Slide #1.

pTrees predicate Trees (pTrees): project each attribute (now 4 files) then vertically slice off each bit position (now 12 files) then compress each bit slice into a tree using the predicate =2 e.g., the compression of R into P goes as follows: 11 11 predicate Tree technologies provide fast, accurate horizontal processing of compressed, data-mining-ready, vertical data structures. 1st, Vertically Processing of Horizontal Data (VPHD) e.g., find the number of occurences of 7 0 1 4 nd 2 , using pTrees find the number of occurences of 7 0 1 4 Base 10 for Horizontally structured, record-oriented data, one must scan vertically R(A1 A2 A3 A4) 2 6 3 2 3 2 7 7 7 7 7 7 2 2 0 0 6 6 5 5 1 1 1 1 1 0 1 7 4 5 4 4 R[A1] R[A2] R[A3] R[A4] Base 2 = 010 011 010 010 011 010 111 111 111 111 110 111 010 010 000 000 110 110 101 101 001 001 001 001 001 000 001 111 100 101 100 100 Record truth of predicate: "pure1" = "all 1s" in a tree, recursively on halves, until the half is pure. 1. Whole thing pure1? false  0 2. Left half pure1? false  0 3. Right half pure1? false  0 4. Left half of rt half ? false0 5. Rt half of right half? true1 0 0 0 0 0 0 1 1 R11 pure1? false=0 pure1? true=1 pure1? false=0 pure1? false=0 pure1? false=0 P11 But it's pure0 so this branch ends 0 0 0 01 010 011 010 010 011 010 111 111 111 111 110 111 010 010 000 000 110 110 101 101 001 001 001 001 001 000 001 111 100 101 100 100 R11 R12 R13 R21 R22 R23 R31 R32 R33 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 1 1 P11 P12 P13 1 0 0 0 0 0 0 0 01 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 P21 P22 P23 1 1 1 1 0 0 0 0 0 1 0 0 1 1 1 1 1 1 P31 P32 P33 0 0 0 0 0 0 0 1 0 1 0 0 01 0 10 10 0001 ^ 10 ^ ^ 01 01 7 1 1 0 0 0 0 0 0 R41 R42 R43 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 P41 P42 P43 0 0 0 1 10 01 0 0 0 0 1 0 0 0 0 00 00 0100 ^ ^ 01 ^ ^ 10 01 01 4 To count (7,0,1,4)s use 111000001100 = P11^P12^P13^P’21^P’22^P’23^P’31^P’32^P33^P41^P’42^P’43 0 *23 0 0 *22 0 1 *21 *20 =2
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Slide #2.

R(A 2 3 2 2 5 2 7 7 # change A2 7 7 7 7 2 2 0 0 1 A3 6 6 5 5 1 1 1 1 A4 ) 1 0 1 7 4 5 4 4 = 010 011 010 010 101 010 111 111 111 111 110 111 010 010 000 000 110 110 101 101 001 001 001 001 R11 0 0 0 0 1 0 1 1 001 000 001 111 100 101 100 100 R12 1 1 1 1 0 1 1 1 R13 R21 R22 R23 R31 R32 R33 R41 0 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 P11 P12 P13 This (terminal) 0 makes entire left branch 0 There is no need to look at the other operands. These 0s make this node 0 7 P21 P22 P23 P31 P32 P33 0 0 0 0 0 0 0 0 0 1 0 1 0 0 01 0 0 0 1 0 10 10 01 0001 01 ^ 01 ^ 10 ^ ^ ^ 01 01 10 0 1 0 1 0 1 1 0 1 0 0 P41 P42 P43 0 0 0 1 10 01 0 0 0 0 1 0 0 0 0 00 00 0100 ^ ^ 01 ^ ^ 10 01 01 These 1s and these 0s (which when complemented are 1's) make node 1 4 To count occurrences of 7,0,1,4 use 111000001100: P11^P12^P13^P’21^P’22^P’23^P’31^P’32^P33^P41^P’42^P’43 = 0 R42 R43 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 0 1 ^ The 2 -level has 01 the only 1-bit so 1-count=1*2 1 =2 Top-down construction of basic pTrees is best for understanding, bottom-up is much faster (once across). R11 P11 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 Siblings are pure0 so collapse! 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: R11 R12 R13 R21 R22 R23 R31 R32 R33 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 R41 R42 R43 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0
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Slide #3.

