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Randomized

Randomized. Algorithms. Fundame n tals of Algorit h mics. Gilles Brassard and Paul. Bratley. Chapter 1 0. Rando m iized Algorith m s. SC440/CSC401/CPE401. 1. Introduction (‘Traditional’ algorithms). Traditional algorithms have the following properties:. • •. They are always correct

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Randomized

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  1. Randomized Algorithms FundamentalsofAlgorithmics. GillesBrassardandPaul Bratley. Chapter10. Randomiized Algorithms SC440/CSC401/CPE401 1

  2. Introduction (‘Traditional’ algorithms) Traditional algorithms have the following properties: • • They are always correct They are deterministic --- there may be more than one correct output, but the same instance of a problem always produces the same output They are always precise --- the answer is not given as a range • • Each of them operates at the same efficiency for the same instance of a problem →2 24 → Prime factor Randomiized Algorithms SC440/CSC401/CPE401 2

  3. Introduction (‘Randomized’ algorithms) Randomized (probabilistic) algorithms can be • Nondeterministic --- they can make random but correct decisions : the same algorithm may behave differently when it is applied twice to the same instance Not very is given, of a problem. precise sometimes --- usually the more the better precision can be obtained • time • operating at different efficiencies for the runs. same instance of a problem in different 2 or 3 → 24 → Prime factor Randomiized Algorithms SC440/CSC401/CPE401 3

  4. • Incorrect sometimes: hope we have a high, known probability of being correct. Nonterminating sometimes : do not produce an answer at all (hope fore a low probability for that) • because • A randonmized algorithm is one that makes random choices during the execution Randomiized Algorithms SC440/CSC401/CPE401 4

  5. Why do we need randomized (probabilistic algorithms) • When an algorithm is confronted by a choice, it is sometimes preferable to choose a course of action at random, rather than spending time to work out which alternative is the best. Sometimes we do not have a better method than making random choices • Randomiized Algorithms SC440/CSC401/CPE401 5

  6. Classes of randomized (probabilistic) algorithms 1. Numerical probabilistic algorithms --- give an approximation to the correct answer. Monte Carlo Algorithms --- always give an answer, but there is a probability of being completely wrong. 2. 3. Las Vegas Algorithms --- sometimes fail to give an answer, but if an answer is given, it is correct. These are what we cover in this chapter. Randomiized Algorithms SC440/CSC401/CPE401 6

  7. Expected versus average time • The average time of a deterministic algorithm refers to the average time taken by the algorithm when each possible instance of a given size is considered equally likely. E.g sorting 3 integers T123 T132 T213 T231 T312 T321 1, 1, 2, 2, 3, 3, 2, 3, 1, 3, 1, 2, 3 2 3 1 2 1 A sorting algorithm + 6 Randomiized Algorithms SC440/CSC401/CPE401 7

  8. • The expected time of a probabilistic algorithm is defined on each individual instance. It is the mean time over that and it would take over again. to solve the same instance T1 T2 T3 T4 T5 T6 1, 2, 3 A sorting algorithm Compute the mean Randomiized Algorithms SC440/CSC401/CPE401 8

  9. • The worst case expected time of a probabilistic algorithm refers to the expected time taken by the worst possible instance of a given size, not the time incurred if the worst possible probabilistic choices are unfortunately taken. Randomiized Algorithms SC440/CSC401/CPE401 9

  10. Numerical probabilistic algorithms • For certain real life problems, computation of an exact solution is not possible, maybe – because of uncertainties in the experimental data to be used Because a digital computer cannot represent an irrational number exactly Because a precise answer will take too long to compute – – • A numerical probabilistic algorithm will give an approximate answer (a confidence interval can be provided). The precision of the answer increases when more time is given to the algorithm to work on • Randomiized Algorithms SC440/CSC401/CPE401 10

  11. Buffon’s needle 18th In the century, Georges de Buffon, proved that if you throw a needle at random (in a random position and at a random angle, with uniform distribution) on a floor made of planks of constant width, if the needle is exactly half as long as the planks in the floor are wide and if the width of the cracks between the planks is zero, the probability that the needle will fall across 1/π. a crack is Randomiized Algorithms SC440/CSC401/CPE401 11

  12. Monte Carlo Algorithms • There are problems for which no efficient algorithm is known, and approximate solutions do not make sense. A Monte Carlo algorithm always gives an answer but occasionally makes a mistake. But it finds a correct solution with high probability whatever the instance is processed, i.e. there is no instance on which the probability of error is high. However, no warning is usually given when the algorithm gives a wrong solution. • • • Example: to find the mode (most common element) in a bag of values. Pick an element at random: the probability that it is the mode is high, but there is a Randomiized Algorithms SC440/CSC401/CPE401 17

  13. non-zero probability that it is completely wrong. Making several random choices and combining the results improves the probability of correctness. Randomiized Algorithms SC440/CSC401/CPE401 18

  14. Primality testing The problem is to decide whether a given odd integer is prime or composite. The problem is important for public-key cryptography. • • A Naive method Given an input numberN, we check whether it is divisible by any integer greater than 1 and less than or equal to the square root of N. If the answer is NO, then N is a prime, otherwise not. This relies on the not so trivial fact that if there are any factors other than 1 orN, they cannot all be greater than this square root. Composite Divisible by N → prime Randomiized Algorithms SC440/CSC401/CPE401 19

  15. Probabilistic tests Most popular primality tests are probabilistic tests. In a certain sense, those tests are not really primality • tests -- they do not determine with certainty whether number is prime or not. The basic idea is as follows: a • 1. 2. Randomly pick a number x called a witness. Check some formula involving x and the given number N. If the formula is no fit, then N is a composite number and the test stops. Repeat step 1 unless the required certainty is achieved. 3. After several iterations, if N is not composite number, then it can be prime. found to be a declared probably Randomiized Algorithms SC440/CSC401/CPE401 20

  16. • The simplest probabilistic primality test is the Fermat primality test. It is sometimes used if a rapid screening of numbers • is needed, for instance in the key generation phase of the RSA public key cryptographical algorithm. Fermat’s little Theorem If nis a prime, then an-1 mod n = 1 1≤ a≤n For any integer a such that - 1. E.g. n n n = = = 7, a = 8, a = 1 or 2 or 3 or … 6 27 mod 8 = 0 ≠ 1 2, we have 29 mod 10 = 2 ≠1 10, a = 2, we have Randomiized Algorithms SC440/CSC401/CPE401 21

  17. The contrapositive: If there is a value a such that 1≤ a≤n – 1and an-1 mod n ≠1, then n is not a prime. Fermat (n) a = uniform(1, n-1); (an-1 mod n≠1) then return “n is composite” else return “n is prime” if an-1 • However, there are composite numbers such that 414 mod n = 1. For example, n= 15, mod 15 = 1. • The Miller-Rabin test is a more sophisticated variant of the Fermat test but with a lower error rate; it is often the method of choice. Randomiized Algorithms SC440/CSC401/CPE401 22

  18. • Suppose n > 4 is an odd integer which we want to test for primality. Choose a random integera with2 ≤a≤n- 2. Finds and t such that t is odd andn- 1 = 2st . • • • 2rt If (atmod n = 1) or (a mod n = n- 1) for at least one r = 0, ..., s-1, then nis declared "prime"; If not, thennis definitely composite. • MillerRabin(n, k) for (i = 1 to k do) a = uniform(2, n – 2); if (Btest(a, n) == false) then return return “prime” “composite” Randomiized Algorithms SC440/CSC401/CPE401 23

