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But it's pure0 so this branch ends

pTrees p redicate Tree technologie s provide fast, accurate horizontal processing of compressed, data-mining-ready, vertical data structures. predicate Trees (pTrees) : project each attribute (now 4 files). 1 st , Vertically Processing of Horizontal Data (VPHD).

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But it's pure0 so this branch ends

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  1. pTreespredicateTreetechnologies provide fast, accurate horizontal processing of compressed, data-mining-ready, vertical data structures. predicate Trees (pTrees): project each attribute (now 4 files) 1st, Vertically Processing of Horizontal Data (VPHD) R[A1] R[A2] R[A3] R[A4] R(A1 A2 A3 A4) 2 7 6 1 6 7 6 0 3 7 5 1 2 7 5 7 3 2 1 4 2 2 1 5 7 0 1 4 7 0 1 4 010 111 110 001 011 111 110 000 010 110 101 001 010 111 101 111 011 010 001 100 010 010 001 101 111 000 001 100 111 000 001 100 010 111 110 001 011 111 110 000 010 110 101 001 010 111 101 111 011 010 001 100 010 010 001 101 111 000 001 100 111 000 001 100 for Horizontally structured, record-oriented data, one must scan vertically = pure1? true=1 pure1? false=0 R11 R12 R13 R21 R22 R23 R31 R32 R33 R41 R42 R43 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 pure1? false=0 pure1? false=0 pure1? false=0 0 0 0 1 0 01 0 1 0 1 0 1 1. Whole thing pure1? false  0 P11 P12 P13 P21 P22 P23 P31 P32 P33 P41 P42 P43 2. Left half pure1? false  0 P11 0 0 0 0 0 01 3. Right half pure1? false  0 0 0 0 0 1 0 0 10 01 0 0 0 1 0 0 0 0 0 0 0 1 01 10 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 ^ ^ ^ ^ ^ ^ ^ 0 0 1 0 1 4. Left half of rt half? false0 7 0 1 4 0 0 1 0 0 01 5. Rt half of right half? true1 0 *23 0 0 *22 =2 0 1 *21 *20 0 1 0 To count (7,0,1,4)s use 111000001100P11^P12^P13^P’21^P’22^P’23^P’31^P’32^P33^P41^P’42^P’43 = then vertically slice off each bit position (now 12 files) then compress each bit slice into a treeusing the predicate e.g., the compression of R11 into P11 goes as follows: =2 e.g., find the number of occurences of 7 0 1 4 2nd, using pTreesfind the number of occurences of 7 0 1 4 Base 10 Base 2 R11 0 0 0 0 0 0 1 1 Record truth of predicate: "pure1" = "all 1s" in a tree, recursively on halves, until the half is pure. P11 But it's pure0 so this branch ends

  2. R(A1 A2 A3 A4) R11 R12 R13 R21 R22 R23 R31 R32 R33 R41 R42 R43 0 0 0 0 1 0 0 0 0 0 Top-down construction of basic pTrees is best for understanding, bottom-up is much faster (once across). 1 0 0 1 0 1 0 0 0 0 1 0 01 1 1 0 1 0 0 1 01 R11 R12 R13 R21 R22 R23 R31 R32 R33 R41 R42 R43 This (terminal) 0 makes entire left branch 0 There is no need to look at the other operands. 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 7 0 1 4 These 0s make this node 0 P11 P12 P13 P21 P22 P23 P31 P32 P33 P41 P42 P43 These 1s and these 0s(which when complemented are 1's)make node1 0 0 0 0 0 01 0 0 0 0 1 0 0 10 01 0 0 0 1 0 0 0 0 0 0 0 1 01 10 0 0 0 0 1 10 0 1 0 ^ ^ ^ ^ ^ ^ ^ ^ ^ 0 0 1 0 1 0 0 1 0 0 01 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 010 111 110 001 011 111 110 000 010 110 101 001 010 111 101 111 101 010 001 100 010 010 001 101 111 000 001 100 111 000 001 100 2 7 6 1 3 7 6 0 2 7 5 1 2 7 5 7 5 2 1 4 2 2 1 5 7 0 1 4 7 0 1 4 = # change To count occurrences of 7,0,1,4 use 111000001100: 0 P11^P12^P13^P’21^P’22^P’23^P’31^P’32^P33^P41^P’42^P’43 = 0 0 01 The 21-level has the only 1-bit so 1-count=1*21 =2 ^ R11 0 0 0 0 1 0 1 1 Bottom-up construction of 1-Dim, P11, is done using in-order tree traversal, collapsing of pure siblings as we go: P11 0 Siblings are pure0 so collapse!

  3. level_3 level_2 level_1 level_0 P11 P11 P11 P23 P41 P12 P31 P12 P13 P43 P12 P22 P23 P41 P42 P43 P21 P42 P31 P21 P13 P22 P31 P22 P13 P21 P43 P42 P41 P23 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ Store complements or derive them when needed? Or process complement set with separate code? P11^P12^P13^P’21^P’22^P’23^P’31^P’32^P33^P41^P’42^P’43 PureOne level 2 (Pure1_lev2)  7,0,1,4=111000001100 PureZero level2 (Pure0_lev2)  Mixed level 2 (Mixed_lev2)  P32 P32 P33 P33 P33 P32 P11 P41 P21 P22 P23 P31 P32 P42 P43 P12 P13 P33 0 1 0 1 0 0 0 0 1 0 01 0 0 0 0 0 01 0 0 0 1 0 01 0 1 0 1 0 0 0 0 0 0 01 0 1 0 1 0 0 0 1 0 0 01 0 0 1 0 0 01 0 0 1 0 0 01 0 0 0 1 0 01 0 0 0 0 0 01 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 01 0 1 0 0 1 01 0 1 0 0 1 01 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 Retrieve level_1 vectors : lev1_P112 lev1_P122 lev1_P132 lev1_P’222 lev1_P’432 0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 1 0 1 1 0 PureOne_level1 (Pure1_lev1)  Derive comps: Mix of comp-no change. Swap p1, p0 0 1 0 1 0 1 0 1 0 1 P11 P41 P’21 P’22 P’23 P’31 P’32 P’42 P’43 P12 P13 P33 7 0 1 4 0 0 PureZero_level1 (Pure0_lev1)  0 0 0 0 1 0 0 1 PureOne level 2 (Pure1_lev2)  0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 PureZero level 2 (Pure0_lev2)  1 0 0 0 0 0 0 0 0 0 0 0 1 01 10 0 0 0 0 1 0 0 10 01 0 0 0 0 1 0 0 10 01 0 0 0 0 0 0 1 01 10 0 0 0 0 0 0 1 01 10 0 0 0 0 1 0 0 10 01 0 0 0 0 1 10 0 0 0 0 1 10 0 0 0 0 1 10 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 Mixed level 2 (Mixed_lev2)  0 1 1 0 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 A Mixed pTree is the AND of the corresponding complements of the pure1 and pure0. Derive it as needed? Or derive once and store? Any one of Pure1, Pure0 or Mixed is derivable from the other two. Store 2? Which 2? Store all 3? Note: level and stage used interchangeably Retrieve level 1 only if orPure0_lev2 (=10) has a 0-bit. The contribution to the result 1-bit count from lev2: 22 * Count( & Pure1_lev2 ) = 4 * Count( 0 0 ) = 4 * 0 = 0 And then for each individual pTree, retrieve that level 1 vector only if Pure1_lev2 has a 0-bit there. So retrieve: P112 P122 P132 P’222 P’432 The contribution to the result 1-bit count from level 1: 21 * Count( & Pure1_lev1 ) = 2 * Count( 0 1 ) = 2* 1 = 2 Retrieve level 0 vector only if orPure0_lev1 (=11) has a 0-bit in that position. And then for each individual pTree, retrieve that level_0 vector only if Pure1_lev1 has a 0-bit there. Since orPure0_lev1 )=(11) has no zero-bits, no level_0 vectors need to be retrieved. The answer, then, is 0 + 2 = 2. Binary pTrees are used for better understanding. In reality, the "coverage stride" of each level would be tuned to processor architecture. E.g., on 64-bit processors, it would make sense to have only 2 levels, where each level_1 bit "covers" or "is the predicate truth of" a string of 64 consecutive level_0 bits. Alternatively, tune coverages to cache strides. If there are GPUs (e.g., Invidia Teslas), have only level 0. In any case, we suggest storing level_0 in its entirety and building level_1 pTrees using various predicates (e.g., pure1, pure0, mixed, gte50%ones). Then only if the tables are very very deep, build level_2 pTrees on top of that... Last point: Level_1 and Level_2 pTrees can be used independently of the level_0 pTrees to do "approximate but very fast" data mining (especially using the gte50%ones predicate).

  4. R11 0 0 0 0 1 0 1 1 pTrees construction [one-time] 0 0 0 1 1 0 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 (changed R11 so this issue of recording 1-counts as you go is pertinent) • 1-child is pure1 and 0-child is just below 50% (so parent_node=1) • 1-child is just above 50% and the 0-child has almost no 1-bits (so that the parent node=0). (example on next slide). R11 R12 R13 R21 R22 R23 R31 R32 R33 R41 R42 R43 1 1 0 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 Can be done as 1 pass thru each bit slice required for bottom-up construction of pure1 pTrees. R11 R12 R13 R21 R22 R23 R31 R32 R33 R41 R42 R43 8_4_2_1_gte50%ones_pTree11 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 binary_pure1 _pTree11 = 8_4_2_1_gte100%ones_pTree11 node_name: (Level, offset left-to-right) E.g., lower left corner node (0,0). Array of nodes at level=L is [L, *] pTree name: Sn-1_..._S1_S0_gteX%onesi,j is a n-level pTree with predicate gteX%ones. S=Stride=number of leaf bits strided by the node. Subscripts, i and j, specify attribute and bitslice. 0 Must record 1-count of stride of each inode (e.g., In binary trees, if a child=1 and the other=0, it could be the 1-child is pure1 and the 0-child is just below 50% (so parent_node=1) or the 1-child is just above 50% and the 0-child has almost no 1-bits (so parent node=0). R11 1 0 0 0 1 0 1 1 8_4_2_1_gte50%_pTree11 1 0 or 1? Need to know left branch OneCount=1, and right branch Onecount=3. So this stride=8 subtree OneCount=4 ( 50%). 0 or 1? OneCount of left branch=1, of right branch=0. So stride=4 subtree OneCount=1 (< 50%). OneCount of right branch=0 (pure0), but OneCount of left branch=?. Finally, recording the OneCounts as we build the tree upwards is a near-zero-extra-cost step.

