Analytic Urns Philippe Flajolet, INRIA Rocquencourt, F . With Kim - PDF document

January 20, 2003 Analytic Urns Philippe Flajolet, INRIA Rocquencourt, F . With Kim Gabarro and Helmut Pekari, Barcelona 1

General: BALLS and one or more URNS. Two kinds of models � “Balls-and-bins”: Throw balls at random into a number of urns. = Random allocations. Basic in the analysis of hashing algorithms; also SAT problem, cf V . Puyhaubert. = Techniques: Exponential generating functions and saddle point. Poissonization &c. Kolchin et al., Random Allocations , 1978. � “Urn models”: One urn contains balls whose nature may randomly change according to ball drawn and finite set of rules. 2

✟ ✠ ✡ ✠ ☛ ☞ ✁ ✂ ✡ ✝ ✌ ✆ ☎ Here: URNS with BALLS of TWO COLOURS Type I Type II Black White ✁ Matrix RULES are given by a The composition of the urn at time 0 is fixed. At time ✄ , a ball in the urn is randomly chosen and its colour is inspected (thus the ball is drawn, looked at and then placed back in the urn): if it is black, then black and white balls are subsequently inserted; if it is white, then, black balls and ✞ white balls are inserted. drawn added ✍✏✎ 3

✒ ✓ ✓ ✑ ✓ ✑ ✑ ✒ ✓ ✓ ✓ ✒ ✔ ✑ ✑ ✓ ✒ ✒ ✓ ✑ ✑ ✑ ✕ � Drawing with replacement = . � Drawing without replacement = . � Laplace’s “melancholic” model [1811]: if a ball is drawn, it is repainted black no matter what its colour is. � Ehrenfest & Ehrenfest = ¨ Uber zwei bekannte -Theorem , Einw ¨ ande gegen das Boltzmannsche 1923. Irreversibility contradicts Ergodicity. Exchanges of basic balls (“atoms of heat”) between two urns, one cold and one hot Bernoulli [1768], Laplace [1812]. 4

✖ ✑ ✗ ✒ ✘ ✖ ✑ ✑ ✖ ✗ ✁ ✑ ✖ ✒ � P´ olya Eggenberger model. A ball is drawn at random and then replaced, together with ✖ balls of the same colour. A model of positive influence. Closed form. � “Adverse influence” model Used in epidemiology, etc. � The special search tree model Yao [1978]; Bagchi and Pal [1985]; Aldous et al [1988]; Prodinger & Panholzer [1998] 5

✢ ✖ ✤ ✖ ✢ ✌ ✛ ☞ ✢ ☛ ✪ ✙ ✛ ✙ ✍ ✌ ☞ ☛ ✂ ✁ Johnson & Kotz, Urn Models and their Application , Wiley 1973 ✁ -matrix Here case of a with constant row-sum ✍✣✎ ✚✜✛ ✥✧✦ satisfies ✥✧✦ ✥✧★ ✖✩✤ . At time size Constant increment A problem with three parameters + two initial conditions. Kotz, Mahmoud, Robert [2000] show “pathologies’ in some of the other cases. Huge literature: Math. Reviews TITLE=urn : Number of Matches=186’’ 6

✤ ✢ ✓ ✛ ✬ ✮ Lead to amazingly wide variety of behaviours, special functions, and limit distributions. Methods � Difference equations and explicit solutions. � Same but with probability generating functions. � Connection with branching processes. � Stochastic differential equations [KMR] � Martingales [Gouet] Here: A frontal attack: — PDE of snapshots at time — Usual solution for quasilinear PDE — Bivariate GF and singularity perturbation Conformal mapping argument, Abelian integrals ✫✭✬ over Fermat curves + Special solutions with elliptic functions 7

✲✳ ✒ ✁ ✗ ✘ ✒ ✗ ✲✳ Part I ✁✰✯✱✗ model—basic equations The � Insertions in a 2–3 tree: 2–node 3–node; 3–node (2–node + 2–node). � Fringe-balanced 2–3 tree analogous to median-of-three quicksort. Mahmoud [1998]; Panholzer–Prodinger [1998] 8

