Advances in algorithms based on CbO Petr Krajca, Jan Outrata, Vilem - - PowerPoint PPT Presentation

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Advances in algorithms based on CbO Petr Krajca, Jan Outrata, Vilem - - PowerPoint PPT Presentation

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Advances in algorithms based on CbO Petr Krajca, Jan Outrata, Vilem Vychodil Palacky University, Olomouc, Czech Republic Concept Lattices and Their Applications, Sevilla, 2010 Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms

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Advances in algorithms based on CbO

Petr Krajca, Jan Outrata, Vilem Vychodil

Palacky University, Olomouc, Czech Republic

Concept Lattices and Their Applications, Sevilla, 2010

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 1 / 21

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Contribution

Three topics of interest:

1

improved canonicity test

additional canonicity test for Close-by-One result: reduction of the number of concepts computed multiple times

2

parallelization

simultanous computation of disjoint sets of concepts focus on various workload distributions

3

data preprocessing

role of attribute permutations experimental observations: efficiency of algorithms w. r. t. number of inversions

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 2 / 21

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Related Work

Next Closure, Close-by-One: Ganter B.: Two basic algorithms in concept analysis. (Technical Report FB4-Preprint No. 831). TH Darmstadt, 1984. Kuznetsov S.: A fast algorithm for computing all intersections of objects in a finite semi-lattice. Autom. Docum. Math. Ling., 27(5)(1993), 11–21. Parallel and Distributed CbO: Krajca, Outrata, Vychodil: Parallel algorithm for computing fixpoints of Galois

  • connections. AMAI (to appear), DOI 10.1007/s10472-010-9199-5.

Krajca, Vychodil: Distributed algorithm for computing formal concepts using map-reduce framework. In: Proc. IDA 2009, LNCS 5772(2009), 333–344. Krajca, Vychodil: Comparison of data structures for computing formal concepts. In: Proc. MDAI 2009, LNAI 5861(2009), 114–125.

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 3 / 21

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Inside CbO: The Canonicity Test

Assumptions: X = {0, 1, . . . , m} (objects); Y = {0, 1, . . . , n} (attributes); A, B . . . current formal concept; attribute y ∈ Y such that y ∈ B; put C = (B ∪ {y})↓ and D = (B ∪ {y})↓↑ .

The Test

For A, B and C, D check whether D ∩ {0, 1, . . . , y − 1} = B ∩ {0, 1, . . . , y − 1} is true: yes (success) = ⇒ proceed with C, D no (failure) = ⇒ skip C, D

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 4 / 21

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CbO Represented by Recursive Procedure GenerateFrom

Procedure GenerateFrom

input: formal concept A, B, first attribute y to be added to A, B

  • utput: all formal concepts with intents containing B

procedure GenerateFrom(A, B, y)

  • utput A, B

if B = Y then for j from y upto n do if j ∈ B then C ← A ∩ {y}↓ D ← C↑ if D ∩ {0, 1, . . . , j − 1} = B ∩ {0, 1, . . . , j − 1} then call GenerateFrom(C, D, j + 1) Initially called with ∅↓, ∅↓↑ and y = 0.

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 5 / 21

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Performance Issues

Canonicity Test: the same closure computed multiple times (returned once) canonicity test is performed after computing of closure Proposed Solution: reuse information about canonicity test failure perform additional test before computing of closure (if possible) CbO Tree – a call tree for procedure GenerateFrom nodes – represent computed closures, two types:

1

Bi, y: represents invocation of GenerateFrom with arguments Bi and y

2

+ Bi: Bi is computed but fails the canonicity test

edges – between nodes labeled by attributes

edge between Bi and Bj is labeled by y whenever (Bi ∪ {y})↓↑ = Bj

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 6 / 21

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Example (CbO Tree)

5 4 3 5 4 3 5 5 4 5 5 4 5 3 5 3 5 4 3 5 4 3 2 3 3 4 2 4 2 1 1 2

  • B8
  • B9
  • B8
  • B8
  • B8
  • B8

B12, 3

  • B5
  • B6
  • B5
  • B5
  • B5
  • B5

B11, 3 B10, 2

  • B8
  • B8

B9, 5

  • B8
  • B8
  • B8

B8, 4 B7, 3

  • B5
  • B5

B6, 5

  • B5
  • B5
  • B5

B5, 4 B4, 3 B3, 2 B2, 1 B1, 0

1 2 3 4 5 × × × × × × × × × × × × ×

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 7 / 21

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Additional Test

Under the notation B j =

  • (B ∪ {j})↓↑ \ B
  • ∩ {0, 1, . . . , j − 1}

we have:

Lemma (On Test Failure Propagation)

Let B ⊆ Y , j ∈ B, and B j = ∅. Then, for each B′ ⊇ B such that j ∈ B′ and B j ⊆ B′, we have B′ j = ∅. Consequences: if the original test fails for intent B, attribute j ∈ B and D = (B ∪ {j})↓↑, then it fails for any intent B′ ⊇ B and D′ = (B′ ∪ {j})↓↑ provided that j ∈ B′ and D ∩ {0, 1, . . . , j − 1} B′ ∩ {0, 1, . . . , j − 1} closure D′ = (B′ ∪ {j})↓↑ need not be computed (!!)