  5. Given a table and a row-set-predicate, p, a raw pTree is a truth map of the rsp applied to each row as a singleton row-set. Bit slices of numeric attributes and bit map of categorical attributes are typical examples of raw pTrees. Multi-level pTree w fanout f: Divide the raw pTree into f equal segments (final segment short? - the remainder segment). Represent each segment by a 0 or 1 bit placed as f children of the root, depending on the truth of p applied to the segment. Recursively, divide each of those child nodes into f equal segments (with, possibly, a smaller final remainder segment). Represent each segment by a 0 or 1 bit placed as f children of that child depending on the truth of p applied to the segment The canonical example row-set-predicate is pure1. For example, a table (1 column) with 11 values and f=3. Note: with respect to a remainder node pure0 for f=3 means three 0s. Value 3-Bit representation 7 1 1 1 7 1 1 1 2 0 1 0 0 0 0 0 0 0 0 0 7 1 1 1 0 0 0 0 6 1 1 0 5 1 0 1 1 0 0 1 4 1 0 0 For the left-most (high order) raw-bit-slice: f=3: 0 pure1: 0 0 0 0 . 1 1 0 0 0 1 0 1 1 0 1 f=3: 0 pure0: 0 0 0 0 . 0 0 1 1 1 0 1 0 0 1 0 f=2: 0 pure1: 1 0 0 0 0 1 0 1 01 10 1 f=2: 0 pure0: 0 1 0 0 0 1 . 1 0 10 01 0 Thm: (Tnode= Terminal node, Lnode= bottom level node). Each Tnode of the Complement of a pure1pTree = Complement of that Tnode of the pure0pTree. Each Inode of the Complement of a pure1pTree = that Inode of the pure0pTree.

  6. E s c gr |0 |1 |2| |0 |0 |3| |3 |1 |3| |3 |3 |0| |1 |3 |0| |1 |0 |2| |2 |2 |2| |2 |3 |3| |4 |0 |2| |5 |1 |2| E.c0 1 0 1 1 1 0 0 1 0 1 E.g1 1 1 1 0 0 1 1 1 1 1 E.g0 0 1 1 0 0 0 0 1 0 0 E.s2 0 0 0 0 0 0 0 0 1 1 E.s1 0 0 1 1 0 0 1 1 0 0 E.s0 0 0 1 1 1 1 0 0 0 1 E.c1 0 0 0 1 1 0 1 1 0 0 S.s=0 1 1 0 0 S.s=2 0 0 1 1 S.s=3 0 1 0 1 C.c=0 1 1 0 0 1 0 C.c=1 1 0 0 1 0 1 C.c=2 0 0 1 0 0 0 C.c=3 0 1 1 1 0 0 S.s=1 1 0 0 1 S.s=4 1 0 0 0 S.s=5 0 1 0 0 3 2 1 3 Suppose we think of this as a labeled graph, E, between entities S and C; S  E(Label:gr)  C ; then we can do ARM two ways, on StudentSets or on CourseSets. CourseSet ARM: Let C={0,1} be a CourseSet. Supp(C)=|{s | s takes each of c=0,1}| = OneCount(ANDk=0,1 C.c=k) = 1 Conf({0}{1})= [OneCount(C.c=0 AND C.c=1) / OneCount(E.c=0] = 1/3 Assuming minsup=3 and minconf=1/3 APRIORI ARM would go as follow. C.c=1 1 0 0 1 0 1 C.c=3 0 1 1 1 0 0 C.c=1&3 0 0 0 1 0 0 Find all large 1CourseSets (c=1,3). Form all candidate 2CourseSets ({1,3} only). Of those form large 2CourseSets (none)

  7. R:r cap |0 00|30 11| |1 01|20 10| |2 10|30 11| |3 11|10 01| C:c ncred |0 00|B|1 01| |1 01|D|3 11| |2 10|M|3 11| |3 11|S|2 10| S:s ngn |0 000|A|M| |1 001|T|M| |2 010|S|F| |3 011|B|F| |4 100|C|M| |5 101|J|F| O :o c r |0 000|0 00|0 01| |1 001|0 00|1 01| |2 010|1 01|0 00| |3 011|1 01|1 01| |4 100|2 10|0 00| |5 101|2 10|2 10| |6 110|2 10|3 11| |7 111|3 11|2 10| E:s o grade |0 000|1 001|2 10| |0 000|0 000|3 11| |3 011|1 001|3 11| |3 011|3 011|0 00| |1 001|3 011|0 00| |1 001|0 000|2 10| |2 010|2 010|2 10| |2 010|7 111|3 11| |4 100|4 100|2 10| |5 101|5 101|2 10| C.n B D M S S.n A T S B C J S.g M M F F M F O.o2 0 0 0 0 1 1 1 1 C.c1 0 0 1 1 R.r1 0 0 1 1 R.r0 0 1 0 1 R.c1 1 1 1 0 R.c0 1 0 1 1 O.o1 0 0 1 1 0 0 1 1 O.o0 0 1 0 1 0 1 0 1 O.c1 0 0 0 0 1 1 1 1 O.c0 0 0 1 1 0 0 0 1 O.r1 0 0 0 0 0 1 1 1 O.r0 1 1 0 1 0 0 1 0 S.s2 0 0 1 1 0 0 S.s1 0 0 0 0 1 1 S.s0 0 1 0 1 0 1 C.c0 0 1 0 1 C.r1 0 1 1 1 C.r0 1 1 1 0 E.s2 0 0 0 0 0 0 0 0 1 1 E.s1 0 0 1 1 0 0 1 1 0 0 E.s0 0 0 1 1 1 1 0 0 0 1 E.o2 0 0 0 0 0 0 0 1 1 1 E.o1 0 0 0 1 1 0 1 1 0 0 E.o0 1 0 1 1 1 0 0 1 0 1 E.g1 1 1 1 0 0 1 1 1 1 1 E.g0 0 1 1 0 0 0 0 1 0 0 S  E(label=gr)  C  O(no label)  R AS, BR SupportEO(AB)=|{cC| sA, (s,c)E and rB, (c,r)O}|? confEO(AB)=Supp(AB)/supp(A) A,DS EC BR SupEO(A)=|{cC| sA, (s,c)E and rB, (c,r)O}|? (all A-stus who take courses in every B-room) ConfEO(AD)=SupEO(AD) / sup(A) (high conf means: if A-stu then that student likely takes a course in each B-room

  8. Current examples of column-oriented DBMSs: Commercial Oracle Exadata Database Machine Oracle Retail Predictive Application Server (RPAS) EMC Greenplum SAND CDBMS SenSage SAP HANA Sybase IQ SADAS Vertica its acad open-source cousin C-Store Valentina (Database)|Valentina Database KDB FAME Kickfire Addamark, now Sensage Scalabl Log Server 1010data's Tenbase database DataProbe EXASolution Infobright Enterprise Edition, integrates with MySQL (formerly Brighthouse) Skytide XOLAP Server SuperSTAR from Space-Time Research ParAccel Analytic Database Aster Data Systems FluidDB Ingres & Vectorwise initiative smartFOCUS smartSERVER ADS Hive Intelligence Hex Engine Microsoft SQL Server 2011 (Denali) has a new feature "Columnstore Index" which can be created on top of a traditional row-based table. They are not updatable, and must be dropped and recreated to insert or update data in the actual table. Therefore it cannot be classified as a true column-oriented DBMS Vizubi columnar DB with Excel graphic interface Open-source (proprietary software) Calpont's InfiniDB Enterprise Ed, MySQL-front end RC21 commercial open source project Xplain Semantic DB (called transposed files) (dead). Open-source (free software) Calpont's InfiniDB Com Ed MySQL-front GPLv2 C-Store No new release since 11/06 GenoByte Col storage sys and API for genotype data Lemur Bitmap Index C++ Library FastBit Infobright Community Edition, regular updates LucidDB and Eigenbase MonetDB academic project Metakit ?? The S programming language and GNU R col-data for stat anal See also Apache CassandraNoSQL References A decomposition storage model, Copeland etal SIGMOD'85 C-Store: A column-oriented DBMS, Stonebraker, VLDB05 The Star Schema Benchmark and Augmented Fact Table Indexing, O’Neis et al TPC Tech Conf 8/24/09 D. J. Abadi et al, Column vs. row-stores, SIGMOD08, p967 Bruno, Teach an old elephant new tricks CIDR09 D Lemire, et al, Sorting improves word-aligned bitmap indexes. DKE69, 2010. " " Reordering Cols for Smaller Indexes (arXiv:0909.1346) Brighthouse: an analytic data warehouse for ad-hoc queries, Slezak et al., VLDB, Auckland 08 ^A DBMS for Large Statistical Databases, Turner, Hammond, Cotton, VLDB 1979, Rio de Janeiro, Brazil.

  9. 86 34 22 30 17 17 11 11 13 17 53333 42443 31331 43121 32152 34334 11111 01110 11001 00111 10101 10111 10010 11011 11110 10011 11100 00100 11100 10011 00100 11011 11001 01000 01 010 00010 01011 00110 01000 11111 10010 11100 10111 10110 01110 11011 s3_s2_s5_s5_gt60_PPW,1 IRIS (3,2,5,5)-leveled 60% rough pTrees (level_4 each bit strides 3 bits at level_3) 0 s2_s5_s5_gt60_PPW,1 (level_3 each bit strides 2 bits at level_2) 100 s5_s5_gt60_PPW,1 (level_2 each bit strides 5 bits at level_1) 11 10 01 s5_gt60_PPW,1 (level_1 each bit strides 5bits at level_0) 11111 10111 10110 11000 10010 11111 PPW,1 11111 01110 11001 00111 10101 10111 10010 11011 11110 10011 11100 00100 11100 10011 00100 11011 11001 01000 01 010 00010 01011 00110 01000 11111 10010 11100 10111 10110 01110 11011 11 1's out of 30, not 15 (>=60%) Consider Node 2.2, which is a s5_s5_gt60 node as described above. Note that there are only 11 1-bits out of 30 at the leaf (level-0) of its subtree which is well short of the 15 required for gt60% thus the node truth value is at least misleading (It does correctly indicate that gt60% of the next level bits are 1-bits, but it incorrectly suggests that gt60% of the raw level-0 bits are 1-bits.). One way around this problem is to use pure1 above level-1. That way, a 2.2 1-bit would indicate that all 5 level-1 bits are 1's and thus all level-0 5-bit strings have a majority of 1-bits or at least 3. Thus the level-0 stride of 2.2 has at least 15 1-bits and thus the "true" at 2.2 correctly indicates that there are a majority of 1-bits strided by it at level-0 (as well as at level-1). However, what do we do if (as is the case above) 2.2 strides a majority of level-1 1-bits but a minority of level-0 1-bits? The use of either a 0 or a 1 bit at 2.2 is misleading. I suggest: residualize all rough pTree bit-vectors (as done for gt60 predicate above) and then for each inode, residualize the level count arrays (for level-1 and up):

  10. 86 69 17 17 17 11 11 13 17 53333 42443 31331 43121 32152 34334 11111 01110 11001 00111 10101 10111 10010 11011 11110 10011 11100 00100 11100 10011 00100 11011 11001 01000 01 010 00010 01011 00110 01000 11111 10010 11100 10111 10110 01110 11011 86 41 31 14 11 11 10 09 08 06 07 10 10 04 4331 3233 2341 4113 1223 2211 1222 1423 1423 22 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 11 From the discussion on the previous slide, it seem practical to have the same fanout through out the tree?. Otherwise it is very difficult to even identify inodes (e.g., what does 2.2 mean). global_fanout= 4 makes sense for images. global_fanout= 8 makes sense for solids, global_fanout=16 for numeric data columns that are not spatial? global_fanout=64 for sparse numeric non-spatial data columns???? global_fanout=1024 for very sparse numeric data columns and for high cardinality bitmapped categorical columns????? On the other hand, maybe a database_global_fanout so that the processing code is simpler??? global_fanout=5: global_fanout=4:

  11. 89 41 31 17 11 11 10 09 08 06 07 10 10 07 4331 3233 2341 4113 1223 2211 1222 1423 1423 223 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 94 41 31 22 11 11 10 09 08 06 07 10 10 11 01 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1 86 41 31 14 11 11 10 09 08 06 07 10 10 04 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 01 95 4331 3233 2341 4113 1223 2211 1222 1423 1423 22 41 31 23 11 11 10 09 08 06 07 10 10 11 02 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 11 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 11 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 As the table grows:

  12. 103 41 31 31 11 11 10 09 08 06 07 10 10 11 07 03 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 108 41 31 36 11 11 10 09 08 06 07 10 10 11 07 08 119 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 44 41 31 36 11 11 11 10 09 08 06 07 10 10 11 07 08 10 01 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1 95 41 31 23 11 11 10 09 08 06 07 10 10 11 02 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 11 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 As the table continues to grow:

  13. As the table continues to grow: 125 41 31 36 17 11 11 10 09 08 06 07 10 10 11 07 08 10 06 01 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1041 1 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 0000 1111 0100 0001 131 41 31 36 23 119 11 11 10 09 08 06 07 10 10 11 07 08 10 06 05 02 41 31 36 11 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1041 1040 2 11 11 10 09 08 06 07 10 10 11 07 08 10 01 132 131 01 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 0000 1111 0100 0001 0000 1111 0000 0110 41 31 36 23 01 11 11 10 09 08 06 07 10 10 11 07 08 10 06 05 02 01 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1041 1040 2000 1 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 1100 1011 1111 0100 0001 1111 0001 1110 0011 1101 0000 1111 1111 0101 0000 1000 0000 1111 0100 0001 0000 1111 0000 0110 0000 0000 0000 0100

  14. 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1110 1111 1110 1001 0111 1100 0111 0111 0111 11 1111 1011 1011 0010 0111 1010 1101 1110 0101 1011 1111 0100 1111 1000 0100 1110 0100 1100 1001 1011 1100 1010 0001 0100 0010 0101 1001 10 01 0001 1111 1001 0111 0010 1111 0110 0111 0110 11 At this point, I see no reason that rough pTree, e.g., node 2.2 truth value, should be the truth of its child but should instead be the truth of the leaf-segment it strides. So for global_fanout=4 and gt50:

  15. 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 1 7 2 0 0 0 0 0 0 0 0 1 0 8 3 0 0 0 0 0 0 0 0 1 1 9 4 0 0 0 0 0 0 0 1 0 0 10 4 0 0 0 0 0 0 0 1 0 0 11 8 0 0 0 0 0 0 1 0 0 0 12 11 0 0 0 0 0 0 1 0 1 1 13 13 0 0 0 0 0 0 1 1 0 1 14 14 0 0 0 0 0 0 1 1 1 0 15 15 0 0 0 0 0 0 1 1 1 1 16 15 0 0 0 0 0 0 1 1 1 1 17 19 0 0 0 0 0 1 0 0 1 1 18 20 0 0 0 0 0 1 0 1 0 0 19 21 0 0 0 0 0 1 0 1 0 1 20 23 0 0 0 0 0 1 0 1 1 1 21 25 0 0 0 0 0 1 1 0 0 1 22 25 0 0 0 0 0 1 1 0 0 1 23 27 0 0 0 0 0 1 1 0 1 1 24 27 0 0 0 0 0 1 1 0 1 1 25 27 0 0 0 0 0 1 1 0 1 1 26 27 0 0 0 0 0 1 1 0 1 1 27 28 0 0 0 0 0 1 1 1 0 0 28 30 0 0 0 0 0 1 1 1 1 0 29 31 0 0 0 0 0 1 1 1 1 1 30 32 0 0 0 0 1 0 0 0 0 0 31 32 0 0 0 0 1 0 0 0 0 0 32 33 0 0 0 0 1 0 0 0 0 1 33 35 0 0 0 0 1 0 0 0 1 1 34 35 0 0 0 0 1 0 0 0 1 1 35 37 0 0 0 0 1 0 0 1 0 1 36 37 0 0 0 0 1 0 0 1 0 1 37 37 0 0 0 0 1 0 0 1 0 1 38 37 0 0 0 0 1 0 0 1 0 1 39 37 0 0 0 0 1 0 0 1 0 1 40 38 0 0 0 0 1 0 0 1 1 0 41 38 0 0 0 0 1 0 0 1 1 0 42 38 0 0 0 0 1 0 0 1 1 0 43 39 0 0 0 0 1 0 0 1 1 1 44 40 0 0 0 0 1 0 1 0 0 0 45 40 0 0 0 0 1 0 1 0 0 0 46 40 0 0 0 0 1 0 1 0 0 0 47 41 0 0 0 0 1 0 1 0 0 1 48 41 0 0 0 0 1 0 1 0 0 1 49 41 0 0 0 0 1 0 1 0 0 1 50 41 0 0 0 0 1 0 1 0 0 1 51 41 0 0 0 0 1 0 1 0 0 1 52 42 0 0 0 0 1 0 1 0 1 0 53 42 0 0 0 0 1 0 1 0 1 0 54 42 0 0 0 0 1 0 1 0 1 0 55 43 0 0 0 0 1 0 1 0 1 1 56 45 0 0 0 0 1 0 1 1 0 1 57 46 0 0 0 0 1 0 1 1 1 0 58 46 0 0 0 0 1 0 1 1 1 0 59 46 0 0 0 0 1 0 1 1 1 0 60 46 0 0 0 0 1 0 1 1 1 0 61 47 0 0 0 0 1 0 1 1 1 1 62 47 0 0 0 0 1 0 1 1 1 1 63 48 0 0 0 0 1 1 0 0 0 0 64 48 0 0 0 0 1 1 0 0 0 0 65 48 0 0 0 0 1 1 0 0 0 0 66 48 0 0 0 0 1 1 0 0 0 0 67 48 0 0 0 0 1 1 0 0 0 0 68 48 0 0 0 0 1 1 0 0 0 0 69 49 0 0 0 0 1 1 0 0 0 1 70 49 0 0 0 0 1 1 0 0 0 1 71 49 0 0 0 0 1 1 0 0 0 1 72 49 0 0 0 0 1 1 0 0 0 1 73 50 0 0 0 0 1 1 0 0 1 0 74 50 0 0 0 0 1 1 0 0 1 0 75 50 0 0 0 0 1 1 0 0 1 0 76 50 0 0 0 0 1 1 0 0 1 0 77 50 0 0 0 0 1 1 0 0 1 0 78 50 0 0 0 0 1 1 0 0 1 0 79 51 0 0 0 0 1 1 0 0 1 1 80 52 0 0 0 0 1 1 0 1 0 0 81 52 0 0 0 0 1 1 0 1 0 0 82 52 0 0 0 0 1 1 0 1 0 0 83 52 0 0 0 0 1 1 0 1 0 0 84 52 0 0 0 0 1 1 0 1 0 0 85 53 0 0 0 0 1 1 0 1 0 1 86 53 0 0 0 0 1 1 0 1 0 1 87 53 0 0 0 0 1 1 0 1 0 1 88 53 0 0 0 0 1 1 0 1 0 1 89 53 0 0 0 0 1 1 0 1 0 1 90 53 0 0 0 0 1 1 0 1 0 1 91 54 0 0 0 0 1 1 0 1 1 0 92 54 0 0 0 0 1 1 0 1 1 0 93 54 0 0 0 0 1 1 0 1 1 0 94 54 0 0 0 0 1 1 0 1 1 0 95 55 0 0 0 0 1 1 0 1 1 1 96 56 0 0 0 0 1 1 1 0 0 0 97 56 0 0 0 0 1 1 1 0 0 0 98 56 0 0 0 0 1 1 1 0 0 0 99 56 0 0 0 0 1 1 1 0 0 0 100 57 0 0 0 0 1 1 1 0 0 1 101 57 0 0 0 0 1 1 1 0 0 1 102 57 0 0 0 0 1 1 1 0 0 1 103 57 0 0 0 0 1 1 1 0 0 1 104 58 0 0 0 0 1 1 1 0 1 0 105 58 0 0 0 0 1 1 1 0 1 0 106 58 0 0 0 0 1 1 1 0 1 0 107 58 0 0 0 0 1 1 1 0 1 0 108 58 0 0 0 0 1 1 1 0 1 0 109 58 0 0 0 0 1 1 1 0 1 0 110 58 0 0 0 0 1 1 1 0 1 0 111 58 0 0 0 0 1 1 1 0 1 0 112 59 0 0 0 0 1 1 1 0 1 1 113 59 0 0 0 0 1 1 1 0 1 1 114 60 0 0 0 0 1 1 1 1 0 0 115 60 0 0 0 0 1 1 1 1 0 0 116 60 0 0 0 0 1 1 1 1 0 0 117 60 0 0 0 0 1 1 1 1 0 0 118 60 0 0 0 0 1 1 1 1 0 0 119 61 0 0 0 0 1 1 1 1 0 1 120 61 0 0 0 0 1 1 1 1 0 1 121 61 0 0 0 0 1 1 1 1 0 1 122 61 0 0 0 0 1 1 1 1 0 1 123 61 0 0 0 0 1 1 1 1 0 1 124 61 0 0 0 0 1 1 1 1 0 1 125 62 0 0 0 0 1 1 1 1 1 0 126 62 0 0 0 0 1 1 1 1 1 0 127 62 0 0 0 0 1 1 1 1 1 0 128 62 0 0 0 0 1 1 1 1 1 0 129 63 0 0 0 0 1 1 1 1 1 1 130 63 0 0 0 0 1 1 1 1 1 1 131 63 0 0 0 0 1 1 1 1 1 1 132 63 0 0 0 0 1 1 1 1 1 1 133 64 0 0 0 1 0 0 0 0 0 0 134 65 0 0 0 1 0 0 0 0 0 1 135 65 0 0 0 1 0 0 0 0 0 1 136 65 0 0 0 1 0 0 0 0 0 1 137 65 0 0 0 1 0 0 0 0 0 1 138 65 0 0 0 1 0 0 0 0 0 1 139 65 0 0 0 1 0 0 0 0 0 1 140 65 0 0 0 1 0 0 0 0 0 1 141 66 0 0 0 1 0 0 0 0 1 0 142 66 0 0 0 1 0 0 0 0 1 0 143 66 0 0 0 1 0 0 0 0 1 0 144 67 0 0 0 1 0 0 0 0 1 1 145 67 0 0 0 1 0 0 0 0 1 1 146 67 0 0 0 1 0 0 0 0 1 1 147 67 0 0 0 1 0 0 0 0 1 1 148 67 0 0 0 1 0 0 0 0 1 1 149 67 0 0 0 1 0 0 0 0 1 1 150 67 0 0 0 1 0 0 0 0 1 1 151 67 0 0 0 1 0 0 0 0 1 1 152 68 0 0 0 1 0 0 0 1 0 0 153 68 0 0 0 1 0 0 0 1 0 0 154 68 0 0 0 1 0 0 0 1 0 0 155 68 0 0 0 1 0 0 0 1 0 0 156 68 0 0 0 1 0 0 0 1 0 0 157 68 0 0 0 1 0 0 0 1 0 0 158 68 0 0 0 1 0 0 0 1 0 0 159 69 0 0 0 1 0 0 0 1 0 1 160 69 0 0 0 1 0 0 0 1 0 1 161 69 0 0 0 1 0 0 0 1 0 1 162 69 0 0 0 1 0 0 0 1 0 1 163 69 0 0 0 1 0 0 0 1 0 1 164 70 0 0 0 1 0 0 0 1 1 0 165 70 0 0 0 1 0 0 0 1 1 0 166 70 0 0 0 1 0 0 0 1 1 0 167 70 0 0 0 1 0 0 0 1 1 0 168 70 0 0 0 1 0 0 0 1 1 0 169 70 0 0 0 1 0 0 0 1 1 0 170 71 0 0 0 1 0 0 0 1 1 1 171 71 0 0 0 1 0 0 0 1 1 1 172 71 0 0 0 1 0 0 0 1 1 1 173 71 0 0 0 1 0 0 0 1 1 1 174 71 0 0 0 1 0 0 0 1 1 1 175 71 0 0 0 1 0 0 0 1 1 1 176 71 0 0 0 1 0 0 0 1 1 1 177 71 0 0 0 1 0 0 0 1 1 1 178 71 0 0 0 1 0 0 0 1 1 1 179 71 0 0 0 1 0 0 0 1 1 1 180 71 0 0 0 1 0 0 0 1 1 1 181 72 0 0 0 1 0 0 1 0 0 0 182 72 0 0 0 1 0 0 1 0 0 0 183 72 0 0 0 1 0 0 1 0 0 0 184 72 0 0 0 1 0 0 1 0 0 0 185 72 0 0 0 1 0 0 1 0 0 0 186 72 0 0 0 1 0 0 1 0 0 0 187 72 0 0 0 1 0 0 1 0 0 0 188 73 0 0 0 1 0 0 1 0 0 1 189 73 0 0 0 1 0 0 1 0 0 1 190 73 0 0 0 1 0 0 1 0 0 1 191 73 0 0 0 1 0 0 1 0 0 1 192 73 0 0 0 1 0 0 1 0 0 1 193 73 0 0 0 1 0 0 1 0 0 1 194 74 0 0 0 1 0 0 1 0 1 0 195 74 0 0 0 1 0 0 1 0 1 0 196 74 0 0 0 1 0 0 1 0 1 0 197 74 0 0 0 1 0 0 1 0 1 0 198 74 0 0 0 1 0 0 1 0 1 0 199 74 0 0 0 1 0 0 1 0 1 0 200 74 0 0 0 1 0 0 1 0 1 0 301 90 0 0 0 1 0 1 1 0 1 0 302 90 0 0 0 1 0 1 1 0 1 0 303 90 0 0 0 1 0 1 1 0 1 0 304 90 0 0 0 1 0 1 1 0 1 0 305 90 0 0 0 1 0 1 1 0 1 0 306 90 0 0 0 1 0 1 1 0 1 0 307 91 0 0 0 1 0 1 1 0 1 1 308 91 0 0 0 1 0 1 1 0 1 1 309 91 0 0 0 1 0 1 1 0 1 1 310 91 0 0 0 1 0 1 1 0 1 1 311 91 0 0 0 1 0 1 1 0 1 1 312 91 0 0 0 1 0 1 1 0 1 1 313 91 0 0 0 1 0 1 1 0 1 1 314 91 0 0 0 1 0 1 1 0 1 1 315 91 0 0 0 1 0 1 1 0 1 1 316 91 0 0 0 1 0 1 1 0 1 1 317 91 0 0 0 1 0 1 1 0 1 1 318 92 0 0 0 1 0 1 1 1 0 0 319 92 0 0 0 1 0 1 1 1 0 0 320 92 0 0 0 1 0 1 1 1 0 0 321 92 0 0 0 1 0 1 1 1 0 0 322 92 0 0 0 1 0 1 1 1 0 0 323 92 0 0 0 1 0 1 1 1 0 0 324 92 0 0 0 1 0 1 1 1 0 0 325 92 0 0 0 1 0 1 1 1 0 0 326 92 0 0 0 1 0 1 1 1 0 0 327 92 0 0 0 1 0 1 1 1 0 0 328 92 0 0 0 1 0 1 1 1 0 0 329 93 0 0 0 1 0 1 1 1 0 1 330 93 0 0 0 1 0 1 1 1 0 1 331 93 0 0 0 1 0 1 1 1 0 1 332 93 0 0 0 1 0 1 1 1 0 1 333 93 0 0 0 1 0 1 1 1 0 1 334 93 0 0 0 1 0 1 1 1 0 1 335 93 0 0 0 1 0 1 1 1 0 1 336 93 0 0 0 1 0 1 1 1 0 1 337 93 0 0 0 1 0 1 1 1 0 1 338 93 0 0 0 1 0 1 1 1 0 1 339 93 0 0 0 1 0 1 1 1 0 1 340 94 0 0 0 1 0 1 1 1 1 0 341 94 0 0 0 1 0 1 1 1 1 0 342 94 0 0 0 1 0 1 1 1 1 0 343 94 0 0 0 1 0 1 1 1 1 0 344 95 0 0 0 1 0 1 1 1 1 1 345 95 0 0 0 1 0 1 1 1 1 1 346 95 0 0 0 1 0 1 1 1 1 1 347 95 0 0 0 1 0 1 1 1 1 1 348 96 0 0 0 1 1 0 0 0 0 0 349 96 0 0 0 1 1 0 0 0 0 0 350 96 0 0 0 1 1 0 0 0 0 0 351 96 0 0 0 1 1 0 0 0 0 0 352 96 0 0 0 1 1 0 0 0 0 0 353 96 0 0 0 1 1 0 0 0 0 0 354 96 0 0 0 1 1 0 0 0 0 0 355 96 0 0 0 1 1 0 0 0 0 0 356 97 0 0 0 1 1 0 0 0 0 1 357 97 0 0 0 1 1 0 0 0 0 1 358 97 0 0 0 1 1 0 0 0 0 1 359 97 0 0 0 1 1 0 0 0 0 1 360 97 0 0 0 1 1 0 0 0 0 1 361 97 0 0 0 1 1 0 0 0 0 1 362 97 0 0 0 1 1 0 0 0 0 1 363 97 0 0 0 1 1 0 0 0 0 1 364 98 0 0 0 1 1 0 0 0 1 0 365 98 0 0 0 1 1 0 0 0 1 0 366 98 0 0 0 1 1 0 0 0 1 0 367 98 0 0 0 1 1 0 0 0 1 0 368 98 0 0 0 1 1 0 0 0 1 0 369 98 0 0 0 1 1 0 0 0 1 0 370 98 0 0 0 1 1 0 0 0 1 0 371 98 0 0 0 1 1 0 0 0 1 0 372 98 0 0 0 1 1 0 0 0 1 0 373 98 0 0 0 1 1 0 0 0 1 0 374 98 0 0 0 1 1 0 0 0 1 0 375 98 0 0 0 1 1 0 0 0 1 0 376 98 0 0 0 1 1 0 0 0 1 0 377 99 0 0 0 1 1 0 0 0 1 1 378 99 0 0 0 1 1 0 0 0 1 1 379 99 0 0 0 1 1 0 0 0 1 1 380 99 0 0 0 1 1 0 0 0 1 1 381 99 0 0 0 1 1 0 0 0 1 1 382 99 0 0 0 1 1 0 0 0 1 1 383 99 0 0 0 1 1 0 0 0 1 1 384 99 0 0 0 1 1 0 0 0 1 1 385 99 0 0 0 1 1 0 0 0 1 1 386 99 0 0 0 1 1 0 0 0 1 1 387 100 0 0 0 1 1 0 0 1 0 0 388 100 0 0 0 1 1 0 0 1 0 0 389 100 0 0 0 1 1 0 0 1 0 0 390 100 0 0 0 1 1 0 0 1 0 0 391 100 0 0 0 1 1 0 0 1 0 0 392 100 0 0 0 1 1 0 0 1 0 0 393 101 0 0 0 1 1 0 0 1 0 1 394 101 0 0 0 1 1 0 0 1 0 1 395 101 0 0 0 1 1 0 0 1 0 1 396 101 0 0 0 1 1 0 0 1 0 1 397 101 0 0 0 1 1 0 0 1 0 1 398 101 0 0 0 1 1 0 0 1 0 1 399 101 0 0 0 1 1 0 0 1 0 1 400 101 0 0 0 1 1 0 0 1 0 1 401 102 0 0 0 1 1 0 0 1 1 0 402 102 0 0 0 1 1 0 0 1 1 0 403 102 0 0 0 1 1 0 0 1 1 0 404 102 0 0 0 1 1 0 0 1 1 0 405 102 0 0 0 1 1 0 0 1 1 0 406 102 0 0 0 1 1 0 0 1 1 0 407 102 0 0 0 1 1 0 0 1 1 0 408 103 0 0 0 1 1 0 0 1 1 1 409 103 0 0 0 1 1 0 0 1 1 1 410 103 0 0 0 1 1 0 0 1 1 1 411 103 0 0 0 1 1 0 0 1 1 1 412 103 0 0 0 1 1 0 0 1 1 1 413 103 0 0 0 1 1 0 0 1 1 1 414 103 0 0 0 1 1 0 0 1 1 1 415 103 0 0 0 1 1 0 0 1 1 1 416 104 0 0 0 1 1 0 1 0 0 0 417 104 0 0 0 1 1 0 1 0 0 0 418 105 0 0 0 1 1 0 1 0 0 1 419 105 0 0 0 1 1 0 1 0 0 1 420 105 0 0 0 1 1 0 1 0 0 1 421 105 0 0 0 1 1 0 1 0 0 1 422 105 0 0 0 1 1 0 1 0 0 1 423 105 0 0 0 1 1 0 1 0 0 1 424 105 0 0 0 1 1 0 1 0 0 1 425 105 0 0 0 1 1 0 1 0 0 1 426 105 0 0 0 1 1 0 1 0 0 1 427 105 0 0 0 1 1 0 1 0 0 1 428 106 0 0 0 1 1 0 1 0 1 0 429 106 0 0 0 1 1 0 1 0 1 0 430 106 0 0 0 1 1 0 1 0 1 0 431 106 0 0 0 1 1 0 1 0 1 0 432 106 0 0 0 1 1 0 1 0 1 0 433 106 0 0 0 1 1 0 1 0 1 0 434 106 0 0 0 1 1 0 1 0 1 0 435 106 0 0 0 1 1 0 1 0 1 0 436 106 0 0 0 1 1 0 1 0 1 0 437 106 0 0 0 1 1 0 1 0 1 0 438 107 0 0 0 1 1 0 1 0 1 1 439 107 0 0 0 1 1 0 1 0 1 1 440 107 0 0 0 1 1 0 1 0 1 1 441 108 0 0 0 1 1 0 1 1 0 0 442 108 0 0 0 1 1 0 1 1 0 0 443 108 0 0 0 1 1 0 1 1 0 0 444 108 0 0 0 1 1 0 1 1 0 0 445 108 0 0 0 1 1 0 1 1 0 0 446 108 0 0 0 1 1 0 1 1 0 0 447 108 0 0 0 1 1 0 1 1 0 0 448 108 0 0 0 1 1 0 1 1 0 0 449 109 0 0 0 1 1 0 1 1 0 1 450 109 0 0 0 1 1 0 1 1 0 1 451 109 0 0 0 1 1 0 1 1 0 1 452 109 0 0 0 1 1 0 1 1 0 1 453 109 0 0 0 1 1 0 1 1 0 1 454 109 0 0 0 1 1 0 1 1 0 1 455 109 0 0 0 1 1 0 1 1 0 1 456 109 0 0 0 1 1 0 1 1 0 1 457 109 0 0 0 1 1 0 1 1 0 1 458 109 0 0 0 1 1 0 1 1 0 1 459 110 0 0 0 1 1 0 1 1 1 0 460 110 0 0 0 1 1 0 1 1 1 0 461 110 0 0 0 1 1 0 1 1 1 0 462 110 0 0 0 1 1 0 1 1 1 0 463 110 0 0 0 1 1 0 1 1 1 0 464 110 0 0 0 1 1 0 1 1 1 0 465 110 0 0 0 1 1 0 1 1 1 0 466 110 0 0 0 1 1 0 1 1 1 0 467 110 0 0 0 1 1 0 1 1 1 0 468 110 0 0 0 1 1 0 1 1 1 0 469 111 0 0 0 1 1 0 1 1 1 1 470 111 0 0 0 1 1 0 1 1 1 1 471 111 0 0 0 1 1 0 1 1 1 1 472 111 0 0 0 1 1 0 1 1 1 1 473 111 0 0 0 1 1 0 1 1 1 1 474 111 0 0 0 1 1 0 1 1 1 1 475 111 0 0 0 1 1 0 1 1 1 1 476 111 0 0 0 1 1 0 1 1 1 1 477 112 0 0 0 1 1 1 0 0 0 0 478 112 0 0 0 1 1 1 0 0 0 0 479 112 0 0 0 1 1 1 0 0 0 0 480 112 0 0 0 1 1 1 0 0 0 0 481 112 0 0 0 1 1 1 0 0 0 0 482 112 0 0 0 1 1 1 0 0 0 0 483 112 0 0 0 1 1 1 0 0 0 0 484 112 0 0 0 1 1 1 0 0 0 0 485 112 0 0 0 1 1 1 0 0 0 0 486 113 0 0 0 1 1 1 0 0 0 1 487 113 0 0 0 1 1 1 0 0 0 1 488 113 0 0 0 1 1 1 0 0 0 1 489 113 0 0 0 1 1 1 0 0 0 1 490 113 0 0 0 1 1 1 0 0 0 1 491 113 0 0 0 1 1 1 0 0 0 1 492 113 0 0 0 1 1 1 0 0 0 1 493 113 0 0 0 1 1 1 0 0 0 1 494 114 0 0 0 1 1 1 0 0 1 0 495 114 0 0 0 1 1 1 0 0 1 0 496 114 0 0 0 1 1 1 0 0 1 0 497 114 0 0 0 1 1 1 0 0 1 0 498 114 0 0 0 1 1 1 0 0 1 0 499 114 0 0 0 1 1 1 0 0 1 0 500 114 0 0 0 1 1 1 0 0 1 0 201 75 0 0 0 1 0 0 1 0 1 1 202 75 0 0 0 1 0 0 1 0 1 1 203 75 0 0 0 1 0 0 1 0 1 1 204 75 0 0 0 1 0 0 1 0 1 1 205 75 0 0 0 1 0 0 1 0 1 1 206 75 0 0 0 1 0 0 1 0 1 1 207 76 0 0 0 1 0 0 1 1 0 0 208 76 0 0 0 1 0 0 1 1 0 0 209 76 0 0 0 1 0 0 1 1 0 0 210 76 0 0 0 1 0 0 1 1 0 0 211 76 0 0 0 1 0 0 1 1 0 0 212 76 0 0 0 1 0 0 1 1 0 0 213 76 0 0 0 1 0 0 1 1 0 0 214 76 0 0 0 1 0 0 1 1 0 0 215 77 0 0 0 1 0 0 1 1 0 1 216 77 0 0 0 1 0 0 1 1 0 1 217 77 0 0 0 1 0 0 1 1 0 1 218 77 0 0 0 1 0 0 1 1 0 1 219 77 0 0 0 1 0 0 1 1 0 1 220 77 0 0 0 1 0 0 1 1 0 1 221 77 0 0 0 1 0 0 1 1 0 1 222 78 0 0 0 1 0 0 1 1 1 0 223 78 0 0 0 1 0 0 1 1 1 0 224 78 0 0 0 1 0 0 1 1 1 0 225 78 0 0 0 1 0 0 1 1 1 0 226 78 0 0 0 1 0 0 1 1 1 0 227 78 0 0 0 1 0 0 1 1 1 0 228 79 0 0 0 1 0 0 1 1 1 1 229 79 0 0 0 1 0 0 1 1 1 1 230 79 0 0 0 1 0 0 1 1 1 1 231 79 0 0 0 1 0 0 1 1 1 1 232 80 0 0 0 1 0 1 0 0 0 0 233 80 0 0 0 1 0 1 0 0 0 0 234 80 0 0 0 1 0 1 0 0 0 0 235 80 0 0 0 1 0 1 0 0 0 0 236 81 0 0 0 1 0 1 0 0 0 1 237 81 0 0 0 1 0 1 0 0 0 1 238 81 0 0 0 1 0 1 0 0 0 1 239 81 0 0 0 1 0 1 0 0 0 1 240 81 0 0 0 1 0 1 0 0 0 1 241 81 0 0 0 1 0 1 0 0 0 1 242 81 0 0 0 1 0 1 0 0 0 1 243 81 0 0 0 1 0 1 0 0 0 1 244 82 0 0 0 1 0 1 0 0 1 0 245 82 0 0 0 1 0 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0 0 1 1 0 1604 198 0 0 1 1 0 0 0 1 1 0 1605 198 0 0 1 1 0 0 0 1 1 0 1606 198 0 0 1 1 0 0 0 1 1 0 1607 198 0 0 1 1 0 0 0 1 1 0 1608 199 0 0 1 1 0 0 0 1 1 1 1609 199 0 0 1 1 0 0 0 1 1 1 1610 200 0 0 1 1 0 0 1 0 0 0 1611 200 0 0 1 1 0 0 1 0 0 0 1612 200 0 0 1 1 0 0 1 0 0 0 1613 201 0 0 1 1 0 0 1 0 0 1 1614 201 0 0 1 1 0 0 1 0 0 1 1615 201 0 0 1 1 0 0 1 0 0 1 1616 202 0 0 1 1 0 0 1 0 1 0 1617 202 0 0 1 1 0 0 1 0 1 0 1618 203 0 0 1 1 0 0 1 0 1 1 1619 203 0 0 1 1 0 0 1 0 1 1 1620 203 0 0 1 1 0 0 1 0 1 1 1621 203 0 0 1 1 0 0 1 0 1 1 1622 203 0 0 1 1 0 0 1 0 1 1 1623 204 0 0 1 1 0 0 1 1 0 0 1624 204 0 0 1 1 0 0 1 1 0 0 1625 205 0 0 1 1 0 0 1 1 0 1 1626 205 0 0 1 1 0 0 1 1 0 1 1627 205 0 0 1 1 0 0 1 1 0 1 1628 206 0 0 1 1 0 0 1 1 1 0 1629 207 0 0 1 1 0 0 1 1 1 1 1630 207 0 0 1 1 0 0 1 1 1 1 1631 207 0 0 1 1 0 0 1 1 1 1 1632 207 0 0 1 1 0 0 1 1 1 1 1633 208 0 0 1 1 0 1 0 0 0 0 1634 208 0 0 1 1 0 1 0 0 0 0 1635 208 0 0 1 1 0 1 0 0 0 0 1636 209 0 0 1 1 0 1 0 0 0 1 1637 209 0 0 1 1 0 1 0 0 0 1 1638 210 0 0 1 1 0 1 0 0 1 0 1639 211 0 0 1 1 0 1 0 0 1 1 1640 212 0 0 1 1 0 1 0 1 0 0 1641 213 0 0 1 1 0 1 0 1 0 1 1642 213 0 0 1 1 0 1 0 1 0 1 1643 213 0 0 1 1 0 1 0 1 0 1 1644 213 0 0 1 1 0 1 0 1 0 1 1645 215 0 0 1 1 0 1 0 1 1 1 1646 215 0 0 1 1 0 1 0 1 1 1 1647 215 0 0 1 1 0 1 0 1 1 1 1648 215 0 0 1 1 0 1 0 1 1 1 1649 215 0 0 1 1 0 1 0 1 1 1 1650 215 0 0 1 1 0 1 0 1 1 1 1651 217 0 0 1 1 0 1 1 0 0 1 1652 217 0 0 1 1 0 1 1 0 0 1 1653 218 0 0 1 1 0 1 1 0 1 0 1654 219 0 0 1 1 0 1 1 0 1 1 1655 219 0 0 1 1 0 1 1 0 1 1 1656 220 0 0 1 1 0 1 1 1 0 0 1657 221 0 0 1 1 0 1 1 1 0 1 1658 222 0 0 1 1 0 1 1 1 1 0 1659 222 0 0 1 1 0 1 1 1 1 0 1660 223 0 0 1 1 0 1 1 1 1 1 1661 223 0 0 1 1 0 1 1 1 1 1 1662 224 0 0 1 1 1 0 0 0 0 0 1663 224 0 0 1 1 1 0 0 0 0 0 1664 226 0 0 1 1 1 0 0 0 1 0 1665 227 0 0 1 1 1 0 0 0 1 1 1666 229 0 0 1 1 1 0 0 1 0 1 1667 229 0 0 1 1 1 0 0 1 0 1 1668 230 0 0 1 1 1 0 0 1 1 0 1669 230 0 0 1 1 1 0 0 1 1 0 1670 231 0 0 1 1 1 0 0 1 1 1 1671 232 0 0 1 1 1 0 1 0 0 0 1672 234 0 0 1 1 1 0 1 0 1 0 1673 234 0 0 1 1 1 0 1 0 1 0 1674 236 0 0 1 1 1 0 1 1 0 0 1675 236 0 0 1 1 1 0 1 1 0 0 1676 237 0 0 1 1 1 0 1 1 0 1 1677 238 0 0 1 1 1 0 1 1 1 0 1678 247 0 0 1 1 1 1 0 1 1 1 1679 254 0 0 1 1 1 1 1 1 1 0 1680 265 0 1 0 0 0 0 1 0 0 1 1681 265 0 1 0 0 0 0 1 0 0 1 1682 268 0 1 0 0 0 0 1 1 0 0 1683 269 0 1 0 0 0 0 1 1 0 1 1684 270 0 1 0 0 0 0 1 1 1 0 1685 295 0 1 0 0 1 0 0 1 1 1 1686 346 0 1 0 1 0 1 1 0 1 0 1687 415 0 1 1 0 0 1 1 1 1 1 1688 426 0 1 1 0 1 0 1 0 1 0 1689 435 0 1 1 0 1 1 0 0 1 1 1690 777 1 1 0 0 0 0 1 0 0 1 1691 780 1 1 0 0 0 0 1 1 0 0 The pTrees for the 1691 pixel crop yield data (in value asc order). [Line_numbers, values, pTrees]

