A Spatial Data and Sensor Network Application:

1
A Spatial Data and Sensor Network Application
CubE for Active Situation Replication (CEASR)
Nano-sensors dropped into the Situation space
Wherever threshold level is sensed (chem, bio,
thermal...) a ping is registered in 1 compressed
Ptree for that location.
Using Alien Technologys Fluidic Self-assembly
(FSA) technology, clear layers are laminated into
a cube, with a embedded nano-LED at each voxel.
........ .. . . .
..... . ..... ........
.. . . . ..... . ..... ......
.. .. . . . ..... .
.....
The Ptree is transmitted to the cube, where the
pattern is reconstructed (uncompress Ptree,
display on the cube).
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.
Soldier sees replica of sensed situation prior to
entering space
2
Spatial Data

• Pixel a point in a space
• Band feature attribute of the pixels
• Value usually one byte (0255)
• 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
3
RSI dataset example
RSI data can be viewed as collection of pixels.
Each pixel has a value for each feature attribute
TIFF image
Yield Map
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.
4
Spatial Data Formats

• Existing formats
• BSQ (Band Sequential)
• BIL (Band Interleaved by Line)
• BIP (Band Interleaved by Pixel)
• New format
• bSQ (bit Sequential)

• BAND-1
• 54 127
• (1111 1110) (0111 1111)
• 4 193
• (0000 1110) (1100 0001)

• BAND-2
• 7 240
• (0010 0101) (1111 0000)
• 00 19
• (1100 1000) (0001 0011)

BSQ format (2 files) Band 1 254 127 14
193 Band 2 37 240 200 19
5
Spatial Data Formats (Cont.)

• BAND-1
• 54 127
• (1111 1110) (0111 1111)
• 4 193
• (0000 1110) (1100 0001)

• BAND-2
• 7 240
• (0010 0101) (1111 0000)
• 00 19
• (1100 1000) (0001 0011)

BSQ format (2 files) 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
• 54 127
• (1111 1110) (0111 1111)
• 4 193
• (0000 1110) (1100 0001)

• BAND-2
• 7 240
• (0010 0101) (1111 0000)
• 00 19
• (1100 1000) (0001 0011)

BSQ format (2 files) 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
6
Spatial Data Formats (Cont.)

• BAND-1
• 54 127
• (1111 1110) (0111 1111)
• 4 193
  
• (0000 1110) (1100 0001)

• BAND-2
• 7 240
• (0010 0101) (1111 0000)
• 00 19
• (1100 1000) (0001 0011)

BSQ format (2 files) 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
7
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.)

8
An example of PC-tree
1 1 1 1 1 1 0 0 1 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
Given a bSQ file, Bij, (shown in spatial
positions also) we create its basic PC-tree, Pij
as follows.

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

• Level
• Fan-out
• QID (Quadrant ID)

9
Our example of PC-tree (again)
?Level-3
?Level-2
?Level-1
?Level-0

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

• Level
• Fan-out
• QID (Quadrant ID)

10
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 1s)

11

• Ptree Algebra
• And
• Or
• Complement
• Other

Depth-first Pure 1 path code
0 100 101 102 12 132 20 21 220 221 223 23 3
0 20 21 22 231 ? RESULT 0

0 ? 0
20
20 ?
20 21
21
? 21
220 221 223
22 ? 220 221 223
23
231 ? 231
12
Basic, Value and Tuple Ptrees
13
Creating Peano-Count-trees (PC-trees) from
Spatial Relations

• Take any spatial relation, R(K1,..,Kk, A1, A2,
  , An) (Kistructure, Aifeature 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
  RAi.
• 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)

