Use of patterns in n-tuple combinatorial generation

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1 Use of patterns in n-tuple combinatorial generation Dominik Zalewski Faculty of Mathematics and Computer Science UAM Otwarte Wykłady dla Doktorantów May 2007

2 Agenda 1 Applications Peer-to-peer networks Calendaring 2 Combinatorial generation Model Data structures Algorithms 3 Running time Theoretical analysis Experimental results

3 Bibliografia D. E. Knuth, The art of computer programming: Volume 4, Addison Wesley, D. L. Kreher and D. R. Stinson, Combinatorial algorithms: Generation, enumeration and search, CRC press LTC, Boca Raton, Florida, Dmitri Loguinov, Juan Casas, and Xiaoming Wang, Graph-theoretic analysis of structured peer-to-peer systems: routing distances and fault resilience, IEEE/ACM Trans. Netw. 13 (2005), no. 5, C. D. Savage, A survey of combinatorial gray codes, Paul Vixie, Cron manual, 1993.

4 Agenda 1 Applications Peer-to-peer networks Calendaring 2 Combinatorial generation Model Data structures Algorithms 3 Running time Theoretical analysis Experimental results

5 Chord definition Choose key space S, such that S = 2 m. Assign a random key k i S to every node in the network Order nodes such that they form cyclic list and k i < k i+1 Every node v keeps its routing table R v of size m, such that R i [j] stores address of node in distance 2 j from v

6 Chord searching

7 Distributed Hashtables Other structured peer-to-peer networks based on DHT use minimal-change topologies like n-cube Butterfly de Brujin graph What if someone wanted to visit some subset of nodes?

8 Distributed Hashtables Other structured peer-to-peer networks based on DHT use minimal-change topologies like n-cube Butterfly de Brujin graph What if someone wanted to visit some subset of nodes?

9 Agenda 1 Applications Peer-to-peer networks Calendaring 2 Combinatorial generation Model Data structures Algorithms 3 Running time Theoretical analysis Experimental results

10 Software Personal Information Mangement (PIM): Palm Desktop, Mozilla Google Calendar CRON

11 PIM pattern specification choose exclusively one of the patterns (granularity) choose start and end date

12 PIM views

13 CRON pattern specification 6 fields: year, month, dom, dow, hour, minute example:,, 1 15,, 0 23/2, 0 runs in background checking every minute if there is an event matching a pattern How to generate monthly view for CRON?

14 CRON pattern specification 6 fields: year, month, dom, dow, hour, minute example:,, 1 15,, 0 23/2, 0 runs in background checking every minute if there is an event matching a pattern How to generate monthly view for CRON?

15 Agenda 1 Applications Peer-to-peer networks Calendaring 2 Combinatorial generation Model Data structures Algorithms 3 Running time Theoretical analysis Experimental results

16 Sequence spaces Definition (Consecutive sequence space) Space Ω n is called consecutive sequence space iff for some n N, s j N, e j N we have Ω n = {(a n ) : s j a j e j, 1 j n, a j N}. Definition (Pattern sequence space) Space Ψ n is called pattern sequence space iff for some n N and characteristic functions χ = (χ n ), where χ j : N {0, 1} for all 1 j n, we have Ψ n = {(a n ) : χ j (a j ) = 1, 1 j n, a j N}.

17 Sequence spaces Definition (Consecutive sequence space) Space Ω n is called consecutive sequence space iff for some n N, s j N, e j N we have Ω n = {(a n ) : s j a j e j, 1 j n, a j N}. Definition (Pattern sequence space) Space Ψ n is called pattern sequence space iff for some n N and characteristic functions χ = (χ n ), where χ j : N {0, 1} for all 1 j n, we have Ψ n = {(a n ) : χ j (a j ) = 1, 1 j n, a j N}.