  19. s20_gt50 gives perfect classification of "very low yield" (which was defined as < 44 bpa). The last 5 "very low yield pixels fall into 4th s20_gt50 pixel and thus are misclassified (as expected) The significance of this information is that s20_gt50 works but that s20_gte100 (pure1) doesn't. It fails on others (5 pixels fall into the "very low yield class that are not supposed to be there). s3_s20_gt50 is also perfect. s20_gte100 9 8 7 6 5 4 3 2 1 0 s20_gt50 9 8 7 6 5 4 3 2 1 0 sort, s20_gt50, values sort (1st 55 <44) sort, s20_gte100, vals sort (1st 55 <44) s3_s20_gt50 9 8 7 6 5 4 3 2 1 0 sort, s3_s20_gt50 vals (1st 55 <44) 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 1 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 1 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 0 1 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 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 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 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 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 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 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 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 1 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 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 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 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 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 0 33 42 48 52 56 61 66 71 72 77 76 82 86 88 91 93 96 98 101 103 106 108 110 112 114 117 118 121 123 120 125 126 128 129 130 131 132 134 135 136 137 138 140 141 142 143 144 146 147 148 149 150 151 152 153 154 155 156 157 158 160 160 161 163 164 167 168 169 170 173 173 175 177 179 183 184 184 191 193 202 205 215 230 271 1 2 3 4 5 6 7 8 9 01 1 2 3 4 5 6 7 8 9 0 2 1 2 3 4 5 6 7 8 9 0 3 1 2 3 4 5 6 7 8 9 0 4 1 2 3 4 5 6 7 8 9 0 5 1 2 3 4 5 6 7 8 9 0 6 1 2 3 4 5 6 7 8 9 0 7 1 2 3 4 5 6 7 8 9 0 8 1 2 3 4 5 6 7 8 9 0 9 1 2 3 4 5 6 7 0 0 32 32 48 56 0 64 68 72 72 64 80 84 88 88 92 64 96 96 96 104 108 96 112 112 116 118 112 120 120 124 124 0 128 128 131 132 132 134 128 136 138 136 140 140 128 144 144 146 144 148 148 150 144 152 152 154 152 156 158 128 160 160 160 164 164 160 169 168 168 172 160 176 176 180 176 184 128 192 192 192 208 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 32 48 71 76 82 89 103 110 112 123 124 128 134 137 140 142 146 151 152 157 160 161 169 173 179 184 203 199

