|\ 0001 1101 0010

Vector Implemented Ptrees (Vector Ptrees have 1 row for each mixed quadrant, with that quadrant’s (qid, P-vector) P1V

Qid PgVc[] 1001[1] 0010[1.0] 1110[1.3] 0010[2] 1101[2.2] 1101

P0VQid PgVc[] 0000[1] 0100[1.0] 0001[1.3] 1101[2] 0000[2.2] 0010

NP0VQid PgVc[] 1111[1] 1011[1.0] 1110[1.3] 0010[2] 1111[2.2] 1101

NP1VQid PgVc[] 0110[1] 1101[1.0] 0001[1.3] 1101[2] 0010[2.2] 0010

PeanoMixed (PM) .---- 1 ----. / / \ \0 1 1 0 // \ \ // \ \ 1 0 0 1 00 1 0 //|\ //|\ //|\0000 0000 0000

PMVQid PgVc[] 0110[1] 1001[2] 0010

Leaf-vectors always 0000 Can be omitted.

We may need Peano Mixed (PM) trees (e.g., distributed P-trees).

Note:

PM= P1 xor NP0

The Peano Cube of a relation (P-cube)Suppose we have R(K, A1, A2, A3 ) with each Ai a 2-bit numberConstruct the cube of all tuple-P-trees for R

Form the cube of all RootCountP(t)

P-Cube of R

P-Cube(A1, A2, A3, rcP(A1,A2,A3))

(rootcounts form the feature attributesand Ai’s form the structure attributes)

We can intervalize the RCs, (eg, 4 intervals, [0,0], [1,8], [9,63], [64,), labelled, 00, 01, 10 ,11 respectively).

Meta-P-trees of R, by forming basic Ptrees over the P-Cube of R(1 feature attribute and, if we intervalize as above, 4 basic Ptrees).

- |HR| |R| and = iff (A1, A2, A3 ) candidate key for R - what is the relationship to the Haar wavelet low-pass tree?

0 0

0 0

0 0

0 0

0 0

1 5

0 0

0 0

1100 01 10

00

01

10

11

0 0

1 0

0 1

0 0

0 0

14 5

0 0

3 0

1000 01 10

00

01

10

11

0 0

1 0

0 0

0 0

0 0

5 5

0 0

17 0

0100

01

10

11rc

P(0,0,0)

00 01 10 11

11

10

01

00

00

A1

A2

A 3

rcP(1,0,0)

rcP(0,2,0)

rcP(1,2,0)

rcP(2,2,0)

rcP(3,2,0)

rcP(0,3,0)

rcP(1,3,0)

rcP(2,3,0)

rcP(3,3,0)

rcP(0,0,0)

rcP(1,1,0)

rcP(2,1,0)

rcP(3,1,0)

rcP(3,0,0)

rcP(2,0,0)

rcP(0,0,1)

rcP(1,0,1)

rcP(2,0,1)

rcP(3,0,1)

rcP(0,0,2)

rcP(1,0,2)

rcP(2,0,2)

rcP(3,0,2)

rcP(2,0,3)

rcP(1,0,3)

rcP(0,0,3)

rcP(3,0,3)

rcP313

rcP312

rcP311

rcP323

rcP333

rcP322

rcP321

rcP331

rcP332

The P-tree Algebra (Complement, AND, OR, …) Complement Tree = the Ptree for the bit-complement of the bSQ file) (‘)

– We will use the “prime” notation.– PC-tree of a complement formed by purity-complementing each count.– Truth-tree of a complement: by bit-complementing leaves only.

Tree Complement = Complement of the tree - each tree entry is complemented. (“)– Not the same as the Complement Tree!– We will use”double prime” notation.

P1 = P0’ .---- 0 ---. / / \ \1 0 0 1 // \ \ // \ \ 0 0 1 0 11 0 1 //|\ //|\ //|\1110 0010 1101

P0 = P1’ .---- 0 ----. / / \ \0 0 0 0 // \ \ // \ \ 0 1 0 0 00 0 0 //|\ //|\ //|\0001 1101 0010

NP0 = NP1’ .---- 1 ----. / / \ \1 1 1 1 // \ \ // \ \ 1 0 1 1 11 1 1 //|\ //|\ //|\1110 0010 1101

NP0VQid PgVc[] 1111[1] 1011[1.0] 1110[1.3] 0010[2] 1111[2.2] 1101

NP1=NP0’=P1” .---- 1 ----. / / \ \0 1 1 0 // \ \ // \\ 1 1 0 1 00 10 //|\ //|\ //|\ 0001 1101 0010

NP1VQid PgVc[] 0110[1] 1101[1.0] 0001[1.3] 1101[2] 0010[2.2] 0010

P1VQid PgVc[] 1001 [1] 0010 [1.0] 1110 [1.3] 0010 [2] 1101 [2.2] 1101

P0VQid PgVc[] 0000 [1] 0100 [1.0] 0001 [1.3] 1101 [2] 0000 [2.2] 0010

P1” .---- 1 ---. / / \ \0 1 1 0 // \ \ // \ \ 1 1 0 1 00 1 0 //|\ //|\ //|\0001 1101 0010

P0” .---- 1 ----. / / \ \1 1 1 1 // \ \ // \ \ 1 0 1 1 11 1 1 //|\ //|\ //|\1110 0010 1101

NP0” = P0 .---- 0 ----. / / \ \0 0 0 0 // \ \ // \ \ 0 1 0 0 00 0 0 //|\ //|\ //|\0001 1101 0010

NP0V”Qid PgVc[] 0000[1] 0100[1.0] 0001[1.3] 1101[2] 0000[2.2] 1101

NP1” = P1 .---- 0 ----. / / \ \1 0 0 1 // \ \ // \\ 0 0 1 0 11 01 //|\ //|\ //|\ 1110 0010 1101

NP1V”Qid PgVc[] 1001[1] 0010[1.0] 0001[1.3] 0010[2] 1101[2.2] 1101

P1V”Qid PgVc[] 0110 [1] 1101 [1.0] 0001 [1.3] 1101 [2] 0010 [2.2] 0010

P0V”Qid PgVc[] 1111 [1] 1011 [1.0] 1110[1.3] 1101 [2] 1111 [2.2] 1101

ANDing (for all Truth-trees, just AND bit-wise)

0 0 100 101 102 12 132 2020 2121 220 221 223220 221 223 2323 3 AND 00 20 20 2121 2222 231231 00 20 20 2121 220 221 223220 221 223 231231

Pure1-quad-list method: For each operand, list the qids of the pure1 quad’s in depth-first order. Do one multi-cursor scan across the operand lists , for every pure1 quad common to all operands, install it in the result.

