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d 1. F rrn m 1 m 2 (m 21 m 22 ). F rrn m 1 m 2 (m 21 m 22 ). 0 1 2 3. f 200 30 1 1 0 1 f 201 31 1 1 0 1 f 210 32 0 2 1 0 f 211 33 0 3 1 1 f 300 34 0 0 0 0 f 301 35 0 0 0 0 f 310 36 0 3 1 1 f 311 37 0 3 1 1

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D 3

d1

F rrn m1 m2(m21 m22)

F rrn m1 m2(m21 m22)

0 1 2 3

f200 30 1 1 0 1

f201 31 1 1 0 1

f210 32 0 2 1 0

f211 33 0 3 1 1

f300 34 0 0 0 0

f301 35 0 0 0 0

f310 36 0 3 1 1

f311 37 0 3 1 1

f202 38 1 3 1 1

f212 39 0 2 1 0

f302 40 1 0 0 0

f312 41 0 3 1 1

f220 42 1 3 1 1

f221 43 1 0 0 0

f230 44 1 0 0 0

f231 45 1 0 0 0

f320 46 0 0 0 0

f321 47 0 3 1 1

f330 48 0 3 1 1

f331 49 1 3 1 1

f222 50 1 3 1 1

f232 51 1 3 1 1

f322 52 0 0 0 0

f332 53 0 0 1 0

f240 54 0 2 1 0

f241 55 0 2 0 1

f340 56 0 2 1 0

f341 57 1 2 1 0

f242 58 1 1 0 1

f342 59 1 1 0 1

f000 0 1 3 1 1

f001 1 0 1 0 1

f010 2 0 2 1 0

f011 3 1 0 0 0

f100 4 0 0 0 0

f101 5 0 3 1 1

f110 6 0 3 1 1

f111 7 1 2 0 1

f002 8 1 3 1 1

f012 9 0 2 1 0

f102 10 0 0 0 0

f112 11 0 3 1 1

f020 12 1 3 1 1

f021 13 0 1 0 1

f030 14 1 2 1 0

f031 15 1 0 0 0

f120 16 0 0 0 0

f121 17 1 3 1 1

f130 18 0 3 1 1

f131 19 1 2 0 1

f022 20 1 3 1 1

f032 21 0 2 1 0

f122 22 0 0 0 0

f132 23 0 0 0 0

f040 24 0 3 1 1

f041 25 1 1 0 1

f140 26 0 2 1 0

f141 27 1 2 1 0

f042 28 0 0 0 0

f142 29 1 1 0 1

040

140

240

340

4

3

2

1

0

041

141

241

341

142

042

242

342

030

130

230

330

031

131

231

331

032

132

232

322

020

120

220

320

021

121

221

321

022

122

222

322

010

110

310

201

011

111

311

210

012

112

212

312

000

100

200

300

d2

001

101

210

301

002

102

202

302

0

1

d3

2

D1 rrn1 a11 a12

d10 0 1 0

d11 1 0 1

d12 2 0 1

d13 3 1 0

D2 rrn2 a21 a22 a23(a231a232)

d20 0 1 1 2 1 0

d21 1 0 1 2 1 0

d22 2 0 0 0 0 0

d23 3 0 0 3 1 1

d24 4 1 0 3 1 1

D3 rrn3 a31 a32 a33

d30 0 0 1 0

d31 1 1 1 0

d32 2 0 1 1


D 3

3

2

1

0

rrn1

F rrn m1m2(m21 m22)

m1

1

0

0

0

0

0

0

1

1

0

0

0

1

0

0

1

0

1

1

1

1

0

0

0

0

1

0

1

0

1

m21 1

0

1

0

0

1

1

0

1

1

0

1

1

0

1

0

0

1

1

0

1

1

0

0

1

0

1

1

0

0

m22

1

1

0

0

0

1

1

1

1

0

0

1

1

1

0

0

0

1

1

1

1

0

0

0

1

1

0

0

0

1

0 1

f000 0 1 31 1

f001 1 0 10 1

f010 2 0 2 1 0

f011 3 0 00 0

f100 4 0 00 0

f101 5 0 31 1

f110 6 0 31 1

f111 7 1 20 1

f002 8 1 31 1

f012 9 0 21 0

f102 10 0 00 0

f112 11 0 3 1 1

f020 12 1 31 1

f021 13 0 10 1

f030 14 0 21 0

f031 15 1 00 0

f120 16 0 00 0

f121 17 1 3 1 1

f130 18 1 3 1 1

f131 19 1 20 1

f022 20 1 31 1

f032 21 0 2 1 0

f122 22 0 00 0

f132 23 0 00 0

f040 24 0 31 1

f041 25 1 1 0 1

f140 26 0 21 0

f141 27 1 21 0

f042 28 0 00 0

f142 29 1 10 1

030

130

031

131

032

132

020

120

021

121

022

122

010

110

011

111

012

112

000

100

rrn2

001

101

002

102

0

1

rrn3

2

D2 rrn2 a21 a22 a23(a231 a232)

d20 0 1 1 2 1 0

d21 1 0 1 2 1 0

d22 2 0 0 0 0 0

d23 3 0 0 3 1 1

a21

1

0

0

0

a22

1

1

0

0

a231

1

1

0

1

a232

0

0

0

1

D1rrn1 a11 a12

d10 0 1 0

d11 1 0 1

a11

1

0

a12

0

1

a31

0

1

0

D3rrn3 a31 a32 a33

d30 0 0 1 0

d31 1 1 1 0

d32 2 0 1 1

a32

1

1

1

a33

0

0

1


D 3

Pattern=JI

1 1

1 0

0 0

1 0

0 1

0 1

0 1

0 0

0 0

1 0

0 0

1 0

3

2

1

0

rrn1

F rrn m1m2(m21 m22)

m1

1

0

0

0

0

0

0

1

1

0

0

0

1

0

0

1

0

1

1

1

1

0

0

0

0

1

0

1

0

1

m21 1

0

1

0

0

1

1

0

1

1

0

1

1

0

1

0

0

1

1

0

1

1

0

0

1

0

1

1

0

0

m22

1

1

0

0

0

1

1

1

1

0

0

1

1

1

0

0

0

1

1

1

1

0

0

0

1

1

0

0

0

1

0 1

Pattern is the Join-Index (JI) of the star join

necessary to produce the materialized view (MV).

As a 0-dim P-tree (P-sequence), it is m1 .

What we want to do is create the basic Ptrees

for the MV without having to create the MV itself.

(directly from the basic Ptrees for the dimension

relations).

Note that we already have the basic Ptrees for each

measurement in the fact file. What we need is

the other basic MV Ptrees (corresponding to the

feature attributes of the dimension files) and we

need to be able to build those MV Ptrees without

having to construct MV itself.

If m1 were pure1.

f000 0 1 31 1

f001 1 0 10 1

f010 2 0 2 1 0

f011 3 0 00 0

f100 4 0 00 0

f101 5 0 31 1

f110 6 0 31 1

f111 7 1 20 1

f002 8 1 31 1

f012 9 0 21 0

f102 10 0 00 0

f112 11 0 3 1 1

f020 12 1 31 1

f021 13 0 10 1

f030 14 0 21 0

f031 15 1 00 0

f120 16 0 00 0

f121 17 1 3 1 1

f130 18 1 3 1 1

f131 19 1 20 1

f022 20 1 31 1

f032 21 0 2 1 0

f122 22 0 00 0

f132 23 0 00 0

f040 24 0 31 1

f041 25 1 1 0 1

f140 26 0 21 0

f141 27 1 21 0

f042 28 0 00 0

f142 29 1 10 1

030

130

031

131

032

132

020

120

021

121

022

122

010

110

011

111

012

112

000

100

rrn2

001

101

002

102

0

1

rrn3

2

D2 rrn2 a21 a22 a23(a231 a232)

d20 0 1 1 2 1 0

d21 1 0 1 2 1 0

d22 2 0 0 0 0 0

d23 3 0 0 3 1 1

a21

1

0

0

0

a22

1

1

0

0

a231

1

1

0

1

a232

0

0

0

1

D1rrn1 a11 a12

d10 0 1 0

d11 1 0 1

a11

1

0

a12

0

1

a31

0

1

0

D3rrn3 a31 a32 a33

d30 0 0 1 0

d31 1 1 1 0

d32 2 0 1 1

a32

1

1

1

a33

0

0

1


D 3

Join Index

1 1

1 0

0 0

1 0

0 1

0 1

0 1

0 0

0 0

1 0

0 0

1 0

3

2

1

0

rrn1

F rrn m1m2(m21 m22)

m1

1

0

0

0

0

0

0

1

1

0

0

0

1

0

0

1

0

1

1

1

1

0

0

0

0

1

0

1

0

1

m21 1

0

1

0

0

1

1

0

1

1

0

1

1

0

1

0

0

1

1

0

1

1

0

0

1

0

1

1

0

0

m22

1

1

0

0

0

1

1

1

1

0

0

1

1

1

0

0

0

1

1

1

1

0

0

0

1

1

0

0

0

1

0 1

If m1 were pure1. Ptree(MV.a1i) we can think of

creating a “a1i–pattern” and then AND that pattern

with JI to give Ptree(MV.a1i).