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 8 / 21

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Example (Pruned CbO Tree)

5 4 3 1 2

  • B8
  • B9
  • B8

B10, 2 B2, 1 B1, 0 B12, 3

1 2 3 4 5 × × × × × × × × × × × × ×

M3 = M5 = {0, 2, 3, 4, 5} = B8; M4 = {0, 4} = B9

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 9 / 21

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Example (Pruned CbO Tree)

5 3 4 2 1 5 4 3 1 2

  • B8
  • B8
  • B8
  • B9
  • B8

B9, 5 B7, 3 B3, 2 B10, 2 B2, 1 B1, 0 B12, 3

1 2 3 4 5 × × × × × × × × × × × × ×

M3 = M5 = {0, 2, 3, 4, 5} = B8; M4 = {0, 4} = B9 2 ∈ M3 and 2 ∈ B2 = {0} = ⇒ test fails for j = 3

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 9 / 21

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Example (Pruned CbO Tree)

5 4 3 5 4 3 5 5 5 5 5 3 5 3 5 4 3 4 4 5 4 3 2 3 3 4 2 4 2 1 1 2

  • B8
  • B9
  • B8
  • B8
  • B8
  • B8

B12, 3

  • B5
  • B6
  • B5
  • B5
  • B5
  • B5

B11, 3 B10, 2

  • B8
  • B8

B9, 5

  • B8
  • B8
  • B8

B8, 4 B7, 3

  • B5
  • B5

B6, 5

  • B5
  • B5
  • B5

B5, 4 B4, 3 B3, 2 B2, 1 B1, 0

1 2 3 4 5 × × × × × × × × × × × × ×

M3 = M5 = {0, 2, 3, 4, 5} = B8; M4 = {0, 4} = B9 2 ∈ M3 and 2 ∈ B2 = {0} = ⇒ test fails for j = 3

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 9 / 21

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Procedure FastGenerateFrom

addtional input: (pointers to) sets of attributes {Ny | y ∈ Y } (initially empty) procedure FastGenerateFrom(A, B, y, {Ny | y ∈ Y })

  • utput A, B

if B = Y then for j from y upto n do Mj ← Nj if j ∈ B and Nj ∩ {0, 1, . . . , j − 1} ⊆ B ∩ {0, 1, . . . , j − 1} then C ← A ∩ {y}↓ D ← C↑ if D ∩ {0, 1, . . . , j − 1} = B ∩ {0, 1, . . . , j − 1} then put C, D, j to queue else Mj ← D while get C, D, j from queue call FastGenerateFrom(C, D, j + 1, {My | y ∈ Y })

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 10 / 21

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Soundness, Complexity and Relationship to Other Algorithms

soundness: procedure FastGenerateFrom is sound can be proved using unique derivations (paths in FCbO tree) complexity: polynomial time-delay O(|X| · |Y |3) relationship to other algorithms: lists concepts in the same order as CbO (FCbO = CbO in the worst case) lists as NextClosure (but faster) if the main loop is “from n downto y” Outrata, Vychodil: Fast algorithm for computing fixpoints of Galois connections induced by object-attribute relational data (in preparation).

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 11 / 21

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Performance evaluation I

Test Conditions: algorithms: CbO, FCbO performance indicator: computed closures Results: concepts closures closures ratio ratio (CbO) (FCbO) (CbO) (FCbO) mushroom 238,710 4,006,498 426,563 5.9 % 55.9 %

  • anon. web

129,009 27,949,552 1,475,341 0.4 % 8.7 % debian tags 38,977 12,045,680 679,911 0.3 % 5.7 % tit-tac-toe 59,505 221,608 128,434 26.8 % 46.3 %

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 12 / 21

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Performance evaluation II

Test Conditions: algorithms: CbO, FCbO, NextClosure performance indicator: computation time in seconds Results: mushroom tic-tac-toe debian tags

  • anon. web

size 8,124×119 958×29 14,315× 475 32,710×295 density 19 % 34 % < 1 % 1 % FCbO 0.23 0.02 0.10 0.15 CbO 4.34 0.06 5.31 27.14 NextClosure 685.00 1.86 1,432.25 8,236.85

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 13 / 21

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Parallelization of FCbO (and CbO)

Computation in Three Stages:

1

initialization (serial computation):

goes through CbO/FCbO tree up to depth level L, concepts at level L are enqueued.

