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Bijective tree encoding

Bijective tree encoding. Saverio Caminiti. Talk Outline. Domains Prüfer-like codes Prüfer code (1918) Neville codes (1953) Deo and Micikevičius code (2002) Picciotto codes (1999) Applications, Operations and Properties Random trees generation (with constrains) Locality and Heritability

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Bijective tree encoding

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  1. Bijective tree encoding Saverio Caminiti

  2. Talk Outline • Domains • Prüfer-like codes • Prüfer code (1918) • Neville codes (1953) • Deo and Micikevičius code (2002) • Picciotto codes (1999) • Applications, Operations and Properties • Random trees generation (with constrains) • Locality and Heritability • Other operations • Future work

  3. Domains • Labeled trees Tn • n nodes labeled with distinct symbols in  s.t. || = n i.e. indexed with integers in [n] = {1, 2, ..., n} • Both rooted and unrooted • Undirected • No ordered among nodes children • Strings according with Cayley’s theorem • In n-2 for unrooted (i.e. [n]n-2) • In n-1 for rooted (i.e. [n]n-1)

  4. 1 3 6 4 5 4 2 1 3 6 2 5 Examples 4 1 3 3 1 4 3 3 4

  5. Prüfer code • Introduced in 1918 to prove the Cayley’s theorem is the first bijection between Tn and [n]n-2 (T) = adj(u) :: (T-u) • where: • u is the smallest leaf in T, • adj(u) is the only node adjacent to u in T, • T-u is the tree obtained from T removing u, • and the operator :: is the string concatenation.

  6. Example: Prüfer encode unrooted (T) = adj(u) :: (T-u) • S 2 4 1 5 3 • C 4 1 3 3 6 1 3 6 = n 4 5 2 = n n - 2

  7. Example: Prüfer encode rooted (T) = adj(u) :: (T-u) • S 2 1 5 6 3 • C 1 4 3 3 4  4 = n 1 3 6 2 5  = n n - 1

  8. Notes: Prüfer encode (T) = adj(u) :: (T-u) • S 2 1 5 6 3 • C 1 4 3 3 4 Focus on rooted trees • Each node (but the root) is removed exactly once • Each node appear in C once for each children • A node can be removed only after all its children 4 1 3 6 2 5 n - 1

  9. Example: Prüfer decode • C 1 4 3 3 4 • S ? ? ? ? ? • Let l be the length of the string C • n = l + 1 = 6 • First step: the leaves of initial tree are those nodes that do not appear in C: {2, 5, 6} choose the smallest one

  10. Example: Prüfer decode • C 1 4 3 3 4 • S 2 • The remaining code 4 3 3 4 is (T-{2})then we should choose the smallest leaf among {1, 5, 6}

  11. Example: Prüfer decode • C 1 4 3 3 4 • S 2 1 • The remaining code 3 3 4 is (T-{2, 1})then we should choose the smallest leaf among {5, 6}

  12. 4 1 3 6 2 5 Example: Prüfer decode • C 1 4 3 3 4 • S 2 1 5 6 3

  13. Other Prüfer-like codes • Neville (1953) for rooted trees • The first one was indeed the Prüfer code • Moon (1970) • Adapts Neville’s codes to trees • Deo and Micikevičius (2002)

  14. Second Neville code

  15. Third Neville Code

  16. Deo and Micikevičius code

  17. Generalization • It has been proven that any deterministic procedure P able to choose at each stepa non- empty sequence of leaves can be usedto generate a bijective code (T) = adj(P(T)) :: (T-P(T))

  18. Why several codes • Different codes may have different properties and allow different operations • Encoding and Decoding algorithms for different code may have different time (and/or space) complexity

  19. Implementation of Prüfer code • Straightforward implementation: O(n log n) • First linear time algorithm in 1978(left as exercise in Combinatorial algorithms) • Optimal parallel algorithm 2000 • Linear time sequential algorithm rediscovered in 2000 and 2001 • Still unknown in 2003 !!!

  20. Implementation of other codes • Second Neville code 2002 • Third Neville code 1953 (trivial) • Deo and Micikevičius 2002(in the original paper)

  21. A unified approach • The encoding of all four codes can be reduce to sorting pairs integer in [n] • The decoding can be reduced to the computation of the rightmost occurrence of each symbol in the code string

  22. Encoding: Second Neville code • pair 0,3 0,4 0,5 0,8 0,9 1,1 1,6 1,10 2,2 • S 3 4 5 8 9 1 6 10 2 • C 6 10 6 1 7 2 7 7 7 (l(v), v) where l(v) is the level of v from the bottom

  23. Encoding: Third Neville code • pair 3,0 4,0 4,1 5,0 5,1 8,0 8,1 8,2 8,3 • S 3 4 10 5 6 8 1 2 7 • C 6 10 7 6 7 1 2 7 9 ( (v), d(v, (v)) ) where (v) is the greatest leaf in the subtree rooted at v