What is the “a1i–pattern” or M1i (M for “Mask”)?

In terms of 0-D P-sequences in Raster order, it’s

easy. One should be able to do the raster-to-Peano

reordering to get M1i

Then a similar process should yield Mij for all ij

f000 0 1 31 1

f001 1 0 10 1

f010 2 0 2 1 0

f011 3 0 00 0

f100 4 0 00 0

f101 5 0 31 1

f110 6 0 31 1

f111 7 1 20 1

f002 8 1 31 1

f012 9 0 21 0

f102 10 0 00 0

f112 11 0 3 1 1

f020 12 1 31 1

f021 13 0 10 1

f030 14 0 21 0

f031 15 1 00 0

f120 16 0 00 0

f121 17 1 3 1 1

f130 18 1 3 1 1

f131 19 1 20 1

f022 20 1 31 1

f032 21 0 2 1 0

f122 22 0 00 0

f132 23 0 00 0

f040 24 0 31 1

f041 25 1 1 0 1

f140 26 0 21 0

f141 27 1 21 0

f042 28 0 00 0

f142 29 1 10 1

030

130

031

131

032

132

020

120

021

121

022

122

010

110

011

111

012

112

000

100

rrn2

001

101

002

102

0

1

rrn3

2

D2 rrn2 a21 a22 a23(a231 a232)

d20 0 1 1 2 1 0

d21 1 0 1 2 1 0

d22 2 0 0 0 0 0

d23 3 0 0 3 1 1

a21

1

0

0

0

a22

1

1

0

0

a231

1

1

0

1

a232

0

0

0

1

D1rrn1 a11 a12

d10 0 1 0

d11 1 0 1

a11

1

0

a12

0

1

a31

0

1

0

D3rrn3 a31 a32 a33

d30 0 0 1 0

d31 1 1 1 0

d32 2 0 1 1

a32

1

1

1

a33

0

0

1


D 3

pat

a231

a231

a232

a232

a22

a12

a21

a31

a22

a32

a33

a32

a31

a11

a21

a12

a11

a33

1 1

0 0

0 0

0 0

1 0

1 0

1 0

1 0

0 0

0 0

1 1

1 1

1 1

1 1

1 1

1 1

0 0

0 0

0 0

0 0

1 0

1 0

1 0

1 0

0 0

0 0

0 0

0 0

1 1

1 1

1 1

1 1

0 0

0 0

0 0

0 0

0 0

1 1

1 1

1 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 1

0 0

0 0

0 0

1 1

0 0

0 0

0 0

1 1

0 0

0 0

0 0

0 0

0 0

1 1

1 1

1 1

0 0

0 0

1 1

1 1

1 1

0 0

0 0

0 0

1 1

1 0

0 0

1 0

0 0

1 1

1 1

1 1

1 0

1 0

1 0

1 0

1 1

0 1

0 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 1

1 0

1 0

1 0

1 0

0 0

0 0

0 0

0 0

0 0

0 0

1 1

1 1

1 1

0 1

0 1

0 0

0 1

0 1

0 1

0 1

1 1

1 1

1 1

1 1

0 0

1 1

1 1

1 1

0 0

0 0

0 0

1 1

1 1

0 0

0 0

0 0

0 1

0 1

0 1

0 1

0 0

0 0

1 1

1 1

1 1

1 1

1 1

1 1

1 1

0 0

0 0

0 0

0 1

0 1

0 1

0 0

0 1

0 1

0 1

0 1

0 0

0 0

0 0

0 0

1 0

1 0

1 0

1 0

0 0

0 0

0 0

1 1

1 0

1 0

1 0

1 0

0 0

0 0

1 1

1 1

0 1

0 1

0 1

0 1

0 0

1 1

1 1

1 1

0 0

0 0

0 0

1 1

1 1

0 0

0 0

0 0

0 0

0 0

1 1

1 1

1 1

0 0

0 0

0 0

1 1

1 1

1 1

1 1

0 0

0 0

0 0

0 0

0 0

1 1

1 1

1 1

1 1

1 1

1 1

1 1

0 0

1 0

0 0

1 0

0 0

1 0

0 0

1 0

0 0

1 0

0 0

1 0

3

2

1

0

rrn1

F rrn m1m2(m21 m22)

