Dependence between Path Lengths and Size in Random Trees (joint with H.-H. Chern, H.-K. Hwang and R. Neininger)

Transcription

1 Dependence between Path Lengths and Size in Random Trees (joint with H.-H. Chern, H.-K. Hwang and R. Neininger) Michael Fuchs Institute of Applied Mathematics National Chiao Tung University Hsinchu, Taiwan Kraków, July 4, 2016 Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

2 Random m-ary Search Trees Proposed by Muntz and Uzgalis in Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

3 Random m-ary Search Trees Proposed by Muntz and Uzgalis in Input: 6, 2, 4, 8, 7, 1, 5, 3, 10, 9 Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

4 Random m-ary Search Trees Proposed by Muntz and Uzgalis in Input: 6, 2, 4, 8, 7, 1, 5, 3, 10, 9 2, 6 1 4, 5 7, 8 3 9, 10 m = 3 Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

5 Random m-ary Search Trees Proposed by Muntz and Uzgalis in Input: 6, 2, 4, 8, 7, 1, 5, 3, 10, 9 2, 6 2, 4, 6 1 4, 5 7, , 8, 9 3 9, 10 m = 3 m = 4 10 Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

6 Random m-ary Search Trees Proposed by Muntz and Uzgalis in Input: 6, 2, 4, 8, 7, 1, 5, 3, 10, 9 2, 6 2, 4, 6 1 4, 5 7, , 8, 9 3 9, 10 m = 3 m = 4 10 If permutations are equally likely random m-ary search trees Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

7 Size, KPL, and NPL Size (or Storage Requirement) Number of nodes holding keys. Only random if m 3. S n = size of a random m-ary search tree built from n keys. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

8 Size, KPL, and NPL Size (or Storage Requirement) Number of nodes holding keys. Only random if m 3. S n = size of a random m-ary search tree built from n keys. Key Path Length (KPL) Sum of all key-distances to the root. K n = KPL of a random m-ary search tree built from n keys. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

9 Size, KPL, and NPL Size (or Storage Requirement) Number of nodes holding keys. Only random if m 3. S n = size of a random m-ary search tree built from n keys. Key Path Length (KPL) Sum of all key-distances to the root. K n = KPL of a random m-ary search tree built from n keys. Node Path Length (NPL) Sum of all node-distances to the root. N n = NPL of a random m-ary search tree built from n keys. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

10 Size: Mean Knuth (1973): where E(S n ) φn, φ := and H m are the Harmonic numbers. 1 2(H m 1) Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

11 Size: Mean Knuth (1973): where E(S n ) φn, φ := and H m are the Harmonic numbers. Mahmoud and Pittel (1989): 1 2(H m 1) E(S n ) = φ(n + 1) 1 m 1 + O(nα 1 ), where α is the real part of the second largest zero of Λ(z) = z(z + 1) (z + m 2) m!. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

12 Size: Phase Change for Variance Mahmoud and Pittel (1989): { C S n, if m 26; Var(S n ) F 1 (β log n)n 2α 2, if m 27, where λ = α + iβ is the second largest zero of Λ(z). Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

13 Size: Phase Change for Variance Mahmoud and Pittel (1989): { C S n, if m 26; Var(S n ) F 1 (β log n)n 2α 2, if m 27, where λ = α + iβ is the second largest zero of Λ(z). Here, F 1 (z) is the periodic function ( F 1 (z) = 2 A 2 Γ(λ) ( A 2 e 2iz + 2R Γ(λ) 2 with A = 1/(λ(λ 1) 0 j m 2 m!(m 1) Γ(λ) 2 ) Γ(2α + m 2) m!γ(2α 1) ( 1 + m!(m 1)Γ(λ) 2 Γ(2λ + m 2) m!γ(2λ 1) 1 j+λ ). )) Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

14 Size: Phase Change for Limit Law Theorem (Mahmoud & Pittel (1989); Lew & Mahmoud (1994)) For 3 m 26, S n E(S n ) Var(Sn ) d N(0, 1), where N(0, 1) is the standard normal distribution. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

15 Size: Phase Change for Limit Law Theorem (Mahmoud & Pittel (1989); Lew & Mahmoud (1994)) For 3 m 26, S n E(S n ) Var(Sn ) d N(0, 1), where N(0, 1) is the standard normal distribution. Theorem (Chern & Hwang (2001)) For m 27, S n E(S n ) Var(Sn ) does not converge to a fixed limit law. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

16 KPL: Moments Mahmoud (1986): E(K n ) = 2φn log n + c 1 n + o(n), where c 1 is an explicitly computable constant. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