14
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
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
15
The Peano Cube of a relation (P-cube)
Suppose we have R(K, A1, A2, A3 ) with each Ai a
2-bit number Construct 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
attributes and Ais 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-tr
ees 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?
rc P(2,0,3)
rc P(1,0,3)
rc P(0,0,3)
rc P(3,0,3)
rc P(0,0,2)
rc P(1,0,2)
rc P(2,0,2)
rc P(3,0,2)
1
5
0
0
11
rc P(0,0,1)
rc P(1,0,1)
rc P(2,0,1)
rc P(3,0,1)
14
5
3
0
11
rc P313
0
0
0
0
5
5
17
0
10
11
rc P312
0
0
0
0
rc P(0,0,0)
rc P(1,0,0)
rc P(3,0,0)
rc P(2,0,0)
00
10
rc P311
rc P323
0
0
0
0
0
0
0
0
01
10
rc P322
rc P(0,0,0)
rc P(1,1,0)
rc P(2,1,0)
rc P(3,1,0)
1
0
0
0
01
01
rc P321
rc P333
0
0
0
0
1
0
0
0
A2
00
01
rc P332
11
rc P(0,2,0)
rc P(1,2,0)
rc P(2,2,0)
rc P(3,2,0)
0
0
0
1
10
00
rc P331
00
01
10
10
0
0
0
0
00
00
01
10
01
rc P(0,3,0)
rc P(1,3,0)
rc P(2,3,0)
rc P(3,3,0)
A3
11
00
00
01
10
11
A1
16
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 usedouble 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
NP0V Qid PgVc 1111 1 1011 1.0
1110 1.3 0010 2 1111 2.2 1101
NP1NP0P1 .---- 1 ----. / /
\ \ 0 1 1 0 // \ \ //
\\ 1 1 0 1 00 10 //\ //\ //\
0001 1101 0010
NP1V Qid PgVc 0110 1 1101 1.0
0001 1.3 1101 2 0010 2.2 0010
P1V Qid PgVc 1001 1 0010 1.0
1110 1.3 0010 2 1101 2.2 1101
P0V Qid 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
17
ANDing (for all Truth-trees, just AND bit-wise)
Pure1-quad-list method For each operand, list
the qids of the pure1 quads 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.
0 100 101 102 12 132 20 21 220 221 223 23 3
AND 0 20 21 22 231 ? 0 20 21 220
221 223 231
P1operand1 0 1 0
0 1 // \ \ // \\ 0 0 1 0 1 1 01
//\ //\ //\ 1110 0010 1101
P0operand1 0 0 0
0 0 // \ \ // \ \ 0 1 0 0 0 0 00
//\ //\ //\ 0001 1101 0010
NP0operand1 1 1 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
bitwise
Depth first traversal using 111, 100, 000.
AND
P1operand2 0 1 0 0
0 / / \ \ 1 1 1 0
//\ 0100
P0op2 P1op2 0 0 1
0 1 / / \ \ 0 0 0 0
//\ 1011
NP0operand2 1 1 0 1
0 / / \ \ 1 11 1
//\ 0100
NP1operand2 NP0 1 0 1 1
1 / / \ \ 0 0 0 1
//\ 1011

P1op1P1op2 0 1 0 0 0
// \ 11 0 0 //\
//\ 1101 0100
P1op1P0op2 P1op1P1op2 0 0 0 0
1 // \ \ //\ \ 0 0 1 0 000 0 //\
//\ //\ 1110 0010 1011
NP0op1NP0op2 1 1 0 1 0
// \ 11 1 1
//\ //\ 1101 0100
NP0op1NP0op2 1 0 1 1
1 // \ \ /// \ 1 0 1 1 000 1 //\
//\ //\ 1110 0010 1011
18
Example1 One band, B1, with 3-bit precision
B11
B13
B12
B1
PNP0V11 P1V11 (combined into 1
table) qid NP0 P1 1111 1001 01
1011 0010 10 1111 1101 01.00 1110 1110 01.1
1 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, NP0P
19
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).

