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 Einw ¨ ande gegen das Boltzmannsche -Theorem , 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 Here case of a ✣ -matrix 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

⑤ ❤ ♦ ❩ ❜ q ▼ ♦ ❴ ❭ ♥ ❪ ◆ ❍ ▼ ⑤ ▼ ❪ ⑤ ⑤ ⑤ ❩ ▼ ❪ ❻ ❩ ♥ ❩ ⑤ ❤ ♦ ❋ ♥ ❽ ◆ ❭ ❜ ❩ ❽ ♥ ❽ ❭ ❭ ⑥ ❹ ❨ ❻ ❋ ➆ ❯❲ ❋ ❱ ❚❯ ◆ ➆ ❋ ➆ ❷ ◆ ◆ ❤ ❪ ❴ ❤ ❩ ❪ ❴ ❽ ▼ ❍ ❨ ❍ ▼ ❪ ❍ ❻ ❪ ❜ ❭ ❪ Evolution is: ▼▲P❘● ◆❳❏ ●■❍ ❏▲❑ ▼✾◆❖❋ ▼◗P❘●❙❍ ▼◗P❘● with probability ❩❳❬ with probability . ▼▲❫❵❴ ❛❝❜❞❋ ❡❣❢ , ❜✐❤❥❢❧❦❅❍ ▼◗❫♠❴ Let , and ❜♣♦rqs❤t❢✉❦❅❍ ❜✐❤❥❢✇❤ ▼◗❫♠❴ ▼▲❫❵❴ ▼▲✈❘● whose elicitation is our main target. Lemma: PDE satisfied by BGF of probabilities is ❜✐❤②①③♦④◆ ❤t❢✰⑤ ❤⑦⑥⑧❢✰⑤ ❤⑦①⑨♥ ❤⑦⑩✏❍ ❶❸❷ ⑤❺❹ P ROOF . Each is determined from previous one by = a differential recurrence. Gives PDE for bivariate ♥ . generating function ❜✐❤❥❢ that satisfies PDE and write ❪❼❻ ❦❅❍ ❜✐❤t❢ ❜♣♦rq❾❤❥❢ . then, we get a homogeneous PDE . Take ❜❿❤⑦①➀♦➁◆ ❤❥❢ ❤⑦⑥➂❢ ❤⑦① ❶❸❷ ●➅➄ P❃➉➊➄ with ❜✐❤t❢➃❍ ❤✾⑥➂❢ ⑩➇❜ ⑥➈❢ ⑥➌➋ 9

➵ ➭ ➦ ➡ ➸ ➵ ➭ ➼ ➧ ➶ ➤ ➳ ➭ ➶ ➧ ➼ ➾ ➾ ➼ ➳ ➚ ➵ ➶ ➾ ➭ ➧ ➶ ➶ ➭ ➶ ➪ ➚ ➍ ➚ ➧ ➤ ➛ ➑ → ➲ ➓ ➵ ➭ ➙ ➞ ➏ → ➨ ➓ ❰ Ï ➙ ➡ ➸ ➡ ➡ ➹ ➼ ➻ ➤ ➸ ➮ ➵ ➳ ➪ ➡ ➡ ➹ ➠ ➶ Quasilinear first-order PDE’s are reducible to ODEs. ➎➣➏⑦➐↔➑↕➔ ➎✇➏②➐➒➑➝➔ ➎✇➏②➐➒➑✉➐➀➓ ➔❸→ ➎➣➏⑦➐↔➑➜➐⑨➓ ➔◗→ ➎➣➏⑦➐↔➑➜➐⑨➓ ➔■➟ ➢➥➤r➦❾➧❝➦✴➨➫➩➯➭ 1. Look for a solution in implicit form ➲ . ➢♣➤r➦s➧❝➦❾➵➁➩➺➸ ➢➥➤r➦➽➧❝➦❾➵➁➩✰➸ ➢♣➤❣➦➽➧❝➦❾➵➁➩➺➸ ➲❸➪ 2. Consider the ordinary differential system The solution of two “independent” ordinary differential equations, e.g., and leads to two families of integral curves, ➢✐➧➘➦➴➤r➦❾➵➁➩➃➭ ➢✐➧❝➦➱➤❣➦➽➵④➩➃➭ ➚➬➷ ➚❙✃ and 3. The generic solution of the PDE is provided by ➢♣➤r➦❾➧➘➦➽➵④➩➃➭ ❐❒➢ ➢✐➧➘➦➱➤r➦➽➵➁➩✴➦➒➮ ➢✐➧➘➦➱➤r➦➽➵➁➩➽➩✴➦ ➢♣➤r➦❾➧➘➦❾➵④➩❮➭ for arbitrary bivariate ❐ . Solving for in ❰④Ï➯➢♣➤r➦❾➧❥➩ . General solution is provides a relation ➢➥➤r➦❾➧t➩❧Ð❅➭ ➢♣➤r➦➽➧❥➩✴➪ 10

î è Ù ï ã ð Ö õ Þ Þ ì á Ü Ó Ù × Ü ì ê ì ë ê Ú Ü ë ì î õ Ü Þ ð ß Þ ì á Ü è Ó × Ü ì ì Ò ë ê Ú Ü ó ò Ñ ó Ú × Ü ì á Ù ó Ó Ò Ô á Þ Ü Ò ì Ò á ð Þ Ö Ù Þ Ú Ò Ø Ò Ü Ú Ô Ø × Ô Ò å Ó Ò á ä à ç é Þ Ü × Ü Ò ã Þ è Ü å Ò á ä à ã Ó Ò á Ö ð Ñ➥Ò↕ÓÕÔ Ò➝×◗Ø↕Ù Ñ➴Û➁Ö Ò❝ÜÝ×❸Ø↕Ù Ò↕Ó ß❃à árã Consider ÛâÖ árã first integral by separation: Ñ➴ÛâÖ ×➀æ➜ç➊è Ô variation of constant: Ô❘Ñ➴Û➁Ö Ò➌ÜÝ× ç➊è é➁í❺à Ñ➴Û➁Ö Bind the two integrals by arbitrary & Ô❥Ñ➊Û➁Ö Ò❝ÜÝ× ç➴è Ñ➊Û➁Ö Ñ➊Û➁Ö ç➴è Üñð Ù , introducing arbitrary Solve for ò : Ñ✇Ô Ò➝× ÑôÒ↕× Ñ➥Ò➝×➊Ô Ñ➥Ò➝×➒× ÑôÒ➝×❳ö Ñ➴Û➁Ö Ñ➥Ò➝×❳ö Ñ➴Û➁Ö ç➴è Ü . with 11

✂ ÷ ✓ ÿ ✠ ü ÷ ✟ ✞ ü ✝ ☞ ☎ ÿ ✂ ÿ ÿ ✠ ✚ ✠ ✟ ✞ ü ú ✝ ÷ ù ✝ ü ü ✑ ✏ Initial conditions identify ÷ . Theorem 1. Define the quantities ø✾ù➥ú➝û ù➊ýâþ ú❝ÿÝû ✁�✄✂ ÿ ✆☎ ùôú↕û ùôú➝û ✠✍✌ ù➴ý➁þ ù➴ýâþ ✠✒✑ û ☛✡ û ☛✎ (1) Then, the bivariate generating function of the probabilities is ù ✕✔✖☎ ➒ú➝û ø✾ùôú↕û ù ✕✔ ❣ø✾ù➥ú➝û ✘✗ ù➥ú➝û↔û ✙☎ (2) where is the function defined parametrically for ý by ✚✜✛ ùôú➝û➒û ☞ ✏ù➥ú➝û (3) 12

❢ ❬ ❬ ▲ ❝ ❬ ❉ ♠ ♥ ❬ ❜ ◆ ❢ ❴ ♥ ❴ ❢ ❬ ♥ ♥ q ♥ ♠ ❬ ❱ ♠ ❣ ❱ ❍ ❝ ♠ ❣ ❍ ❢ ♠ ▲ ❳ ◗ ❂ ✢ ✣ ✤ ✥ ✦ ✧ ✮ ✢ ✢ ✸ ✧ ✿ ❁ ✯ ❀ ❇ ❁ ✻ Dominant singularities of ✢ ? The diagram that summarizes is ★✪✩✫✣✭✬ ✩✕✦✲✬✳✧ ✴✵✩✫✣✭✬✷✶ ✯✱✰ The radius of analyticity of is ★✪✩✍✹✺✬✷✻ ★✪✩✼✣✭✬✵✽✾✧ ❁❃❂ ✩✍✹ ✬❃❄❆❅ Proof: There is local (analytic) invertibility of ❈❊❉●❋■❍ along ❉❑❏▼▲❖◆P❍ . Thus is analytic along ❉✫❏▼▲❘◆P❍ . We have ❉✫❚❨▲❖❏❩❍ which has nonnegative coeffs ◗❙❉❑❚❯❍❲❱ and is Pringsheim. We have while . Thus ◆ is a singularity. ❈❊❉❘❬❭❍❫❪ ❵❛❉❘❬❭❍❛❱ By Eulerian Beta integrals: ❉❘❬❭❤ ❉❘❬❭❤ ❈❊❉❘❬❭❍✘❱ ❍❛❱ ❏♦❥P❬✒♣✘❥P❬✷❏♦q ❥✺q ❝❡❞ ❝✵❣ ❉✐❬❭❤❦❥❧❍ ①✐②④③ ◗sr☛t ❏♦❥P❬✒♣❛❥P❬✷❏♦q ❥✺q q⑥⑤ ◗✈✉❘✇ 13