17 KPL: Moments Mahmoud (1986): E(K n ) = 2φn log n + c 1 n + o(n), where c 1 is an explicitly computable constant. Mahmoud (1992): where with H (2) m = 1 j m 1/j2. Var(K n ) C K n 2, ( ) C K = 4φ 2 (m + 1)H m (2) 2 π2 m 1 6 Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

18 KPL: Moments Mahmoud (1986): E(K n ) = 2φn log n + c 1 n + o(n), where c 1 is an explicitly computable constant. Mahmoud (1992): where with H (2) m = 1 j m 1/j2. Var(K n ) C K n 2, ( ) C K = 4φ 2 (m + 1)H m (2) 2 π2 m 1 6 So, no phase change here for the variance! Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

19 KPL: Limit Law Theorem (Neininger & Rüschendorf (1999)) We have, K n E(K n ) n where K is the unique solution of X d = 1 r m V r X (r) + 2φ with X (r) an independent copy of X and d K, 1 r m V r = U (r) U (r 1), V r log V r where U (r) is the r-th order statistic of m i.i.d. uniform RVs. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

20 Node Path Length (NPL) N n = sum of all node-distances in an m-search tree built from n keys. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

21 Node Path Length (NPL) N n = sum of all node-distances in an m-search tree built from n keys. Broutin and Holmgren (2012): E(N n ) = 2φ 2 n log n + c 2 n + o(n), where c 2 is an explicitly computable constant. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

22 Node Path Length (NPL) N n = sum of all node-distances in an m-search tree built from n keys. Broutin and Holmgren (2012): E(N n ) = 2φ 2 n log n + c 2 n + o(n), where c 2 is an explicitly computable constant. We have, { d (1) S n = S I S (m) N n d = N (1) I 1 I m + 1, + + N (m) I m + S (1) I S (m) I m. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

23 Node Path Length (NPL) N n = sum of all node-distances in an m-search tree built from n keys. Broutin and Holmgren (2012): E(N n ) = 2φ 2 n log n + c 2 n + o(n), where c 2 is an explicitly computable constant. We have, { d (1) S n = S I S (m) N n d = N (1) I 1 I m + 1, + + N (m) I m + S (1) I S (m) I m. So, one expects a strong positive dependence between S n and N n! Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

24 Size and NPL: Correlation (i) Theorem (Chern, F., Hwang, Neininger) We have, Cov(S n, N n ) { C R n log n, if 3 m 13; φf 2 (β log n)n α, if m 14, where C R is a constant and F 2 (z) is a periodic function. Moreover, Var(N n ) φ 2 C K n 2., Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

25 Size and NPL: Correlation (i) Theorem (Chern, F., Hwang, Neininger) We have, Cov(S n, N n ) { C R n log n, if 3 m 13; φf 2 (β log n)n α, if m 14, where C R is a constant and F 2 (z) is a periodic function. Moreover, Var(N n ) φ 2 C K n 2., Thus (!), 0, if 3 m 26; ρ(s n, N n ) F 2(β log n), if m 27. CK F 1 (β log n) Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

26 Size and NPL: Correlation (ii) Periodic function of ρ(s n, N n ) for m = 27, 54,..., 270. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

27 Pearson s Correlation Coefficient Pearson: for RVs X and Y ρ(x, Y ) = Cov(X, Y ) Var(X)Var(Y ). Measures linear dependence between X and Y! Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

28 Pearson s Correlation Coefficient Pearson: for RVs X and Y ρ(x, Y ) = Cov(X, Y ) Var(X)Var(Y ). Measures linear dependence between X and Y! Refined correlation measures: Distance correlation, Brownian covariance, mutual information, total correlation, dual total correlation, etc. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

29 Pearson s Correlation Coefficient Pearson: for RVs X and Y ρ(x, Y ) = Cov(X, Y ) Var(X)Var(Y ). Measures linear dependence between X and Y! Refined correlation measures: Distance correlation, Brownian covariance, mutual information, total correlation, dual total correlation, etc. Question: Can our counterintuitive result for ρ(s n, N n ) be ascribed to the weakness of Pearson s correlation coefficient? Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

30 Pearson s Correlation Coefficient Pearson: for RVs X and Y ρ(x, Y ) = Cov(X, Y ) Var(X)Var(Y ). Measures linear dependence between X and Y! Refined correlation measures: Distance correlation, Brownian covariance, mutual information, total correlation, dual total correlation, etc. Question: Can our counterintuitive result for ρ(s n, N n ) be ascribed to the weakness of Pearson s correlation coefficient? NO! Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

