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Randomized Algorithms Randomized Algorithms Prabhakar Raghavan IBM Almaden Research Center San Jose CA Typ eset byFoilT X E Deterministic Algorithms INPUT ALGORITHM OUTPUT Goal To prove that the algorithm solves the problem correctly always and quickly typically the number of steps should be p olynomial in the size of the input Typ eset byFoilT X E Randomized Algorithms INPUT ALGORITHM OUTPUT RANDOM NUMBERS In addition to input algorithm takes a source of random numbers and makes random choices during execution Behavior can vary even on a xed input Typ eset byFoilT X E Randomized Algorithms INPUT ALGORITHM OUTPUT RANDOM NUMBERS Design algorithm analysis to show that this b ehavior is likely to be good on every input The likeliho o d is over the random numbers only Typ eset byFoilT X E Not to be confused with the Probabilistic Analysis of Algorithms RANDOM INPUT ALGORITHM OUTPUT DISTRIBUTION Here the input is assumed to be from a probability distribution Show that the algorithm works for most inputs Typ eset byFoilT X E Monte Carlo and Las Vegas A Monte Carlo algorithm runs for a xed number of steps and pro duces an answer that is correct with probability A Las Vegas algorithm always pro duces the correct answer its running time is a random variable whose exp ectation is bounded say by a polynomial Typ eset byFoilT X E Monte Carlo and Las Vegas These probabilitiesexp ectations are only over the random choices made by the algorithm indep endent of the input Thus indep endent rep etitions of Monte Carlo algorithms drive down the failure probability exp onentially Typ eset byFoilT X E Advantages of randomized algorithms Simplicity Performance For many problems a randomized algorithm is the simplest the fastest or b oth Typ eset byFoilT X E Scop e Numb ertheoretic algorithms Primality testing Monte Carlo Data structures Sorting order statistics searching computational geometry Algebraic identities Polynomial and matrix identity verication Interactive pro of systems Typ eset byFoilT X E Mathematical programming Faster algorithms for linear programming Rounding linear program solutions to integer program solutions Graph algorithms Minimum spanning trees shortest paths minimum cuts Counting and enumeration Matrix p ermanent Counting combinatorial structures Typ eset byFoilT X E Parallel and distributed computing Deadlo ck avoidance distributed consensus Probabilistic existence pro ofs Show that a combinatorial object arises with nonzero probability among objects drawn from a suitable probability space Typ eset byFoilT X E Derandomization First devise a randomized algorithm then argue that it can be derandomized to yield a deterministic algorithm Typ eset byFoilT X E Gametree evaluation 9 MAX 93 MIN MIN 79 317 MAX MAX MAX MAX 1 7 9 5 3 2 17 12 Tree with alternating MAXMIN no des Each leaf has a real value Goal To evaluate the ro ot Typ eset byFoilT X E Gametree evaluation 9 MAX 93 MIN MIN 79 317 MAX MAX MAX MAX 1 7 9 5 3 2 17 12 Cost The number of leaves insp ected Algorithm Computes leaf insp ection sequence Typ eset byFoilT X E Simple sp ecial case Uniform binary tree height h h with n leaves All values are b o olean so MAXOR and MINAND Exercise Every deterministic algorithm can be forced to read n leaves Typ eset byFoilT X E Randomized algorithm AND 10 OR OR 01 00 AND AND AND AND 1011 1010 Evaluate recursively a random child of the current no de If this do es not determine the value of the current no de evaluate the other child Typ eset byFoilT X E Analysis of tree evaluation ANDOR trees of depth k k levels of AND and k levels of OR on any path will prove by induction on k Exp ected cost for an OR subtree evaluating to k k k Typ eset byFoilT X E Analysis of tree evaluation For an OR subtree evaluating to we must evaluate b oth of its subtrees incurring cost k Now consider the ro ot AND a If it is both OR subtrees must evaluate to exp ected k cost paid is twice ab ove b If it is at least one OR subtree evaluates to Typ eset byFoilT X E Game tree analysis Exp ected cost for ro ot AND evaluating to k k k k Number of leaves in tree Thus exp ected cost of randomized algorithm n Typ eset byFoilT X E Lower bounds and the minimax principle Basic Idea For a lower bound on the performance of all