P1operand1 01 0 0 1 // \ \ // \\ 0 0 1 0 1 1 01 //|\ //|\ //|\1110 0010 1101

P0operand1 00 0 0 0 // \ \ // \ \ 0 1 0 0 0 0 00 //|\ //|\ //|\0001 1101 0010

NP0operand1 11 1 1 1 // \ \ // \\ 1 0 1 1 1 1 11 //|\ //|\ //|\ 1110 0010 1101

NP1operand1 NP0’ 1 0 1 1 0 // \ \ // \\ 1 1 0 1 0 0 10 //|\ //|\ //|\ 0001 1101 0010

1 1 1 1 1 1 0 01 1 1 1 1 0 0 01 1 1 1 1 1 0 01 1 1 1 1 1 1 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 1 1 1 1 1 1 1

P1operand2 01 0 0 0 / / \ \ 1 1 1 0 //|\ 0100

P0op2 = P1’op2 00 1 0 1 / / \ \ 0 0 0 0 //|\ 1011

NP0operand2 11 0 1 0 / / \ \ 1 11 1 //|\ 0100

NP1operand2 NP0’ 10 1 1 1 / / \ \ 0 0 0 1 //|\ 1011

P1op1^P1op2 01 0 0 0 // | \ 11 0 0 //|\ //|\ 1101 0100

P1op1^P0op2 = P1op1^P1’op2 00 0 0 1 // \ \ //\ \ 0 0 1 0 000 0 //|\ //|\ //|\1110 0010 1011

NP0op1^NP0op2

11 0 1 0 // | \ 11 1 1 //|\ //|\ 1101 0100

NP0op1^NP0’op2

10 1 1 1 // \ \ /// \ 1 0 1 1 000 1 //|\ //|\ //|\ 1110 0010 1011

1 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 0 1 0 0 0 01 1 0 0 0 0 0 0

1 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 1 1 0 0 0 01 1 0 1 0 0 0 00 1 0 0 0 0 0 0

AND

=

Depth first traversal using1^1=1, 1^0=0, 0^0=0.

bitwise

Example1: One band, B1, with 3-bit precision

PNP0V11 P1V11 (combined into 1 table)

qid NP0 P1[ ] 1111 1001[01] 1011 0010[10] 1111 1101[01.00] 1110 1110[01.11] 0010 0010[10.10] 1101 1101

P12

qid NP0 P1[ ] 1010 1000[10] 1111 1110[10.11] 0111

P13

qid NP0 P1[ ] 0111 0001[01] 1111 1110[10] 1110 0110[01.11] 0110[10.00] 1000

Redundant! Since, at leaf, NP0=P

1 1 1 1 1 1 0 01 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1

B11 B13B12

1 1 1 1 0 0 0 01 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0

0 0 0 0 1 1 1 10 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1

6 6 6 6 5 5 1 16 6 6 6 5 1 1 1 6 6 6 6 5 5 0 1 6 6 6 6 5 5 5 0 7 6 7 7 5 5 5 5 6 6 7 7 5 5 5 5 7 7 4 6 5 5 5 5 3 7 6 6 5 5 5 5

B1

Data Mining in Genomics

• There is (will be?) an explosion of gene expression data.

• Current emphasis is on extracting meaningful information from huge raw data sets.

•Methods employed are Clustering and Classification

Microarray data is most often represented as a relation G(Gid, T1, T2, ., Tn) where Gid is the gene identifier; T1…. Tn are the various treatments (or conditions) and the data values are gene expression levels. We will call this the " Gene Table”.

Currently, data-mining techniques concentrate on the Gene table, G(Gid, T1, T2, ., Tn) - specifically, on finding clusters of genes that exhibit similar expression patterns under selected treatments (clustering the gene table).

Gene Table

….….….….G4

….….….….G3

….….….….G2

….….….….G1

T4T3T2T1 Treatmt-ID

Gene-ID .

Using the Universal Relation approach to mining across different Microarray datasets, one can use a consistent Gene-id. Each Microarray will be embedded in a subquadrant. Therefore the data will be sparse and can be handled by Vector Implemented P-trees in which the prefix of the subquadrant can be listed only once:

P13 [01.10.11.11.01.00]qid NP0 P1[ ] 0111 0001[01] 1111 1110[10] 1110 0110[01.11] 0110[10.00] 1000

Example1: ANDing to get rc P1(6)

P1(6) = P1(110) = P111^P112^P013 = P11^P12^NP0”13

PM1(110)= P1(110) xor NP01(110) = P11^P12^NP0”13 xor NP011^NP012^P1”13

At [ ]: CNT[ ]=1-cnt*4level =1*42=16 since P1(110)[ ] = 1001^1000^1000=1000

PM1(110)[ ] = P11 ^ P12 ^NP0”13 xor NP011^NP012^P1”13

=1001^1000^ 1000 xor 1111 ^ 1010 ^1110 = 0010

At [10]: CNT[10]= 1-cnt*4level=0*41=0 since P1(110)[10]= 1101^1110^0001=0000

PM1(110)[10] = P11^P 12 ^NP0”13 xor NP011^NP012^P1”13

=1101^1110^0001 xor 1111^1111^1001= 0000 xor 1001=1001

At [10.00]: CNT=[10.00]1-cnt*4level=3*40=3 since P1(110)[10.00]= 1111^1111^0111=0111

At [10.11]: CNT=[10.11]1-cnt*4level=3*40=3 since P1(110)[10.11]= 1111^0111^1111=0111

Thus, rcP1(6) = 16 + 0 + 3 + 3 = 22

[10] only mixed child

[10.00], [10.11] mixed children

BpQid NP0 P111[ ] 1111 100112[ ] 1010 100013[ ] 0111 0001 11[01] 1011 001013[01] 1111 1110

11[01.00] 1110

11[01.11] 001013[01.11]

0110

11[10] 1111 110112[10] 1111 111013[10] 1110 0110

13[10.00] 1000

11[10.10] 1101

12[10.11] 0111

For P(p)= P(100- ---- , … , 011- ---- ): At each [..]1. swap and take bit comp of each [..]NP0V [..]P1V pair corresponding to 0-bits.2. AND the resulting vector-pairs. Result: [..]NP0V(p)[..]P1V(p). To get PMV(p) for the next level, 3. xor the two vectors.

ANDing in the NP0V-P1V Vector-Pair Format

For P(p)= P(110- ---- , … , ---- ---- ) (previous example, P1(6) at qid[ ] )

At each [..]1. swap and complement each [..]NP0V [..]P1V pair corresponding to 0-bits. Result denoted with *2. AND the resulting vector-pairs. Result: [..]NP0V(p)[..]P1V(p). To get PMV(p) for the next level, 3. xor the two vectors to get [..]PMV(p)

bit NP0V* P1V*1 1 1 1 1 1 0 0 11 1 0 1 0 1 0 0 00 1 1 1 0 1 0 0 0-----

…

-…-_____________________ 1 0 1 0 1 0 0 0

pos NP0V P1V1 1 1 1 1 1 0 0 12 1 0 1 0 1 0 0 03 0 1 1 1 0 0 0 1-----

…

-…-

NP0V P1V

p 1 0 1 0 1 0 0 0

PMV(p) = 0 0 1 0

Striping P-trees?

Assume 5-computer cluster; NodeC, Node00, Node01, Node10, Node11

Send to Nij if qid ends in ij:

BpQid NP0 P1 0011[01.00] 111013[10.00] 1000

BpQid NP0 P1 C11[ ] 1111 100112[ ] 1010 100013[ ] 0111 0001

BpQid NP0 P1 0111[01] 1011 001013[01] 1111 1110

BpQid NP0 P1 1011[10] 1111 110111[10.10] 110112[10] 1111 111013[10] 1110 0110

BpQid NP0 P1 1111[01.11] 001012[10.11] 011113[01.11] 0110

BpQid NP0 P111[ ] 1111 100112[ ] 1010 100013[ ] 0111 0001 11[01] 1011 001013[01] 1111 1110

11[01.00] 1110

11[01.11] 001013[01.11]

0110

11[10] 1111 110112[10] 1111 111013[10] 1110 0110

13[10.00] 1000

11[10.10] 1101

12[10.11] 0111

P11(110) = P111^P112^P013 = P11^P12^NP0”13 PM1(110) = P11^P12^NP0”13 xor NP011^NP012^P1”13

At NC: CNT[ ]=1-cnt*4level =1*42=16 since P1(110)[ ]= 1001^1000^1000=1000

PM1(110)[ ] =1001^1000^1000 xor 1111^1010^1110= 0010

At N10: CNT[10]= 1-cnt*4level=0*41=0 since P1(110)[10]= 1101^1110^0001=0000

PM1(110)[10] = 1101^1110^0001 xor 1111^1111^1001= 0000 xor 1001=1001

At N00: CNT=[10.00]1-cnt*4level=3*40=3 since P1(110)[10.00]= 1111^1111^0111=0111

At N11: CNT=[10.11]1-cnt*4level=3*40=3 since P1(110)[10.11]= 1111^0111^1111=0111

Every node sends accumulated CNT to C, where rcP1(6) = 16 + 0 + 3 + 3 = 22 calculated.