31 Size and NPL: Limit Law for 3 m 26 Theorem (Chern, F., Hwang, Neininger) Consider Then, Q n = (S n, N n ). Cov(Q n ) 1/2( ) ( ) d Q n E(Q n ) N, C 1/2 K K, where N has a standard normal distribution. Moreover, N and K are independent! Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

32 Size and NPL: Limit Law for 3 m 26 Theorem (Chern, F., Hwang, Neininger) Consider Then, Q n = (S n, N n ). Cov(Q n ) 1/2( ) ( ) d Q n E(Q n ) N, C 1/2 K K, where N has a standard normal distribution. Moreover, N and K are independent! Thus, asymptotic independence for 3 m 26 is also observed in the bivariate limit law! Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

33 Size and NPL: Limit Law for m 27 Theorem (Chern, F., Hwang, Neininger) Consider Then, Y n = ( Sn φn n α 1, N ) n E(N n ). n l 2 (Y n, (R(n iβ Λ), φk)) 0, where l 2 is the minimal L 2 -metric and Λ is the unique solution of W d = 1 r m with W (r) independent copies of W. Vr λ 1 W (r) Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

34 Size and KPL Same results hold for size and KPL, e.g., { 0, if 3 m 26; ρ(s n, K n ) ρ(s n, N n ), if m 27. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

35 Size and KPL Same results hold for size and KPL, e.g., { 0, if 3 m 26; ρ(s n, K n ) ρ(s n, N n ), if m 27. Theorem (Chern, F., Hwang, Neininger) We have ρ(k n, N n ) 1 and N n φk n (E(N n φk n )) 2 = o(n). In particular, ( Kn E(K n ) n, N ) n E(N n ) d (K, φk). n Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

36 Trie Proposed by René de la Briandais in Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

37 Trie Proposed by René de la Briandais in Name from the word data retrieval (suggested by Fredkin). Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

38 Trie Proposed by René de la Briandais in Name from the word data retrieval (suggested by Fredkin). Example: Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

39 Trie Proposed by René de la Briandais in Name from the word data retrieval (suggested by Fredkin). Example: Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

40 Trie Proposed by René de la Briandais in Name from the word data retrieval (suggested by Fredkin). Example: Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

41 Trie Proposed by René de la Briandais in Name from the word data retrieval (suggested by Fredkin). Example: Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

42 Trie Proposed by René de la Briandais in Name from the word data retrieval (suggested by Fredkin). Example: Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

43 Trie Proposed by René de la Briandais in Name from the word data retrieval (suggested by Fredkin). Example: Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

44 Notation and Random Model Two types of nodes: Internal nodes: only used for branching; External nodes: nodes which hold data. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

45 Notation and Random Model Two types of nodes: Internal nodes: only used for branching; External nodes: nodes which hold data. Random Model: Bits are independent Bernoulli random variables with mean p. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

46 Notation and Random Model Two types of nodes: Internal nodes: only used for branching; External nodes: nodes which hold data. Random Model: Bits are independent Bernoulli random variables with mean p. p = 1/2: symmetric trie; p 1/2: asymmetric trie. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

47 Notation and Random Model Two types of nodes: Internal nodes: only used for branching; External nodes: nodes which hold data. Random Model: Bits are independent Bernoulli random variables with mean p. p = 1/2: symmetric trie; p 1/2: asymmetric trie. Question: correlation between size and path-length in random tries? Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

48 Size, EPL, and IPL Size Number of internal nodes. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

49 Size, EPL, and IPL Size Number of internal nodes. External Path Length (EPL) Sum of all distances between external nodes and root. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

50 Size, EPL, and IPL Size Number of internal nodes. External Path Length (EPL) Sum of all distances between external nodes and root. Internal Path Length (IPL) Sum of all distances between internal nodes and root. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

51 Size, EPL, and IPL Size Number of internal nodes. External Path Length (EPL) Sum of all distances between external nodes and root. Internal Path Length (IPL) Sum of all distances between internal nodes and root. We again use S n, K n, N n and EPL KPL; IPL NPL. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

52 Some Notation We use the following notation: Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

53 Some Notation We use the following notation: Entropy: h = p log p q log q; Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

54 Some Notation We use the following notation: Entropy: h = p log p q log q; If log p/ log q Q, then log p log q = r, gcd(r, l) = 1. l Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

55 Some Notation We use the following notation: Entropy: h = p log p q log q; If log p/ log q Q, then log p log q = r, gcd(r, l) = 1. l and χ k = 2rkπi log p, (k Z). Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