20
Gene Table
P13 01.10.11.11.01.00 qid NP0 P1
0111 0001 01 1111 1110 10 1110 0110 01.11
0110 10.00 1000
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
21
Example1 ANDing to get rc P1(6)
BpQid NP0 P1 11 1111
1001 12 1010 1000 13
0111 0001 1101 1011
0010 1301 1111 1110 1101.00
1110 1101.11
0010 1301.11 0110 1110 1111
1101 1210 1111 1110 1310
1110 0110 1310.00 1000 1110.10
1101 1210.11 0111
P1(6) P1(110) P111P112P013
P11P12NP013 PM1(110) P1(110) xor NP01(110)
P11P12NP013 xor NP011NP012P113 At
CNT 1-cnt4level 14216 since P1(110)
1001100010001000 PM1(110)
P11 P12 NP013 xor NP011NP012P113
10011000 1000 xor
1111 1010 1110 0010 At 10 CNT10
1-cnt4level0410 since P1(110)10
1101111000010000 PM1(110)10
P11P 12 NP013 xor NP011NP012P113
110111100001 xor
111111111001 0000 xor 10011001 At 10.00
CNT10.001-cnt4level3403 since
P1(110)10.00 1111111101110111 At 10.11
CNT10.111-cnt4level3403 since
P1(110)10.11 1111011111110111
Thus, rcP1(6) 16 0 3 3 22
10 only mixed child
10.00, 10.11 mixed children
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.
22
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 1 1 1 0 1 0 1 0 0 0 0 1
1 1 0 1 0 0 0 - - - - - - - ____________
_________ 1 0 1 0 1 0 0 0
pos NP0V P1V 1 1 1 1 1 1
0 0 1 2 1 0 1 0 1 0 0 0 3 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
23
Striping P-trees?
BpQid NP0 P1 00 1101.00
1110 1310.00 1000
BpQid NP0 P1 01 1101 1011 0010 1301
1111 1110
BpQid NP0 P1 11 1111
1001 12 1010 1000 13
0111 0001 1101 1011
0010 1301 1111 1110 1101.00
1110 1101.11
0010 1301.11 0110 1110 1111
1101 1210 1111 1110 1310
1110 0110 1310.00 1000 1110.10
1101 1210.11 0111
BpQid NP0 P1 C 11 1111 1001 12
1010 1000 13 0111 0001
BpQid NP0 P1 10 1110 1111
1101 1110.10 1101 1210 1111
1110 1310 1110 0110
BpQid NP0 P1 11 1101.11 0010 1210.11
0111 1301.11 0110
Assume 5-computer cluster NodeC, Node00, Node01,
Node10, Node11 Send to Nij if qid ends in ij
P11(110) P111P112P013 P11P12NP013
PM1(110) P11P12NP013 xor NP011NP012P113
At NC CNT 1-cnt4level 14216 since
P1(110) 1001100010001000
PM1(110) 100110001000 xor 111110101110
0010 At N10 CNT10 1-cnt4level0410
since P1(110)10 1101111000010000
PM1(110)10 110111100001 xor
111111111001 0000 xor 10011001 At N00
CNT10.001-cnt4level3403 since
P1(110)10.00 1111111101110111 At N11
CNT10.111-cnt4level3403 since
P1(110)10.11 1111011111110111 Every node
sends accumulated CNT to C, where rcP1(6) 16
0 3 3 22 calculated.
24
Striping P-trees?
P11
P13
P12
qid NP0 P1 1111 1001 01 1011 0010 10
1111 1101 01.00 1110 01.11 0010 10.10 11
01
qid NP0 P1 1010 1000 10
1111 1110 10.11 0111
qid NP0 P1 0111 0001 01 1111 1110 10 111
0 0110 01.11 0110 10.00 1000
Bp qid NP0 P1 00
Bp qid NP0 P1 01 1101 1011 0010 1101.00
1110 1101.11 0010 1301 1111 1110 1301.11 0
110
Bp qid NP0 P1 C 11 1111 1001 12
1010 1000 13 0111 0001
Bp qid NP0 P1 10 1110 1111 1101 1110.10
1101 1210 1111 1110 1210.11 0111 1310 1110
0110 1310.00 1000
Bp qid NP0 P1 11
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 p4
0-gt0 0.3-gt0 0.3.2-gt0 0.3.2.2-gt2
0.3.2.2.3-gt2 0.3.2.2.3.1-gt2
0.3.2.2.3.1.0-gt2 0.3.2.2.3.1.0-gt2
0.3.2.2.3.1.0.1-gt1 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.s2sk --gt Node
?k? p
25
Example 1 (bottom-up)
B11
Bp qid NP0 P1 1100.00
1111
Band, B1, with 3-bit values
Bp qid NP0 P1 1100.00
1111 1100.01 1111
Bp qid NP0 P1 1100.00
1111 1100.01 1111 1100.10
1111
Bp qid NP0 P1 1100.00
1111 1100.01 1111 1100.10
1111 1100.11 1111
Bp qid NP0 P1 1100 0000
1111
This ends the possibility of a larger pure1
quad. So 00 can be installed in parent as a pure1.
Node-C Bp qid NP0 P1 11
01__ 10__
Bp qid NP0 P1 1100 0000
1111 1101.00 1110
Node-00 Bp qid NP0 P1 1101.00
1110
Mixed leaf quad sent. Also ends
possibility parent is pure so it all siblings
are installed as bits in parent.
Bp qid NP0 P1 1101.00
1110 1101.01 0000
Node-01 Bp qid NP0 P1 1101
1011 0010
1101.10 1111
1101.11 0001
Mixed leaf quad sent. Ends parent so install bits
in grandparent also
Node-10 Bp qid NP0 P1
Node-11 Bp qid NP0 P1 1101.11
0001
26
Example 1 (bottom-up)
B11
Bp qid NP0 P1 1110.00
1111
Band, B1, with 3-bit values
Bp qid NP0 P1 1110.00
1111 1110.01 1111
Bp qid NP0 P1 1110.00
1111 1110.01 1111 1110.10
1101 1110.11 1111
Ends the possibility of a larger pure1 quad. All
can be installed in parent/grandparent as a
1-bit. 10.10 can be installed.
Bp qid NP0 P1 1111.00
1111 1111.01 1111 1111.10
1111 1111.11 1111
Node-C Bp qid NP0 P1 11
0111 1001
Bp qid NP0 P1 1111 0000
1111
Node-00 Bp qid NP0 P1 1101.00
1110
Node-01 Bp qid NP0 P1 1101
1011 0010
Ends quad-11. All can be installed in Parent as a
1-bit.
Node-10 Bp qid NP0 P1 1110.10
1101 1110 1111 1101
Node-11 Bp qid NP0 P1 1101.11
0001
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.
27
Example2
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
B1
B11
B12
B13
6 6 6 6 5 5 1 1 6 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 0 1 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 0 1 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 1 0 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 1 4 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 0 1 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 0 0 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 1 0 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