Striping P-trees?

qid NP0 P1[ ] 1111 1001[01] 1011 0010[10] 1111 1101[01.00] 1110[01.11] 0010[10.10] 1101

qid NP0 P1[ ] 1010 1000[10] 1111 1110[10.11] 0111

qid NP0 P1[ ] 0111 0001[01] 1111 1110[10] 1110 0110[01.11] 0110[10.00] 1000

P11 P12 P13

Alternatively, Send to Nodeij if qid starts with qid segment, ij. Is this better? How would the AND code be revised? AND performance?

OR: Send to Nodeij if the largest qid segment divisible by p is ij eg if p=4: [0]->0; [0.3]->0; [0.3.2]->0; [0.3.2.2]->2; [0.3.2.2.3]->2; [0.3.2.2.3.1]->2; [0.3.2.2.3.1.0]->2; [0.3.2.2.3.1.0]->2; [0.3.2.2.3.1.0.1]->1 etc.Similar to fanout 4. Implement by multicasting externally only every 4th segment. More generally, choose any increasing sequence, p=(p1..pL), define x p = {max pi x},then multicast [s1.s2…sk] --> Node k p

Bp qid NP0 P1 00

Bp qid NP0 P1 C11[ ] 1111 100112[ ] 1010 100013[ ] 0111 0001

Bp qid NP0 P1 0111[01] 1011 001011[01.00] 111011[01.11] 001013[01] 1111 111013[01.11] 0110

Bp qid NP0 P1 1011[10] 1111 110111[10.10] 110112[10] 1111 111012[10.11] 011113[10] 1110 011013[10.00] 1000 Bp qid NP0 P1 11

Example 1 (bottom-up)1 1 1 1 1 1 0 01 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1

B11

6 6 6 6 5 5 1 16 6 6 6 5 1 1 1 6 6 6 6 5 5 0 1 6 6 6 6 5 5 5 0 7 6 7 7 5 5 5 5 6 6 7 7 5 5 5 5 7 7 4 6 5 5 5 5 3 7 6 6 5 5 5 5

Band, B1, with 3-bit values

Bp qid NP0 P111[00.00] 1111

Bp qid NP0 P111[00.00] 111111[00.01] 1111

Bp qid NP0 P111[00.00] 111111[00.01] 111111[00.10] 1111

Bp qid NP0 P111[00.00] 111111[00.01] 111111[00.10] 111111[00.11] 1111

Bp qid NP0 P111[00] 0000 1111

Bp qid NP0 P111[00] 0000 111111[01.00] 1110

This ends the possibilityof a larger pure1 quad.So 00 can be installed inparent as a pure1.

Bp qid NP0 P111[01.00] 111011[01.01] 0000

Mixed leaf quad sent.Also ends possibilityparent is pure so it &all siblings are installedas bits in parent.

11[01.10] 1111

11[01.11] 0001

Mixed leaf quad sent.Ends parent so install bits in grandparent also

Node-00Node-00 Bp qid NP0 P111[01.00] 1110

Node-01Node-01 Bp qid NP0 P111[01] 1011 0010

Node-10Node-10 Bp qid NP0 P1


Node-CNode-C Bp qid NP0 P111[] 01__ 10__

Example 1 (bottom-up)1 1 1 1 1 1 0 01 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1

B11

6 6 6 6 5 5 1 16 6 6 6 5 1 1 1 6 6 6 6 5 5 0 1 6 6 6 6 5 5 5 0 7 6 7 7 5 5 5 5 6 6 7 7 5 5 5 5 7 7 4 6 5 5 5 5 3 7 6 6 5 5 5 5

Band, B1, with 3-bit values

Bp qid NP0 P111[10.00] 1111

Bp qid NP0 P111[10.00] 111111[10.01] 1111

Bp qid NP0 P111[10.00] 111111[10.01] 111111[10.10] 110111[10.11] 1111

Bp qid NP0 P111[11.00] 111111[11.01] 111111[11.10] 111111[11.11] 1111

Bp qid NP0 P111[11] 0000 1111


Node-01Node-01 Bp qid NP0 P111[01] 1011 0010

Node-10Node-10 Bp qid NP0 P111[10.10] 110111[10] 1111 1101


Node-CNode-C Bp qid NP0 P111[] 0111 1001

Ends the possibilityof a larger pure1 quad.All can be installed inparent/grandparentas a 1-bit.10.10 can be installed.

Ends quad-11.All can be installed inParent as a 1-bit.

Bottom-up bottom-line: Since it is better to use 2-D than 3-D (higher compression), it should be better to use 1-D than 2-D? This should be investigated.

Example2

B1 B11 B12 B13

6 6 6 6 5 5 1 16 6 6 6 5 1 1 1 6 6 6 6 5 6 6 6 6 5 5 0 7 6 7 7 5 5 5 5 6 6 7 7 5 5 5 7 7 4 6 5

1 1 1 1 1 1 0 01 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 0 0 0 01 1 1 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0

0 0 0 0 1 1 1 10 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1

B2 B21 B22 B23

4 4 4 4 3 2 1 14 4 4 2 3 2 1 1 3 3 2 2 3 3 3 2 2 3 3 2 3 6 6 6 2 2 2 2 6 6 7 7 2 2 2 6 6 5 3 2

1 1 1 1 0 0 0 01 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0

0 0 0 0 1 1 0 00 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1

0 0 0 0 1 0 1 10 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0

X, Y, B1, B2

000 000 6 4 000 001 6 4 000 010 6 4 000 011 6 4 000 100 5 3 000 101 5 2 000 110 1 1 000 111 1 1 001 000 6 4 001 001 6 4 001 010 6 4 001 011 6 2 001 100 5 3 001 101 1 2 001 110 1 1 001 111 1 1 010 000 6 3 010 001 6 3 010 010 6 2 010 011 6 2 010 100 5 3 011 000 6 3 011 001 6 3 011 010 6 2 011 011 6 2 011 100 5 3 011 101 5 3 011 111 0 2 100 111 5 2 100 000 7 3 100 001 6 6 100 010 7 6 100 011 7 6 100 100 5 2 100 101 5 2 100 110 5 2 101 000 6 6 101 001 6 6 101 010 7 7 101 011 7 7 101 100 5 2 101 101 5 2 101 110 5 2 110 000 7 6 110 001 7 6 110 010 4 5 110 011 6 3 110 100 5 2

Example2

Example2: StripingX, Y, B1, B2

000 000 6 4 000 001 6 4 000 010 6 4 000 011 6 4 000 100 5 3 000 101 5 2 000 110 1 1 000 111 1 1 001 000 6 4 001 001 6 4 001 010 6 4 001 011 6 2 001 100 5 3 001 101 1 2 001 110 1 1 001 111 1 1 010 000 6 3 010 001 6 3 010 010 6 2 010 011 6 2 010 100 5 3 011 000 6 3 011 001 6 3 011 010 6 2 011 011 6 2 011 100 5 3 011 101 5 3 011 111 0 2 100 111 5 2 100 000 7 3 100 001 6 6 100 010 7 6 100 011 7 6 100 100 5 2 100 101 5 2 100 110 5 2 101 000 6 6 101 001 6 6 101 010 7 7 101 011 7 7 101 100 5 2 101 101 5 2 101 110 5 2 110 000 7 6 110 001 7 6 110 010 4 5 110 011 6 3 110 100 5 2