  24. Linear time implementation • All the information appearing in pairs can be computer with a simple tree traversal O(n) • To sort the set of pairs it is enough to execute twice a stable integer sort O(n)

  25. Decoding: Third Neville code • C 6 10 7 6 7 1 2 7 9 • S ? ? ? ? ? ? ? ? ? • Compute the rightmost occurrence of eachv [n] into C: v 1 2 3 4 5 6 7 8 9 10 v 6 7 0 0 0 4 8 0 9 2

  26. Decoding: Third Neville code • C 6 10 7 6 7 1 2 7 9 • S ? ? ? ? ? ? ? ? ? • Compute the rightmost occurrence of eachv [n] into C: v 1 2 3 4 5 6 7 8 9 10 v 6 7 0 0 0 4 8 0 9 2

  27. Decoding: Third Neville code • C 6 10 7 6 7 1 2 7 9 • S ? ? 10 ? 6 ? 1 2 7 • Compute the rightmost occurrence of eachv [n] into C: v 1 2 3 4 5 6 7 8 9 10 v 6 7 0 0 0 4 8 0 9 2

  28. Decoding: Third Neville code • C 6 10 7 6 7 1 2 7 9 • S 3 4 10 5 6 8 1 2 7

  29. Parallel results • These techniques allow us to efficiently encode and decode on EREW PRAM: • Integer Sorting require O(log n) timeand O(n √ log n) operations • The rightmost occurrence computation can be reduced to Integer Sorting

  30. Talk Outline • Domains • Prüfer-like codes • Prüfer code (1918) • Neville codes (1953) • Deo and Micikevičius code (2002) • Picciotto codes (1999) • Applications, Operations and Properties • Random trees generation (with constrains) • Locality and Heritability • Other operations • Future work

  31. Picciotto’s codes • In her PhD thesis Picciotto proposed three codes for unrooted trees: • Blob code • Happy code • Dandelion code • Easily adapted to rooted tree (T, r) c1 c2 ... cn-2r n - 1

  32. Happy code 0 6 2 3 5 4 7 1

  33. Happy code 0 6 2 3 5 4 7 1

  34. Happy code 0 6 3 2 5 4 7 1

  35. Happy code 0 3 2 6 4 5 7 1

  36. Happy code 0 3 2 6 4 5 7 1 Node 2 3 4 5 6 7 C 0 4 3 6 6 5

  37. Happy code x f(x) 0 0 1 0 2 0 3 4 4 3 5 6 6 6 7 7 0 3 2 6 4 5 7 1 Node 2 3 4 5 6 7 C 0 4 3 6 6 5

  38. Happy code • Create a bijection between Tn and a subset of the endofunctions on [n] {ƒ:[n][n] s.t. ƒ(0) = ƒ(1) = 0} • The code string is ƒ(2) :: ƒ(3) :: ... :: ƒ(n) • Linear time encoding and decoding(identify and break cycles, reconstruct the original path from 1 to 0)

  39. Blob code 0 5 2 3 1 4 Node 1 2 3 4 5 C

  40. Blob code 0 5 2 3 1 4 Node 1 2 3 4 5 C -

  41. Blob code 0 5 2 3 1 4 Node 1 2 3 4 5 C 0 -

  42. Blob code path(3, 0)  Blob 3 is stable 0 5 2 3 1 4 Node 1 2 3 4 5 C 5 0 -

  43. Blob code 0 5 2 3 1 4 Node 1 2 3 4 5 C 2 5 0 -

  44. Blob code path(1, 0)  Blob 1 is stable 0 5 2 3 1 4 Node 1 2 3 4 5 C 2 2 5 0 -

  45. Blob code • Straight forward implementation leads to O(n2)(used in 2003) • Can be reduced to the transformation of the tree in a functional digraph • Linear time encoding and decoding algorithm

  46. Blob code path(v, 0) contains u > vv is stable 0 5 2 3 1 4 Node 1 2 3 4 5 C 2 5 -

  47. Blob code 0 5 2 3 1 4 Node 1 2 3 4 5 C 2 2 5 0 -

  48. Blob code x f(x) 0 0 1 2 2 2 3 5 4 0 5 0 0 5 2 3 1 4 Node 1 2 3 4 5 C 2 2 5 0 - ƒ(1) ƒ(2) ƒ(3) ƒ(4)

  49. Dandelion code Node 2 3 4 5 6 7 8 9 10 11 C 5 6 10 2 4 2 1 0 3 9

  50. Dandelion code Node 2 3 4 5 6 7 8 9 10 11 C 5 6 10 2 4 2 1 0 3 9

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