28
Example2 Striping
__PNP0V_ __P1V__ Band?111 222 111
222 bit-pos?123 123 123 123
110 111 110 000 101
011 000 000 111 111 100 000
101 010 101 010
Raster order
Peano order
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
X, Y, B11B12B13B21B22B23
x1y1x2y2x3y3 B11B12B13B21B22B23
0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 1
0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0
1 1 1 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 1
1 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0
0 1 0 0 1 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1
0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 1 0
0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1
1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1
1 0 1 0 0 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0
1 0 0 1 1 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1
1 0 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0
0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1
0 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1
1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1
0 1 0 1 1 0 0 1 0 0 1 1 0 1 1 1 1 0 0 1
0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1
0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0
0 0 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 1
0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 0 0 1 1 1
1 1 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0
1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1
0 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 1
1 0 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 1
0 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1
1 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 1
0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 0
0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1
0 1 1 0 0 1 0 1 0 0 1 0 1 1 1 0 0 1 1 1
1 0 0 1 1 1 1 0 1 0 0 1 0 1 0 1 0
0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 1 0
1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1
1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0
1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0
0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1
1 0 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0
0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0
1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1
1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0
1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0
1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 0
1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1
0 0 1 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0
1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1
1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1
0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0
1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0
1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 1
0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1
1 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1
1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 1 0 1 0 0 1
1 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1
1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 0 1 0 1
0 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1
0 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0
1 0 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1
1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0
OR for PNP0 AND for P1
00_PNP0V__ __P1V__ 110 111 110 000
Send B21B22B23 to Node00
01_PNP0V__ __P1V__ 101 011 000 000
Send B11B13 B22B23 to Node01
Bp qid NP0 P1 C 11 1111 1011 12
1010 1000 13 0111 0001 21
1010 0000 22 1111 0001 23
1110 0000 Purity Template 16 12 12 8
10_PNP0V__ __P1V__ 111 111 100 000
Send B12B13 B21B22B23 to Node10
11_PNP0V__ __P1V__ 101 010 101 010
Send nothing to Node11
29
Example2 striping at Node 00
Bp qid NP0 P1 00 2100
1100 1000 2200 0111 0011 2300
0010 0010 PurityTemplate 00 4 4 4 4 1101.00
1110 2301.00 1010 1210.00
1111 1310.00 1000 2110.00
0111 2210.00 1111 2310.00 1000
_PNP0V__ __P1___ Band
111 222 111 222 bit-pos 123 123 123 123 00
100 100
110 000 011 011
010 010
x1y1x2y2x3y3B11B12B13 B21B22B23
0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 1 0 1 0 0 0 0 0