7 0 1 4 P11 7,0,1,4=111000001100 P11^P12^P13^P’21^P’22^P’23^P’31^P’32^P33^P41^P’42^P’43 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? 0 0 0 01 ^ 10 P11 level_3 level_2 level_1 level_0 Note: level and stage used interchangeably Store complements or derive them when needed? Or process complement set with separate code? PureOne level 2 (Pure1_lev2)  PureZero level2 (Pure0_lev2)  Mixed level 2 (Mixed_lev2)  Derive comps: Mix of comp-no change. Swap p1, p0 PureOne level 2 (Pure1_lev2)  PureZero level 2 (Pure0_lev2)  Mixed level 2 (Mixed_lev2)  0 0 0 01 ^ 10 P11 P12 0 1 0 01 ^ 01 P12 0 1 0 01 ^ 01 P12 P13 P21 P22 1 P13 0 0 1 P31 0 0 0 P31 0 0 1 0 0 0 10 10 ^ 01 P33 P41 0 P21 P22 P32 P33 0 0 0 0 10 0 1 01 P32 P33 0 0 0 0 10 0 1 01 P23 0 0 0 0 10 10 ^ 01 0 0 P41 P42 P43 0 0 1 00 ^ 01 0 0 0 0 0 0 00 0100 ^ 01 ^ ^ 01 10 P41 P42 P43 0 0 0 0 0 0 0 00 0100 ^ 01 ^ ^ 01 10 0 1 00 ^ 01 P31 0 P32 0 0 0 10 P42 P43 0 0 0 0 0 0 00 0100 ^ 01 ^ ^ 01 10 0 0 0 01 ^ 10 0 1 0 01 ^ 01 0 0 0 0001 ^ 10 ^ 01 0 0 1 01 0 0 1 00 ^ 01 P11 P12 P13 P33 P41 P21 P22 P23 P31 P32 P42 P43 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 1 P11 P12 P13 P33 P41 P’21 P’22 P’23 P’31 P’32 P’42 P’43 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 The contribution to the result 1-bit count from lev2: 2 2 * 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. Retrieve level_1 vectors : 1 P23 0 0 1 P23 0 0 0 10 10 ^ 01 1 P21 P22 0 0 0 0001 ^ 10 ^ 01 P13 0 0 0 0 0 0001 ^ 10 ^ 01 lev1_P 112 lev1_P122 lev1_P132 PureOne_level1 (Pure1_lev1)  0 1 0 1 0 1 0 0 PureZero_level1 (Pure0_lev1)  0 0 0 0 The contribution to the result 1-bit count from level 1: 2 1 * Count( & Pure1_lev1 ) = 2 * Count( 0 1 ) = 2* 1 = 2 1 0 1 1 0 0 1 0 0 1 0 Retrieve level 1 only if orPure0_lev2 (=10) has a 0-bit. So retrieve: P112 P122 P132 P’222 P’432 lev1_P’222 lev1_P’432 0 1 1 0 0 1 0 1 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).
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Slide #4.

pTrees construction [one-time] 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 8_4_2_1_gte50%ones_pTree11 1 0 0 0 0 0 0 1 1 1 0 1 1 binary_pure1 _pTree11 = 8_4_2_1_gte100%ones_pTree11 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 R11 0 0 1 1 0 1 1 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 R41 R42 R43 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 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. 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 th 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). 1 0 0 0 1 0 1 1 8_4_2_1_gte50%_pTree11 R11 (changed R11 so this issue of recording 1-counts as you go is pertinent) 1.1-child is pure1 and 0-child is just below 50% (so parent_node=1) 2.1-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 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 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 1 1 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 0 0 0 1 0 1 1 1 Onecount=3. 1 1 0 this 0 stride=8 0 0 0 OneCount=4 1 1 ( 0 50%). 0 0 or 1? Need to know left branch OneCount=1, and right branch So subtree 1 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.
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Slide #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 . 110 001 011 01 f=3: 0 pure0: 0 0 0 0 . 001 110 100 10 f=2: 0 pure1: 1 0 0 0 0 1 01 01 10 1 f=2: 0 pure0: 0 1 0 0 0 1 . 10 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.
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Slide #6.

s |0 |0 |3 |3 |1 |1 |2 |2 |4 |5 E c |1 |0 |1 |3 |3 |0 |2 |3 |0 |1 gr |2| |3| |3| |0| |0| |2| |2| |3| |2| |2| E.s2 E.s1 E.s0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 E.c1 E.c0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 E.g1 E.g0 1 1 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 1 0 0 S.s=0 S.s=1 S.s=2 S.s=3 S.s=4 S.s=5 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 C.c=0 C.c=1 C.c=2 C.c=3 0 1 0 0 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 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(AND k=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. Find all large 1CourseSets (c=1,3). Form all candidate 2CourseSets ({1,3} only). Of those form large 2CourseSets (none) C.c=1&3 0 0 0 1 0 0 C.c=1 C.c=3 1 0 0 1 0 1 0 1 1 1 0 0
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Slide #7.