randomized algorithms for a problem derive instead a lower bound for any deterministic algorithm for the problem when the inputs are drawn from a probability distribution of your choice Typ eset byFoilT X E Let I denote the set of instances of the problem and A the set of deterministic algorithms for the problem Any randomized algorithm can be viewed as a probability distribution on the algorithms in A Typ eset byFoilT X E Minimax Principle For all distributions p over I and q over A min EC I A max E C I A p q AA I I Typ eset byFoilT X E Lower b ound for game tree evaluation We will sp ecify a probability distribution on instances values for the leaves and lower bound the exp ected running timeof any deterministic algorithm Typ eset byFoilT X E NOR trees instead Exercise Showthat a balanced ANDOR tree of even depth is equivalent to a tree of the same depth all of whose no des are NOR no des We will show the lower b ound for a NOR tree with n leaves Typ eset byFoilT X E The input distribution p Let p Claim The value of any no de is with probability p indep endent of all other no des at the same level Claim Any deterministic algorithm mayas well determine the value of one subtree of a no de before insp ecting any leaf of its sibling subtree Typ eset byFoilT X E The Analysis Let W h be the exp ected number of leaves it insp ects in determining the value of a no de at distance h from the leaves Clearly W h W hpW h Letting h log n this gives a lower b ound of n Typ eset byFoilT X E Exercise Why is this lower bound weak Typ eset byFoilT X E The SAT Problem Given a set of b o olean clauses in CNF each containing two literals nd a satisfying assignment if one exists x x x x x x Start with any tentative assignmen t If there is an unsatised clause pick one of its two literals at random and ip it Typ eset byFoilT X E If no solution found in n steps declare none exists Monte Carlo If a solution exists will nd it with probability If not will always declare none exists Typ eset byFoilT X E Random Walk Analysis What is the probability of missing a satisfying assignment in n steps Fix any particular satisfying assignment A Consider the number of literals on which the algorithms tentative assignment agrees with A Typ eset byFoilT X E This is a random walk on the integers that increases with probability at least at each step Exp ected time to nd an assignment n Markovs inequality probability of missing an assignment in n steps is Typ eset byFoilT X E Binary planar partitions L2 L1 L1 s1 L2 L3 s 3 sss s s 1 3 3 2 2 L3 Given n nonintersecting line segments in the plane build a small linear decision tree that has pieces of at most one segment in each cell Typ eset byFoilT X E Autopartitions Cho ose the input segments in random order Extend the chosen segment if necessary to cut any cells where it is not alone The exp ected size of the resulting tree is n nH n Typ eset byFoilT X E Analysis of autopartition size v 1 u v 2 Dene l u to be the line containing segment u u a v denotes l u cuts v in the constructed tree indexu v is i if l u intersects i other segments before hitting v Typ eset byFoilT X E Autopartitions Pr u a v indexu v Thus the exp ected size of the tree constructed is X X n Pr u a v u v u X X n indexu v u v u Typ eset byFoilT X E For any u i indexu v i for at most two v thus the ab ove sum is n nH n It follows that for any n segments in the plane there always exists an autopartition of size O n log n Typ eset byFoilT X E Matrix pro duct verication Supp ose A B and C are n n matrices with entries from a nite eld F We wish to verify whether AB C Multiplying out takes ab out n steps this can be cut to O n steps using sophisticated algorithms Typ eset byFoilT X E Simple randomized algorithm Pick an nvector x with entries randomly drawn from F Compute z ABx If z Cx output AB C else output AB C Takes O n steps Typ eset byFoilT X E Simple randomized algorithm If AB C will always output AB C If AB C will output AB C with probability at most jF j Typ eset byFoilT X E Sources RM Karp An intro duction to randomized algorithms Discrete Applied Mathematics R Motwani and P Raghavan Randomized Algorithms Cambridge University Press D J A Welsh Randomised algorithms Discrete Applied Mathematics Typ eset byFoilT X E.
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