Digital search trees JASS
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1 Digital search trees Analysis of different digital trees with Rice s integrals. JASS Nicolai v. Hoyningen-Huene JASS 04 - Digital search trees 1
2 content Tree Digital search tree: Definition Average case analysis Tries: Definition Average case analysis General framework JASS 04 - Digital search trees 2
3 content Tree Digital search tree Tries General framework JASS 04 - Digital search trees 3
4 Tree JASS 04 - Digital search trees 4
5 Tree Definition 0.1 A tree is defined in several ways: 1. A connected, undirected, acyclic graph. It is rooted and ordered unless otherwise specified. 2. A data structure accessed beginning at the root node. Each node is either a leaf or an internal node. An internal node has one or more child nodes and is called the parent of its child nodes. All children of the same node are siblings. 3. A tree is either empty (no nodes), or a root and zero or more subtrees. The subtrees are ordered JASS 04 - Digital search trees 5
6 Search tree JASS 04 - Digital search trees 6
7 content Tree Digital search tree: Definition Average case analysis Tries General framework JASS 04 - Digital search trees 7
8 content Tree Digital search tree: Definition Average case analysis Tries General framework JASS 04 - Digital search trees 8
9 Digital search tree JASS 04 - Digital search trees 9
10 Digital search tree A digital search tree is a dictionary implemented as a digital tree which stores strings in internal nodes, so there is no need for extra leaf nodes to store the strings JASS 04 - Digital search trees 10
11 content Tree Digital search tree: Definition Average case analysis: Internal path length External internal nodes Tries General framework JASS 04 - Digital search trees 11
12 content Tree Digital search tree: Definition Average case analysis: Internal path length External internal nodes Tries General framework JASS 04 - Digital search trees 12
13 Internal path length The internal path length of a tree is the sum of the depth of every node of the tree JASS 04 - Digital search trees 13
14 Internal path length Fundamental recurrence relation A N = N 1+ with A 0 := 0. k=0 1 2 N 1 ( ) N 1 k (A k + A N 1 k ), N JASS 04 - Digital search trees 14
15 Internal path length Transformation N=1 A N z N 1 = ze z + 2 (N 1)! k! k=0 ( z A (z) = ze z + 2A 2) A k ( z 2 e z 2 ) k e z JASS 04 - Digital search trees 15
16 Internal path length A (z) = e z B (z) = Substitution by B(z) A N = ( N=0 N k=0 ) ( z N ) z N B N N! N! N=0 ( N k ) B k JASS 04 - Digital search trees 16
17 Internal path length Substitution by B(z) ( z A (z) = ze z + 2A e 2) z 2 ( z B (z) + B (z) = z + 2B 2) B N + B N 1 = 1 2 N 2 B N JASS 04 - Digital search trees 17
18 Internal path length Substitution by B(z) N 2 B N = ( 1) N j=1 (1 12 j ) JASS 04 - Digital search trees 18
19 Internal path length Introduction of Q N Q N = N j=1 (1 12 j ) JASS 04 - Digital search trees 19
20 Internal path length Introduction of Q(x) Q (x) = j=1 (1 x 2 j ) Q (1) = Q JASS 04 - Digital search trees 20
21 Internal path length Q(x) is used for a meromorphic function of Q N Q N = Q (1) Q (2 N ) JASS 04 - Digital search trees 21
22 Internal path length The function Q(x) 4 y x JASS 04 - Digital search trees 22
23 Internal path length Explicit formula for A N A N = N k=2 ( N k ) ( 1) k Q (1) Q (2 k+2 ) JASS 04 - Digital search trees 23
24 Internal path length Rice s method N k=0 ( N k ) ( 1) k f (k) = 1 2πi C B (N + 1, z) f (z) dz JASS 04 - Digital search trees 24
25 Internal path length Application of Rice s method A N = 1 2πi C B (N + 1, z) Q (1) Q (2 z+2 ) dz JASS 04 - Digital search trees 25
26 Internal path length Beta-Function B (p, q) = Γ (p) Γ (q) Γ (p + q) JASS 04 - Digital search trees 26
27 Internal path length Beta-Function 1e+11 5e+10 z e+10 1e JASS 04 - Digital search trees 27
28 Internal path length Gamma-Function y z JASS 04 - Digital search trees 28
29 Internal path length Cauchy s theorem If f(z) is analytic in C except final number of poles a 1, a 2,..., a n inside C, then 1 2πi C f (z) dz = n Res z=ak f (z) k= JASS 04 - Digital search trees 29
30 Internal path length Approximation with Rectangle R XY with ( 1 2 ± iy, X ± iy ) A N = 1 2πi R XY B (N + 1, z) Q (1) Q (2 z+2 ) dz Res R XY /C JASS 04 - Digital search trees 30
31 Internal path length Approximation of the integral 1 2πi R XY B (N + 1, z) Q (1) Q (2 z+2 ) dz = O ( Y Y ) Γ (N + 1) Γ ( N + 1 iy) dy 2 = O ( Y Y ) N 1 2 iy dy = O ( ) N JASS 04 - Digital search trees 31
32 Internal path length Residue at z = 1 B (N + 1, z) = N z 1 N (H N 1 1) + O (z 1) H N 1 = γ + ln N O B (N + 1, z) = N z 1 ( 1 N ) N (γ + ln N 1) + O (z 1) JASS 04 - Digital search trees 32
33 Internal path length Residue at z = 1 Q (1) Q (2 z+1 ) = 1 α ln 2 (z 1) + O ( (z 1) 2) JASS 04 - Digital search trees 33
34 Internal path length Residue at z = = 1 2 z+1 1 eln 2( z+1) 1 = ( z + 1) ln z = (z 1) ln O (z 1) 2 + O ( ( z + 1) 3) JASS 04 - Digital search trees 34
35 Internal path length Residue at z = 1 z=1 = N lg N N ( γ 1 ln 2 α + 1 ) 2 + O (1) JASS 04 - Digital search trees 35
36 Internal path length Residues at z = j ± 2πik ln 2 for Q(2 z+j ) where z=1± 2πIk ln 2 δ (N) = 1 ln 2 Γ k 0 = Nδ (N) + O (1) ( 1 2πik ) 2πik lg N e ln JASS 04 - Digital search trees 36
37 Internal path length Average case A N = N lg N + N ( γ 1 ln 2 α + 1 ) 2 + δ (N) + O ( ) N JASS 04 - Digital search trees 37
38 content Tree Digital search tree: Definition Average case analysis: Internal path length External internal nodes Multiway branching Tries General framework JASS 04 - Digital search trees 38
39 External internal nodes Definition External internal nodes are nodes with both links null JASS 04 - Digital search trees 39
40 External internal nodes Fundamental recurrence relation C N = k=0 1 2 N 1 with C 1 = 1 and C 0 = 0. ( ) N 1 k (C k + C N 1 k ), N JASS 04 - Digital search trees 40
41 External internal nodes Transformation C N z N C (z) = N! N=0 ( z C (z) = 1 + 2C 2) e z JASS 04 - Digital search trees 41
42 External internal nodes Substitution by D (z) = e z C (z) D (z) = N=0 D N z N N! D (z) + D (z) = e z + 2D ( z 2) JASS 04 - Digital search trees 42
43 External internal nodes Recurrence for D N D N + D N 1 = ( 1) N D N 2 N 1 ( D N = ( 1) N ) D 2 N 2 N 1, N 2 with D 1 = 1 and D 0 = JASS 04 - Digital search trees 43
44 External internal nodes Introduction of R N R N = Q N (1 + N k=1 1 Q k ) JASS 04 - Digital search trees 44
45 External internal nodes Explicit formula for C N C N = N k=2 ( N k ) ( 1) k R k JASS 04 - Digital search trees 45
46 External internal nodes Simpler coefficients R N R N = (N + 1 α) qn+1 1 q N q N+1 R N JASS 04 - Digital search trees 46
47 External internal nodes The meromorphic function R (z) R (z) = i=2 (z i α) q z+1+i i j=0 (1 qz+1+j ) JASS 04 - Digital search trees 47
48 External internal nodes The meromorphic function R (z) 4 y z JASS 04 - Digital search trees 48
49 External internal nodes Explicit formula for C N C N = (N 1) (α + 1) k=2 ( N k ) ( 1) k R k JASS 04 - Digital search trees 49
50 External internal nodes Applying Rice s method C N (N 1) (α + 1) = 1 2πi C B (N + 1, z) R (z 2) dz JASS 04 - Digital search trees 50
51 External internal nodes Approximation with Rectangle R XY C N (N 1) (α + 1) = 1 2πi R XY B (N + 1, z) R (z 2) dz R JASS 04 - Digital search trees 51
52 External internal nodes Approximation of the integral O ( Y Y ) Γ (N + 1) Γ ( N + 1 iy) dy 2 = O ( Y Y ) N 1 2 iy dy = O ( ) N JASS 04 - Digital search trees 52
53 External internal nodes Residue at z = 1 z=1 = N ( β Q ( α 2 α 1 )) ln q JASS 04 - Digital search trees 53
54 δ (N) = External internal nodes Residues at z = 1 ± 2πik ln q 2πik Q ln q k 0 ( 1 ln q Γ 1 2πik ) 2πik lg N e ln q JASS 04 - Digital search trees 54
55 External internal nodes Average case C N = N ( β Q ( ) 1 ln 2 + α2 α ) + δ (N) +O ( ) N JASS 04 - Digital search trees 55
56 content Tree Digital search tree: Definition Average case analysis: Internal path length External internal nodes Multiway branching Tries General framework JASS 04 - Digital search trees 56
57 Multiway branching Fundamental recurrence relation for external nodes C [M] N = k 1 +k k M =N 1 ( ) ( M ) 1 N 1 C [M] M N 1 k k 1, k 2,..., k i M i=1 with C [M] 1 = 1 and C [M] 0 = JASS 04 - Digital search trees 57
58 Multiway branching Fundamental recurrence relation for external nodes C [M] N = M k 1 +k k M =N 1 ( ) 1 N 1 C [M] M N 1 k k 1, k 2,..., k 1 M with C [M] 1 = 1 and C [M] 0 = JASS 04 - Digital search trees 58
59 Multiway branching Transformation C [M] (z) = N=0 C [M] (z) = 1 + MC [M] ( z M C [M] N zn N! ) ( e ( ) 1 1 M )z JASS 04 - Digital search trees 59
60 Multiway branching average case for external nodes C [M] N = N ( β [M] Nδ [M] (N) ( ) +O N 1 2 Q [M] ( ) ) 1 ln M + α[m]2 α [M] JASS 04 - Digital search trees 60
61 content Tree Digital search tree Tries Defintions Average case analysis General framework JASS 04 - Digital search trees 61
62 content Tree Digital search tree Tries: Defintion of Digital search trie Patricia trie Average case analysis General framework JASS 04 - Digital search trees 62
63 Digital search trie A digital search trie is a digital tree for storing a set of strings in which there is one node for every prefix of every string in the set JASS 04 - Digital search trees 63
64 Patricia trie A Patricia tree is defined as a compact representation of a digital search trie where all nodes with one child are merged with their parent JASS 04 - Digital search trees 64
65 content Tree Digital search tree: Tries: Defintions Average case analysis: External path length External internal nodes General framework JASS 04 - Digital search trees 65
66 content Tree Digital search tree Tries: Defintions Average case analysis: External path length External internal nodes General framework JASS 04 - Digital search trees 66
67 External path length for digital search trie Fundamental recurrence relation A [T ] N = N + k=0 with A [T ] 0 = A [T ] 1 = N ( N k ) ( ) A [T ] k + A [T ] N k, N JASS 04 - Digital search trees 67
68 External path length for digital search trie Transformation A [T ] (z) = z (e z 1) + 2A [T ] ( z 2 ) e z JASS 04 - Digital search trees 68
69 External path length for digital search trie Substitution by B (z) A (z) = e z B (z) B [T ] (z) = z ( 1 e z) + 2B [T ] ( z 2 ) JASS 04 - Digital search trees 69
70 External path length for digital search trie Explicit formula A [T ] N B [T ] (z) = = k=2 ( N k N ( 1)N ) N 1 1 ( 1 2 ) k ( 1) k 1 ( ) 1 k JASS 04 - Digital search trees 70
71 External path length for digital search trie average case A [T ] N ( γ = N lg N + N ln ) 2 + δ (N) + O(1) JASS 04 - Digital search trees 71
72 External path length for Patricia trie Fundamental recurrence relation ( A [P ] N = N 1 1 ) + 2 N 1 k=0 1 2 N ( N k ) ( ) A [P ] k + A [P ] N k, N JASS 04 - Digital search trees 72
73 External path length for Patricia trie Transformation A [P ] (z) = z ( ) e z e z 2 + 2A [P ] ( z 2) e z JASS 04 - Digital search trees 73
74 External path length for Patricia trie Substitution B [P ] (z) = z ( ) 1 e z 2 + 2B [P ] ( z 2) B [P ] (z) = N ( 1)N 2 N JASS 04 - Digital search trees 74
75 External path length for Patricia trie Explicit formula A [P ] N = k=2 ( N k ) k ( 1) k 2 k 1 1 = A[T ] N N JASS 04 - Digital search trees 75
76 content Tree Digital search tree Tries: Defintions Average case analysis: External path length External internal nodes General framework JASS 04 - Digital search trees 76
77 External internal nodes for Patricia trie Fundamental recurrence relation C [P ] N = 1 2 N ( N k with C [P ] 0 = C [P ] 1 = 0 and C [P ] 2 = 1 ) ( ) C [P ] k + C [P ] N k, N JASS 04 - Digital search trees 77
78 External internal nodes for Patricia trie Transformation C [P ] (z) = ( z 2 ) 2 + 2C [P ] ( z 2) e z JASS 04 - Digital search trees 78
79 External internal nodes for Patricia trie Substitution D [P ] (z) = ( z 2 ) 2 e z + 2D [P ] ( z 2) JASS 04 - Digital search trees 79
80 External internal nodes for Patricia trie Explicit formula C [P ] N = 1 4 N k=2 ( N k ) k (k 1) ( 1) k 1 1 k JASS 04 - Digital search trees 80
81 External internal nodes for Patricia trie average case ( ) C [P ] 1 N = N 4 ln 2 + δ[p ] (N) JASS 04 - Digital search trees 81
82 content Tree Digital search tree Trie General framework JASS 04 - Digital search trees 82
83 General Framework for digital search trees Fundamental recurrence relation X (T ) = subtrees T j of the root of T X (T j ) + x (T ) JASS 04 - Digital search trees 83
84 General Framework for digital search trees Transformation z N X (z) = X N N! N=0 ( z ) X (z) = MX e ( 1 1 M )z + x (z) M JASS 04 - Digital search trees 84
85 General Framework for digital search trees Substitution Y (z) = e z X (z) y (z) = e z x (z) ( z ) Y (z) + Y (z) = MY + y (z) + y (z) M JASS 04 - Digital search trees 85
86 General Framework for digital search trees Explicit formula X N = N k=0 ( N k ) Y k JASS 04 - Digital search trees 86
87 General Framework for digital search trees Find a function Y k which Asymptotic analysis of ( 1) k Y k (i) is simply related to Y k so that N k=0 ( N k easily evaluated, (ii) satisfies a recurrence of the form Y N+1 = (1 g (M, N)) Y N (iii) goes to zero quickly as N. + f (M, N), ) ( ) Y k ( 1) k Yk is JASS 04 - Digital search trees 87
88 General Framework for digital search trees Asymptotic analysis of ( 1) k Y k Turn the recurrence around to extend Y N Evaluate N k=0 ( N k sections. ) ( Y k ( 1) k Y k ) to the complex plane. as detailed in the previous JASS 04 - Digital search trees 88
89 General Framework for tries X (T ) = X (T j ) + x (T ) subtrees T j of the root of T X (z) = MX ( z M ) e z M + x (z) This can be solved by Rice s method or also by Mellin transform techniques JASS 04 - Digital search trees 89
90 Thank you for your attention! JASS 04 - Digital search trees 90
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