0 0 0 0 0 0 1 1 0 1 0 00 0 0 0 0 1 1 1 0 1 0 00 0 0 0 1 0 1 1 0 1 0 00 0 0 0 1 1 1 1 0 1 0 00 0 0 1 0 0 1 0 1 0 1 10 0 0 1 0 1 1 0 1 0 1 00 0 0 1 1 0 0 0 1 0 0 10 0 0 1 1 1 0 0 1 0 0 10 0 1 0 0 0 1 1 0 1 0 00 0 1 0 0 1 1 1 0 1 0 00 0 1 0 1 0 1 1 0 1 0 00 0 1 0 1 1 1 1 0 0 1 00 0 1 1 0 0 1 0 1 0 1 10 0 1 1 0 1 0 0 1 0 1 00 0 1 1 1 0 0 0 1 0 0 10 0 1 1 1 1 0 0 1 0 0 10 1 0 0 0 0 1 1 0 0 1 10 1 0 0 0 1 1 1 0 0 1 10 1 0 0 1 0 1 1 0 0 1 00 1 0 0 1 1 1 1 0 0 1 00 1 0 1 0 0 1 0 1 0 1 10 1 1 0 0 0 1 1 0 0 1 10 1 1 0 0 1 1 1 0 0 1 10 1 1 0 1 0 1 1 0 0 1 00 1 1 0 1 1 1 1 0 0 1 00 1 1 1 0 0 1 0 1 0 1 10 1 1 1 0 1 1 0 1 0 1 10 1 1 1 1 1 0 0 0 0 1 01 0 0 0 0 0 1 1 1 0 1 11 0 0 0 0 1 1 1 0 1 1 01 0 0 0 1 0 1 1 1 1 1 01 0 0 0 1 1 1 1 1 1 1 01 0 0 1 0 0 1 0 1 0 1 01 0 0 1 0 1 1 0 1 0 1 01 0 0 1 1 0 1 0 1 0 1 01 0 0 1 1 1 1 0 1 0 1 01 0 1 0 0 0 1 1 0 1 1 01 0 1 0 0 1 1 1 0 1 1 01 0 1 0 1 0 1 1 1 1 1 11 0 1 0 1 1 1 1 1 1 1 11 0 1 1 0 0 1 0 1 0 1 01 0 1 1 0 1 1 0 1 0 1 01 0 1 1 1 0 1 0 1 0 1 01 1 0 0 0 0 1 1 1 1 1 01 1 0 0 0 1 1 1 1 1 1 01 1 0 0 1 0 1 0 0 1 0 11 1 0 0 1 1 1 1 0 0 1 11 1 0 1 0 0 1 0 1 0 1 0

X, Y, B11B12B13B21B22B23

0 0 0 0 0 0 1 1 0 1 0 00 0 0 0 0 1 1 1 0 1 0 00 0 0 0 1 0 1 1 0 1 0 00 0 0 0 1 1 1 1 0 1 0 00 0 0 1 0 0 1 1 0 1 0 00 0 0 1 0 1 1 1 0 1 0 00 0 0 1 1 0 1 1 0 1 0 00 0 0 1 1 1 1 1 0 0 1 00 0 1 0 0 0 1 1 0 0 1 10 0 1 0 0 1 1 1 0 0 1 10 0 1 0 1 0 1 1 0 0 1 10 0 1 0 1 1 1 1 0 0 1 10 0 1 1 0 0 1 1 0 0 1 00 0 1 1 0 1 1 1 0 0 1 00 0 1 1 1 0 1 1 0 0 1 00 0 1 1 1 1 1 1 0 0 1 00 1 0 0 0 0 1 0 1 0 1 10 1 0 0 0 1 1 0 1 0 1 00 1 0 0 1 0 1 0 1 0 1 10 1 0 0 1 1 0 0 1 0 1 00 1 0 1 0 0 0 0 1 0 0 10 1 0 1 0 1 0 0 1 0 0 10 1 0 1 1 0 0 0 1 0 0 10 1 0 1 1 1 0 0 1 0 0 10 1 1 0 0 0 1 0 1 0 1 10 1 1 0 1 0 1 0 1 0 1 10 1 1 0 1 1 1 0 1 0 1 10 1 1 1 1 1 0 0 0 0 1 01 0 0 0 0 0 1 1 1 0 1 11 0 0 0 0 1 1 1 0 1 1 01 0 0 0 1 0 1 1 0 1 1 01 0 0 0 1 1 1 1 0 1 1 01 0 0 1 0 0 1 1 1 1 1 01 0 0 1 0 1 1 1 1 1 1 01 0 0 1 1 0 1 1 1 1 1 11 0 0 1 1 1 1 1 1 1 1 11 0 1 0 0 0 1 1 1 1 1 01 0 1 0 0 1 1 1 1 1 1 01 0 1 1 0 0 1 0 0 1 0 11 0 1 1 0 1 1 1 0 0 1 11 1 0 0 0 0 1 0 1 0 1 01 1 0 0 0 1 1 0 1 0 1 01 1 0 0 1 0 1 0 1 0 1 01 1 0 0 1 1 1 0 1 0 1 01 1 0 1 0 0 1 0 1 0 1 01 1 0 1 0 1 1 0 1 0 1 01 1 0 1 1 0 1 0 1 0 1 01 1 1 0 0 0 1 0 1 0 1 0

x1y1x2y2x3y3 B11B12B13B21B22B23

__PNP0V_ __P1V__ Band111 222 111 222bit-pos123 123 123 123[ ] === === === === 110 111 110 000 101 011 000 000 111 111 100 000 101 010 101 010

00_PNP0V__ __P1V__ 110 111 110 000

11_PNP0V__ __P1V__ 101 010 101 010

01_PNP0V__ __P1V__ 101 011 000 000

10_PNP0V__ __P1V__ 111 111 100 000

Send B21B22B23 to Node00

Send B11B13 B22B23 to Node01

Send B12B13 B21B22B23 to Node10

Send nothing to Node11

Bp qid NP0 P1 C11[ ] 1111 101112[ ] 1010 100013[ ] 0111 000121[ ] 1010 000022[ ] 1111 000123[ ] 1110 0000

Purity Template[ ] 16 12 12 8

Raster order Peano order

OR for PNP0AND for P1

Example2: striping at Node 00

0 0 0 0 0 0 1 0 00 0 0 0 0 1 1 0 00 0 0 0 1 0 1 0 00 0 0 0 1 1 1 0 0

0 0 0 1 0 0 1 0 00 0 0 1 0 1 1 0 00 0 0 1 1 0 1 0 00 0 0 1 1 1 0 1 0

0 0 1 0 0 0 0 1 10 0 1 0 0 1 0 1 10 0 1 0 1 0 0 1 10 0 1 0 1 1 0 1 1

0 0 1 1 0 0 0 1 00 0 1 1 0 1 0 1 00 0 1 1 1 0 0 1 00 0 1 1 1 1 0 1 0

x1y1x2y2x3y3B11B12B13 B21B22B23

_PNP0V__ __P1V__ 110 100 110 100

_PNP0V__ __P1V__ 110 010 110 010

_PNP0V__ __P1V__ 110 110 110 000

_PNP0V__ __P1V__ 110 011 110 011




_PNP0V__ __P1___Band 111 222 111 222bit-pos 123 123 123 123[00 ] === === === === 100 100 110 000 011 011 010 010

Send [ ]B21B22 to Node01

Bp qid NP0 P1 0021[00 ] 1100 100022[00 ] 0111 001123[00 ] 0010 0010PurityTemplate [00] 4 4 4 411[01.00 ] 111023[01.00 ] 1010

12[10.00 ] 111113[10.00 ] 100021[10.00 ] 011122[10.00 ] 111123[10.00 ] 1000

0 1 0 0 0 0 1 10 1 0 0 0 1 1 00 1 0 0 1 0 1 10 1 0 0 1 1 0 0

x1y1x2y2x3y3 B11 B23

From [01 ]