❢ ❘ ❨ ❉ ▲ ❬ ✻ ❨ ■ ❋ ❳ ● ✼ ✶ ✻ ❢ ✻ ● ❢ ❢ ❢ ❉ ✻ ● q ❉ ❳ ❉ ❢ ❘ ❨ ✴ ❳ s ✼ ❋ ▲ ❉ s ❳ s ❋ ❋ ❣ ♦ ❈ q ✴ ⑤ ❄❆ ✴ ❅ ❃❄ ✼ ⑤ ✴ ⑤ ♠ ✼ ✼ ❘ ● ■ ❘ ❉ ● ■ s ✻ ✶ ❈ ✶ ✻ ● ✶ q ● ▲ ❋ ● Evolution is: ✻✺✿❁✵ ✼❇✸ ✵✷✶ ✸✺✹ ✻✽✼✾✴ ✻❀✿❁✵❂✶ ✻❀✿❁✵ with probability ❉❇❊ with probability . ✻✺❍❏■ ❑▼▲◆✴ ❖◗P , ▲❙❘❚P❱❯✱✶ ✻❀❍❲■ Let , and ▲❩❨❭❬❪❘❫P❴❯✱✶ ▲❙❘❚P❛❘ ✻❀❍❲■ ✻✺❍❏■ ✻✺❵❁✵ whose elicitation is our main target. Lemma: PDE satisfied by BGF of probabilities is ▲❙❘❝❜❞❨❡✼ ❘❫P✣❢ ❘❤❣✐P✣❢ ❘❤❜❥❳ ❘❤❦✏✶ ❧♥♠ ❢♣♦ P ROOF . Each is determined from previous one by = a differential recurrence. Gives PDE for bivariate ❳ . generating function ▲❙❘❚P that satisfies PDE and write ●rq ❯✱✶ ▲❙❘❫P ▲❩❨❭❬t❘❚P . then, we get a homogeneous PDE . Take ▲✉❘❤❜✈❨✇✼ ❘❚P ❘❤❣①P ❘❤❜ ❧♥♠ ✵④③ ✿✰⑧⑨③ with ▲❙❘❫P②✶ ❘✽❣①P ❦⑥▲ ❣⑦P ❣❶⑩ 9

➑ ❢ ↕ ✶ ➅ ➙ ❷ ✶ ➆ ✴ ✶ ➊ ✼ ❘ ❢ ✴ ➈ ❉ ➈ ✼ ❢ ➊ ❬ ➈ ✴ ✶ ➅ ➑ ➆ ✴ s ✼ ✶ ➈ ❧ ➆ ✶ ♠ ➊ ➍ ❨ ✴ ✢ ✫ ❸ ❻ ✫ ❸ ✠ ✫ ✢ ❽ ✫ ❸ ❻ ✫ ❸ ❹ ❽ ➂ ✛ ✫ ➅ s ✴ ❸ ❼ ✑ Quasilinear first-order PDE’s are reducible to ODEs. ✯❾❹❿❼ ✯❺❹➁❼ ✯❺❹❴✯✈❻ ❼♥❽ ✯❾❹➀✯❥❻ ❼❀❽ ✯❾❹➀✯❥❻ ▲➃❨❭❬t❘▼❬ P➄✶ ❧ . 1. Look for a solution in implicit form ▲❩❨❭❬❪❘▼❬t➆✇P➇❢ ▲➃❨❭❬➉❘▼❬t➆✇P✣❢ ▲❩❨◗❬➉❘▼❬t➆✇P➇❢ ❧♥♠ 2. Consider the ordinary differential system ⑩➇❨ ⑩➋❘ ⑩➋➆ The solution of two “independent” ordinary differential equations, e.g., ⑩➌❘ ⑩➋➆ ⑩➇❨ ⑩➋❘ and leads to two families of integral curves, ▲❙❘➎❬➏❨❭❬t➆✇P②✶ ➊➐✵ ▲❙❘▼❬➒❨◗❬➉➆❡P②✶ ➊❂➓✩♠ and 3. The generic solution of the PDE is provided by ▲❩❨❭❬t❘➎❬➉➆❡P②✶ ➔→▲➣➍ ▲❙❘➎❬➒❨❭❬➉➆✇P✧❬ ▲❙❘➎❬➒❨❭❬➉➆✇P➉P✧❬ ▲❩❨❭❬t❘➎❬t➆❡P↔✶ ➔ . Solving for ↕❡➙➄▲❩❨❭❬t❘❚P . General solution is for arbitrary bivariate in provides a relation ▲➃❨❭❬t❘❫P❱❯✱✶ ▲❩❨❭❬➉❘❚P✧♠ 10

