Approximate Counting via the Poisson-Laplace-Mellin Method (joint with Chung-Kuei Lee and Helmut Prodinger)
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1 Approximate Counting via the Poisson-Laplace-Mellin Method (joint with Chung-Kuei Lee and Helmut Prodinger) Michael Fuchs Department of Applied Mathematics National Chiao Tung University Hsinchu, Taiwan Montreal, June 18, 2012 Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
2 Approximate Counting (Morris 1978) Space needed for counting n objects: Θ(log n). Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
3 Approximate Counting (Morris 1978) Space needed for counting n objects: Θ(log n). Problem: What if space is very limited? Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
4 Approximate Counting (Morris 1978) Space needed for counting n objects: Θ(log n). Problem: What if space is very limited? Answer: Allow an error tolerance: approximate counting. Counter C n with C 0 = 0 and (0 < q < 1) { C n + 1, with probability q Cn ; C n+1 = C n, with probability 1 q Cn. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
5 Approximate Counting (Morris 1978) Space needed for counting n objects: Θ(log n). Problem: What if space is very limited? Answer: Allow an error tolerance: approximate counting. Counter C n with C 0 = 0 and (0 < q < 1) { C n + 1, with probability q Cn ; C n+1 = C n, with probability 1 q Cn. Easy to show: E(q Cn ) = n ( q 1 1 ) + 1. Now, only Θ(log log n) space is needed. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
6 Applications Approximate counting has found many applications: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
7 Applications Approximate counting has found many applications: Analysis of the Webgraph. Monitoring network traffic. Finding patterns in protein and DNA sequencing. Computing frequency moments of data streams. Data storage in flash memory. Etc. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
8 Applications Approximate counting has found many applications: Analysis of the Webgraph. Monitoring network traffic. Finding patterns in protein and DNA sequencing. Computing frequency moments of data streams. Data storage in flash memory. Etc. Many refinements have been proposed. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
9 The Markov Chain C n Other problems leading to the same Markov chain: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
10 The Markov Chain C n Other problems leading to the same Markov chain: Width of greedy decomposition of random acyclic digraphs into node-disjoint paths. Size of greedy independent set in random graphs. Size of greedy clique in random graphs. Length of leftmost path in random digital search trees. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
11 The Markov Chain C n Other problems leading to the same Markov chain: Width of greedy decomposition of random acyclic digraphs into node-disjoint paths. Size of greedy independent set in random graphs. Size of greedy clique in random graphs. Length of leftmost path in random digital search trees. Variations of this Markov chain were also studied: Simon (1988); Crippa and Simon (1997); Bertoin, Biane and Yor (2003); Guillemin, Robert and Zwart (2004); Louchard and Prodinger (2008) Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
12 Analysis of Approximate Counting Flajolet (1985): E(C n ) log 1/q n + C mean + F (log 1/q n), where F (z) is a 1-periodic function Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
13 Analysis of Approximate Counting Flajolet (1985): E(C n ) log 1/q n + C mean + F (log 1/q n), where F (z) is a 1-periodic function and Var(C n ) C var + G(log 1/q n), where G(z) is a 1-periodic function and C var = π 2 6 log 2 (1/q) α β log(1/q) l sinh(2lπ 2 / log(1/q)) l 1 with α = l 1 ql /(1 q l ) and β = l 1 q2l /(1 q l ) 2. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
14 Methods Many different methods have been used: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
15 Methods Many different methods have been used: Mellin Transform: Flajolet (1985); Prodinger (1992) Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
16 Methods Many different methods have been used: Mellin Transform: Flajolet (1985); Prodinger (1992) Rice Method: Kirschenhofer and Prodinger (1991) Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
17 Methods Many different methods have been used: Mellin Transform: Flajolet (1985); Prodinger (1992) Rice Method: Kirschenhofer and Prodinger (1991) Euler Transform: Prodinger (1994) Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
18 Methods Many different methods have been used: Mellin Transform: Flajolet (1985); Prodinger (1992) Rice Method: Kirschenhofer and Prodinger (1991) Euler Transform: Prodinger (1994) Analysis of Extreme Value Distributions: Louchard and Prodinger (2006) Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