56 Some Notation We use the following notation: Entropy: h = p log p q log q; If log p/ log q Q, then log p log q = r, gcd(r, l) = 1. l and For a sequence g k : χ k = 2rkπi log p, (k Z). F [g](z) = { k Z g kz χ k, if log p/ log q Q; g 0, if log p/ log q Q. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

57 Means and Variances 1 Shape parameters n (mean) 1 n (variance) Size S n F [ ](n) F [g (1) ](n) NPL N n E(S n) KPL K n Depth D n log n h n log n h + F [ ](n) E(D n ) = E(Kn) n pq log 2 p q h 2 Var(S n) n log n h (log n) 2 h 2 Var(D n ) = Var(Kn) n + F [g (3) ](n) + O(1) Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

58 Means and Variances 1 Shape parameters n (mean) 1 n (variance) Size S n F [ ](n) F [g (1) ](n) NPL N n E(S n) KPL K n Depth D n log n h n log n h + F [ ](n) E(D n ) = E(Kn) n pq log 2 p q h 2 Var(S n) n log n h (log n) 2 h 2 Var(D n ) = Var(Kn) n + F [g (3) ](n) + O(1) Note that and for p q N n S n log n h Var(D n ) Var(K n). n Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

59 Main Approaches Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

60 Main Approaches Rice Method Exercise 54 in Section of Knuth s book. Developed into a systematic tool by Flajolet and Sedgewick. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

61 Main Approaches Rice Method Exercise 54 in Section of Knuth s book. Developed into a systematic tool by Flajolet and Sedgewick. Two-stage Approach Introduced by Jacquet and Régnier. Further developed by Jacquet and Szpankowski. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

62 Main Approaches Rice Method Exercise 54 in Section of Knuth s book. Developed into a systematic tool by Flajolet and Sedgewick. Two-stage Approach Introduced by Jacquet and Régnier. Further developed by Jacquet and Szpankowski. Poisson Variance and Covariance F., Hwang and Zacharovas. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

63 Main Approaches Rice Method Exercise 54 in Section of Knuth s book. Developed into a systematic tool by Flajolet and Sedgewick. Two-stage Approach Introduced by Jacquet and Régnier. Further developed by Jacquet and Szpankowski. Poisson Variance and Covariance F., Hwang and Zacharovas. Other Approaches Elementary (Schachinger), probabilistic (Devroye, Janson, etc.) Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

64 Size: Variance Theorem (Régnier & Jacquet (1989); Kirschenhofer & Prodinger (1991); F., Hwang, Zacharovas (2014)) We have, where Var(S n ) F [g (1) ](n)n, g (1) k = χ ( kγ( 1 + χ k ) 1 χ ) k + 3 h 2 1+χ k 1 h 2 Γ(χ j + 1)Γ(χ k j + 1) j Z 2 ( 1) j (j χ k )Γ(j + χ k ) ( p j+1 + q j+1) h (j 1)!(j + 1)(1 p j+1 q j+1. ) j 1 Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

65 Size and NPL: Variances and Covariance Theorem (F., Hwang, Zacharovas (2014)) We have, and and Var(S n ) F [g (1) ](n)n, Cov(S n, N n ) F [g (1) ](n) n log n h Var(N n ) F [g (1) ](n) n log2 n h 2. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

66 Size and NPL: Variances and Covariance Theorem (F., Hwang, Zacharovas (2014)) We have, Var(S n ) F [g (1) ](n)n, and and Corollary We have, Cov(S n, N n ) F [g (1) ](n) n log n h Var(N n ) F [g (1) ](n) n log2 n h 2. ρ(s n, N n ) 1. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

67 Size and NPL: Limit Law Theorem (F. & Lee (2015)) We have, ( ) S n E(S n ) Var(Sn ), N n E(N n ) d N (0, E 2 ), Var(Nn ) where E 2 is the 2 2 unit matrix E 2 = ( ) Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

68 Size and NPL: Limit Law Theorem (F. & Lee (2015)) We have, ( ) S n E(S n ) Var(Sn ), N n E(N n ) d N (0, E 2 ), Var(Nn ) where E 2 is the 2 2 unit matrix E 2 = ( ) NPL was also investigated by Nguyen-The (2004) in his PhD-thesis, but his result is incorrect. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

69 Size and NPL: Limit Law Theorem (F. & Lee (2015)) We have, ( ) S n E(S n ) Var(Sn ), N n E(N n ) d N (0, E 2 ), Var(Nn ) where E 2 is the 2 2 unit matrix E 2 = ( ) NPL was also investigated by Nguyen-The (2004) in his PhD-thesis, but his result is incorrect. Question: same result for size and KPL? Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

70 Size and KPL: Covariance Theorem (F. & Hwang) We have, Cov(S n, K n ) F [g (2) ](n)n, where g (2) k = Γ(χ ( k) 1 χ k + 2 ) h 2 χ k+1 1 h 2 Γ(χ k j + 1)(χ j 1)Γ(χ j ) j Z\{0} Γ(χ k + 1) ( h 2 γ ψ(χ k + 1) p log2 p + q log 2 q ) 2h + 1 ( 1) j (2j 2 2j (2j 1)χ k )Γ(j 1χ k )(p j + q j ) h j!(1 p j q j. ) j 2 Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

71 Correlation Coefficient p = q = 1/2: ρ(s n, K n ) Cov(S n,k n) Var(Sn)(Var(K n)+1.046) Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

72 Size and KPL: Correlation Coefficient Theorem (F. & Hwang) We have, ρ(s n, K n ) { 0, if p q; F (n), if p = q, where F (n) = F [g (2) ](n) F [g (1) ](n)f [g (3) ](n) is a periodic function with average value = and amplitude Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

73 Size and KPL: Correlation Coefficient Theorem (F. & Hwang) We have, ρ(s n, K n ) { 0, if p q; F (n), if p = q, where F (n) = F [g (2) ](n) F [g (1) ](n)f [g (3) ](n) is a periodic function with average value = and amplitude Question: is now this behavior due to the weakness of Pearson s correlation coefficient? Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

74 Size and KPL: Correlation Coefficient Theorem (F. & Hwang) We have, ρ(s n, K n ) { 0, if p q; F (n), if p = q, where F (n) = F [g (2) ](n) F [g (1) ](n)f [g (3) ](n) is a periodic function with average value = and amplitude Question: is now this behavior due to the weakness of Pearson s correlation coefficient? Again NO! Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

75 Size and KPL: Limit Laws Theorem (F. & Hwang) p q: ( ) S n E(S n ) Var(Sn ), K n E(K n ) d N (0, I 2 ), Var(Kn ) where I 2 is the 2 2 identity matrix. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

76 Size and KPL: Limit Laws Theorem (F. & Hwang) p q: ( ) S n E(S n ) Var(Sn ), K n E(K n ) d N (0, I 2 ), Var(Kn ) where I 2 is the 2 2 identity matrix. p = q: ( ) Σn 1/2 d S n E(S n ), K n E(K n ) N 2 (0, I 2 ), where Σ n is the (asymptotic) covariance matrix: ( F [g Σ n := n (1) ](n) F [g (2) ) ](n) F [g (2) ](n) F [g (3). ](n) Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

77 Joint Distribution of S n and K n p = 0.1 p = 0.2 p = 0.3 K n S n K n Sn K n Sn p = 0.4 p = 0.5 p = 0.6 K n S n K n Sn K n S n Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

78 Summary trees { ρ(s n, K n ) ρ(s n, N n ) p q : 0 tries 1 p = { q : periodic m-ary 3 m 26 : 0 search trees m 27 : periodic Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

79 Summary trees { ρ(s n, K n ) ρ(s n, N n ) p q : 0 tries 1 p = { q : periodic m-ary 3 m 26 : 0 search trees m 27 : periodic Similar results for fringe-balanced binary search trees and quadtrees: H.-H. Chern, M. Fuchs, H.-K. Hwang, R. Neininger. Dependence and phase changes in random m-ary search trees, arxiv: Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

80 Summary trees { ρ(s n, K n ) ρ(s n, N n ) p q : 0 tries 1 p = { q : periodic m-ary 3 m 26 : 0 search trees m 27 : periodic Similar results for fringe-balanced binary search trees and quadtrees: H.-H. Chern, M. Fuchs, H.-K. Hwang, R. Neininger. Dependence and phase changes in random m-ary search trees, arxiv: Similar results for m-ary tries, m-ary PATRICIA tries and bucket digital search trees. Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30

81 Summary trees { ρ(s n, K n ) ρ(s n, N n ) p q : 0 tries 1 p = { q : periodic m-ary 3 m 26 : 0 search trees m 27 : periodic Similar results for fringe-balanced binary search trees and quadtrees: H.-H. Chern, M. Fuchs, H.-K. Hwang, R. Neininger. Dependence and phase changes in random m-ary search trees, arxiv: Similar results for m-ary tries, m-ary PATRICIA tries and bucket digital search trees. Better explanation of our results? Michael Fuchs (NCTU) Dependence in Random Trees Kraków, July 4, / 30