18 Sequence ranges Definition (Sequence range) A pair ((f n ), (t n )) is called sequence range iff (f n ) Ω n, (t n ) Ω n and (f n ) Ωn (t n ). Definition (Sequence space subrange) Sequence range ((f n ), (t n )) generates sequence space subrange Φn f,t for some sequence space Φ n where Φ f,t n = {(a n ) Φ n : (f n ) (a n ) (t n )} and by Φ n we may take pattern sequence space Ψ n or consecutive sequence space Ω n.

19 Sequence ranges Definition (Sequence range) A pair ((f n ), (t n )) is called sequence range iff (f n ) Ω n, (t n ) Ω n and (f n ) Ωn (t n ). Definition (Sequence space subrange) Sequence range ((f n ), (t n )) generates sequence space subrange Φn f,t for some sequence space Φ n where Φ f,t n = {(a n ) Φ n : (f n ) (a n ) (t n )} and by Φ n we may take pattern sequence space Ψ n or consecutive sequence space Ω n.

20 Not so easy task Definition (Pattern sequence rookie) Sequence (r n ) Ψ f,t n is called pattern sequence rookie iff (a n ) Ψ f,t n : (r n ) (a n ). Solution: find pattern sequence rookie!

21 Not so easy task Definition (Pattern sequence rookie) Sequence (r n ) Ψ f,t n is called pattern sequence rookie iff (a n ) Ψ f,t n : (r n ) (a n ). Solution: find pattern sequence rookie!

22 Agenda 1 Applications Peer-to-peer networks Calendaring 2 Combinatorial generation Model Data structures Algorithms 3 Running time Theoretical analysis Experimental results

23 Pattern Definition (Pattern) Let Γ = (Γ n ) be a pattern defined as (Γ j ) = supp{χ j } for all 1 j n. Definition (Filtered pattern) For sequence range ((f n ), (t n )) we define Γ f = (Γ f n) to be a filtered pattern, where Γ f j = Γ j {f j,..., } for all 1 j n.

24 Pattern Definition (Pattern) Let Γ = (Γ n ) be a pattern defined as (Γ j ) = supp{χ j } for all 1 j n. Definition (Filtered pattern) For sequence range ((f n ), (t n )) we define Γ f = (Γ f n) to be a filtered pattern, where Γ f j = Γ j {f j,..., } for all 1 j n.

25 Agenda 1 Applications Peer-to-peer networks Calendaring 2 Combinatorial generation Model Data structures Algorithms 3 Running time Theoretical analysis Experimental results

26 Naive generate Algorithm 2.1: NAIVEGENERATE((f n ), (t n ), (χ n )) global ε (c n ) (f n ) (1) while (c n ) (t n ) (2) if χ((c n )) 1 (3) do then output ((c n )) (4) (c n ) (c n ) ε (5)

27 JSUCCESSOR() Algorithm 2.2: JSUCCESSOR(j, (c n ), (Γ n )) global (λ n ) i j while i > 0 and λ i = Γ i (1) do i i 1 if i = 0 then return ( undefined ) (s n ) (c n ) (2) λ i λ i + 1 s i Γ i,λi for k { i + 1 to j (3) λk = 1 do s k Γ k,1 return ((s n )) (4)

28 More definitions Definition (Favorite subsequence) Subsequence (u j ) = (u 1, u 2,..., u j ) for 1 j n is called favorite subsequence of sequence space subrange Ψ f,t n iff (u j ) is a pattern sequence rookie of sequence space subrange Ψ f,t j.

29 Favorite subsequence lemma Lemma If (u j 1 ) Ψ f,t,t j 1 is a favorite subsequence of Ψf n for 1 < j n and pattern sequence rookie (r n ) Ψ f n,t exists then: (T1) (u 1, u 2,..., u j 1, Γ f j,1, Γ j+1,1, Γ j+2,1,..., Γ n,1 ) is a pattern sequence rookie of Ψ f n,t when f j < Γ f j,1, (T2) (v 1, v 2,..., v j 1, Γ j,1, Γ j+1,1,..., Γ n,1 ) is a pattern sequence rookie of Ψ f n,t when Γ f j,1 does not exist, where (v j 1) Ψ f,t j 1 is a successor of (u j 1 ) Ψ f,t j 1, (R) (u 1, u 2,..., u j 1, Γ f,t j,1 ) Ψf j is a favorite subsequence of Ψ f n,t when f j = Γ f j,1. If j = 1 and pattern sequence rookie (r n ) Ψ f n,t,t ) is a favorite subsequence of Ψf (Γ f 1,1 n. exists, then