m1

1

0

0

0

0

0

0

1

1

0

0

0

1

0

0

1

0

1

1

1

1

0

0

0

0

1

0

1

0

1

m21 1

0

1

0

0

1

1

0

1

1

0

1

1

0

1

0

0

1

1

0

1

1

0

0

1

0

1

1

0

0

a11

1

0

0

0

0

0

0

0

1

0

0

0

1

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

a12

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

1

1

0

0

0

0

0

0

0

0

0

0

a21

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

a22

1

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

a231

1

0

0

0

0

0

0

1

1

0

0

0

1

0

0

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

a232

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

a31

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

0

0

0

0

a32

1

0

0

0

0

0

0

1

1

0

0

0

1

0

0

1

0

1

1

1

1

0

0

0

0

1

0

1

0

1

a33

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

m22

1

1

0

0

0

1

1

1

1

0

0

1

1

1

0

0

0

1

1

1

1

0

0

0

1

1

0

0

0

1

0 1

f000 0 1 31 1

f001 1 0 10 1

f010 2 0 2 1 0

f011 3 0 00 0

f100 4 0 00 0

f101 5 0 31 1

f110 6 0 31 1

f111 7 1 20 1

f002 8 1 31 1

f012 9 0 21 0

f102 10 0 00 0

f112 11 0 3 1 1

f020 12 1 31 1

f021 13 0 10 1

f030 14 0 21 0

f031 15 1 00 0

f120 16 0 00 0

f121 17 1 3 1 1

f130 18 1 3 1 1

f131 19 1 20 1

f022 20 1 31 1

f032 21 0 2 1 0

f122 22 0 00 0

f132 23 0 00 0

f040 24 0 31 1

f041 25 1 1 0 1

f140 26 0 21 0

f141 27 1 21 0

f042 28 0 00 0

f142 29 1 10 1

030

130

031

131

032

132

020

120

021

121

022

122

010

110

011

111

012

112

000

100

rrn2

001

101

002

102

0

1

rrn3

2

How do we build the MV-Ptrees

in practice? E.g., a11:

Form the a11 matrix in raster order.

Sort by Peano order (bit pos 1st).

Form compressed tree. AND with

JoinIndex Ptree (pattern).

Or there may be a more direct

formula as in later slides?

The direct construction may be

necessary in some cases due

to the fact that the cube may

be gigantic!

D3rrn3 a31 a32 a33

d30 0 0 1 0

d31 1 1 1 0

d32 2 0 1 1

D2 rrn2 a21 a22 a231 a232

d20 0 1 1 1 0

d21 1 0 1 1 0

d22 2 0 0 0 0

d23 3 0 0 1 1

D1rrn1 a11 a12

d10 0 1 0

d11 1 0 1


D 3

Example UF with a 2-D Reflexive Fact File (a graph)

Graph G (as Reflexive 2-D

relationship)

t1 t2 t3 t4 t5 t6 t7

t1 0 1 1 0 1 1 0

t2 1 0 0 0 0 0 1

t3 1 1 1 0 1 0 0

t4 0 0 0 0 0 0 0

t5 1 0 1 0 1 0 1

t6 1 0 0 0 0 0 0

t7 0 1 0 0 1 0 0

Tid1 Tid2

ie, 2-D reflexive relationship on a single dimension file

e.g., a Protein-Protein interaction graph. Note, the dimension files are identical copies of the gene table

Graph G (as

Edge Table)

G(Tid1 Tid2)

t1 t2

t1 t3

t1 t5

t1 t6

t2 t1

t2 t7

t3 t1

t3 t2

t3 t3

t3 t5

t5 t1

t5 t3

t5 t5

t5 t7

t6 t1

t7 t2

t7 t5

Single Dimension File, R

Tid a1 a2 a3 a4 a5 a6 a7 a8 a9 C)

t1 1 0 1 0 0 0 1 1 0 1

t2 0 1 1 0 1 1 0 0 0 1

t3 0 1 0 0 1 0 0 0 1 1

t4 1 0 1 1 0 0 1 0 1 1

t5 0 1 0 1 0 0 1 1 0 0

t6 1 0 1 0 1 0 0 0 1 0

t7 0 0 1 1 0 0 1 1 0 0

Note: Given any 2-D Reflexive Fact File (Graph), the standard Universal Fact File will be denoted as, UF1.

UF2 will denote the UF coming from the “2-hop Graph” Fact File (join of G with itself, G2 = ( G Tid1JOINTid’2 G’)[ Tid1, Tid2’].