19 Methods Many different methods have been used: Mellin Transform: Flajolet (1985); Prodinger (1992) Rice Method: Kirschenhofer and Prodinger (1991) Euler Transform: Prodinger (1994) Analysis of Extreme Value Distributions: Louchard and Prodinger (2006) Martingale Theory: Rosenkrantz (1987) Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
20 Methods Many different methods have been used: Mellin Transform: Flajolet (1985); Prodinger (1992) Rice Method: Kirschenhofer and Prodinger (1991) Euler Transform: Prodinger (1994) Analysis of Extreme Value Distributions: Louchard and Prodinger (2006) Martingale Theory: Rosenkrantz (1987) Probability Theory: Robert (2005) Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
21 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
22 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
23 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
24 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
25 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
26 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
27 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
28 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
29 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
30 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
31 Digital Search Tree (DST) Introduced by Coffman and Eve (1970). Example: a digital search tree build from 9 keys: Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
32 Random Model and Leftmost Path Random Model: Bits are generated by independent Bernoulli random variables with mean p. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
33 Random Model and Leftmost Path Random Model: Bits are generated by independent Bernoulli random variables with mean p. Two types of trees: p = 1/2: symmetric digital search tree; Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
34 Random Model and Leftmost Path Random Model: Bits are generated by independent Bernoulli random variables with mean p. Two types of trees: p = 1/2: symmetric digital search tree; p 1/2: asymmetric digital search tree. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
35 Random Model and Leftmost Path Random Model: Bits are generated by independent Bernoulli random variables with mean p. Two types of trees: p = 1/2: symmetric digital search tree; p 1/2: asymmetric digital search tree. Length of the Leftmost Path: X n : number of vertices on leftmost path. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
36 Random Model and Leftmost Path Random Model: Bits are generated by independent Bernoulli random variables with mean p. Two types of trees: p = 1/2: symmetric digital search tree; p 1/2: asymmetric digital search tree. Length of the Leftmost Path: X n : number of vertices on leftmost path. Note that: X n d = Cn. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
37 Distributional Recurrence of X n X n+1 d = XIn + 1 Root 0 1 I n d = Binomial(n, q); X n, I n independent. Size: I n Size: n I n Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
38 Distributional Recurrence of X n X n+1 d = XIn + 1 Root 0 1 I n d = Binomial(n, q); X n, I n independent. Size: I n Size: n I n Recurrence of moments: f n+1 = n j=0 ( ) n q j p n j f j + g n. j Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
39 Other Shape Parameters Depth Konheim, Newman, Knuth, Devroye, Louchard, Szpankowski Total Path Length Flajolet, Sedgewick, Prodinger, Kirschenhofer, Szpankowski, Hubalek Peripheral Path Length Drmota, Gittenberger, Panholzer, Prodinger, Ward # of Occurrences of Patterns Knuth, Flajolet, Sedgewick, Prodinger, Kirschenhofer Colless Index Fuchs, Hwang, Zacharovas Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
40 Methods Rice Method Introduced by Flajolet and Sedgewick. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
41 Methods Rice Method Introduced by Flajolet and Sedgewick. Approach of Flajolet and Richmond Based on Euler transform, Mellin transform, and singularity analysis. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
42 Methods Rice Method Introduced by Flajolet and Sedgewick. Approach of Flajolet and Richmond Based on Euler transform, Mellin transform, and singularity analysis. Approach via Analytic Depoissonization Introduced by Jacquet & Regnier and Jacquet & Szpankowski. Based on saddle point method and Mellin transform. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