  20. s20_gt50 - perfect classification of "high yield" also (which was defined as > 170 bpa). 1401 170 0 0 1 0 1 0 1 0 1 0 1402 170 0 0 1 0 1 0 1 0 1 0 1403 171 0 0 1 0 1 0 1 0 1 1 1404 171 0 0 1 0 1 0 1 0 1 1 1405 171 0 0 1 0 1 0 1 0 1 1 1406 171 0 0 1 0 1 0 1 0 1 1 1407 171 0 0 1 0 1 0 1 0 1 1 1408 172 0 0 1 0 1 0 1 1 0 0 1409 172 0 0 1 0 1 0 1 1 0 0 1410 172 0 0 1 0 1 0 1 1 0 0 1411 172 0 0 1 0 1 0 1 1 0 0 1412 172 0 0 1 0 1 0 1 1 0 0 1413 172 0 0 1 0 1 0 1 1 0 0 1414 173 0 0 1 0 1 0 1 1 0 1 1415 173 0 0 1 0 1 0 1 1 0 1 1416 173 0 0 1 0 1 0 1 1 0 1 1417 173 0 0 1 0 1 0 1 1 0 1 1418 173 0 0 1 0 1 0 1 1 0 1 1419 173 0 0 1 0 1 0 1 1 0 1 1420 173 0 0 1 0 1 0 1 1 0 1 1421 173 0 0 1 0 1 0 1 1 0 1 1422 173 0 0 1 0 1 0 1 1 0 1 1423 173 0 0 1 0 1 0 1 1 0 1 1424 173 0 0 1 0 1 0 1 1 0 1 1425 173 0 0 1 0 1 0 1 1 0 1 1426 173 0 0 1 0 1 0 1 1 0 1 1427 173 0 0 1 0 1 0 1 1 0 1 1428 173 0 0 1 0 1 0 1 1 0 1 1429 173 0 0 1 0 1 0 1 1 0 1 1430 173 0 0 1 0 1 0 1 1 0 1 1431 173 0 0 1 0 1 0 1 1 0 1 1432 174 0 0 1 0 1 0 1 1 1 0 1433 174 0 0 1 0 1 0 1 1 1 0 1434 174 0 0 1 0 1 0 1 1 1 0 1435 174 0 0 1 0 1 0 1 1 1 0 1436 174 0 0 1 0 1 0 1 1 1 0 1437 174 0 0 1 0 1 0 1 1 1 0 1438 174 0 0 1 0 1 0 1 1 1 0 1439 174 0 0 1 0 1 0 1 1 1 0 1440 174 0 0 1 0 1 0 1 1 1 0 1441 174 0 0 1 0 1 0 1 1 1 0 1442 174 0 0 1 0 1 0 1 1 1 0 1443 175 0 0 1 0 1 0 1 1 1 1 1444 175 0 0 1 0 1 0 1 1 1 1 1445 175 0 0 1 0 1 0 1 1 1 1 1446 175 0 0 1 0 1 0 1 1 1 1 1447 175 0 0 1 0 1 0 1 1 1 1 1448 175 0 0 1 0 1 0 1 1 1 1 1449 175 0 0 1 0 1 0 1 1 1 1 1450 175 0 0 1 0 1 0 1 1 1 1 1451 175 0 0 1 0 1 0 1 1 1 1 1452 175 0 0 1 0 1 0 1 1 1 1 1453 175 0 0 1 0 1 0 1 1 1 1 1454 176 0 0 1 0 1 1 0 0 0 0 1455 176 0 0 1 0 1 1 0 0 0 0 1456 176 0 0 1 0 1 1 0 0 0 0 1457 176 0 0 1 0 1 1 0 0 0 0 1458 176 0 0 1 0 1 1 0 0 0 0 1459 176 0 0 1 0 1 1 0 0 0 0 1460 176 0 0 1 0 1 1 0 0 0 0 1461 176 0 0 1 0 1 1 0 0 0 0 1462 176 0 0 1 0 1 1 0 0 0 0 1463 176 0 0 1 0 1 1 0 0 0 0 1464 176 0 0 1 0 1 1 0 0 0 0 1465 177 0 0 1 0 1 1 0 0 0 1 1466 177 0 0 1 0 1 1 0 0 0 1 1467 177 0 0 1 0 1 1 0 0 0 1 1468 177 0 0 1 0 1 1 0 0 0 1 1469 177 0 0 1 0 1 1 0 0 0 1 1470 177 0 0 1 0 1 1 0 0 0 1 1471 177 0 0 1 0 1 1 0 0 0 1 1472 177 0 0 1 0 1 1 0 0 0 1 1473 177 0 0 1 0 1 1 0 0 0 1 1474 177 0 0 1 0 1 1 0 0 0 1 1475 177 0 0 1 0 1 1 0 0 0 1 1476 178 0 0 1 0 1 1 0 0 1 0 1477 178 0 0 1 0 1 1 0 0 1 0 1478 178 0 0 1 0 1 1 0 0 1 0 1479 178 0 0 1 0 1 1 0 0 1 0 1480 178 0 0 1 0 1 1 0 0 1 0 1481 178 0 0 1 0 1 1 0 0 1 0 1482 178 0 0 1 0 1 1 0 0 1 0 1483 178 0 0 1 0 1 1 0 0 1 0 1484 179 0 0 1 0 1 1 0 0 1 1 1485 179 0 0 1 0 1 1 0 0 1 1 1486 179 0 0 1 0 1 1 0 0 1 1 1487 179 0 0 1 0 1 1 0 0 1 1 1488 179 0 0 1 0 1 1 0 0 1 1 1489 179 0 0 1 0 1 1 0 0 1 1 1490 179 0 0 1 0 1 1 0 0 1 1 1491 179 0 0 1 0 1 1 0 0 1 1 1492 179 0 0 1 0 1 1 0 0 1 1 1493 179 0 0 1 0 1 1 0 0 1 1 1494 179 0 0 1 0 1 1 0 0 1 1 1495 179 0 0 1 0 1 1 0 0 1 1 1496 180 0 0 1 0 1 1 0 1 0 0 1497 180 0 0 1 0 1 1 0 1 0 0 1498 180 0 0 1 0 1 1 0 1 0 0 1499 180 0 0 1 0 1 1 0 1 0 0 1500 180 0 0 1 0 1 1 0 1 0 0 1501 180 0 0 1 0 1 1 0 1 0 0 1502 180 0 0 1 0 1 1 0 1 0 0 1503 181 0 0 1 0 1 1 0 1 0 1 1504 181 0 0 1 0 1 1 0 1 0 1 1505 181 0 0 1 0 1 1 0 1 0 1 1506 181 0 0 1 0 1 1 0 1 0 1 1507 181 0 0 1 0 1 1 0 1 0 1 1508 182 0 0 1 0 1 1 0 1 1 0 1509 182 0 0 1 0 1 1 0 1 1 0 1510 182 0 0 1 0 1 1 0 1 1 0 1511 182 0 0 1 0 1 1 0 1 1 0 1512 182 0 0 1 0 1 1 0 1 1 0 1513 182 0 0 1 0 1 1 0 1 1 0 1514 182 0 0 1 0 1 1 0 1 1 0 1515 183 0 0 1 0 1 1 0 1 1 1 1516 183 0 0 1 0 1 1 0 1 1 1 1517 183 0 0 1 0 1 1 0 1 1 1 1518 183 0 0 1 0 1 1 0 1 1 1 1519 183 0 0 1 0 1 1 0 1 1 1 1520 183 0 0 1 0 1 1 0 1 1 1 1521 183 0 0 1 0 1 1 0 1 1 1 1522 183 0 0 1 0 1 1 0 1 1 1 1523 183 0 0 1 0 1 1 0 1 1 1 1524 184 0 0 1 0 1 1 1 0 0 0 1525 184 0 0 1 0 1 1 1 0 0 0 1526 184 0 0 1 0 1 1 1 0 0 0 1527 184 0 0 1 0 1 1 1 0 0 0 1528 184 0 0 1 0 1 1 1 0 0 0 1529 184 0 0 1 0 1 1 1 0 0 0 1530 184 0 0 1 0 1 1 1 0 0 0 1531 184 0 0 1 0 1 1 1 0 0 0 1532 184 0 0 1 0 1 1 1 0 0 0 1533 185 0 0 1 0 1 1 1 0 0 1 1534 185 0 0 1 0 1 1 1 0 0 1 1535 185 0 0 1 0 1 1 1 0 0 1 1536 185 0 0 1 0 1 1 1 0 0 1 1537 185 0 0 1 0 1 1 1 0 0 1 1538 186 0 0 1 0 1 1 1 0 1 0 1539 186 0 0 1 0 1 1 1 0 1 0 1540 186 0 0 1 0 1 1 1 0 1 0 1541 186 0 0 1 0 1 1 1 0 1 0 1542 186 0 0 1 0 1 1 1 0 1 0 1543 186 0 0 1 0 1 1 1 0 1 0 1544 186 0 0 1 0 1 1 1 0 1 0 1545 186 0 0 1 0 1 1 1 0 1 0 1546 187 0 0 1 0 1 1 1 0 1 1 1547 187 0 0 1 0 1 1 1 0 1 1 1548 187 0 0 1 0 1 1 1 0 1 1 1549 187 0 0 1 0 1 1 1 0 1 1 1550 187 0 0 1 0 1 1 1 0 1 1 1551 188 0 0 1 0 1 1 1 1 0 0 1552 188 0 0 1 0 1 1 1 1 0 0 1553 188 0 0 1 0 1 1 1 1 0 0 1554 188 0 0 1 0 1 1 1 1 0 0 1555 188 0 0 1 0 1 1 1 1 0 0 1556 188 0 0 1 0 1 1 1 1 0 0 1557 189 0 0 1 0 1 1 1 1 0 1 1558 189 0 0 1 0 1 1 1 1 0 1 1559 189 0 0 1 0 1 1 1 1 0 1 1560 189 0 0 1 0 1 1 1 1 0 1 1561 189 0 0 1 0 1 1 1 1 0 1 1562 190 0 0 1 0 1 1 1 1 1 0 1563 190 0 0 1 0 1 1 1 1 1 0 1564 190 0 0 1 0 1 1 1 1 1 0 1565 190 0 0 1 0 1 1 1 1 1 0 1566 191 0 0 1 0 1 1 1 1 1 1 1567 191 0 0 1 0 1 1 1 1 1 1 1568 191 0 0 1 0 1 1 1 1 1 1 1569 191 0 0 1 0 1 1 1 1 1 1 1570 191 0 0 1 0 1 1 1 1 1 1 1571 191 0 0 1 0 1 1 1 1 1 1 1572 191 0 0 1 0 1 1 1 1 1 1 1573 191 0 0 1 0 1 1 1 1 1 1 1574 191 0 0 1 0 1 1 1 1 1 1 1575 192 0 0 1 1 0 0 0 0 0 0 1576 192 0 0 1 1 0 0 0 0 0 0 1577 192 0 0 1 1 0 0 0 0 0 0 1578 192 0 0 1 1 0 0 0 0 0 0 1579 193 0 0 1 1 0 0 0 0 0 1 1580 193 0 0 1 1 0 0 0 0 0 1 1581 193 0 0 1 1 0 0 0 0 0 1 1582 193 0 0 1 1 0 0 0 0 0 1 1583 194 0 0 1 1 0 0 0 0 1 0 1584 194 0 0 1 1 0 0 0 0 1 0 1585 195 0 0 1 1 0 0 0 0 1 1 1586 195 0 0 1 1 0 0 0 0 1 1 1587 195 0 0 1 1 0 0 0 0 1 1 1588 195 0 0 1 1 0 0 0 0 1 1 1589 195 0 0 1 1 0 0 0 0 1 1 1590 195 0 0 1 1 0 0 0 0 1 1 1591 195 0 0 1 1 0 0 0 0 1 1 1592 196 0 0 1 1 0 0 0 1 0 0 1593 196 0 0 1 1 0 0 0 1 0 0 1594 196 0 0 1 1 0 0 0 1 0 0 1595 196 0 0 1 1 0 0 0 1 0 0 1596 196 0 0 1 1 0 0 0 1 0 0 1597 197 0 0 1 1 0 0 0 1 0 1 1598 197 0 0 1 1 0 0 0 1 0 1 1599 197 0 0 1 1 0 0 0 1 0 1 1600 197 0 0 1 1 0 0 0 1 0 1 1601 197 0 0 1 1 0 0 0 1 0 1 1602 197 0 0 1 1 0 0 0 1 0 1 1603 198 0 0 1 1 0 0 0 1 1 0 1604 198 0 0 1 1 0 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0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 Note there are many more "true" here than there were for pure pTrees (more information at the upper levels). IRIS (next): On this and the next few slides, we do a similar thing for the IRIS dataset and apply the simplest FAUST algorithm for clustering the IRIS classes, setosa, versicolor and virginica. We find that rough pTrees reveal massive amounts of data mining information at their upper levels! Do you agree? It would be great to have the mid-season RGB for this yield dataset. Then we could really test whether rough upper levels hold enough data to do good classification. Should we try on Netflix data?