UF3 will come from the “3-hop Graph” Fact File, G3= G1 Tid1JOINTid2’ G’[ …


D 3

For this example: UF = UF1= R THETAJOIN R’(THETAJOIN using THETA=G)

UF1

d1 d2 a1 a2 a3 a4 a5 a6 a7 a8 a9 C a1'a2'a3'a4'a5'a6'a7'a8'a9‘ C'

t1 t2 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 0 0 1

t1 t3 1 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 0 0 1 1

t1 t5 1 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0

t1 t6 1 0 1 0 0 0 1 1 0 1 1 0 1 0 1 0 0 0 1 0

t2 t1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1

t2 t7 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0

t3 t1 0 1 0 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 0 1

t3 t2 0 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 0 0 1

t3 t3 0 1 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1

t3 t5 0 1 0 0 1 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0

t5 t1 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 1

t5 t3 0 1 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1

t5 t5 0 1 0 1 0 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0

t5 t7 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0

t6 t1 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1

t7 t2 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 0 1

t7 t5 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0

Recursively, for k > 1 (letting G1=G)

Gk =(Gk-1 gkJOINg1’ G’)(g1,…,gk+1) where gk+1 = g2’

UFk= R Gk-join R’ where Gk-join is ThetaJoin using Gk[g1,gk+1]


D 3

PF

Dimension File, R

Tid a1 a2 a3 a4 a5 a6 a7 a8 a9 C)

t1 1 0 1 0 0 0 1 1 0 1

t2 0 1 1 0 1 1 0 0 0 1

t3 0 1 0 0 1 0 0 0 1 1

t4 1 0 1 1 0 0 1 0 1 1

t5 0 1 0 1 0 0 1 1 0 0

t6 1 0 1 0 1 0 0 0 1 0

t7 0 0 1 1 0 0 1 1 0 0

t15

0

0

0

0

0

0

0

0

0

0

1

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1 1 1 1

1 1

1 1 1 1

1 1 1 1

1

1 1

UF1[a1]

t13

t16

t12

0

1

0

0

1

0

1

0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

Replicate R[a1] columns:

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

0 1 0 0 1 0 1 0

t61

t16

0 1 0 0 1 0 1 0

t21

0

0

1

1

0

1

1

0

0

0

1

1

0

1

1

0

0

0

1

1

0

1

1

0

0

0

1

1

0

1

1

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

1

0

0

1

0

0

0

0

1

0

0

1

0

0

0

0

1

0

0

1

0

0

Replicate R’[a1]=R[a1]tr rows:

t31

t51

t61

UF1[a1’ ]

From R and F

Ptrees, create

Ptrees for UF?

F (Edge Tbl)

t1

t2

1

2

1

3

1

5

1

6

2

1

For UF1[a1] AND with PF

2

7

3

1

3

2

3

3

3

5

5

1

5

3

5

5

5

7

6

1

7

2

For UF1[a1’] AND with PF

7

5


D 3

PR[a1]

R[a1]replicated

0

0 0 0 0

01 10 10

012

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

PG-pattern

0

0

0011

0011

0011

0011

0011

0011

0011

0011

0001

0010

0101

0011

0100

0001

0011

0

013

0

0

0

0

0

0

0

0

221

0

0

0

0

R[a1]

0

0

0

0

1

0

0

1

0

1

0

0

112

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1100

0001

0010

1100

0001

0001

0001

1100

1100

1100

0100

1100

0001

1100

1100

1100

1100

1100

1100

1100

1100

1100

1100

103

0

0

P R[a1]-replicated

0

0

0

0

Direct development of MV-Ptrees:Develop the algorithm and code for creating

the basic Ppattern PR[ai]-replicated Ptrees and (therefore) PUF[ai] Ptrees from PF and R Ptrees.


D 3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

1

0

1

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

RG1[a2]

t21

t27

t31

t32

t33

t35

0

0

1

1

0

1

0

0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

0

0

1

1

0

1

0

0

Replicate R[a2] as cols of matrix

For UF1[a2] AND with pat

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

0 0 1 1 0 1 0 0

t53

t55

t57

t51

0

0

1

1

0

1

1

0

0

0

1

1

0

1

1

0

0

0

1

1

0

1

1

0

0

0

1

1

0

1

1

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

0

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

1

0

0

1

0

0

0

0

1

0

0

1

0

0

0

0

1

0

0

1

0

0

Replicate R[a2]tr as rows of matrix:

For UF1[a2’] AND with pat

UF1[a2’ ]

t12 t13 t15 t32 t33 t35 t53 t55 t72 t75


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