R:r cap |0 00|30 11| |1 01|20 10| |2 10|30 11| |3 11|10 01| C:c n cred |0 00|B|1 01| |1 01|D|3 11| |2 10|M|3 11| |3 11|S|2 10| C.n B D M S R.r1 0 0 1 1 R.r0 0 1 0 1 R.c1 1 1 1 0 R.c0 1 0 1 1 C.c1 0 0 1 1 C.c0 0 1 0 1 C.r1 0 1 1 1 S.n A T S B C J S.g M M F F M F C.r0 1 1 1 0 S.s2 0 0 1 1 0 0 S:s n gn |0 000|A|M| |1 001|T|M| |2 010|S|F| |3 011|B|F| |4 100|C|M| |5 101|J|F| S.s1 0 0 0 0 1 1 S  E(label=gr)  C  O(no label)  R S.s0 0 1 0 1 0 1 O.o2 0 0 0 0 1 1 1 1 O :o |0 000|0 |1 001|0 |2 010|1 |3 011|1 |4 100|2 |5 101|2 |6 110|2 |7 111|3 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 c 00|0 00|1 01|0 01|1 10|0 10|2 10|3 11|2 O.c0 0 0 1 1 0 0 0 1 O.r1 0 0 0 0 0 1 1 1 E:s o |0 000|1 |0 000|0 |3 011|1 |3 011|3 |1 001|3 |1 001|0 |2 010|2 |2 010|7 |4 100|4 |5 101|5 r 01| 01| 00| 01| 00| 10| 11| 10| O.r0 1 1 0 1 0 0 1 0 AS, BR SupportEO(AB)=|{cC | sA, (s,c)E and rB, (c,r)O}|? 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 grade 001|2 10| 000|3 11| 001|3 11| 011|0 00| 011|0 00| 000|2 10| 010|2 10| 111|3 11| 100|2 10| 101|2 10| 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 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
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Slide #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 Cassandra NoSQL 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.
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Slide #9.

IRIS (3,2,5,5)-leveled 60% rough pTrees (level_4 each bit strides 3 bits at level_3) (level_3 each bit strides 2 bits at level_2) (level_2 each bit strides 5 bits at level_1) (level_1 each bit strides 5 bits at level_0) s3_s2_s5_s5_gt60_PPW,1 0 s2_s5_s5_gt60_PPW,1 100 s5_s5_gt60_PPW,1 11 10 01 11111 10111 10110 11000 10010 11111 s5_gt60_PPW,1 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 1bits 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): 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
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Slide #10.

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??? 86 global_fanout=5: 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 global_fanout=4: 41 11 11 10 09 31 14 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
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Slide #11.

86 As the table grows: 41 11 11 10 09 31 14 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 89 41 11 11 10 09 31 17 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 11 11 10 09 31 22 08 06 07 10 10 11 01 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 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 01 95 41 11 11 10 09 31 23 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 1100 1011 1111 0100 000
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Slide #12.

95 As the table continues to grow:11 41 11 10 09 31 23 08 06 07 10 10 11 02 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 11 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 103 41 11 11 10 09 31 31 08 06 07 10 10 11 07 03 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3 0 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 11 11 10 09 31 36 08 06 07 10 10 11 07 08 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 44 0 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 119 41 11 11 10 09 31 36 08 06 07 10 11 10 11 07 08 10 01 4331 3233 2341 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1 0 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
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Slide #13.

1 119 31 36 As the table continues to grow: 11 08 06 07 10 10 11 07 08 10 01 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1 10 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 1 125 31 36 17 08 06 07 10 10 11 07 08 10 06 01 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1041 1 10 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 1 131 31 36 23 08 06 07 10 10 11 07 08 10 06 05 02 4113 1223 2211 1222 1423 1423 2234 1141 3230 4420 1041 1040 2 010 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 1 132 131 31 01 36 08 06 07 10 23 01 10 11 07 08 10 06 05 02 4113 1223 2211 1222 1423 1423 2234 01 1141 3230 4420 1041 1040 2000 1 010 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
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Slide #14.