  22. Multi-level pTree theory and examples (2011_07_09) The attached table (crop yields) will be used for discussing multi-level pTrees today.  There are about 1700 pixels in the field.  The pTrees were created using the given raster ordering (not the best choice for spatial datasets of course, but easiest, since the table came so ordered (my bad – laziness ;-(  ).  The pTree compression level is much lower that it would be if the ordering had been Peano or Hilbert.) Anyone want to convert to Peano Ordering (look at lat-lon or x-y values and [roughly] re-order based on it)? I did the following (and need your help in evaluation) – I created separate 3-level pTrees using a segment lengths of 20, 10, 5 respectively. I need to explain how I’m using the term “segment”. There are [more than] two ways to create pTrees. Method-1 is to decide upon a global pTree fanout, f, (e.g., f = 2,4,8,16,32,64,…) and let the segmentation [segment lengths] be determined from that.  This works for datawarehouse tables (non-volatile). Method-2 is to decide upon a standard segment size base and let the fanouts vary as the table grows (I am assuming, in our new age of infinite storage, that we use the “historical database” approach – i.e., never update anything in place, but keep all old [timestamped] version forever.  How should we handle the timestamp column?  Future research! e.g., Uncompressed_bit_vector_length=2011, 3 levels, level-0 (leaf), level-1 and level-2 (not including the root which would be a single inode at level-3). (Note that an uncompressed pTree then is simply a 1-level pTree in which the fanout=segment length, and there is the single bit at the root (level-1)) the [starting] level-2-fanout=4; the standard segment length for level-2 (top level) is then floor(2011/4) = 502.  So the level-2 segment lengths are 502,502,502,505. Note: I have chosen to make the last segment longer (length=505), rather than using standard segments length=roof(2011/4)=503 and having a last {remainder segment] of length=502.  The final “remainder” segment length and fanout of each inode has to be recorded separately anyway and, in using floor, as the table grows, we can split off a standard segment as soon as the remainder segment length equals or exceeds twice the standard segment length. Level-1 standard segment length is floor(502/4)=125, and the segment lengths are 125,125,125,127. Level-0 standard segment lengths are always 1. QUESTIONS: Is the roof or floor method better? How do we tune the “standard segment length” choices to best fit a table? The table type?, the data area (e.g., spatial, RSI, Netflix ratings, precision ag, comp aided medical decisioning, commodity algo-trading, …)? Other questions?

  23. 20p1 @min(a1..a20) 9 8 7 6 5 4 3 2 1 0 20p0 @if(@max(a1..a20)=0,1,0) 9 8 7 6 5 4 3 2 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 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 1 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 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 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 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 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 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 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 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 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 1 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 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 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 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 0 0 1 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 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 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 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 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 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 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 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0

  24. 10p0 L-1 continued-1 9 8 7 6 5 4 3 2 1 0 10p0 L-1 continued-2 9 8 7 6 5 4 3 2 1 0 10p1 level-1 continued-1 9 8 7 6 5 4 3 2 1 0 10p1 level-1 continued-2 9 8 7 6 5 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 0 0 1 1 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 0 1 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 1 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 1 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 1 0 0 0 0 0 0 0 0 0 1 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 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 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 1 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 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 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1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 10p1 level-2 9 8 7 6 5 4 3 2 1 0 10p0 Level-2 9 8 7 6 5 4 3 2 1 0 10p1 level-1 @min(a1..a10) 9 8 7 6 5 4 3 2 1 0 10p0 L-1 @if(@max(a1..a10)=0,1,0) 9 8 7 6 5 4 3 2 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 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 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 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0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

  27. stage_3 stage_2 stage_1 stage_0 p11ONE-1-2 p11ZRO-1-2 Retrieve stage_1 vectors : P11-1-2 P12-1-2 P13-1-2 P’22-1-2 P’43-1-2 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 P11 P21 P22 P23 P31 P43 P12 P13 P41 P42 ^ ^ ^ ^ ^ ^ ^ ^ ^ The terminology in general is (XYZ can be ONE, ZRO or MIX): P11^P12^P13^P’21^P’22^P’23^P’31^P’32^P33^P41^P’42^P’43 pXYZ 7,0,1,4=111000001100 pXYZ-3 P32 P33 0 0 0 0 0 01 0 0 0 1 0 01 0 1 0 1 0 0 0 1 0 0 01 pXYZ-2-1 pXYZ-2-2 0 1 0 0 1 01 pXYZ-0-2 pXYZ-0-1 7 0 1 4 0 0 1 0 1 0 0 0 1 0 pXYZ-1-1.1 pXYZ-1-2.2 pXYZ-1-1.2 0 0 0 0 1 0 0 10 01 0 0 0 0 0 0 1 01 10 0 0 0 0 1 10 pXYZ-1-2.1 pXYZ-0-1.1.1 pXYZ-0-2.1.1 pXYZ-0-1.1.2 pXYZ-0-1.2.1 pXYZ-0-1.2.2 pXYZ-0-2.1.2 pXYZ-0-2.2.1 pXYZ-0-2.2.2 0 1 0 0 1 0 pXYZ-0 pXYZ-2 The contribution to the result 1-bit count from stage1: 21 * Count( &ij(pijONE-1-2) ) = 2*Count(0 1) = 2*1 = 2 Retrieve stage0 vectors only if ( ORij(pijZRO-1-2) )=(1 1) has a 0-bit. And then for each individual Ptree, retrieve that stage_0 vector only if pijONE-1-2 has a 0-bit there. Since (ORij(pijZRO-1-2) = (1 1) with no zero-bits, no stage_0 vectors need to be retrieved. The answer is 0 + 2 = 2. Would a good storage scheme be by type then stage? pONE-3; pONE-2; pONE-1; pONE-0; pZRO-3; pZRO-2; pZRO-1; pZRO-0; pMIX-3; pMIX-2; pMIX-1; pMIX-0; Or by stage, then type? pONE-3; pZRO-3; pMIX-3; pONE-2; pZRO-2; pMIX-2; pONE-1; pZRO-1; pMIX-1; pONE-0; pZRO-0; pMIX-0;

  28. (PL) Storage scheme: by predicate then level pONE-3; pONE-2; pONE-1; pONE-0; pZRO-3; pZRO-2; pZRO-1; pZRO-0; pMIX-3; pMIX-2; pMIX-1; pMIX-0; P11 P43 P42 P31 P12 P41 P22 P13 P21 P23 ^ ^ ^ ^ ^ ^ ^ ^ ^ P32 P33 0 0 0 0 0 01 0 0 0 1 0 01 0 1 0 1 0 0 0 1 0 0 01 0 1 0 0 1 01 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 01 10 0 0 0 0 1 0 0 10 01 0 0 0 0 1 10 0 1 0 0 1 0 0000 0010 0001 0110 1011 0011 1101 0010 1001 0000 0000 0 Note that for this storage scheme, we need to pad back in all compressed out bits The savings is in the fact that we can implement the algorithm in the previous slides by accessing segments via offsets (that is we don't ever have to read the re-padded bits or the legitimate bit strings that are not needed for the computation. Another advantage might be, e.g., to AND the complement of 0000 0100 1000 1110 0001 0000 0101 1111 1010 0000 0000 0 For this storage scheme, we also need to pad back in all compressed out bits. There may be some advantage in contiguity of stages (retrieving a full stage at a time as an extent) (LP) Stored by level, then predicate? pONE-3; pZRO-3; pMIX-3; pONE-2; pZRO-2; pMIX-2; pONE-1; pZRO-1; pMIX-1; pONE-0; pZRO-0; pMIX-0; These alternatives require research. The only thought I have at this point is that in either case, the representation of and individual pTree is contiguous and therefore the appropriate parts needed can be accessed by offset into the bit string representing that pTree. However, advantages with respect to prefetching, ANDing and COUNTing speed, etc., must be studied.