❸ ✓ ✒ ➜ ✢ ➤ ★ ✥ ❸ ✒ ❸ ✥ ➜ ❼ ➛ ➢ ➜ ➝ ✓ ✫ ✛ ❻ ❼ ➛ ➢ ➜ ➝ ➞ ➧ ➦ ✯ ❻ ❸ ✓ ✒ ❹ ➜ ❼ ➜ ✥ ❻ ✥ ✯ ➜ ❼ ➛ ➢ ➜ ➝ ✥ ✍ ✒ ❸ ❸ ✓ ✒ ❹ ➜ ❼ ✥ ✓ ❸ ❸ ✫ ✛ ✍ ❸ ➨ ❸ ✍ ✫ ❸ ✢ ➩ ❸ ➩ ❸ ➤ ★ ✥ ➜ ➦ ✒ ✛ ✫ ❹ ➛ ❽ ✫ ✒ ❹ ✒ ❻ ✒ ✓ ✫ ❹ ➛ ➞ ✎ ❸ ➝ ✛ ❹ ✑ ✒ ❽ ❻ ❽ ❹ ✢ ✛ ✎ ➜ ➝ ✢ ✫ ❹ ✓ ❽ ✒ ❹ ➝ � ➟ ✓ � ➝ ❹ ➝ ✫ ❸ ✒ ➡ ➜ ✢ ➤ ★ ✥ ❸ ✎ ➟ ➂ ➜ ➞ ❸ ✓ ✒ ❹ ✛ ✓ ➜ ❹❿➛ ❹➁❼ ❹▼➜➌❼ ❹❿➛➋❻ ➝❭➞ Consider ➝❭➞ first integral by separation: ❼✈➠➀➡⑨➢ ✫ variation of constant: ➂✇➥♣✎ ❹❶➜➌❼ ➡⑨➢ ✥✾✛ Bind the two integrals by arbitrary & ❹▼➜➌❼ ➡➏➢ ✑✰✯ ➡➏➢ ❻ , introducing arbitrary Solve for ➨ : ✯❺❹➁❼ ❹❿❼ ❹➁❼ ❹➁❼❺❼①✯ ❹➁❼ ✚✜✛ ❹➁❼ ➡➏➢ ✚➫✛ ➜ . with 11

➢ ✎ ✛ ➤ ★ ➩ ❸ ✓ ✒ ✥ ➜ ❸ ➨ ➜ ➝ ✥ ❹ ❸ ➵ ➨ ❻ ❸ ✫ ❸ ✛ ✍ ❸ ➩ ➨ ❸ ✫ ✍ ❸ ➭ ✢ ➩ ✎ ✍ ❸ ❸ ✛ ❸ ✓ ✒ ❹ ➜ ❼ ➭ ➜ ✯ ❸ ✯ ✛ ✛ ➤ ★ ✥ ❸ ✓ ✒ ✥ ➜ ❼ ➛ ➢ ➜ ❸ Initial conditions identify ➨ . Theorem 1. Define the quantities ❹➁❼ ➡➏➢ ❹❿❼ ❹➁❼ ✥⑨➯ ❼❺➲ (1) Then, the bivariate generating function of the probabilities is ✯❺❹➁❼ ❹❿❼ ❹➁❼ ❹➁❼❾❼➳✯ (2) where is the function defined parametrically for ➵➺➸ ✓ by ❹➁❼❺❼ ❹➁❼ (3) 12

Ï ➮ ❋ ❬ ✃ ❋ ▲ ➈ ✃ ❋ ❊ ❋ ▲ ➪ ➶ ♠ ✸ ❋ ❐ ➷ ❋ ▲ ➬ ❊ ➷ ❋ ▲ ➪ ❐ ❋ ❊ ♠ ❊ ♠ ❐ ⑧ ❋ ✸ ➹ ❜ ❊ ♠ ❋ ✶ q ❋ ❦ ▲ ❒ P ✃ ♠ ▲ ❒ P ❐ ✶ ▲ ❬ s ➹ ❸ ❸ ➩ ✛ ➾ ➨ ✎ ❸ ➭ ✛ ❼ ✫ ➨ ➤ ✳ ✒ ➽ ❸ ➩ ✛ ✫ ➼ ➻ ❹ ➨ ❸ ➩ ★ ➜ ➹ ➪ ✯ ✥ ➜ ➢ ➛ ❼ ➝ ✥ ✥ ✒ ✓ ❸ Dominant singularities of ➨ ? The diagram that summarizes is ❹➁❼ ❹➁❼ The radius of analyticity of is ✓⑥❼➚✯ ❹➁❼ ✚✜✛ ▲❙❘❚P along ▲❩❧♥❬➒➶❀P . Thus ▲➃❧♥❬➉➶❀P . Proof: There is local (analytic) invertibility of is analytic along ▲❩❨♣P②✶ ▲➃❨❭❬➒❧➘P which has nonnegative coeffs We have and is Pringsheim. P❱➴ P➄✶ ➶ is a singularity. We have while . Thus By Eulerian Beta integrals: P➱✶ P➄✶ ❋⑦❮ ❋⑦❮ ❧✣✸ ❧✣Ï ✸⑥Ï ✃✏❒ ✸➇P ❋①❰ ❋⑦❮ ✵t③ ❧✣✸ ❧✣Ï ✸⑥Ï ÏÓÒ ➹ÑÐ ❋①❰ 13