0 1 1 1 0 0 0 0 0 1 0 0
1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 0
1 0 0 0 0 0 1 1 1 0 1
0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1
0 1 1 0 0 1 0 1 0 0 1 1 0 0
1 0 1 1 0 1 1 0 0 1 1 0 0
0 1 0 0 0 1 1 0 1 0 1 0 0 0 1 1 1
0 0 1 0 0 0 1 1 1 1 0 1
0
_PNP0V__ __P1V__ 110 100 110 100
Send nothing to Node00
_PNP0V__ __P1V__ 110 110 110 000
Send B21B22 to Node01
_PNP0V__ __P1V__ 110 011 110 011
Pages on disk
Send nothing to Node10
_PNP0V__ __P1V__ 110 010 110 010
Send nothing to Node11
P1 Band 12 bit-pos
13 01.00 11 10
11 00
P1 Band 11 222 bit-pos
23 123 10.00 11 011
10 110 10 110 10 110
x1y1x2y2x3y3 B11 B23
x1y1x2y2x3y3 B12B12 B23B23B23
0 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0
1 1 0 1 0 0 1 1 0 0
1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1
1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 1 1 0
1 1 0
From 01
From 10
To 01
30
Example2 striping at Node 01
_PNP0V__ __P1___ Band
111 222 111 222 bit-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
Bp qid NP0 P1 01 1101
1010 0010 1301 1110 1110 2201
1010 1010 2301 1110 0110 PurityTemplate
01 4 4 3 1 2100.01 1110 2200.01
0001 2310.01 0011
To 00
x1y1x2y2x3y3 B11 B13 B22B23
0 1 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1
1 1 0 0 1 0 0 1 0 1 1 1 1 0 1 0 0
1 1 0 1 1 0 0 1 0 1 0 0 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0
1 0 1 0 1 0 1 1 1 0 1 0 1 0 1
1 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1
1 1 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1
0 0 1 0
_PNP0V__ __P1V__ 1 1 11 0 1 10
Send 01B11B23 to Node00
_PNP0V__ __P1V__ 0 1 01 0 1 01
Send nothing to Node01
_PNP0V__ __P1V__ 1 1 11 1 1 11
Send nothing to Node10
Pages on disk
_PNP0V__ __P1V__ 0 0 10 0 0 10
Bp qid NP0 P1 01 1101 1010 0010
Send nothing to Node11
Bp qid NP0 P1 01 1301 1110 1110
Bp qid NP0 P1 01 2100.01 1110
P1 Band 22 bit-pos
12 00.01 10 10
10 01
P1 Band 2 bit-pos
3 10.01 0 0 1
1
x1y1x2y2x3y3 B21B22
x1y1x2y2x3y3 B23
Bp qid NP0 P1 01 2201
1010 1010 2200.01 0001
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
1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 1 0
1 1 0 0 1 1 1 1
Bp qid NP0 P1 01 2301
1110 0110 2310.01 0011
From 00
From 10
31
Example2 striping at Node 10
_PNP0V__ __P1___ Band
111 222 111 222 bit-pos 123 123 123 123 10
11 111 10 010
11 111 11 110 11 110 11 110
10 111 00 001
Bp qid NP0 P1 10 1210
1111 1110 1310 1110 0110 2110
1111 0110 2210 1111 1110 2310
1101 0001 PurityTemplate 10 4 4 2 2
To 00
To01
x1y1x2y2x3y3 B12B13B21B22B23
1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 0
1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1
1 1 0 1 1 0 1 0 0 1 0 0 1 1 1 1
0 1 0 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1
1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1
0 0 0 1 1 1 1 0 1 0 1 0 0 1 1 1 1
1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 1 0 1
1 0 0 1 1
_PNP0V__ __P1V__ 11 111 10 010
Send 10B13B21B23 to Node00
_PNP0V__ __P1V__ 11 111 11 110
Send 10 B23 to Node01
_PNP0V__ __P1V__ 11 110 11 110
Send nothing to Node10
Pages on disk
_PNP0V__ __P1V__ 10 111 00 001
Send 10B12B21B22 to Node11
To 11
32
Example2 striping at Node11
Bp qid NP0 P1 11 1210.11
01 2210.11 10 2310.11 01
Pages on disk
Bp qid NP0 P1 11 1210.11 01
P1 Band 122 bit-pos
223 10.11 010 101
x1y1x2y2x3y3 B12 B21B22
Bp qid NP0 P1 11 2210.11 10
1 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1
0 1
Bp qid NP0 P1 11 2310.11 01
From 10
33
Example2.1AND at NodeC or
RC(P 101,010) P11 P12 P13 P21 P22 P23
NP0-pattern NP0 P1 11
xxxx 12 prime 13
xxxx 21 prime 22
xxxx 23 prime
NP0 1111 0111 0111 1111 1111 1111 ------AND 0111
Sum 8 so far. Invocation 101,010 send
to Nodes 01, 10
P1-pattern NP0 P1 11
xxxx 12 prime 13
xxxx 21 prime 22
xxxx 23 prime
P1 1011 0101 0001 0101 0001 0001 ------AND 0001
Bp qid NP0 P1 1101 1010 0010
Bp qid NP0 P1 1101.00 1110
Bp qid NP0 P1 C 11 1111 1011 12
1010 1000 13 0111 0001 21
1010 0000 22 1111 0001 23
1110 0000
Bp qid NP0 P1 1210 1111 1110
Bp qid NP0 P1 1301 1110 1110
Bp qid NP0 P1 1210.00 1111
Bp qid NP0 P1 1310 1110
0110
Bp qid NP0 P1 2100.01
1110
Bp qid NP0 P1 1310.00 1000
Bp qid NP0 P1 2110 1111 0110
Bp qid NP0 P1 2201 1010
1010 2200.01 0001
Bp qid NP0 P1 2100
1100 1000 2110.00 0111
Bp qid NP0 P1 2210 1111 1110
Bp qid NP0 P1 2300 0010
0010 2301.00 1010 2310.00