43 Methods Rice Method Introduced by Flajolet and Sedgewick. Approach of Flajolet and Richmond Based on Euler transform, Mellin transform, and singularity analysis. Approach via Analytic Depoissonization Introduced by Jacquet & Regnier and Jacquet & Szpankowski. Based on saddle point method and Mellin transform. Schachinger s Approach Largely elementary. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
44 Poisson-Laplace-Mellin Method Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
45 Poisson-Laplace-Mellin Method Poissonized mean and variance satisfy differential-function equations. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
46 Poisson-Laplace-Mellin Method Poissonized mean and variance satisfy differential-function equations. Due to Jacquet-Szpankowski s theory of depoissonization only the Poisson model has to be analyzed. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
47 Poisson-Laplace-Mellin Method Poissonized mean and variance satisfy differential-function equations. Due to Jacquet-Szpankowski s theory of depoissonization only the Poisson model has to be analyzed. Laplace transform to get rid of the differential operator. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
48 Poisson-Laplace-Mellin Method Poissonized mean and variance satisfy differential-function equations. Due to Jacquet-Szpankowski s theory of depoissonization only the Poisson model has to be analyzed. Laplace transform to get rid of the differential operator. A normalization factor simplifies the functional equation satisfied by Laplace transform. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
49 Poisson-Laplace-Mellin Method Poissonized mean and variance satisfy differential-function equations. Due to Jacquet-Szpankowski s theory of depoissonization only the Poisson model has to be analyzed. Laplace transform to get rid of the differential operator. A normalization factor simplifies the functional equation satisfied by Laplace transform. Mellin transform is applied which can be computed explicitly. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
50 Poisson-Laplace-Mellin Method Poissonized mean and variance satisfy differential-function equations. Due to Jacquet-Szpankowski s theory of depoissonization only the Poisson model has to be analyzed. Laplace transform to get rid of the differential operator. A normalization factor simplifies the functional equation satisfied by Laplace transform. Mellin transform is applied which can be computed explicitly. We use inverse Mellin transform and inverse Laplace transform to obtain asymptotic expansions in the Poisson model. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
51 Poissonization Moments satisfy the recurrence: f n+1 = n j=0 ( ) n q j p n j f j + g n. j Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
52 Poissonization Moments satisfy the recurrence: f n+1 = n j=0 ( ) n q j p n j f j + g n. j Consider Poisson-generating function of f n and g n, i.e., f(z) := e z n 0 z n f n n!, g(z) := z n e z g n n!. n 0 Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
53 Poissonization Moments satisfy the recurrence: f n+1 = n j=0 ( ) n q j p n j f j + g n. j Consider Poisson-generating function of f n and g n, i.e., Then, f(z) := e z n 0 z n f n n!, g(z) := z n e z g n n!. n 0 f(z) + f (z) = f(qz) + g(z). This is a differential-functional equation. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
54 Poisson Heuristic Poisson Heuristic: f n sufficiently smooth = f n f(n). Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
55 Poisson Heuristic Poisson Heuristic: f n sufficiently smooth = f n f(n). More precisely: if f n is smooth enough, f n j 0 f (j) (n) τ j (n) = n! f(n) n 2 f (n) +..., where τ j (n) := n j e z This is called Poisson-Charlier expansion (can be already found in Ramanujan s notebooks). Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
56 Jacquet-Szpankowski-admissibility (JS-admissibility) f(z) is called JS-admissible if (I) Uniformly for arg(z) ɛ, f(z) = O (O) Uniformly for ɛ < arg(z) π, ( ) z α log β z, f(z) := e z f(z) = O (e (1 ɛ) z ). Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
57 Jacquet-Szpankowski-admissibility (JS-admissibility) f(z) is called JS-admissible if (I) Uniformly for arg(z) ɛ, f(z) = O (O) Uniformly for ɛ < arg(z) π, Theorem (Jacquet and Szpankowski) If f(z) is JS-admissible, then ( ) z α log β z, f(z) := e z f(z) = O (e (1 ɛ) z ). f n f(n) n 2 f (n) +. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
58 Depoissonization JS-admissibility satisfies closure properties: (i) f, g JS-admissible, then f + g JS-admissible. (ii) f JS-admissible, then f JS-admissible. Etc. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
59 Depoissonization JS-admissibility satisfies closure properties: (i) f, g JS-admissible, then f + g JS-admissible. (ii) f JS-admissible, then f JS-admissible. Etc. Proposition Consider f(z) + f (z) = f(qz) + g(z). We have, g(z) JS-admissible f(z) JS-admissible. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
