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The Node Profile of Symmetric Digital Search Trees (joint with M. Drmota, H.-K. Hwang and R. Neininger) Michael Fuchs Department of Applied Mathematics National Chiao Tung University June 8th, 2015 Michael Fuchs (NCTU) Node Profile of DSTs

  1. The Node Profile of Symmetric Digital Search Trees (joint with M. Drmota, H.-K. Hwang and R. Neininger) Michael Fuchs Department of Applied Mathematics National Chiao Tung University June 8th, 2015 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 1 / 28

  2. Node Profile of (Rooted) Trees B n,k = number of external nodes at level k ; I n,k = number of internal nodes at level k . Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

  3. Node Profile of (Rooted) Trees B n,k = number of external nodes at level k ; I n,k = number of internal nodes at level k . Example: Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

  4. Node Profile of (Rooted) Trees B n,k = number of external nodes at level k ; I n,k = number of internal nodes at level k . Example: B 5 , 0 = 0 , B 5 , 1 = 0 , B 5 , 2 = 2 , B 5 , 3 = 4 , Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

  5. Node Profile of (Rooted) Trees B n,k = number of external nodes at level k ; I n,k = number of internal nodes at level k . Example: B 5 , 0 = 0 , I 5 , 0 = 1 ; B 5 , 1 = 0 , I 5 , 1 = 2 ; B 5 , 2 = 2 , I 5 , 2 = 2 ; B 5 , 3 = 4 , I 5 , 3 = 0 . Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

  6. Relations to Other Shape Parameters Many shape parameters can by analyzed through the profile. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 3 / 28

  7. Relations to Other Shape Parameters Many shape parameters can by analyzed through the profile. Depth: P ( D n = k ) = B n,k / ( n + 1) ; Width: max { B n,k : k ≥ 0 } ; Total Path Length: � k kB n,k ; Height: max { k : B n,k > 0 } ; Shortest Path: min { k : B n,k > 0 } ; Fill-up Level: max { k : I n,k = 2 k } ; Etc. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 3 / 28

  8. Profile of Random Trees √ n -Trees: Aldous (1991); Drmota and Gittenberger (1997); Kersting (1998); Pitman (1999); etc. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 4 / 28

  9. Profile of Random Trees √ n -Trees: Aldous (1991); Drmota and Gittenberger (1997); Kersting (1998); Pitman (1999); etc. log n -Trees: Binary Search Trees: Chauvin, Drmota, Jabbour-Hattab (2001); Drmota and Hwang (2005); F., Hwang, Neininger (2006). Recursive Trees: Drmota and Hwang (2005); F., Hwang, Neininger (2006). Plane-oriented Recursive Trees: Hwang (2007). m -ary Seach Trees: Drmota, Janson, Neininger (2008). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 4 / 28

  10. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  11. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 011011 010101 101110 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  12. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 011011 010101 101110 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  13. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 011011 010101 1 101110 010000 0 1 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  14. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 1 011011 010101 1 101110 010000 0 1 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  15. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 1 011011 010101 1 101110 010000 0 1 101010 001100 0 1 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  16. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 1 011011 010101 1 0 101110 010000 0 1 1 101010 001100 0 1 0 1 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  17. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 1 011011 010101 0 1 0 101110 010000 0 1 1 101010 001100 0 1 0 1 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  18. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  19. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 010101 101110 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  20. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  21. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  22. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 0 1 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  23. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 0 1 010000 101010 0 1 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  24. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 0 1 010000 101010 0 1 0 1 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  25. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 0 1 010000 101010 0 1 0 1 0 1 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  26. Random Model Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

  27. Random Model Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Two types: p = 1 / 2 : symmetric digital trees; p � = 1 / 2 : asymmetric digital trees. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

  28. Random Model Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Two types: p = 1 / 2 : symmetric digital trees; p � = 1 / 2 : asymmetric digital trees. Question: What can be said about the profile? Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

  29. Random Model Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Two types: p = 1 / 2 : symmetric digital trees; p � = 1 / 2 : asymmetric digital trees. Question: What can be said about the profile? In this talk, we are interested in mean, variance and limit laws of the profile for symmetric DSTs. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

  30. Profile of Digital Trees Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

  31. Profile of Digital Trees Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). PATRICIA tries: Mean: Magner, Knessl, Szpankowski (2014); Variance & limit laws: Szpankowkski & Magner ( → Thursday). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

  32. Profile of Digital Trees Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). PATRICIA tries: Mean: Magner, Knessl, Szpankowski (2014); Variance & limit laws: Szpankowkski & Magner ( → Thursday). Asymmetric DSTs: Mean: Drmota and Szpankowski (2011); Variance: Kazemi and Vahidi-Asl (2011); so far no limit laws. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

  33. Profile of Digital Trees Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). PATRICIA tries: Mean: Magner, Knessl, Szpankowski (2014); Variance & limit laws: Szpankowkski & Magner ( → Thursday). Asymmetric DSTs: Mean: Drmota and Szpankowski (2011); Variance: Kazemi and Vahidi-Asl (2011); so far no limit laws. Symmetric DSTs: Variance & limit laws: Drmota, F., Hwang, Neininger ( → this talk). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

  34. Profile of Tries Hwang, Nicod´ eme, Park, Szpankowski (2009) Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 9 / 28

  35. Plot of Mean Profile of Symmetric Tries Hwang, Nicod´ eme, Park, Szpankowski (2009): Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 10 / 28

  36. Symmetric Tries: Mean We have, � n (1 − 2 − k ) n − 1 , if 2 − k n → ∞ ; µ n,k := E ( B n,k ) ∼ ˜ if 4 − k n → 0 , M k, 1 ( n ) , where M k, 1 ( z ) = z ( e − z/ 2 k − e − z/ 2 k − 1 ) . ˜ Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 11 / 28

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