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: 1 1 1 1 1 1 0 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
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Slide #15.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 0 0 0 0 0 1 2 3 4 4 8 11 13 14 15 15 19 20 21 23 25 25 27 27 27 27 28 30 31 32 32 33 35 35 37 37 37 37 37 38 38 38 39 40 40 40 41 41 41 41 41 42 42 42 43 45 46 46 46 46 47 47 48 48 48 48 48 48 49 49 49 49 50 50 50 50 50 50 51 52 52 52 52 52 53 53 53 53 53 53 54 54 54 54 55 56 56 56 56 57 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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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Slide #16.

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1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 1 0 The pTrees for the 1691 pixel crop yield data (in value asc order). [Line_numbers, values, pTrees]
More slides like this


Slide #19.

s20_gte100 sort, s20_gte100, vals s20_gt50 9 8 7 6 5 4 3 2 1 0 sort (1st 55 <44) 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 1 0 1 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 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 0 0 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 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 1 1 0 1 1 0 1 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) sort, s20_gt50, values sort (1st 55 <44) 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 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. s3_s20_gt50 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 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1 1 1 0 1 1 sort, s3_s20_gt50 vals (1st 55 <44) 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
More slides like this


Slide #20.

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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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 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 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 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 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 1 1 1 1 1 1 0 0 0 0 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 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 1 0 s20_gt50 - perfect classification of "high yield" also (which was defined as > 170 bpa). s20gt50 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 0 0 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 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 1 1 0 1 1 0 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 s3_s20_gt50 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 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Slide #21.

s20_gt75 (level_1) s20_lt25 (level_1) s10_gt75 (level_1) s10_gt75 (level_1 cont.) s10_lt25 (level_1) s10_lt25 (level_1 cont.) 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 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 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 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 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 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 0 0 0 0 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 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 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 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 1 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 1 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 1 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 1 0 0 0 0 1 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 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?
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Slide #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?
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Slide #23.

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Slide #26.

5p0 level-2 5p0 level-3 9 8 7 6 5 4 3 2 1 09 8 7 6 5 4 3 2 1 0 5p0 level-1 @min(a1..a10) 9 8 7 6 5 4 3 2 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 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 1 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 1 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 1 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 1 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 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 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 1 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 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
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Slide #27.

7 0 1 4 P11 7,0,1,4=111000001100 P11^P12^P13^P’21^P’22^P’23^P’31^P’32^P33^P41^P’42^P’43 stage_3 stage_2 stage_1 stage_0 p11ONE-1-2 Retrieve stage_1 vectors : 0 0 0 01 ^ 10 P13 P33 P41 0 0 0 0001 ^ 10 ^ 01 0 0 P12 0 1 0 01 ^ 01 P 11-1-2 P12-1-2 p11ZRO-1-2 0 1 0 0 The contribution to the result 1-bit count from stage1: 0 1 01 0 1 00 ^ 01 P13-1-2 0 1 0 0 P21 P22 0 0 1 0 1 P23 0 0 0 0 10 10 ^ 01 P31 P32 P42 P43 0 0 0 0 0 0 0 0 00 0100 ^ 01 ^ ^ 01 10 1 0 0 0 10 P’22-1-2 0 1 0 0 P’43-1-2 0 1 1 0 0 1 0 0 2 1 * 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 terminology in general is (XYZ can be ONE, ZRO or MIX): 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; The answer is 0 + 2 = 2. pXYZ 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; pXYZ-3 pXYZ-2-1 pXYZ-2-2 pXYZ-2 pXYZ-0-1 pXYZ-0-2 pXYZ-1-1.1 pXYZ-0-1.1.1 pXYZ-0 pXYZ-0-1.1.2 pXYZ-1-1.2 pXYZ-0-1.2.1 pXYZ-0-1.2.2 pXYZ-1-2.2 pXYZ-1-2.1 pXYZ-0-2.1.1 pXYZ-0-2.1.2 pXYZ-0-2.2.1 pXYZ-0-2.2.2
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Slide #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; 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 P11 0 0 0 01 ^ 10 P12 0 1 0 01 ^ 01 P13 P33 P41 0 0 0 0001 ^ 10 ^ 01 0 0 0 1 01 0 1 00 ^ 01 P21 P22 0 0 1 0 1 P23 0 0 0 0 10 10 ^ 01 P31 P32 P42 P43 0 0 0 0 0 0 0 0 00 0100 ^ 01 ^ ^ 01 10 1 0 0 0 10 0000 0010 0001 0110 1011 0011 1101 0010 1001 0000 0000 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) (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.
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Slide #29.

