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(BQ) Part 1 book Algorithms unplugged has contents: Numbers into English words, winning strategies for a matchstick game, scheduling of tournaments or sports leagues, eulerian circuits, shortest paths, marriage broker, the knapsack problem, simulated annealing,.... and other contents.
Part III Planning, Coordination and Simulation Overview Helmut Alt and Ră udiger Reischuk Freie Universită at Berlin, Berlin, Germany Universită at zu Lă ubeck, Lă ubeck, Germany Strategic thinking and planning are commonly regarded as typically human capabilities Ever since computer programs demonstrated that they can beat chess grand masters, however, one can see that some of these skills can be successfully managed by machines On the other hand, some games can be won with very simple strategies, one must have the right knowledge In a chapter in this part of the book we see this demonstrated impressively by the match game Nim In many games, it is important that we don’t allow the enemy to anticipate our moves A simple strategy – referred to in computer science jargon as deterministic – can, however, be predicted This can be avoided if you include random decisions – without this many games such as rock–paper–scissors would be quite boring Many algorithms can be improved or be speeded up in this way – these are called probabilistic or randomized algorithms Now we need to ask ourselves how a computer could toss a coin, given that we would expect only full precision? The chapter here on random numbers provides an answer A strategic and algorithmic approach makes sense even with everyday problems, and not just during games For example, if we wish to disseminate a message to a broad group of people through phone calls or to many computers via an electronic network, then we need a good plan in order to achieve this objective quickly and reliably We see this in the chapter on broadcasting In a further chapter we see a clever approach to determining the winner of an election Some tasks require careful long-term planning An example is the game schedule for the Bundesliga, the German soccer league, which requires us to consider various constraints Two chapters in this section deal with simulations, i.e., simulating natural processes using computers First we consider a problem from physics We see how to calculate the heat distribution in a metal rod or plate using so-called Gauss–Seidel iteration In the other chapter we consider a theme from biology We see how one can determine how closely two organisms are related to B Vă ocking et al (eds.), Algorithms Unplugged, DOI 10.1007/978-3-642-15328-0, c Springer-Verlag Berlin Heidelberg 2011 222 Helmut Alt and Ră udiger Reischuk each other from their genetic information (DNA); and we see from mutations, minimal changes in the genetic heritage, how far apart they are from each other or from a common ancestor The famous mathematician Leonhard Euler posed the Kă onigsberg Bridges Problem: Can you cross all seven bridges exactly once on a walk and then return to the starting point? This playful question – by the way, the answer is “No”! – has important applications, such as in route planning, covered in the chapter on Eulerian circuits In vehicle navigation we are now accustomed to a friendly voice that offers directions or tells us the distance to travel before the next turn For a long time natural speech was an unsolved problem for computers In this part of the book we see that even pronouncing long numbers involves considerable computational effort Finally, we consider a problem in computer graphics Draw a circle as round as possible on a screen, realized using a grid of individual pixels Strictly speaking we cannot draw a slanted or curved line, as we could with paper and pencil However, a detailed analysis of the problem leads to surprisingly easy and fast solution algorithms 22 Broadcasting – How Can I Quickly Disseminate Information? Christian Scheideler Universită at Paderborn, Paderborn, Germany In the Middle Ages, there was no mass media like TV or radio As most people were not able to write or read, information was mostly disseminated on a mouth-to-mouth basis, and since the travel speed of humans was quite restricted at those times, the spreading of information was mostly bounded by the speed of horses (though other means like pigeons and smoke or light signals were also used occasionally) Nowadays, the telephone and other media like the Internet allow anyone to spread information very quickly around the world Let us consider a specific example here Steffi has just been given the task to organize a party for her class, and this at a time when the school holidays have just started! Now, she has to try to reach all fellow students by phone or email Unfortunately, Steffi does not know their email addresses, but she has a list of all 121 students with their phone numbers which was recently given to every student (and that hopefully no one has thrown away yet!) Now, Steffi could try to 120 phone calls, which would consume a lot of time Hence, she thinks about an alternative approach to reach all students as quickly and cheaply as possible Strategy 1: Call everybody directly B Vă ocking et al (eds.), Algorithms Unplugged, DOI 10.1007/978-3-642-15328-0 22, c Springer-Verlag Berlin Heidelberg 2011 224 Christian Scheideler The first strategy that comes to her mind is the silent post game: she just calls the first person on the list and asks him or her to call the next one on the list, who will then call the next one, and so on, until everybody on the list has been reached Strategy 2: Silent post The advantage of this strategy is that every student only has to make one call However, since the calls have to be performed one after the other, a very long time can go by until all students have been reached In fact, if just 10% of the students not reach the next one on the list within the same day they were called, it takes at least 12 days until everybody has been informed Even worse: if someone does not bother to call the next one on the list, the whole system will break down! Thus, Steffi thinks about an alternative approach Since she is interested in computer science, she recalls a sorting method that has also been presented in Chap There, a master uses two helpers to cut a sorting problem into two smaller sorting problems, who themselves use two helpers each to cut their sorting problems into even smaller problems, and so on, until just one element is left Something similar to that should also work for the distribution of calls! For example, Steffi could divide the phone list into two halves and call the first person on each of the two halves Each of them will then be asked to cut their list into two further halves and call the first person on these halves This is continued until everybody has been called, i.e., we reach a level in which people are called who just have to take care of an empty list In this way, the students can be reached much quicker Strategy 3: Partitioning the phone list into sublists 22 Broadcasting 225 Strategy 4: Everybody at list position i calls positions 2i + and 2i + Indeed, Steffi determines that just seven rounds of calls are sufficient to reach all 120 fellow students This is much better than 120 rounds of calls! However, her strategy sounds very technical, so it’s questionable whether the other students can be made to adhere to the rules without errors Thus, she thinks about an alternative strategy Suppose that she calls the first two people on the list, Andi and Berthold, and asks Andi to call the students at positions and while Berthold is asked to call the students at positions and In general, the rule would be that everybody at position i in the list will call the students at positions 2i + and 2i + (if they exist) Then the information spreads at the same speed as in the previous strategy, but the calling rule sounds now much more natural and easy to understand Nevertheless, Steffi is not quite happy with her calling strategy What if one of her fellow students does not count right and calls a wrong pair of students on the list? Moreover, there can still be a couple of students who just forget or not bother about calling their pair on the list In this case, some students would not be informed, who would then be mad at Steffi! Therefore, Steffi thinks about a more robust strategy One possibility would be that everyone at list position i would call the four students at positions 2i + to 2i + In this case, all students (except for the first four on the list who will directly be called by Steffi) will be called by exactly two students in the ideal case Thus, as long as for each such pair at most one of the students is unreliable (by not being reachable or forgetting to make the call), all of the reliable students will still be informed Intuitively, this can be argued as follows: If one can select a caller for each student who works reliably, then everybody who is reliable has a reliable call chain from himself or herself back to Steffi (see also Fig 22.1) Steffi quickly realizes that this strategy can be made even more robust, so that she can be really sure to reach everybody who is reachable: If every student at position i calls the students at positions 2i + to 2i + 2r for some 226 Christian Scheideler Fig 22.1 Chain of reliable students for Alex, if the positions 2i + to 2i + are called fixed r, then every student (except for the first 2r ones who are directly called by Steffi) will be called by exactly r many students in the ideal case Hence, as long as at most r − of these are not calling, all reliable students will still be reached Now, we have come to a point where it would be helpful to conduct some experiments For a given number x (e.g., 10) of unreliable students, who are assumed to be randomly distributed over the list, we want to determine the minimum value of r for which the probability that all reliable students are reached is still above, say, 90% In order to determine this r, one can use the algorithm presented below This algorithm does not emulate the dissemination of information (that runs concurrently in reality) but just determines whether under the given communication rule all reliable students can be reached For this it suffices to run the for-loop in line till N/2 since students with larger list positions will not call any other student The algorithm is based on an array A that is defined as follows: • A: array [1 N ] of integers; A[i] counts, for a reliable student at position i, the number of calls that student would get from other reliable students N is the total number of students Fig 22.2 The r students that will call Alex in the ideal case for r = 22 Broadcasting 227 • For every reliable student, A[i] is initially set to • For all unreliable students at position i, A[i] will initially be set to −r (so that even after r calls there will not be a positive value in A[i]) Algorithm for r-fold information dissemination 10 11 12 13 14 15 16 17 18 19 20 procedure Broadcast (r) begin for j := to ∗ r // Steffi calls students to 2r A[j] := A[j] + endfor for i := to N/2 // Student i calls 2i + to 2i + 2r if A[i] > then // if call has been received for j := ∗ i + to ∗ i + ∗ r if j ≤ N then A[j] := A[j] + endif endfor endif endfor // Did it work? for i := to N if A[i] = then output “not everybody reached”, stop endif endfor output “everybody reached” end After all this thinking, Steffi has more and more fun in inventing new rules Next, she considers the more challenging case that every student has a different list of all the other students, so all of her prior strategies are not applicable any more Is there still a fast and robust strategy to reach all of the reliable students if, say, an arbitrary quarter of the students is unreliable? After some thinking, Steffi has the idea that she, like everybody else who is called the first time, just randomly picks r students on the list and calls them (see Strategy 5) If Steffi starts with this strategy, then she will certainly inform r students who have not already been called (if all of them are reachable) In the ideal case, all of them are reachable and reliable, so each of them will call r other students Hence, at best, r2 students will then be informed In reality, however, it can happen that a student is called more than once Since every student will become active only once (otherwise, the calls would never terminate!), this harms the dissemination of Steffi’s information Also, it can happen that unreliable students are called, which will further lower the dissemination of the information Nevertheless, one can verify through experiments that Steffi’s information will reach all reliable students with high probability if r is sufficiently large (but still reasonably small) In order to determine this r, one can use the algorithm below It is based on two arrays A and C that are defined as follows: 228 Christian Scheideler Strategy 5: Every student, including Steffi, calls r random students for r = Algorithm for random r-fold information dissemination 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 procedure RandomBroadcast (r) begin for j := to r // calls from Steffi if A[C[0][j]] = then A[C[0][j]] := endif endfor continue := // indicator for newly called students while continue = continue := for i := to N // search for newly called students if A[i] = then continue := 1; A[i] := for j := to r if A[C[i][j]] = then A[C[i][j]] := endif endfor endif endfor endwhile // Did it work? for i := to N if A[i] = then output “not everybody reached”, stop endif endfor output “everybody reached” end • A: array [1 N ] of integers; initially A[i] = −1 if student i is unreliable and otherwise A[i] = N is the number of students 22 Broadcasting 229 • If a reliable student i is called for the first time, then A[i] is set to 1, and once he or she has finished all calls, A[i] is set to • C: array [0 N ][1 r] of integers; C[i][j] gives the number of the jth student who is called by student i (Steffi counts here as student 0) C is chosen at random Of course, one can think of many other strategies to disseminate information in a group of people, and everybody is encouraged to so Which strategy would you have chosen if you were Steffi? References http://en.wikipedia.org/wiki/Broadcasting (computing) This Wikipedia article gives an introduction to broadcasting and to standard strategies used in this area C Diot, W Dabbous, and J Crowcroft: Multipoint Communication: A Survey of Protocols, Functions and Mechanisms IEEE Journal on Selected Areas in Communications 15(3), pp 277–290, 1997 K Obraczka: Multicast Transport Protocols: A Survey and Taxonomy IEEE Communications Magazine 36(1), pp 94–102, 1998 M Hosseini, D.T Ahmed, S Shirmohammadi, and N.D Georganas: A Survey of Application-Layer Multicast Protocols IEEE Communications Surveys & Tutorials 9(3), pp 58–74, 2007 These articles are recommended for an introduction to the scientific literature on broadcasting R Karp, S Shenker, C Schindelhauer, and B Văocking: Randomized Rumor Spreading In: IEEE Symposium on Foundations of Computer Science (FOCS), pp 565–574, 2000 This article contains advanced broadcasting methods that are more effective but also more complex than the strategies presented here It is recommended to everyone interested in learning about the newest results in this field and who is not afraid of mathematical formulas ... positions 2i + to 2i + 2r for some 22 6 Christian Scheideler Fig 22 .1 Chain of reliable students for Alex, if the positions 2i + to 2i + are called fixed r, then every student (except for the first 2r... Call everybody directly B Vă ocking et al (eds.), Algorithms Unplugged, DOI 10.1007/978-3-6 42- 15 328 -0 22 , c Springer-Verlag Berlin Heidelberg 20 11 22 4 Christian Scheideler The first strategy that... reached much quicker Strategy 3: Partitioning the phone list into sublists 22 Broadcasting 22 5 Strategy 4: Everybody at list position i calls positions 2i + and 2i + Indeed, Steffi determines that