1000
Bp qid NP0 P1 2301 1110
0110 2310.01 0011
Bp qid NP0 P1 2200 0111
0011 2210.00 1111
Bp qid NP0 P1 2310 1101 0001
34
Example2.1AND at Node01
NP0-pattern NP0 P1 11
xxxx 12 prime 13
xxxx 21 prime 22
xxxx 23 prime
01 NP0 11 1010 12 13 1110 21 22
1010 23 1001 AND------ 1000
Invocation 01 101,010 Sent to Node00
101,010 received
P1-pattern NP0 P1 11
xxxx 12 prime 13
xxxx 21 prime 22
xxxx 23 prime
01 P1 11 0010 12 13 1110 21 22
1010 23 0001 AND------ 0000
Bp qid NP0 P1 1101 1010 0010
Bp qid NP0 P1 1101.00 1110
Bp qid NP0 P1 C 11 1111 1011 12
1010 1000 13 0111 0001 21
1010 0000 22 1111 0001 23
1110 0000
Bp qid NP0 P1 1210 1111 1110
Bp qid NP0 P1 1301 1110 1110
Bp qid NP0 P1 1210.00 1111
Bp qid NP0 P1 1310 1110
0110
Bp qid NP0 P1 2100.01
1110
Bp qid NP0 P1 1310.00 1000
Bp qid NP0 P1 2110 1111 0110
Bp qid NP0 P1 2201 1010
1010 2200.01 0001
Bp qid NP0 P1 2100
1100 1000 2110.00 0111
Bp qid NP0 P1 2210 1111 1110
Bp qid NP0 P1 2300 0010
0010 2301.00 1010 2310.00
1000
Bp qid NP0 P1 2301 1110
0110 2310.01 0011
Bp qid NP0 P1 2200 0111
0011 2210.00 1111
Bp qid NP0 P1 2310 1101 0001
35
Example2.1AND at Node10
NP0-pattern NP0 P1 11 xxxx 12
prime 13 xxxx 21 prime 22
xxxx 23 prime
10 NP0 11 12 0001 13 1110 21 1001 22
1111 23 1110 AND------ 0000
Invocation 10 101,010 Sent nowhere (no
mixed)
P1-pattern NP0 P1 11
xxxx 12 prime 13 xxxx 21 prime 22
xxxx 23 prime
10 P1 11 12 13 21 22 23
AND------
Bp qid NP0 P1 1101 1010 0010
Bp qid NP0 P1 1101.00 1110
Bp qid NP0 P1 C 11 1111 1011 12
1010 1000 13 0111 0001 21
1010 0000 22 1111 0001 23
1110 0000
Bp qid NP0 P1 1210 1111 1110
Bp qid NP0 P1 1301 1110 1110
Bp qid NP0 P1 1210.00 1111
Bp qid NP0 P1 1310 1110
0110
Bp qid NP0 P1 2100.01
1110
Bp qid NP0 P1 1310.00 1000
Bp qid NP0 P1 2110 1111 0110
Bp qid NP0 P1 2201 1010
1010 2200.01 0001
Bp qid NP0 P1 2100
1100 1000 2110.00 0111
Bp qid NP0 P1 2210 1111 1110
Bp qid NP0 P1 2300 0010
0010 2301.00 1010 2310.00
1000
Bp qid NP0 P1 2301 1110
0110 2310.01 0011
Bp qid NP0 P1 2200 0111
0011 2210.00 1111
Bp qid NP0 P1 2310 1101 0001
36
Example2.1AND at Node00
Sum1, sent to NodeC gives a sum total of 8 1
9
P1-pattern P1 11 xxxx 12 prime 13
xxxx 21 prime 22 xxxx 23 prime
01.00 P1 11 1110 12 13 21 22
23 0101 AND------ 0100
Bp qid NP0 P1 1101 1010 0010
Bp qid NP0 P1 1101.00 1110
Bp qid NP0 P1 C 11 1111 1011 12
1010 1000 13 0111 0001 21
1010 0000 22 1111 0001 23
1110 0000
Bp qid NP0 P1 1210 1111 1110
Bp qid NP0 P1 1301 1110 1110
Bp qid NP0 P1 1210.00 1111
Bp qid NP0 P1 1310 1110
0110
Bp qid NP0 P1 2100.01
1110
Bp qid NP0 P1 1310.00 1000
Bp qid NP0 P1 2110 1111 0110
Bp qid NP0 P1 2201 1010
1010 2200.01 0001
Bp qid NP0 P1 2100
1100 1000 2110.00 0111
Bp qid NP0 P1 2210 1111 1110
Bp qid NP0 P1 2300 0010
0010 2301.00 1010 2310.00
1000
Bp qid NP0 P1 2301 1110
0110 2310.01 0011
Bp qid NP0 P1 2200 0111
0011 2210.00 1111
Bp qid NP0 P1 2310 1101 0001
37
Example2.2AND at NodeC or
RC(P 100,101) P11 P12 P13 P21 P22 P23
NP0 ------AND 0010
Sum 0 so far. Invocation 100, 101 send
to Node 10
P1 ------AND 0000
Bp qid NP0 P1 1101 1010 0010
Bp qid NP0 P1 1101.00 1110
Bp qid NP0 P1 C 11 1111 1011 12
1010 1000 13 0111 0001 21
1010 0000 22 1111 0001 23
1110 0000
Bp qid NP0 P1 1210 1111 1110
Bp qid NP0 P1 1301 1110 1110
Bp qid NP0 P1 1210.00 1111
Bp qid NP0 P1 1310 1110
0110
Bp qid NP0 P1 2100.01
1110
Bp qid NP0 P1 1310.00 1000
Bp qid NP0 P1 2110 1111 0110
Bp qid NP0 P1 2201 1010
1010 2200.01 0001
Bp qid NP0 P1 2100
1100 1000 2110.00 0111
Bp qid NP0 P1 2210 1111 1110
Bp qid NP0 P1 2300 0010
0010 2301.00 1010 2310.00
1000
Bp qid NP0 P1 2301 1110
0110 2310.01 0011
Bp qid NP0 P1 2200 0111
0011 2210.00 1111
Bp qid NP0 P1 2310 1101 0001
38
Example2.2AND at Node10
10 NP0 11 12 13 21 22 23
AND------ 0001
Invocation 10 100, 101 Sent to Node 11
10 P1 11 12 13 21 22 23
AND------ 0000
Bp qid NP0 P1 1101 1010 0010
Bp qid NP0 P1 1101.00 1110
Bp qid NP0 P1 C 11 1111 1011 12
1010 1000 13 0111 0001 21
1010 0000 22 1111 0001 23
1110 0000
Bp qid NP0 P1 1210 1111 1110
Bp qid NP0 P1 1301 1110 1110
Bp qid NP0 P1 1210.00 1111
Bp qid NP0 P1 1310 1110
0110
Bp qid NP0 P1 2100.01
1110
Bp qid NP0 P1 1310.00 1000
Bp qid NP0 P1 2110 1111 0110
Bp qid NP0 P1 2201 1010
1010 2200.01 0001
Bp qid NP0 P1 2100
1100 1000 2110.00 0111
Bp qid NP0 P1 2210 1111 1110
Bp qid NP0 P1 2300 0010