What's the best multi-level storage scheme compatible with the data security scheme above? 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; 0000 0010 pZRO-3; pZRO-2; pZRO-1; pZRO-0; 0001 pMIX-3; pMIX-2; pMIX-1; pMIX-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; 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) 0110 1011 0011 1101 0010 1001 0000 0000 0 0000 0100 1000 1110 0001 0000 0101 1111 1010 0000 0000 0 P11 0 0 0 01 ^ 10
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Slide #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
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Slide #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'.
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Slide #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: 1 0.0 0.1.0 (This is the simplest form of the representation)
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Slide #33.

====================== Anding of two P-Trees: ====================== Consider 2 P-Trees: P1 -- Representation -------------0 1.0 0 / \ 1 0 / \ 1 0 P2 -- Representation -------------0.0 0.1.0 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 0.0 0.1.0
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Slide #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 0000 0011 0000 0110 1111 1100 1111 0011 0000 0001 0001 1111 1111 0000 1010 0000 0000 1010 1010 1100 0000 1111 1111 1111 1111 1111 1111 0001 0000 0000 0000 1100 0000 0000 1001 1100 1111 1110 1110 0000 0000 1111 0101 1111 1111 0101 0101 0011 1111 0000 0000 0000 0000 0000 0000 1110 1111 1111 1111 0011 1111 1111 0110 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 AND 0 0 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 0 0 0 0 1 0 0 0 OR 0 0 0 1 1 0 1 0 seg_offset0 1001 0011 0000 0011 0001 0000 0001 0000 seg_offset 1 0000 0001 1111 1111 seg_offset 2 0000 1010 1010 1010 seg_offset 5 1111 1111 0001 0000 seg_offset 7 1100 1001 0000 0001 0000 1010 0001 0000 1000 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.
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Slide #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). 1001 0011 0000 0001 0000 0011 0001 1111 1111 0000 0 0000 1010 0000 0000 1010 1010 1100 0000 0000 1100 10010011 00000001 00001010 11000011 0 1111 1111 1111 1111 1111 1111 0001 0000 4 5 level 0 mixed: 0=10010011 1=00000001 2=00001010 7=11000011 1 4 level 0 mixed: 0=00000001 2=10101010 3=11000000 5=00010000 7=10010000 0100 1000 00000001 10101010 11000000 00010000 10010000 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 0=10010011 and 2=10101010 2=00001010 7=10010000 7=11000011 = 00000001 =1 00001010 =2 10000000 =1 4+8+1+1 = 14 0000 0000 1100 0000 0000 1001
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Slide #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: Grass Black Parking Lot URBAN 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).
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Slide #37.

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 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 R2 000 01001100 01010011 11100101 01011010 R1 000 00110100 11001100 01010011 10010101 G1 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 R 2.2 2 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 2.4 2.5 2.0: 01010011 2.3: 11100101 2.6: 01011010 R1 2.2 2.3 2.5 2.1: 11001100 2.4: 01010011 2.6: 10010101 B1 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 G2 G1 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 G2 0 G1 0 2.2 100 2.2 To AND G2 with B2 for example, 2.4 2.4 2.5 2.4 (since it is a common pure1) 00101100 2.5 2.0: 10101011 2.0: 11001100 2.0: 10101011 (since 2.0 is pure1 in B2) 2.1: 00001010 10101011 00001010 00110010 11110011 10010100 2.4: 01010011 2.2: 10010110 (since 2.2 is pure1 in G2) 2.3: 00110010 2.6: 10010101 100 2.5: 11110101 (since 2.5 is pure1 in G2) 2.6: 11110011 2.7: 10010100 2.6: 11110011 (mixed in G2) 00110100 11001100 01010011 10010101 B2 000 10001000 10010110 11110101 10101001 00100100 B1 000 10101001 (mixed in B2) 2.7: 10100001 10010100 (mixed in G2) B2 2.0 B1 2.2 00100100 (mixed in B2) 2.4 2.0: 10100111 00000100 2.2: 10010110 2.1: 10000101 2.5: 11110101 2.6: 10101001 2.7: 00100100 00100000 10100111 10000101 01001101 01011010 10101001 01100001 2.4: 2.5: 2.6: 2.7: 01001101 01011010 10011001 01100001
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