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It covers randomized geometric sampling for approximation guarantees, hitting/ruin probabilities in a fair ±1 euro game, and stationary distributions via detailed balance and time reversal. Additional tasks apply reverse-sequence Markov reasoning and develop probabilistic method techniques, including coloring arguments and intermediate “sample and modify” constructions.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/randomized-algorithms-and-probabilistic-analysis-of-algorithms-exercise-sheet-45-winter-202223/266552/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/randomized-algorithms-and-probabilistic-analysis-of-algorithms-exercise-sheet-45-winter-202223/266552.png","ImageObject",300,407,{"name":92,"@type":93},"Evangeline","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-20","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What topics are covered in Exercise Sheet 4/5?","Question",{"text":112,"@type":113},"The sheet covers randomized approximation via sampling, gambling/ruin probabilities, cover time bounds for complete graphs, and Markov chain results including stationary distributions and time reversal. It also introduces probabilistic method techniques such as averaging and sample-and-modify.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the sheet approach randomized approximation in the first exercise?",{"text":117,"@type":113},"It samples a random point from a specified square region and uses the probability that the sample lies inside the unit circle to derive the required approximation guarantee.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the sheet require to use probabilistic method arguments?",{"text":121,"@type":113},"It asks to justify answers and to apply probabilistic reasoning such as bounding event probabilities (e.g., clique colorings), using averaging/expectations, and sometimes constructing an intermediate random structure before modifying it to achieve the desired properties.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},266552,1789408855,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":81,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":39},13056703019662,"https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188","Philip Wellnitz, Barış Can Esmer Winter 2022/23  \nRandomized Algorithms and Probabilistic Analysis of Algorithms, Exercise Sheet 4/5  \n[https://www.mpi-inf.mpg.de/departments/algorithms-complexity/teaching/winter22/random](https://www.mpi-inf.mpg.de/departments/algorithms-complexity/teaching/winter22/random)  \nTotal Points: 100 + 100 Due: Wednesday, January 11, 2023  \nYou are allowed to collaborate on the exercise sheets. Justify your answers . Cite all external sources that you use (books, websites, research papers, etc.) . You need to collect at least 50% of all points on exercise sheets to be admitted to the exam.  \nPlease hand in your solutions before the lecture on the day of the deadline.  \n\n| .1 points For 0 \u003C \"; 􀀎 \u003C 1, demonstrate a randomized algorithm that gives an (\"; 􀀎)-approximation of 􀀙 .\u003Cbr>(Hint: Sample a random point from the square (􀀀1; 􀀀1)(1 ; 􀀀1)(1 ; 1)(􀀀1; 1) and consider the event that said sample lies inside of the unit circle.)\u003Cbr> Exercise 4 2  10 + 20 |  |  |\n| --- | --- | --- |\n| .  points \u003Cbr>Your favorite game center offers a new game where in each round, you can equally likely win 1€ or loose 1€ . You start with an initial budget of 200€ and you may play as long as your balance is positive (that is, you can play as long as you are not bankrupt) .\u003Cbr>a) What is the probability that you win 200€ before going bankrupt?\u003Cbr>b) For positive ` 1 ; `2 , what is the probability that you own 200€ + `1 € before you own 200€ 􀀀 `2 €?\u003Cbr> Exercise 4 3  10 |  |  |\n| Show that the\u003Cbr>Exercise | .\u003Cbr>cover time of the complete graph on n vertices is 􀀂(n lnn) .