P1Band 12bit-pos 13[01.00 ] == 11 10 11 00

To [01 ]

1 0 0 0 0 0 1 1 0 1 11 0 0 0 0 1 1 0 1 1 01 0 0 0 1 0 1 0 1 1 01 0 0 0 1 1 1 0 1 1 0

x1y1x2y2x3y3 B12B12 B23B23B23

From [10 ]

P1Band 11 222bit-pos 23 123[10.00 ] == === 11 011 10 110 10 110 10 110

Bp qid NP0 P1 0012[10.00 ] 1111

Bp qid NP0 P1 0013[10.00 ] 1000

Bp qid NP0 P1 0021[00 ] 1100 100021[10.00 ] 0111

Bp qid NP0 P1 0022[00 ] 0111 001122[10.00 ] 1111

Bp qid NP0 P1 0023[00 ] 0010 001023[01.00 ] 101023[10.00 ] 1000

Bp qid NP0 P1 0011[01.00 ] 1110

Pages on disk


0 1 0 0 0 0 1 1 1 10 1 0 0 0 1 1 1 1 00 1 0 0 1 0 1 1 1 10 1 0 0 1 1 0 1 1 0

0 1 0 1 0 0 0 1 0 10 1 0 1 0 1 0 1 0 10 1 0 1 1 0 0 1 0 10 1 0 1 1 1 0 1 0 1

0 1 1 0 0 0 1 1 1 10 1 1 0 1 0 1 1 1 10 1 1 0 1 1 1 1 1 1

0 1 1 1 1 1 0 0 1 0

x1y1x2y2x3y3 B11 B13 B22B23

_PNP0V__ __P1V__ 1 1 11 0 1 10

_PNP0V__ __P1V__ 0 0 10 0 0 10

_PNP0V__ __P1V__ 0 1 01 0 1 01

_PNP0V__ __P1V__ 1 1 11 1 1 11

Send [01]B11B23 to Node00




_PNP0V__ __P1___Band 111 222 111 222bit-pos 123 123 123 123[01 ] === === === === 1 1 11 0 1 10 0 1 01 0 1 01 1 1 11 1 1 11 0 0 10 0 0 10

0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0 1

x1y1x2y2x3y3 B21B22

From [00 ]

P1Band 22bit-pos 12[00.01 ] == 10 10 10 01

To [00 ]

1 0 0 1 0 0 01 0 0 1 0 1 01 0 0 1 1 0 11 0 0 1 1 1 1

x1y1x2y2x3y3 B23

From [10 ]

P1Band 2bit-pos 3[10.01 ] == 0 0 1 1

Bp qid NP0 P1 0121[00.01 ] 1110

Bp qid NP0 P1 0123[01 ] 1110 011023[10.01 ] 0011

Bp qid NP0 P1 0122[01 ] 1010 101022[00.01 ] 0001

Bp qid NP0 P1 0113[01 ] 1110 1110

Bp qid NP0 P1 0111[01 ] 1010 0010

Bp qid NP0 P1 0111[01 ] 1010 001013[01 ] 1110 111022[01 ] 1010 101023[01 ] 1110 0110PurityTemplate [01] 4 4 3 121[00.01 ] 111022[00.01 ] 0001

23[10.01 ] 0011

Pages on disk


1 0 0 0 0 0 1 1 0 1 11 0 0 0 0 1 1 0 1 1 01 0 0 0 1 0 1 0 1 1 01 0 0 0 1 1 1 0 1 1 0

1 0 0 1 0 0 1 1 1 1 01 0 0 1 0 1 1 1 1 1 01 0 0 1 1 0 1 1 1 1 11 0 0 1 1 1 1 1 1 1 1

1 0 1 0 0 0 1 1 1 1 01 0 1 0 0 1 1 1 1 1 0

1 0 1 1 0 0 0 0 1 0 11 0 1 1 0 1 1 0 0 1 1

x1y1x2y2x3y3 B12B13B21B22B23

_PNP0V__ __P1V__ 11 111 10 010

_PNP0V__ __P1V__ 10 111 00 001

_PNP0V__ __P1V__ 11 111 11 110

_PNP0V__ __P1V__ 11 110 11 110

Send [10]B13B21B23 to Node00


Send [10]B12B21B22 to Node11

Send [10] B23 to Node01

_PNP0V__ __P1___Band 111 222 111 222bit-pos 123 123 123 123[10 ] === === === === 11 111 10 010 11 111 11 110 11 110 11 110 10 111 00 001

To [00 ] To[01 ]

To [11 ]

Pages on diskBp qid NP0 P1 1012[10 ] 1111 1110

Bp qid NP0 P1 1013[10 ] 1110 0110

Bp qid NP0 P1 1021[10 ] 1111 0110

Bp qid NP0 P1 1022[10 ] 1111 1110

Bp qid NP0 P1 1023[10 ] 1101 0001

Bp qid NP0 P1 1012[10 ] 1111 111013[10 ] 1110 011021[10 ] 1111 011022[10 ] 1111 111023[10 ] 1101 0001PurityTemplate [10] 4 4 2 2

Example2: striping at Node11

1 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1 0 1

x1y1x2y2x3y3 B12 B21B22

From [10 ]

P1Band 122bit-pos 223[10.11 ] === 010 101

Bp qid NP0 P1 1112[10.11 ] 0122[10.11 ] 1023[10.11 ] 01

Bp qid NP0 P1 1112[10.11 ] 01

Bp qid NP0 P1 1123[10.11 ] 01

Bp qid NP0 P1 1122[10.11 ] 10

Pages on disk

Example2.1AND at NodeC or [ ]

Bp qid NP0 P112[10.11 ] 01

Bp qid NP0 P123[10.11 ] 01

Bp qid NP0 P122[10.11 ] 10

Disk 11

Bp qid NP0 P112[10 ] 1111 1110

Bp qid NP0 P1 13[10 ] 1110 0110

Bp qid NP0 P1 21[10 ] 1111 0110

Bp qid NP0 P122[10 ] 1111 1110

Bp qid NP0 P123[10 ] 1101 0001

Bp qid NP0 P1 21[00.01 ] 1110

Bp qid NP0 P1 23[01 ] 1110 011023[10.01 ] 0011

Bp qid NP0 P1 22[01 ] 1010 101022[00.01 ] 0001

Bp qid NP0 P1 13[01 ] 1110 1110

Bp qid NP0 P1 11[01 ] 1010 0010

Bp qid NP0 P112[10.00 ] 1111

Bp qid NP0 P113[10.00 ] 1000

Bp qid NP0 P121[00 ] 1100 100021[10.00 ] 0111

Bp qid NP0 P122[00 ] 0111 001122[10.00 ] 1111

Bp qid NP0 P123[00 ] 0010 001023[01.00] 101023[10.00] 1000

Bp qid NP0 P111[01.00 ] 1110

Disk 10 PT[10] 4 4 2 2Disk 01 PT[01] 4 4 3 1Disk 00 PT[00] 4 4 4 4Disk C PT[ ] 16 12 12 8

RC(P 101,010) = P11^ P’12^ P13^ P’21^ P22^ P’23

Bp qid NP0 P1 C11[ ] 1111 101112[ ] 1010 100013[ ] 0111 000121[ ] 1010 000022[ ] 1111 000123[ ] 1110 0000