0010 2301.00 1010 2310.00
1000
Bp qid NP0 P1 2301 1110
0110 2310.01 0011
Bp qid NP0 P1 2200 0111
0011 2210.00 1111
Bp qid NP0 P1 2310 1101 0001
39
Example2.2AND at Node11
Sum1, sent to NodeC gives a sum total of 1
10 P1 11 01 12 13 21 22 01 23
01 AND------ 01
Bp qid NP0 P1 1101 1010 0010
Bp qid NP0 P1 1101.00 1110
Bp qid NP0 P1 C 11 1111 1011 12
1010 1000 13 0111 0001 21
1010 0000 22 1111 0001 23
1110 0000
Bp qid NP0 P1 1210 1111 1110
Bp qid NP0 P1 1301 1110 1110
Bp qid NP0 P1 1210.00 1111
Bp qid NP0 P1 1310 1110
0110
Bp qid NP0 P1 2100.01
1110
Bp qid NP0 P1 1310.00 1000
Bp qid NP0 P1 2110 1111 0110
Bp qid NP0 P1 2201 1010
1010 2200.01 0001
Bp qid NP0 P1 2100
1100 1000 2110.00 0111
Bp qid NP0 P1 2210 1111 1110
Bp qid NP0 P1 2300 0010
0010 2301.00 1010 2310.00
1000
Bp qid NP0 P1 2301 1110
0110 2310.01 0011
Bp qid NP0 P1 2200 0111
0011 2210.00 1111
Bp qid NP0 P1 2310 1101 0001
40
Example2, bottom-up
Bp qid NP0 P1 1100.00
1111 1200.00 1111 1300.00
0000 2100.00 1111 2200.00
0000 2300.00 0000
Peano order
x1y1x2y2x3y3 B11B12B13B21B22B23
0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 1 0
1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1
1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0
1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0
0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1
1 0 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0
0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0
1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1
1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0
1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0
1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 0
1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1
0 0 1 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0
1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1
1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1
0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0
1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0
1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 1
0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1
1 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1
1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 1 0 1 0 0 1
1 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1
1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 0 1 0 1
0 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1
0 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0
1 0 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1
1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0
41
Example2, bottom-up
Bp qid NP0 P1 1100.00
1111 1100.01 1111 1200.00
1111 1200.01 1111 1300.00
0000 1300.01 0000 2100.00
1111 2100.01 1110 2200.00
0000 2200.01 0001 2300.00
0000 2300.01 0000
Peano order
x1y1x2y2x3y3 B11B12B13B21B22B23
0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 1 0
1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1
1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0
1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0
0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1
1 0 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0
0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0
1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1
1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0
1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0
1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 0
1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1
0 0 1 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0
1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1
1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1
0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0
1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0
1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 1
0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1
1 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1
1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 1 0 1 0 0 1
1 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1
1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 0 1 0 1
0 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1
0 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0