  19. Btest(a, n) s = 0; t = n – 1; Repeat s = s + 1; t = t/2; Until t is odd; at x = mod n; If (x == 1 or x == n – 1) then return For (i = 1 to s – 1) true x2 x = mod n; if (x == n – 1) then return true Return false The Miller-Rabin algorithm always answer when n > 4 is a prime. • returns the correct • When n > 4 is an odd composite, each call on Btest has probability at most ¼ of making a mistake. Randomiized Algorithms SC440/CSC401/CPE401 24

  20. Las Vegas Algorithms • Las Vegas algorithms make probabilistic choices to help guide them more quickly to a correct solution. • Unlike Monte Carlo algorithms, they never return wrong answer. Two main categories of Las Vegas algorithms. a • First category • The first kind is often used when a known deterministic algorithm to solve a problem runs much faster on the average than in the worst case, e.g. quicksort. [See previous lecture] Randomiized Algorithms SC440/CSC401/CPE401 26

  21. Second category • The second kind of Las Vegas algorithms now and then make choices that bring the problem solving process to a dead end. For example, Eight-queen problem 1st • When we use backtracking, we got the solution after building 114 nodes of the 2057 nodes in the state-space tree. • A greedy Las Vegas algorithm places the queens randomly on successive rows, taking care not to put a queen in an illegal position. Being a greedy algorithm, we do not backtrack. We either succeed if we manage to place all the queens, or we run into a dead end --- failure. • • Randomiized Algorithms SC440/CSC401/CPE401 33

  22. Q Q Q Q Q Q Q Using a computer, it is calculated that the probability of success, p, is approximately 0.1293. Randomiized Algorithms SC440/CSC401/CPE401 34

  23. After failing, try, try again • If we fail to place all the queens in one attempt, we restart from the beginning. • The expected number of attempts ∞ until success is ∑i*P[X=i] i=1 E[X] = = ∞ ∑i* (1− p)i-1p i=1 1 p *1−p ∞ p ∑i*(1− p)i = = = 2 1−p p 1−p p i=1 So for the randomized 8-queen, attempts to reach success. we expect 8 Randomiized Algorithms SC440/CSC401/CPE401 35

  24. Finding Hamiltonian paths • Given a graph, is there a path, any path, that passes through all the vertices of the graph exactly once. Such a path is called Hamiltonian path. There are n vertices in graph G. Try to build the path vertex by vertex. • • • Start from an arbitrary vertex, v0in G and select a neighbour of v0to get a path v0v1 randomly . • Continue to randomly choose the next vertex stop when all vertices are chosen and Randomiized Algorithms SC440/CSC401/CPE401 36

  25. • choose the next vertex randomly from the neighbours of the current vertex which are not already on the path Randomiized Algorithms SC440/CSC401/CPE401 37

  26. • If at a certain vi, all its neighbours are already on the path, we pick one neighbour,vj, change the path from v0v1,…,vjvj+1,…, tov0v1,…,vjvivi-1,…,vj+1and vi-1vi continue to choose the next vertex for the path v5 v0 The path built: v0v1v2v3v4 v5 v3 v => vv vvv v (v = v) 4 0 1 2 5 4 3 j 2 v1 v2 What is the new path if vj= v0? Randomiized Algorithms SC440/CSC401/CPE401 38

  27. Randomized_hamiltonian_path(G) v0= Random vertex in G; i = 0; { do {N = if N N(vi) – {v0, …, is not empty { i = i + 1; vi-1} vi= Random vertex in N; } ≤j < i - 1) if (there is vjis in N(vi) for 0 { } else (v0, …, vi) = (v0, …, return no_solution; vjvivi-1, …, vj+1); else } while (i < |V| return (v0, …, Randomiized Algorithms - 1); vi); } SC440/CSC401/CPE401 39

  28. • If randomized_hamiltonian_path returns true, a Hamiltonian path for graph G has indeed been found. It is possible that the algorithm runs into a dead end and fails to find a path, although such a path exists. The algorithm is good at finding a Hamiltonian path in large graphs. The algorithm is almost always successful for • • • graphs of sufficiently large minimum degree Randomiized Algorithms SC440/CSC401/CPE401 40

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