[]NP0111101110111111111111111------AND0111

[]P1101101010001010100010001------AND0001

Sum= 8 so far. Invocation= [ ] 101,010 send to Nodes 01, 10

P1-pattern NP0 P111 xxxx12 prime13 xxxx21 prime22 xxxx23 prime

NP0-pattern NP0 P111 xxxx12 prime13 xxxx21 prime22 xxxx23 prime

Example2.1AND at Node01

Bp qid NP0 P112[10.11 ] 01

Bp qid NP0 P123[10.11 ] 01

Bp qid NP0 P122[10.11 ] 10

Disk 11

Bp qid NP0 P112[10 ] 1111 1110

Bp qid NP0 P1 13[10 ] 1110 0110

Bp qid NP0 P1 21[10 ] 1111 0110

Bp qid NP0 P122[10 ] 1111 1110

Bp qid NP0 P123[10 ] 1101 0001

Bp qid NP0 P1 21[00.01 ] 1110

Bp qid NP0 P1 23[01 ] 1110 011023[10.01 ] 0011

Bp qid NP0 P1 22[01 ] 1010 101022[00.01 ] 0001

Bp qid NP0 P1 13[01 ] 1110 1110

Bp qid NP0 P1 11[01 ] 1010 0010

Bp qid NP0 P112[10.00 ] 1111

Bp qid NP0 P113[10.00 ] 1000

Bp qid NP0 P121[00 ] 1100 100021[10.00 ] 0111

Bp qid NP0 P122[00 ] 0111 001122[10.00 ] 1111

Bp qid NP0 P123[00 ] 0010 001023[01.00] 101023[10.00] 1000

Bp qid NP0 P111[01.00 ] 1110

Bp qid NP0 P1 C11[ ] 1111 101112[ ] 1010 100013[ ] 0111 000121[ ] 1010 000022[ ] 1111 000123[ ] 1110 0000

Invocation= [01] 101,010Sent to Node00

[01] NP011 101012 13 111021 22 101023 1001AND------ 1000

[01] P111 001012 13 111021 22 101023 0001AND------ 0000



[ ] 101,010 received



Bp qid NP0 P112[10.11 ] 01

Bp qid NP0 P123[10.11 ] 01

Bp qid NP0 P122[10.11 ] 10

Disk 11

Bp qid NP0 P112[10 ] 1111 1110

Bp qid NP0 P1 13[10 ] 1110 0110

Bp qid NP0 P1 21[10 ] 1111 0110

Bp qid NP0 P122[10 ] 1111 1110

Bp qid NP0 P123[10 ] 1101 0001

Bp qid NP0 P1 21[00.01 ] 1110

Bp qid NP0 P1 23[01 ] 1110 011023[10.01 ] 0011

Bp qid NP0 P1 22[01 ] 1010 101022[00.01 ] 0001

Bp qid NP0 P1 13[01 ] 1110 1110

Bp qid NP0 P1 11[01 ] 1010 0010

Bp qid NP0 P112[10.00 ] 1111

Bp qid NP0 P113[10.00 ] 1000

Bp qid NP0 P121[00 ] 1100 100021[10.00 ] 0111

Bp qid NP0 P122[00 ] 0111 001122[10.00 ] 1111

Bp qid NP0 P123[00 ] 0010 001023[01.00] 101023[10.00] 1000

Bp qid NP0 P111[01.00 ] 1110

Bp qid NP0 P1 C11[ ] 1111 101112[ ] 1010 100013[ ] 0111 000121[ ] 1010 000022[ ] 1111 000123[ ] 1110 0000

Invocation= [10] 101,010Sent nowhere (no mixed)

[10] NP011 12 0001 13 111021 100122 111123 1110AND------ 0000

[10] P111 12 13 21 22 23 AND------






Bp qid NP0 P112[10.11 ] 01

Bp qid NP0 P123[10.11 ] 01

Bp qid NP0 P122[10.11 ] 10

Disk 11

Bp qid NP0 P112[10 ] 1111 1110

Bp qid NP0 P1 13[10 ] 1110 0110

Bp qid NP0 P1 21[10 ] 1111 0110

Bp qid NP0 P122[10 ] 1111 1110

Bp qid NP0 P123[10 ] 1101 0001

Bp qid NP0 P1 21[00.01 ] 1110

Bp qid NP0 P1 23[01 ] 1110 011023[10.01 ] 0011

Bp qid NP0 P1 22[01 ] 1010 101022[00.01 ] 0001

Bp qid NP0 P1 13[01 ] 1110 1110

Bp qid NP0 P1 11[01 ] 1010 0010

Bp qid NP0 P112[10.00 ] 1111

Bp qid NP0 P113[10.00 ] 1000

Bp qid NP0 P121[00 ] 1100 100021[10.00 ] 0111

Bp qid NP0 P122[00 ] 0111 001122[10.00 ] 1111

Bp qid NP0 P123[00 ] 0010 001023[01.00] 101023[10.00] 1000

Bp qid NP0 P111[01.00 ] 1110

Bp qid NP0 P1 C11[ ] 1111 101112[ ] 1010 100013[ ] 0111 000121[ ] 1010 000022[ ] 1111 000123[ ] 1110 0000

Sum=1, sent to NodeC gives a

sum total of 8 + 1 = 9

[01.00] P111 111012 13 21 22 23 0101AND------ 0100


[01] 101,010 received

P1-pattern P111 xxxx12 prime13 xxxx21 prime22 xxxx23 prime

Example2.2AND at NodeC or [ ]

Bp qid NP0 P112[10.11 ] 01

Bp qid NP0 P123[10.11 ] 01

Bp qid NP0 P122[10.11 ] 10

Disk 11

Bp qid NP0 P112[10 ] 1111 1110

Bp qid NP0 P1 13[10 ] 1110 0110

Bp qid NP0 P1 21[10 ] 1111 0110

Bp qid NP0 P122[10 ] 1111 1110

Bp qid NP0 P123[10 ] 1101 0001

Bp qid NP0 P1 21[00.01 ] 1110

Bp qid NP0 P1 23[01 ] 1110 011023[10.01 ] 0011

Bp qid NP0 P1 22[01 ] 1010 101022[00.01 ] 0001

Bp qid NP0 P1 13[01 ] 1110 1110

Bp qid NP0 P1 11[01 ] 1010 0010

Bp qid NP0 P112[10.00 ] 1111

Bp qid NP0 P113[10.00 ] 1000

Bp qid NP0 P121[00 ] 1100 100021[10.00 ] 0111

Bp qid NP0 P122[00 ] 0111 001122[10.00 ] 1111

Bp qid NP0 P123[00 ] 0010 001023[01.00] 101023[10.00] 1000

Bp qid NP0 P111[01.00 ] 1110


RC(P 100,101) = P11^ P’12^ P’13^ P21^ P’22^ P23

Bp qid NP0 P1 C11[ ] 1111 101112[ ] 1010 100013[ ] 0111 000121[ ] 1010 000022[ ] 1111 000123[ ] 1110 0000

[]NP0------AND0010

[]P1------AND0000

Sum= 0 so far. Invocation= [ ] 100, 101 send to Node 10

P1-pattern NP0 P111 xxxx12 prime13 prime21 xxxx22 prime23 xxxx

NP0-pattern NP0 P111 xxxx12 prime13 prime21 xxxx22 prime23 xxxx


Bp qid NP0 P112[10.11 ] 01

Bp qid NP0 P123[10.11 ] 01

Bp qid NP0 P122[10.11 ] 10

Disk 11

Bp qid NP0 P112[10 ] 1111 1110

Bp qid NP0 P1 13[10 ] 1110 0110

Bp qid NP0 P1 21[10 ] 1111 0110

Bp qid NP0 P122[10 ] 1111 1110

Bp qid NP0 P123[10 ] 1101 0001

Bp qid NP0 P1 21[00.01 ] 1110

Bp qid NP0 P1 23[01 ] 1110 011023[10.01 ] 0011

Bp qid NP0 P1 22[01 ] 1010 101022[00.01 ] 0001

Bp qid NP0 P1 13[01 ] 1110 1110

Bp qid NP0 P1 11[01 ] 1010 0010

Bp qid NP0 P112[10.00 ] 1111

Bp qid NP0 P113[10.00 ] 1000

Bp qid NP0 P121[00 ] 1100 100021[10.00 ] 0111

Bp qid NP0 P122[00 ] 0111 001122[10.00 ] 1111

Bp qid NP0 P123[00 ] 0010 001023[01.00] 101023[10.00] 1000

Bp qid NP0 P111[01.00 ] 1110

Bp qid NP0 P1 C11[ ] 1111 101112[ ] 1010 100013[ ] 0111 000121[ ] 1010 000022[ ] 1111 000123[ ] 1110 0000

Invocation= [10] 100, 101Sent to Node 11

[10] NP011 12 13 21 22 23 AND------ 0001

[10] P111 12 13 21 22 23 AND------ 0000



P1-pattern NP0 P111 xxxx12 prime13 prime21 xxxx22 prime23 xxxx

NP0-pattern NP0 P111 xxxx12 prime13 prime21 xxxx22 prime23 xxxx


Bp qid NP0 P112[10.11 ] 01

Bp qid NP0 P123[10.11 ] 01

Bp qid NP0 P122[10.11 ] 10

Disk 11

Bp qid NP0 P112[10 ] 1111 1110

Bp qid NP0 P1 13[10 ] 1110 0110

Bp qid NP0 P1 21[10 ] 1111 0110

Bp qid NP0 P122[10 ] 1111 1110

Bp qid NP0 P123[10 ] 1101 0001

Bp qid NP0 P1 21[00.01 ] 1110

Bp qid NP0 P1 23[01 ] 1110 011023[10.01 ] 0011

Bp qid NP0 P1 22[01 ] 1010 101022[00.01 ] 0001

Bp qid NP0 P1 13[01 ] 1110 1110

Bp qid NP0 P1 11[01 ] 1010 0010

Bp qid NP0 P112[10.00 ] 1111

Bp qid NP0 P113[10.00 ] 1000

Bp qid NP0 P121[00 ] 1100 100021[10.00 ] 0111

Bp qid NP0 P122[00 ] 0111 001122[10.00 ] 1111

Bp qid NP0 P123[00 ] 0010 001023[01.00] 101023[10.00] 1000

Bp qid NP0 P111[01.00 ] 1110

Bp qid NP0 P1 C11[ ] 1111 101112[ ] 1010 100013[ ] 0111 000121[ ] 1010 000022[ ] 1111 000123[ ] 1110 0000

[10] P111 0112 13 21 22 0123 01AND------ 01

[10] 100,101 received


Sum=1, sent to NodeC gives a sum total of 1

Example2, bottom-up

0 0 0 0 0 0 1 1 0 1 0 00 0 0 0 0 1 1 1 0 1 0 00 0 0 0 1 0 1 1 0 1 0 00 0 0 0 1 1 1 1 0 1 0 00 0 0 1 0 0 1 1 0 1 0 00 0 0 1 0 1 1 1 0 1 0 00 0 0 1 1 0 1 1 0 1 0 00 0 0 1 1 1 1 1 0 0 1 00 0 1 0 0 0 1 1 0 0 1 10 0 1 0 0 1 1 1 0 0 1 10 0 1 0 1 0 1 1 0 0 1 10 0 1 0 1 1 1 1 0 0 1 10 0 1 1 0 0 1 1 0 0 1 00 0 1 1 0 1 1 1 0 0 1 00 0 1 1 1 0 1 1 0 0 1 00 0 1 1 1 1 1 1 0 0 1 00 1 0 0 0 0 1 0 1 0 1 10 1 0 0 0 1 1 0 1 0 1 00 1 0 0 1 0 1 0 1 0 1 10 1 0 0 1 1 0 0 1 0 1 00 1 0 1 0 0 0 0 1 0 0 10 1 0 1 0 1 0 0 1 0 0 10 1 0 1 1 0 0 0 1 0 0 10 1 0 1 1 1 0 0 1 0 0 10 1 1 0 0 0 1 0 1 0 1 10 1 1 0 1 0 1 0 1 0 1 10 1 1 0 1 1 1 0 1 0 1 10 1 1 1 1 1 0 0 0 0 1 01 0 0 0 0 0 1 1 1 0 1 11 0 0 0 0 1 1 1 0 1 1 01 0 0 0 1 0 1 1 0 1 1 01 0 0 0 1 1 1 1 0 1 1 01 0 0 1 0 0 1 1 1 1 1 01 0 0 1 0 1 1 1 1 1 1 01 0 0 1 1 0 1 1 1 1 1 11 0 0 1 1 1 1 1 1 1 1 11 0 1 0 0 0 1 1 1 1 1 01 0 1 0 0 1 1 1 1 1 1 01 0 1 1 0 0 1 0 0 1 0 11 0 1 1 0 1 1 1 0 0 1 11 1 0 0 0 0 1 0 1 0 1 01 1 0 0 0 1 1 0 1 0 1 01 1 0 0 1 0 1 0 1 0 1 01 1 0 0 1 1 1 0 1 0 1 01 1 0 1 0 0 1 0 1 0 1 01 1 0 1 0 1 1 0 1 0 1 01 1 0 1 1 0 1 0 1 0 1 01 1 1 0 0 0 1 0 1 0 1 0

x1y1x2y2x3y3 B11B12B13B21B22B23

Bp qid NP0 P111[00.00] 111112[00.00] 111113[00.00] 000021[00.00] 111122[00.00] 000023[00.00] 0000

Peano order

Example2, bottom-up

0 0 0 0 0 0 1 1 0 1 0 00 0 0 0 0 1 1 1 0 1 0 00 0 0 0 1 0 1 1 0 1 0 00 0 0 0 1 1 1 1 0 1 0 00 0 0 1 0 0 1 1 0 1 0 00 0 0 1 0 1 1 1 0 1 0 00 0 0 1 1 0 1 1 0 1 0 00 0 0 1 1 1 1 1 0 0 1 00 0 1 0 0 0 1 1 0 0 1 10 0 1 0 0 1 1 1 0 0 1 10 0 1 0 1 0 1 1 0 0 1 10 0 1 0 1 1 1 1 0 0 1 10 0 1 1 0 0 1 1 0 0 1 00 0 1 1 0 1 1 1 0 0 1 00 0 1 1 1 0 1 1 0 0 1 00 0 1 1 1 1 1 1 0 0 1 00 1 0 0 0 0 1 0 1 0 1 10 1 0 0 0 1 1 0 1 0 1 00 1 0 0 1 0 1 0 1 0 1 10 1 0 0 1 1 0 0 1 0 1 00 1 0 1 0 0 0 0 1 0 0 10 1 0 1 0 1 0 0 1 0 0 10 1 0 1 1 0 0 0 1 0 0 10 1 0 1 1 1 0 0 1 0 0 10 1 1 0 0 0 1 0 1 0 1 10 1 1 0 1 0 1 0 1 0 1 10 1 1 0 1 1 1 0 1 0 1 10 1 1 1 1 1 0 0 0 0 1 01 0 0 0 0 0 1 1 1 0 1 11 0 0 0 0 1 1 1 0 1 1 01 0 0 0 1 0 1 1 0 1 1 01 0 0 0 1 1 1 1 0 1 1 01 0 0 1 0 0 1 1 1 1 1 01 0 0 1 0 1 1 1 1 1 1 01 0 0 1 1 0 1 1 1 1 1 11 0 0 1 1 1 1 1 1 1 1 11 0 1 0 0 0 1 1 1 1 1 01 0 1 0 0 1 1 1 1 1 1 01 0 1 1 0 0 1 0 0 1 0 11 0 1 1 0 1 1 1 0 0 1 11 1 0 0 0 0 1 0 1 0 1 01 1 0 0 0 1 1 0 1 0 1 01 1 0 0 1 0 1 0 1 0 1 01 1 0 0 1 1 1 0 1 0 1 01 1 0 1 0 0 1 0 1 0 1 01 1 0 1 0 1 1 0 1 0 1 01 1 0 1 1 0 1 0 1 0 1 01 1 1 0 0 0 1 0 1 0 1 0

x1y1x2y2x3y3 B11B12B13B21B22B23

Bp qid NP0 P111[00.00] 111111[00.01] 1111

12[00.00] 111112[00.01] 1111

13[00.00] 000013[00.01] 0000

21[00.00] 111121[00.01] 1110

22[00.00] 000022[00.01] 0001

23[00.00] 000023[00.01] 0000

Peano order

Mixed quads (can be sent to node01)

Bp qid NP0 P121[00.01] 111022[00.01] 0001

Example2, bottom-up

0 0 0 0 0 0 1 1 0 1 0 00 0 0 0 0 1 1 1 0 1 0 00 0 0 0 1 0 1 1 0 1 0 00 0 0 0 1 1 1 1 0 1 0 00 0 0 1 0 0 1 1 0 1 0 00 0 0 1 0 1 1 1 0 1 0 00 0 0 1 1 0 1 1 0 1 0 00 0 0 1 1 1 1 1 0 0 1 00 0 1 0 0 0 1 1 0 0 1 10 0 1 0 0 1 1 1 0 0 1 10 0 1 0 1 0 1 1 0 0 1 10 0 1 0 1 1 1 1 0 0 1 10 0 1 1 0 0 1 1 0 0 1 00 0 1 1 0 1 1 1 0 0 1 00 0 1 1 1 0 1 1 0 0 1 00 0 1 1 1 1 1 1 0 0 1 00 1 0 0 0 0 1 0 1 0 1 10 1 0 0 0 1 1 0 1 0 1 00 1 0 0 1 0 1 0 1 0 1 10 1 0 0 1 1 0 0 1 0 1 00 1 0 1 0 0 0 0 1 0 0 10 1 0 1 0 1 0 0 1 0 0 10 1 0 1 1 0 0 0 1 0 0 10 1 0 1 1 1 0 0 1 0 0 10 1 1 0 0 0 1 0 1 0 1 10 1 1 0 1 0 1 0 1 0 1 10 1 1 0 1 1 1 0 1 0 1 10 1 1 1 1 1 0 0 0 0 1 01 0 0 0 0 0 1 1 1 0 1 11 0 0 0 0 1 1 1 0 1 1 01 0 0 0 1 0 1 1 0 1 1 01 0 0 0 1 1 1 1 0 1 1 01 0 0 1 0 0 1 1 1 1 1 01 0 0 1 0 1 1 1 1 1 1 01 0 0 1 1 0 1 1 1 1 1 11 0 0 1 1 1 1 1 1 1 1 11 0 1 0 0 0 1 1 1 1 1 01 0 1 0 0 1 1 1 1 1 1 01 0 1 1 0 0 1 0 0 1 0 11 0 1 1 0 1 1 1 0 0 1 11 1 0 0 0 0 1 0 1 0 1 01 1 0 0 0 1 1 0 1 0 1 01 1 0 0 1 0 1 0 1 0 1 01 1 0 0 1 1 1 0 1 0 1 01 1 0 1 0 0 1 0 1 0 1 01 1 0 1 0 1 1 0 1 0 1 01 1 0 1 1 0 1 0 1 0 1 01 1 1 0 0 0 1 0 1 0 1 0

x1y1x2y2x3y3 B11B12B13B21B22B23

Bp qid NP0 P111[00.00] 111111[00.01] 111111[00.10] 1111

12[00.00] 111112[00.01] 111112[00.10] 1111

13[00.00] 000013[00.01] 000013[00.10] 0000

21[00.00] 111121[00.01] 111021[00.10] 0000

22[00.00] 000022[00.01] 000122[00.10] 1111

23[00.00] 000023[00.01] 000023[00.10] 1111

Peano order

Bp qid NP0 P1 at 0023[00] 001- 001-

Mixed quads (sent to node00)

Bp qid NP0 P1 at 0121[00.01] 111022[00.01] 0001

Example2, bottom-up

0 0 0 0 0 0 1 1 0 1 0 00 0 0 0 0 1 1 1 0 1 0 00 0 0 0 1 0 1 1 0 1 0 00 0 0 0 1 1 1 1 0 1 0 00 0 0 1 0 0 1 1 0 1 0 00 0 0 1 0 1 1 1 0 1 0 00 0 0 1 1 0 1 1 0 1 0 00 0 0 1 1 1 1 1 0 0 1 00 0 1 0 0 0 1 1 0 0 1 10 0 1 0 0 1 1 1 0 0 1 10 0 1 0 1 0 1 1 0 0 1 10 0 1 0 1 1 1 1 0 0 1 10 0 1 1 0 0 1 1 0 0 1 00 0 1 1 0 1 1 1 0 0 1 00 0 1 1 1 0 1 1 0 0 1 00 0 1 1 1 1 1 1 0 0 1 00 1 0 0 0 0 1 0 1 0 1 10 1 0 0 0 1 1 0 1 0 1 00 1 0 0 1 0 1 0 1 0 1 10 1 0 0 1 1 0 0 1 0 1 00 1 0 1 0 0 0 0 1 0 0 10 1 0 1 0 1 0 0 1 0 0 10 1 0 1 1 0 0 0 1 0 0 10 1 0 1 1 1 0 0 1 0 0 10 1 1 0 0 0 1 0 1 0 1 10 1 1 0 1 0 1 0 1 0 1 10 1 1 0 1 1 1 0 1 0 1 10 1 1 1 1 1 0 0 0 0 1 01 0 0 0 0 0 1 1 1 0 1 11 0 0 0 0 1 1 1 0 1 1 01 0 0 0 1 0 1 1 0 1 1 01 0 0 0 1 1 1 1 0 1 1 01 0 0 1 0 0 1 1 1 1 1 01 0 0 1 0 1 1 1 1 1 1 01 0 0 1 1 0 1 1 1 1 1 11 0 0 1 1 1 1 1 1 1 1 11 0 1 0 0 0 1 1 1 1 1 01 0 1 0 0 1 1 1 1 1 1 01 0 1 1 0 0 1 0 0 1 0 11 0 1 1 0 1 1 1 0 0 1 11 1 0 0 0 0 1 0 1 0 1 01 1 0 0 0 1 1 0 1 0 1 01 1 0 0 1 0 1 0 1 0 1 01 1 0 0 1 1 1 0 1 0 1 01 1 0 1 0 0 1 0 1 0 1 01 1 0 1 0 1 1 0 1 0 1 01 1 0 1 1 0 1 0 1 0 1 01 1 1 0 0 0 1 0 1 0 1 0

x1y1x2y2x3y3 B11B12B13B21B22B23

Bp qid NP0 P111[00.00] 111111[00.01] 111111[00.10] 111111[00.11] 1111

12[00.00] 111112[00.01] 111112[00.10] 111112[00.11] 1111

13[00.00] 000013[00.01] 000013[00.10] 000013[00.11] 0000

21[00.00] 111121[00.01] 111021[00.10] 000021[00.11] 0000

22[00.00] 000022[00.01] 000122[00.10] 111122[00.11] 1111

23[00.00] 000023[00.01] 000023[00.10] 111123[00.11] 0000

Peano order

00 quads that are pure are:

Bp qid NP0 P111[00] 1111 111112[00] 1111 111113[00] 0000 0000

At 00Bp qid NP0 P123[00] 0010 0010

At 01Bp qid NP0 P121[00.01] 111022[00.01] 0001

a spatial data and sensor network application:

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