1 0 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1
1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0
Mixed quads (can be sent to node01)
Bp qid NP0 P1 2100.01
1110 2200.01 0001
42
Example2, bottom-up
Bp qid NP0 P1 1100.00
1111 1100.01 1111 1100.10
1111 1200.00 1111 1200.01
1111 1200.10 1111 1300.00
0000 1300.01 0000 1300.10
0000 2100.00 1111 2100.01
1110 2100.10 0000 2200.00
0000 2200.01 0001 2200.10
1111 2300.00 0000 2300.01
0000 2300.10 1111
Peano order
x1y1x2y2x3y3 B11B12B13B21B22B23
0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 1 0
1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1
1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0
1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0
0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1
1 0 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0
0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0
1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1
1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0
1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0
1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 0
1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1
0 0 1 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0
1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1
1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1
0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0
1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0
1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 1
0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1
1 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1
1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 1 0 1 0 0 1
1 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1
1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 0 1 0 1
0 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1
0 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0
1 0 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1
1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0
Mixed quads (sent to node00)
Bp qid NP0 P1 at 00 2300
001- 001-
Bp qid NP0 P1 at 01 2100.01
1110 2200.01 0001
43
Example2, bottom-up
Bp qid NP0 P1 1100.00
1111 1100.01 1111 1100.10
1111 1100.11 1111 1200.00
1111 1200.01 1111 1200.10
1111 1200.11 1111 1300.00
0000 1300.01 0000 1300.10
0000 1300.11 0000 2100.00
1111 2100.01 1110 2100.10
0000 2100.11 0000 2200.00
0000 2200.01 0001 2200.10
1111 2200.11 1111 2300.00
0000 2300.01 0000 2300.10
1111 2300.11 0000
Peano order
x1y1x2y2x3y3 B11B12B13B21B22B23
00 quads that are pure are
0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 1 0
1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1
1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0
1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0
0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1
1 0 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0
0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0
1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1
1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 0 0
1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0
1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 0
1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1
0 0 1 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0
1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1
1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1
0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0
1 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0
1 0 1 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1 0 1
0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1
1 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1
1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 1 0 1 0 0 1
1 1 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1
1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 0 1 0 1
0 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1
0 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0
1 0 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1
1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0
Bp qid NP0 P1 1100
1111 1111 1200 1111 1111 1300
0000 0000
At 00 Bp qid NP0 P1 2300
0010 0010
At 01 Bp qid NP0 P1 2100.01
1110 2200.01 0001