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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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  1. 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

  2. 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

  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

  4. 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

  5. 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

  6. 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’[ …

  7. 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]

  8. 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

  9. 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.

  10. 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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