  29. What's the best multi-level storage scheme compatible with the data security scheme above? P11 ^ 0 0 0 0 1 10 It probably doesn't matter as long as the bit pattern repesenting a given pTree is contiguous - then the above scheme can be used as described. (PL) Storage scheme: by predicate then level pONE-3; pONE-2; pONE-1; pONE-0; pZRO-3; pZRO-2; pZRO-1; pZRO-0; pMIX-3; pMIX-2; pMIX-1; pMIX-0; 0000 0010 0001 0110 1011 0011 1101 0010 1001 0000 0000 0 we can implement the algorithm in the previous slides by accessing segments via offsets (that is we don't ever have to read re-padded bits or the legitimate bit strings not needed for the computation. (LP) Stored by level, then predicate? pONE-3; pZRO-3; pMIX-3; pONE-2; pZRO-2; pMIX-2; pONE-1; pZRO-1; pMIX-1; pONE-0; pZRO-0; pMIX-0; 0000 0100 1000 1110 0001 0000 0101 1111 1010 0000 0000 0 For this storage scheme, we also need to pad back in all compressed out bits. There may be some advantage in contiguity of stages (retrieving a full stage at a time as an extent)

  30. =================Storing a P-Tree: =================Suppose we are storing the following P-Tree.        0       / \      0   1     / \    1   0       / \      1   0 We know all the leaf nodes (L-nodes) of a P-Tree are pure and all the internal nodes (i-nodes) are mixed. We only need to store the L-nodes only which will be either 1 or 0. One way to do that is to store the level, node and the value of the node in a table. A level and node is of a P-Tree of 8 bits is defined in the following figure:               (1) ------------------> level 3 : contains only 1 node which is numbered by 1             /     \            /       \          (1)       (2)--------------> level 2 : contains 2 nodes which are numbered by 1,2         /   \      /  \       (1)   (2)   (3) (4)----------> level 1 : contains 4 nodes numbered by 1,2,3,4       / \   / \   / \  / \     (1)(2)(3)(4)(5)(6)(7)(8) -------> level 0 : contains 8 nodes 1,2,3,4,5,6,7,8

  31. So the P-Tree will be represented by the following table:level | Node | Value------+------+------  2   |  2   |  1------+------+------  1   |  1   |  1------+------+------  0   |  3   |  1------+------+------  0   |  4   |  0--------------------Or by following table:level.Node | Value-----------+------    2.2    |  1-----------+------    1.1    |  1-----------+------    0.3    |  1-----------+------    0.4    |  0------------------Another way to represent a P-Tree is to store the path address and its value in a 'depth first order'.

  32. Another way to represent a P-Tree is to store the path address and its value in a 'depth first order'. We define the left path of a binary tree by 0 and the right path by 1. So the 4th node of level 0 will be represented by path address 0.1.1, 1st node of level 1 be 0.0 etc. Now in this way the P-Tree will be represented as follows:path address | Value-------------+------    1        |  1-------------+------    0.0      |  1-------------+------    0.1.0    |  1-------------+------    0.1.1    |  0--------------------We can store a P-Tree only by recording the positions of the pure 1 nodes because other nodes are either mixed or pure 0, in both the cases they are represented by 0. So the P-Tree will be represented as follows:10.00.1.0  (This is the simplest form of the representation)

  33. ======================Anding of two P-Trees:======================Consider 2 P-Trees:P1               Representation--               --------------  0              0 / \             1.01   0    / \  1   0P2               Representation--               --------------      0          0.0     / \         0.1.0    0   0   / \   1   0     / \    1   0 Now R = P1 AND P2 will be computed from their representation as follows:We know in AND operation if both the bits are 1 then the result is 1, otherwise the result is 0. Now in representation of P1, 0 in the first line means that the left sub-tree is pure 1. So the left sub-tree of P2 will be in R (as it appear in P2). That is anything started with 0 (0.*) will appear in R. So 0.0 and 0.1.0 will be in R. Again P1 has 1.0 which mean any thing started with 1.0 (1.0.*) will be in R (there is none in this example) So the result R is 0.00.1.0

  34. Two ways of implementing multi-level pTrees. 1. bit vectors at each level 2. Path addresses of "predicate=true" leaves that occur above level=0 and do required level=0 bit vector processing as was done for uncompressed pTrees. E.g., 64-bit pure1 pTrees, p1, p2, to be ANDed (level0 size=8, fanout=8). Method 1.first: (pure1's then pure0's shown) 1001 0011 0000 0001 0000 1010 0000 0000 1111 1111 1111 1111 0000 0000 1100 0011 0000 0001 1111 1111 1010 1010 1100 0000 1111 1111 0001 0000 0000 0000 1001 0000 0110 1100 1111 1110 1111 0101 1111 1111 0000 0000 0000 0000 1111 1111 0011 1100 1111 1110 0000 0000 0101 0101 0011 1111 0000 0000 1110 1111 1111 1111 0110 1111 AND 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0000 0001 0000 0001 0000 1010 0001 0000 1000 0000 Pure1's Level1: (AND gives 1-count so far =8*1=8). AND also is the mask of already processed (map) level0's 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 OR pure0 level1 indicates (thru 0-bit) which level0's need ANDing (after also ORing w map), so byte offsets 0,1,2,5,7) 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 OR 0 0 0 0 1 0 0 0 seg_offset0 seg_offset 1 seg_offset 2 seg_offset 5 seg_offset 7 1001 0011 0000 0001 0000 1010 1111 1111 1100 0011 0000 0001 1111 1111 1010 1010 0001 0000 1001 0000 Level=0 result gives an additional 1-count (in addition to the 8 from level-1) of 6, so the final 1-count is 14. We note, finally, that there are various ways one could check to determine which level0's are pure1 and thereby avoid retrieving those (e.g. byte_offset1 of p2 and byte_offset5 of p1). One way is to simply check each bit of pure1 level1 before retrieving the indicated level0.

  35. Two ways of implementing multi-level pTrees. 1. bit vectors at each level 2. Path addresses of "predicate=true" leaves that occur above level=0 and do required level=0 bit vector processing as was done for uncompressed pTrees. E.g., 64-bit pure1 pTrees, p1, p2, to be ANDed (level0 size=8, fanout=8). 0 0000 1100 10010011 00000001 00001010 11000011 0 0100 1000 00000001 10101010 11000000 00010000 10010000 1001 0011 0000 0001 0000 1010 0000 0000 1111 1111 1111 1111 0000 0000 1100 0011 0000 0001 1111 1111 1010 1010 1100 0000 1111 1111 0001 0000 0000 0000 1001 0000 1 4 level 0 mixed: 0=00000001 2=10101010 3=11000000 5=00010000 7=10010000 4 5 level 0 mixed: 0=10010011 1=00000001 2=00001010 7=11000011 Pure1's Level1: 4 is common so 8*1=8. 5-p1 is pure1 so retrieve 5-p2 and count =1 1-p2 is pure1 so retrieve 1-p1 and count =1 Done processing level-1 1,4,5. Process (retrieve, AND and count) remaining common mixed level-1's, 0,2,7: 0=00000001 2=10101010 7=10010000 0=10010011 2=00001010 7=11000011 00000001 =1 00001010 =2 10000000 =1 4+8+1+1 = 14 and =

  36. Just to get an accurate picture of how to do multi-level pTrees, I think we should take a real image (one that Dr Karaska sent) Scrape out a few actual bit slices (so we can see what the nature of the typical 1-run or 0-run is). Then from that information try to optimize the level0 size (for a k-level pTree). Then create k-level pTrees for the entire image. Then modify FAUST to use the k-level pTrees (top down, one level at a time?) Then see if there is a speed improvement. Then see if a top-level, level(k-1), FAUST can be devised that gives a pretty good classification and does it blindingly fast (since the bit-vectors would be tiny compared to the full uncompressed pTrees). Then see if a top-few-levels FAUST can be devised that gives a pretty good classification and does it blindingly fast. This kind of work would sell the idea of multi-level pTrees (to us first - and then to the world.) Without this kind of work we are just guessing as to whether multi-level pTrees should be pursued for image analysis. Suppose we have an image of a large grass area (on the left), a black parking lot(in the middle) and a few buildings (on the right) and we have 2 bit RGB color (intensity values are 0,1,2,3) as shown on the next slide-assuming the following ordering: GrassBlack Parking LotURBAN This has 192 pixels. The first 64 are green grass, the next 64 are black pavement, and the final 64 are multi-colors (buildings an sidewalks and blvds etc.). We are going to use 3-level pTrees (3x8x8).

  37. R2 000 01001100 01010011 11100101 01011010 R2 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0011 1111 1111 0000 0000 1110 0101 1111 1111 1111 1111 0101 1010 0000 0000 R1 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 1100 1111 1111 1111 1111 0101 0011 1111 1111 1001 0101 0000 0000 R1 000 00110100 11001100 01010011 10010101 G2 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 1011 0000 1010 1111 1111 0011 0010 1111 1111 1111 1111 1111 0011 1001 0100 G1 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 1011 0000 1010 1111 1100 0011 0010 0100 1111 1111 1111 1111 0011 1001 0100 B2 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 0000 0000 1001 0110 0000 0000 1111 1111 1111 0101 1010 1001 0010 0100 B1 G1 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 0111 1000 0101 1111 1111 0000 0000 0100 1101 0101 1010 1001 1001 0110 0001 0 2.2 2.4 2.5 2.0: 11001100 2.4: 01010011 2.6: 10010101 G2 G2 0 2.2 2.4 2.5 2.0: 10101011 2.1: 00001010 2.3: 00110010 2.6: 11110011 2.7: 10010100 100 00101100 10101011 00001010 00110010 11110011 10010100 R2 R1 2.2 2.4 2.5 2.0: 01010011 2.3: 11100101 2.6: 01011010 2.2 2.3 2.5 2.1: 11001100 2.4: 01010011 2.6: 10010101 G1 100 00110100 11001100 01010011 10010101 B2 B2 2.0 2.4 2.2: 10010110 2.5: 11110101 2.6: 10101001 2.7: 00100100 B1 2.2 2.0: 10100111 2.1: 10000101 2.4: 01001101 2.5: 01011010 2.6: 10011001 2.7: 01100001 000 10001000 10010110 11110101 10101001 00100100 B1 000 00100000 10100111 10000101 01001101 01011010 10101001 01100001 To AND G2 with B2 for example, 2.4 (since it is a common pure1) 2.0: 10101011 (since 2.0 is pure1 in B2) 2.2: 10010110 (since 2.2 is pure1 in G2) 2.5: 11110101 (since 2.5 is pure1 in G2) 2.6: 11110011 (mixed in G2) 10101001 (mixed in B2) 10100001 2.7: 10010100 (mixed in G2) 00100100 (mixed in B2) 00000100

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