60 Poissonized Mean and Second Moment Define f 1 (z) = e z n 0 E(X n ) zn n!, f2 (z)e z E(Xn) 2 zn n! n 0 which are poissonized mean and second moment. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
61 Poissonized Mean and Second Moment Define f 1 (z) = e z n 0 E(X n ) zn n!, f2 (z)e z E(Xn) 2 zn n! n 0 which are poissonized mean and second moment. Then, f 1 (z) + f 1(z) = f 1 (qz) + 1 f 2 (z) + f 2(z) = f 2 (qz) + 2 f 1 (qz) + 1 Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
62 Poissonized Mean and Second Moment Define f 1 (z) = e z n 0 E(X n ) zn n!, f2 (z)e z E(Xn) 2 zn n! n 0 which are poissonized mean and second moment. Then, f 1 (z) + f 1(z) = f 1 (qz) + 1 f 2 (z) + f 2(z) = f 2 (qz) + 2 f 1 (qz) + 1 Previous results show that f 1 (z), f 2 (z) are JS admissible. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
63 Poissonized Variance Important in order to avoid having to cope with cancelations! Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
64 Poissonized Variance Important in order to avoid having to cope with cancelations! Mean, Variance = Θ(log β n): Ṽ (z) = f 2 (z) f 1 (z) 2. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
65 Poissonized Variance Important in order to avoid having to cope with cancelations! Mean, Variance = Θ(log β n): Ṽ (z) = f 2 (z) f 1 (z) 2. Mean, Variance = Θ(n log β n): Ṽ (z) = f 2 (z) f 1 (z) 2 z f 1(z) 2. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
66 Poissonized Variance Important in order to avoid having to cope with cancelations! Mean, Variance = Θ(log β n): Ṽ (z) = f 2 (z) f 1 (z) 2. Mean, Variance = Θ(n log β n): Ṽ (z) = f 2 (z) f 1 (z) 2 z f 1(z) 2. More general: Ṽ (z) = f 2 (z) n 0 f (n) 1 (z) 2 zn n!. Then one obtains even identities! Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
67 Laplace and Mellin Transform (i) We start from, f(z) + f (z) = f(qz) + g(z). Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
68 Laplace and Mellin Transform (i) We start from, f(z) + f (z) = f(qz) + g(z). Applying Laplace transform, (s + 1)L [ f(z); s] = 1 [ q L f(z); 1 ] + L [ g(z); s]. q Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
69 Laplace and Mellin Transform (i) We start from, f(z) + f (z) = f(qz) + g(z). Applying Laplace transform, (s + 1)L [ f(z); s] = 1 [ q L f(z); 1 ] + L [ g(z); s]. q Define, Q(s) := l 1 ( ) 1 q l s and Q := Q(1). Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
70 Laplace and Mellin Transform (ii) Set f(s) := L [ f(z); s] L [ g(z); s], ḡ(s) := Q( s) Q( s/q). Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
71 Laplace and Mellin Transform (ii) Set Then, f(s) := L [ f(z); s] L [ g(z); s], ḡ(s) := Q( s) Q( s/q). f(s) = 1 q f ( ) s + ḡ(s). q Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
72 Laplace and Mellin Transform (ii) Set Then, f(s) := L [ f(z); s] L [ g(z); s], ḡ(s) := Q( s) Q( s/q). f(s) = 1 q f ( ) s + ḡ(s). q Applying Mellin transform, M [ f(s); ω] = M [ḡ(s); ω] 1 q ω 1. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
73 Laplace and Mellin Transform (ii) Set Then, f(s) := L [ f(z); s] L [ g(z); s], ḡ(s) := Q( s) Q( s/q). f(s) = 1 q f ( ) s + ḡ(s). q Applying Mellin transform, M [ f(s); ω] = M [ḡ(s); ω] 1 q ω 1. From this, an asymptotic expansion of f(z) as z is obtained via inverse Mellin transform and inverse Laplace transform. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
74 Inverse Laplace Transform Theorem (F., Hwang, Zacharovas) Let the Laplace transform of f(z) exist and be analytic in C \ (, 0]. Assume that L [ f; s] = O ( s α) uniformly for s 0 and arg(s) π ɛ. Then, f(z) = O ( z α 1) uniformly for z and arg(z) π/2 ɛ. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
75 Our Approach vs. Flajolet-Richmond EGF f(z) Laplace transform of e z f(z) = Euler transform of A(z) OGF A(z) asymptotics of f(z) as z asymptotics of Laplace Q( s) Laplace Q( s) Euler Q( s) asymptotics of Euler Q( s) asymptotics of A(z) as z 1 de-poi by saddle-point Mellin transform singularity analysis Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
76 Main Result Theorem (F., Lee, Prodinger) We have, Var(C n ) k g k n χ k, where g k = Q LΓ(1 + χ k ) h,l,j 0 ( 1) j q h+l+(j+1 2 ) Q h Q l Q j ϕ(χ k ; q h+j + q l+j ). Here, χ k = 2kπi/L, L = log(1/q), Q j = j l=1 (1 ql ) and { π(x χ 1)/(sin(πχ)(x 1)), if x 1; ϕ(χ; x) = πχ/ sin(πχ), if x = 1. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
77 An Identity Corollary (F., Lee, Prodinger) We have, Q L h,l,j 0 ( 1) j q h+l+(j+1 2 ) Q h Q l Q j ψ(q h+j + q l+j ) = π2 6L 2 α β L l sinh(2lπ 2 /L), l 1 where ψ(x) = { log x/(x 1), if x 1; 1, if x = 1. We have a direct proof for this using tools from q-analysis. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
78 Total Path Length (i) T n : total path length in symmetric digital search tree. Theorem (F., Hwang, Zacharovas) We have Var(T n ) n(c var + G(log 2 n)), where G(z) is a 1-periodic function with zero average value and C var = Q L j,h,l 0 ( 1) j 2 (j+1 2 ) Q j Q h Q l 2 h+l δ(2 j h + 2 j l ), where δ(x) := { (x log x 1)/(x 1) 2, if x 1; 1/2, if x = 1. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
79 Total Path Length (ii) Variance of total path length was also derived by Kirschenhofer, Prodinger and Szpankowski with different expression for C var. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
80 Total Path Length (ii) Variance of total path length was also derived by Kirschenhofer, Prodinger and Szpankowski with different expression for C var. To describe their expression we need: Let [F G] 0 denote the 0-th Fourier coefficient of the product of the two Fourier series F (z) and G(z). Put and F (z) = 1 Γ( 1 χ l )e 2lπiz L H(z) = 1 L l 0 ( l 0 1 χ l 2 ) Γ( χ l )e 2lπiz. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
81 C var = 28 3L l 1 l2 l (2 l 1) L l 1 2 ( 1) l+1 (l 5) L (l + 1)l(l 1)(2 l 1) l 3 2Q(1) L + l l Q l 1 2 l 1 + π2 2L L ( ) ( 1) l 2 l+1 2 L(1 2 l+1 )/2 1 L 1 2 l ( 1) r+1 r(r 1)(2 r+l 1) l 1 r 2 ( ) ( 1) r 2 r+1 2 ( j j+r+l+2 1 Q r+l 2 Q r 0 r ( l 1 ( l + 1 ) 2 l+1 2l i i=2 2 (1 2 l r ) 2 + 2l + 2 (1 2 1 l r ) 2 2 L l+1 ( l + 1 ) 2 j j=2 1 2 r+j L j=0 i r+i l r + 2 l+1 ( l + 1 ) L j j=1 l+1 ( l + 1 ) ( 1) i ) j (i + 1)(2 r+j+i 1) + l 1 ( l + 1 ) Qr 2 Q l r 1 1 r 2 l 3 r=2 l Q l 2 j 1 2[F H] 0 [F 2 ] 0. j l+1 ) 1 2 r+j 1 Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
82 Approximate Counting with m Counters (i) Cichoń and Macyna: Consider m counters. When counting the n-th object choose one uniform at random. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
83 Approximate Counting with m Counters (i) Cichoń and Macyna: Consider m counters. When counting the n-th object choose one uniform at random. D n : sum of all counters after counting n objects. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
84 Approximate Counting with m Counters (i) Cichoń and Macyna: Consider m counters. When counting the n-th object choose one uniform at random. D n : sum of all counters after counting n objects. Then, where D n d = C (1) I C (m) I m, P (I 1 = n 1,..., I m = n m ) = 1 ( ) n m n. n 1,..., n m Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
85 Approximate Counting with m Counters (i) Cichoń and Macyna: Consider m counters. When counting the n-th object choose one uniform at random. D n : sum of all counters after counting n objects. Then, where D n d = C (1) I C (m) I m, P (I 1 = n 1,..., I m = n m ) = 1 ( ) n m n. n 1,..., n m Let f D, f C denote Poisson mean of D n and C n. Similar, let ṼD and ṼC denote Poisson variance of D n and C n. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
86 Approximate Counting with m Counters (ii) Poisson model: f D (z) = m f C (z/m), Ṽ D (z) = mṽc(z/m). Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
87 Approximate Counting with m Counters (ii) Poisson model: f D (z) = m f C (z/m), Ṽ D (z) = mṽc(z/m). From this, we obtain the following result. Theorem (F., Lee, Prodinger) We have, E(D n ) m log 1/q (n/m) + mc mean + mf (log 1/q (n/m)), Var(D n ) mc var + mg(log 1/q (n/m)), where C mean, C var and F (z), G(z) are as before. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
88 Approximate Counting with m Counters (iii) Another variant of approximate counting with m counters: Consider m counters. Use a counter until it is increased; then cyclically move on to the next. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
89 Approximate Counting with m Counters (iii) Another variant of approximate counting with m counters: Consider m counters. Use a counter until it is increased; then cyclically move on to the next. D n : sum of all counters after counting n objects. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
90 Approximate Counting with m Counters (iii) Another variant of approximate counting with m counters: Consider m counters. Use a counter until it is increased; then cyclically move on to the next. D n : sum of all counters after counting n objects. Then, where D n d = Xn, X n+m d = XIn + m. X n is the length of the leftmost path in random bucket digital search tree with bucket size m. Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
91 Approximate Counting with m Counters (iv) Recurrence of moments: f n+m = n j=0 ( ) n q j p n j f j + g n. j Michael Fuchs (NCTU) Approximate Counting Montreal, June 18, / 32
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