\u003Cbr>4 4 | points\u003Cbr>10 |\n| . points\u003Cbr>Consider a finite, irreducible, and aperiodic Markov chain with transition matrix P. Show that if there are nonnegative numbers  = (􀀙1 ; : : : ; 􀀙n) that sum to 1 and satisfy for any i; j\u003Cbr>􀀙iPi;j = 􀀙jPj;i ; (1)\u003Cbr>then  is the stationary distribution corresponding to P.\u003Cbr>Note: A Markov chain that satisfies (1) is also called time-reversible.\u003Cbr> Exercise 4 5   10 + 15 + 5  |  |  |\n\n Exercise 4  20  \n.  \npoints  \nConsider a finite Markov chain with n states, transition matrix P, and stationary distribution  . Imagine running the chain for r steps, yielding the sequence X0 ; X1 ; : : : ; Xr and consider the reverse sequence Xr ; : : : ; X0 .  \na) Argue that the reverse sequence is Markovian.  \nb) Argue that for the reverse sequence, the transition probabilities Qi;j are given by  \n􀀙jPj;i Qi;j = :  \n􀀙i  \nc) Argue that for a time-reversible Markov chain, we have Pi;j = Qi;j for all states i; j.  \nIn this exercise sheet, we develop several techniques to prove the existence of objects. Collectively, these methods are also called the probabilistic method.  \n~~  ~~Exercise~~ ~~5.1~~   ~~5~~ ~~+~~ ~~5~~ ~~+~~ ~~5~~ ~~+~~ ~~5~~ ~~points~~  ~~  \nIn this exercise, we show that for integers n and k that satisfy 􀀀 nk􀀁2􀀀􀀀k2􀀁+1 \u003C 1, we can color the edges of the complete graph Kn with two colors such that no complete subgraph Kk is monochromatic.  \na) Define a sample space 􀀊 of all valid colorings of Kn with two colors. What is the size of said sample space?  \nb) Compute the probability that in a coloring sampled uniformly from 􀀊, a specific k-clique Kk is monochromatic.  \n(Hint: Think of uniformly sampling a coloring from 􀀊 as assigning each edge one of the two colors uniformly at random.)  \nc) Assuming 􀀀 nk􀀁2􀀀􀀀k2􀀁+1 \u003C 1, conclude that the probability that all k-cliques in Kn are monochromatic is less than 1 . Further conclude that hence, there is a coloring of Kn that ensures that no k-clique subgraph is monochromatic.  \nd) Discuss how to (efficiently) construct such a coloring of Kn.  \n~~  ~~Exercise~~ ~~5.2~~   ~~10~~ ~~+~~ ~~10~~ ~~points~~   ~~In this exercise, we see an averaging argument that is sometimes easier to apply.  \na) Prove that for a probability space S and a random variable X defined on S with E [ X ] = 􀀖, we have Pr [ X 􀀕 􀀖 ] > 0 and Pr [ X 􀀔 􀀖 ] > 0.  \nb) Prove that in any undirected graph G = (V; E) with m edge","cbCaibvMoioWnZDt","https://ap.wps.com/l/cbCaibvMoioWnZDt","pdf",206591,"English","# Randomized Algorithms and Probabilistic Analysis of Algorithms\n## Exercise Sheet 4/5\n## Approximation by random sampling\n## Gambling and ruin probabilities\n## Cover time of the complete graph\n## Stationary distributions and time reversibility\n## Time-reversal of Markov chains\n## Probabilistic method techniques","[{\"question\":\"What topics are covered in Exercise Sheet 4/5?\",\"answer\":\"The sheet covers randomized approximation via sampling, gambling/ruin probabilities, cover time bounds for complete graphs, and Markov chain results including stationary distributions and time reversal. It also introduces probabilistic method techniques such as averaging and sample-and-modify.\"},{\"question\":\"How does the sheet approach randomized approximation in the first exercise?\",\"answer\":\"It samples a random point from a specified square region and uses the probability that the sample lies inside the unit circle to derive the required approximation guarantee.\"},{\"question\":\"What does the sheet require to use probabilistic method arguments?\",\"answer\":\"It asks to justify answers and to apply probabilistic reasoning such as bounding event probabilities (e.g., clique colorings), using averaging/expectations, and sometimes constructing an intermediate random structure before modifying it to achieve the desired properties.\"}]","Randomized Algorithms and Probabilistic Analysis of Algorithms - Exercise Sheet 4/5 - Winter 2022/23 | PDF"]