30 Auxiliary array (λ f ) Definition { 0, (Γ λ f f j = j ) = z : Γ j,z = Γ f j,1, (Γf j )

31 GETFIRST() Algorithm 2.3: GETFIRST((f n ), (t n ), (Γ n )) for j 1 to n if λ f j = 0 { (rn ) JSUCCESSOR(j 1, (r then n ), (Γ n )) break r j Γ j,λ f do j λ j λ f j if f j < Γ j,λ f { j j j + 1 then break for i j to n do r i Γ i,1 return ((r n ))

32 GETFIRST() validity Theorem Algorithm GETFIRST() returns pattern sequence rookie (r n ) of sequence space subrange Ψ f,t n, if (r n ) exists.

33 Pattern generate Algorithm 2.4: PATTERNGENERATE((f n ), (t n ), (Γ n )) (c n ) GETFIRST((f n ), (t n ), (Γ n )) while { (c n ) undefined output ((cn )) do (c n ) JSUCCESSOR(n, (c n ), (Γ n ))

34 Agenda 1 Applications Peer-to-peer networks Calendaring 2 Combinatorial generation Model Data structures Algorithms 3 Running time Theoretical analysis Experimental results

35 Granularity Definition (Pattern granularity) Let us take sequence range ((f n ), (t n )) and pattern Γ. We define pattern granularity as gran(γ) = Ψf,t n Ω f,t n

36 Naive generate running time Algorithm 3.1: NAIVEGENERATE((f n ), (t n ), (χ n )) global ε (c n ) (f n ) (1) while (c n ) (t n ) (2) if χ((c n )) 1 (3) do then output ((c n )) (4) (c n ) (c n ) ε (5) 1,t O(n Ωf n ) = O(n 1,t Ωf n ) ε ε

37 Pattern generate running time Algorithm 3.2: PATTERNGENERATE((f n ), (t n ), (Γ n )) (c n ) GETFIRST((f n ), (t n ), (Γ n )) while { (c n ) undefined output ((cn )) do (c n ) JSUCCESSOR(n, (c n ), (Γ n )) O(n log n + n gran(γ n ) Ω f,t n ) = O(n gran(γ n ) Ω f,t n ).

38 Agenda 1 Applications Peer-to-peer networks Calendaring 2 Combinatorial generation Model Data structures Algorithms 3 Running time Theoretical analysis Experimental results

40 Ω f n,t T h t h T min t min ms 1.53ms 2.94ms 1.7ms ms 3.12ms 4.33ms 3.35ms ms 4.99ms 6.18ms 5.49ms 1h 2.22ms 8.79ms 10.25ms 10.11ms 2h 2.44ms 16.4ms 17.79ms 18.48ms 4h 2.75ms 31.79ms 33.3ms 35.6ms 8h 3.96ms 62.54ms 63.68ms 70.5ms 16h 4.9ms ms ms ms 1m ms m ms m ms m y y y

41 Summary Define RANK() and UNRANK() for lexicographic ordering Define all algorithms for minimal-change ordering Apply to peer-to-peer structured networks

42 Summary Define RANK() and UNRANK() for lexicographic ordering Define all algorithms for minimal-change ordering Apply to peer-to-peer structured networks

43 Summary Define RANK() and UNRANK() for lexicographic ordering Define all algorithms for minimal-change ordering Apply to peer-to-peer structured networks

Combinatorial algorithms

Combinatorial algorithms computing subset rank and unrank, Gray codes, k-element subset rank and unrank, computing permutation rank and unrank Jiří Vyskočil, Radek Mařík 2012 Combinatorial Generation definition: