Random Walk wih Ani-Correlaed Seps John Noga Dirk Wagner 2 Absrac We conjecure he expeced value of random walks wih ani-correlaed seps o be exacly. We suppor his conjecure wih 2 plausibiliy argumens and experimenal daa. The experimenal analysis includes he compuaion of he expeced values of random walks for seps up o 22. The resul shows he expeced value asympoically converging o. Inroducion Le {µ } be he horizonal random walk 3 ha obeys he following rules:. The walk sars a µ 0 2. An urn conains whie marbles and red marbles. A each sep a marble is seleced from ha urn wihou replacemen. 3. If he marble chosen is red µ increases by one (µ + µ + ) 4. If he marble chosen is whieµ decreases by one (µ + µ -) 5. Le p/3 be he probabiliy o draw a red marble. Accordingly, q2/3 is he probabiliy o draw a whie marble. n is he oal number of marbles in ha urn. Noe ha he absolue maximum posiive is while he walk bounded by in he negaive direcion. Noe also he walk will erminae a - This ype of random walks involve ani-correlaed seps, meaning he more red marbles are drawn (posiive direcion) he more likely are seps ino he negaive direcion (whie marbles are drawn). Ani-correlaed walks are implemened simply be no replacing drawn marbles. We are ineresed in he expeced value E µ ] of {µ }, ha is for he expeced value of a random walk wih ani-correlaed seps. ] proves E µ ] O(). We are going o refine his resul by showing ha he expeced value of random walks wihou replacemen is likely o be exacly, hus E µ ]. Applicaions for random walks wih replacemen can be found in caching algorihms. ] inroduces he RMARK algorihm a random walk which cos is E µ ]. 3] I can be shown ha for random walks wih replacemen E µ ]. Deparmen of Compuer Science, California Sae Universiy of Norhridge, CA 9330. Email: jnoga@ecs.csun.edu 2 Deparmen of Compuer Science, California Sae Universiy of Norhridge, CA 9330. Email: wagiboy@gmail.com 3 Sricly pu we consider random processes because he probabiliy of seps changes wih ime.
2 Conjecure E µ ] We conjecure E µ ] for a random walk wih ani-correlaed sep size. Suppor for his conjecure comes from wo plausibiliy argumen and our experimenal daa. Plausibiliy argumen one: Le } be random walk wih replacemen wih p/3, he probabiliy o draw a red marble. Noe ha p remains consan during he walk. 4] shows he expeced value of random walks wih replacemen is exacly, in shor E ν ]. For random walks wihou replacemen p is changing, in fac p is geing smaller as he walk progresses: A p 0 /3. Afer an equal number of red and whie marbles have been drawn µ reurns o zero. A his poin he probabiliy ha a red marble will be drawn nex is no greaer han /3. Le p he probabiliy o draw a red marble, hen p < p0. The probabiliy of subsequen reurns of µ o zero exhibi an even furher decrease such ha p > p > L. Noe, if whie marbles are drawn subsequen p do increase bu a no 0 p2 effec o E µ ] because µ moves in he negaive direcion. Now when he probabiliy p of drawing reds decreases over he course of a random walk, i is plausible ha he expeced value of random walks wihou replacemen is no greaer han for random walks wih replacemen, or E µ ] E ν ]. max max Plausibiliy argumen wo: As grows large drawing marbles does affec p very lile. In fac, as δ p does no change a all. In oher words he random walk wihou replacemen virually becomes a random walk wih replacemen, hus E µ ] E ν ]. max max 3 Experimenal Mehods In his secion we provide experimenal suppor for our conjecure E max µ ] by compuing E µ ] for random walk of up o 30 seps. We implemened and compared several algorihms o efficienly compue he problem, including exhausive compuing, recursively selecing represenaives, ieraively selecing represenaives and linear opimizaions. As experimenal mehod are implemened and compared on a.6 GHz Penium M machine wih 52 MB of main memory. The final experimenal daa has been compued on a 3.2??? GHz Penium 4 wih hyper hreading and 2 GB??? of main memory. 3. Exhausive Mehod We used he following pseudo code o exhausively compue E µ ]. inpu n 3/2* oal_max_µ0; for each of he n! permuaions µ0; max_µ0; for each marble permuaion if marble is red µ++; if( µ>max_µ ) max_µµ // find maximum µ
E ] This mehod enabled us o comforably compue max µ for he values A he program runime on he (slower).6 GHz Penium M machine amouns o 3 ½ hours. As exhausive compuing represens only a crude firs ake on he problem no furher commens need o be made. 3.2 Recursive Selecion Represenaive Mehod Compuing E µ ] involves generaing permuaions of {µ }, for example µ r, w, w 2, r 2, r 3, w 3, w 4, w 5. These permuaions can be grouped by repeaing subsequences of w s and r s, for example µ r w 2 r 2 w 3. Each group represens permuaions of which each generaes he same value µ max. In oher words he selecing one represenaive from group speeds up he compuaion of E µ ] by a facor of We void developed Selec_k_ouof_n( a recursive algorihm in level, o generae in he loopsar selecion ) of elemens ou of n { E ] which is shown here: Implemenaion of his algorihm enables us o compue max µ for( in iloopsar; i<m_n-m_k+2+level; i++ ) for up { o he wihin a few hours on a.6 GHz Penium machine. m_selecionlevel]i; 3.3 Ieraive Selecion Represenaive Mehod SimulaeRandomWalk( m_selecion ); While he recursive else selecion mehod is shor and concise, i suffers from he overhead incurred by frequen Selec_k_ouof_n( he recursive funcion level+, calls and i+ exiss. ); For his reason we convered } our recursive selecion mehod ino an ieraive. } if( level m_finallevel ) Uilizing an ieraive mehod we here able o push he envelope o for compuaions lasing less han one hour. 3.4 Linear Opimizaions End Walk a There is no need o le he algorihm simulae all drawings of µ. A 3/2 of he walk has been compleed. A his poin here is no chance ha E µ ] can grow, simply because here are no enough red marbles lef o advance µ pas 0 ino he posiive range. This implemenaion opimizaion speeds up he compuaion linearly by /3. Skip E max µ ] 0 Selecions The selecion generaing algorihm produces is sequences in lexicographic order. This fac enables us rim down he selecions from he poin forward when sufficienly whie marbles have been drawn prevening µ o become posiive. More precisely, he firs red marble is drawn a /2 followed by a whie marble. This implemenaion opimizaion speeds up he compuaion by???.
4 Experimenal Resuls We have compued E µ ] as a funcion of he up o 30. As he graph in figure shows E µ ] appears o be converging o he asympoic boundary. Figure also shows he graph of E ν ]. Noe ha for he 2 graphs are geing closer and closer, appearing boh o be converging o. This converging behavior experimenally suppors plausibiliy argumen 2 ha in he limi E µ ] E ν ] max max Expeced Maximum Value of Mu.2 0.8 Emax mu] 0.6 0.4 Emax mu] Emax nu] 0.2 0 2 4 6 8 0 2 4 6 8 20 22 dela Table conains he numerical daa used o creae he graph of E µ ] in figure. I also conains he columns for he number of red and whie marbles. The resuls of E ν ] are augmened wih exac raios are heir respecive run imes. #whie #red Run ime E µ ] E µ ] 2 2 0.30 secs 5/27 0.333333 4 4 2 0.30 secs 524/729 0.466666 6 6 3 0.30 secs 607/9683 0.54769 8 8 4 0.30 secs 0.604040 0 0 5 0.30 secs 0.646354 2 2 6.26 secs 0.679595
4 4 7 3.64 secs 0.706570 6 6 8 2.66 secs 0.728995 8 8 9 69 secs 0.747990 (3 min) 20 20 0 342 secs 0.764323 (22 min) 22 22 930 secs (2 hrs 35 m) 0.77854 Table 2 presen daa for E ν ] #whie #red Run ime E ν ] E ν ] 2 2 0.30 secs /3 0.555556 4 4 2 0.30 secs 7/5 0.78793 6 6 3 0.30 secs 46/84 0.83748 8 8 4 0.30 secs 0.86869 0 0 5 0.30 secs 0.906453 2 2 6.26 secs 0.9339 4 4 7 3.64 secs 0.9499 6 6 8 2.66 secs 0.9667 8 8 9 69 secs (3 min) 0.9707 20 20 0 342 secs (22 min) References ] Rajeev Mowani, Randomized Algorihms, Cambridge Universiy, 995 2] Marek Chrobak, Elias Kousoupias, John Noga. More on Randomized On-line Algorihms for Caching. Theoreical Compuer Science, Denmark, volume 3, pages 997-2008, 2003 3] Yair Baral, Marek Chrobak, John Noga, Prabhakar Raghavan. More on Random Walks, Elecrical Neworks and Harmonic k-server Algorihms. Informaion Processing Leers, volume 84, pages 27-276, 2002 4] Feller. An Inroducion o Probabiliy Theroy and Is Applicaions. Wiley & Sons, Inc. New York, volume, pages 282 o 288, 950