2

parallel execution

create k processes P1, . . . , Pk (threads of execution), concepts from the queue are distributed among P1, . . . , Pk, P1, . . . , Pk compute disjoint sets of formal concepts (in parallel).

3

termination

wait for all processes (trivial synchronizaton) and terminate.

Issues: How to chose L and k? (hints in AMAI paper) How to distribute concepts into k queues? (not discussed before)

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 14 / 21

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Workload Distribution

1

round-robin (original approach, see AMAI paper)

concepts are distributed to queues attached to each processor n-th concept is placed into queuer where r = (n mod k) + 1

2

zig-zag – modification of round-robin

r = min

  • n mod z, z − (n mod z)
  • + 1 and z = 2 × k + 1

3

blocks – assigning blocks of consecutive concepts

the queue of all concepts is divided into k chunks of (approximately) equal size each block is assigned to a single processor

4

fair – idle processors get assigned concepts

single shared queue of concepts processors get concepts from the queue on demand pros: achieves best utilization of processors (in theory) cons: additional synchronization needed (overhead)

5

random – random assignment of processors

considered for testing purposes

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 15 / 21

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Performace under various workload distributions

Test Conditions: algorithm: parallel FCbO; processors: 8 performance indicator: computation time in seconds Results: round-r. blocks fair zig-zag random debian tags 0.0974 0.0988 0.0938 0.0984 0.0986

  • anon. web

0.1518 0.1590 0.1500 0.1528 0.1522 mushroom 0.1772 0.2158 0.1550 0.1788 0.1820 tic-tac-toe 0.0172 0.0198 0.0168 0.0174 0.0180 random 5000×100; 10% 0.0806 0.1194 0.0796 0.0820 0.0876 random 10000×100; 15% 1.1380 2.1326 0.8698 1.0974 1.1670

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 16 / 21

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Data Preprocessing Issues

Does the order of attributes matter?

Notion (permutation resistance)

Algorithm is permutation resistant whenever all isomorphic copies of X, Y, I with Y = {0, 1, . . . , n} require the same number of elementary computation steps (computation of a single fixpoint of ↑I and ↓I) in order to compute all concepts. Lindig’s algorithm (UpperNeighbor) is permutation resistant CbO, FCbO are not permutation resistant

  • rdered formal context – a formal context X, Y, I where

|{0}↓I| ≤ |{1}↓I| ≤ · · · ≤ |{n}↓I|

Fu H., Mephu Nguifo E.: A Parallel Algorithm to Generate Formal Concepts for Large Data. In: Proc. ICFCA 2004, LNCS 2961, pp. 394–401.

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 17 / 21

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Computing Formal Concepts in Ordered Contexts

Theorem (canonicity test in ordered contexts)

Let X, Y, I be an ordered formal context where Y = {0, . . . , n} and {a}↓I = {b}↓I for any a, b ∈ Y . Then for each j ∈ Y such that j ∈ ∅↓I↑I, ∅↓I↑I ∩ {0, . . . , j − 1} = {j}↓I↑I ∩ {0, . . . , j − 1}. Notes: under the conditions of Theorem 1 (pairwise distinct columns):

canonicity test succeeds in the first level (of CbO / FCbO tree) beneficial for parallel variants (sufficient to put L = 1)

general tendency (observed experimetally):

  • rdered contexts −

→ reduced number of computed closures increasing number of inversions − → increasing number of computed closures (inversion: a pair y1, y2 ∈ Y × Y such that |{y1}↓I| ≤ |{y2}↓I|)

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 18 / 21

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Reduction of computed closures by input data ordering

Test Conditions: algorithms: FCbO with ordered / unordered input data performance indicator: computed closures Results: concepts closures closures ratio ratio (unordered) (ordered) (unordered) (ordered) mushroom 238,710 426,563 299,201 55.9 % 79.7 %

  • anon. web

129,009 1,475,341 398,147 8.7 % 32.4 % debian tags 38,977 679,911 298,641 5.7 % 13.0 % tit-tac-toe 59,505 128,434 89,930 46.3 % 66.1 %

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 19 / 21

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Impact of the number of inversions

0.0 · 100 2.0 · 106 4.0 · 106 6.0 · 106 8.0 · 106 1.0 · 107 1.2 · 107 1.4 · 107 1.6 · 107 1000 2000 3000 4000 5000 6000 7000 computed closures number of inversions Close-by-One Fast Close-by-One

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 20 / 21

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Availability

Implementation: ANSI C binaries for various platforms (i686, x86 64, sparc64) and OS (linux, windows) hybrid data representation: pointer lists (extents), bitarrays (intents) test machine: Apple MacPro (Intel Xeon, 2.8 GHz, 16 GB RAM) collection of stand-alone programs, available at:

http:/ /fcalgs.sourceforge.net

Krajca, Outrata, Vychodil (UP Olomouc; CZ) Advances in algorithms based on CbO CLA 2010 21 / 21