Cycle detection - Wikipedia. In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any functionf that maps a finite set. S to itself, and any initial value x. S, the sequence of iterated function valuesx. Once this happens, the sequence must continue periodically, by repeating the same sequence of values from xi to xj . Cycle detection is the problem of finding i and j, given f and x. Several algorithms for finding cycles quickly and with little memory are known. Floyd's tortoise and the hare algorithm moves two pointers at different speeds through the sequence of values until they both point to equal values. Alternatively, Brent's algorithm is based on the idea of exponential search. Disney's The Tortoise and the Hare was released on the Disney Treasures Collection of Silly Symphonies Vol. Oceny, recenzje, obsada, dyskusje wiadomo Watch full length The Tortoise and the Hare Movie for Free Online. The Tortoise and the Hare (1967) is a Black-and-white and Short Film feature film. Both Floyd's and Brent's algorithms use only a constant number of memory cells, and take a number of function evaluations that is proportional to the distance from the start of the sequence to the first repetition. Several other algorithms trade off larger amounts of memory for fewer function evaluations. The applications of cycle detection include testing the quality of pseudorandom number generators and cryptographic hash functions, computational number theory algorithms, detection of infinite loops in computer programs and periodic configurations in cellular automata, and the automated shape analysis of linked list data structures. Example. If one starts from x. The cycle in this value sequence is 6, 3, 1. Definitions. For any i > 0, let xi = f(xi . The cycle detection problem is the task of finding . The set of vertices reachable from starting vertex x. Greek letter rho (. Rather, a cycle detection algorithm may be given access either to the sequence of values xi, or to a subroutine for calculating f. Typically, also, the space complexity of an algorithm for the cycle detection problem is of importance: we wish to solve the problem while using an amount of memory significantly smaller than it would take to store the entire sequence. In some applications, and in particular in Pollard's rho algorithm for integer factorization, the algorithm has much more limited access to S and to f. In Pollard's rho algorithm, for instance, S is the set of integers modulo an unknown prime factor of the number to be factorized, so even the size of S is unknown to the algorithm. To allow cycle detection algorithms to be used with such limited knowledge, they may be designed based on the following capabilities. Initially, the algorithm is assumed to have in its memory an object representing a pointer to the starting value x. At any step, it may perform one of three actions: it may copy any pointer it has to another object in memory, it may apply f and replace any of its pointers by a pointer to the next object in the sequence, or it may apply a subroutine for determining whether two of its pointers represent equal values in the sequence. The equality test action may involve some nontrivial computation: for instance, in Pollard's rho algorithm, it is implemented by testing whether the difference between two stored values has a nontrivial greatest common divisor with the number to be factored. However, the space complexity of this algorithm is proportional to . Additionally, to implement this method as a pointer algorithm would require applying the equality test to each pair of values, resulting in quadratic time overall. Thus, research in this area has concentrated on two goals: using less space than this naive algorithm, and finding pointer algorithms that use fewer equality tests. Tortoise and hare. It is also called the . Floyd, who was credited with its invention by Donald Knuth. PRIMAL ZODIAC SIGN OF TORTOISE. Like the famous children’s fable of the Tortoise and the Hare tells us. Your Primal Zodiac sign represents your animal. THE HARE AND THE TORTOISE J.P. Chapter 5 TORTOISE BECOMES HARE (1964-1969). Finally in December 1967 the NRX-A6 reactor ran for one hour at full power, twice the time achieved before. In fact, Knuth's statement (in 1. Floyd, without citation, is the first known appearance in print, and it thus may be a folk theorem, not attributable to a single individual. Thus, the algorithm only needs to check for repeated values of this special form, one twice as far from the start of the sequence as the other, to find a period . Finally, once the value of . At each step of the algorithm, it increases i by one, moving the tortoise one step forward and the hare two steps forward in the sequence, and then compares the sequence values at these two pointers. The smallest value of i > 0 for which the tortoise and hare point to equal values is the desired value . So hare moving in circle one step at a time, # and tortoise (reset to x. Because the # distance between them is constant at 2. The algorithm uses O(. Brent described an alternative cycle detection algorithm that, like the tortoise and hare algorithm, requires only two pointers into the sequence. For i = 0, 1, 2, .., the algorithm compares x. It has two advantages compared to the tortoise and hare algorithm: it finds the correct length . It is not difficult to show that the number of function evaluations can never be higher than for Floyd's algorithm. Brent claims that, on average, his cycle finding algorithm runs around 3. Floyd's and that it speeds up the Pollard rho algorithm by around 2. He also performs an average case analysis for a randomized version of the algorithm in which the sequence of indices traced by the slower of the two pointers is not the powers of two themselves, but rather a randomized multiple of the powers of two. Although his main intended application was in integer factorization algorithms, Brent also discusses applications in testing pseudorandom number generators. In general these methods store several previously- computed sequence values, and test whether each new value equals one of the previously- computed values. In order to do so quickly, they typically use a hash table or similar data structure for storing the previously- computed values, and therefore are not pointer algorithms: in particular, they usually cannot be applied to Pollard's rho algorithm. Where these methods differ is in how they determine which values to store. By choosing R to be a number close to one, and storing the sequence values at indices that are near a sequence of consecutive powers of R, a cycle detection algorithm can use a number of function evaluations that is within an arbitrarily small factor of the optimum . The technique involves maintaining a numerical parameter d, storing in a table only those positions in the sequence that are multiples of d, and clearing the table and doubling d whenever too many values have been stored. Several authors have described distinguished point methods that store function values in a table based on a criterion involving the values, rather than (as in the method of Sedgewick et al.) based on their positions. For instance, values equal to zero modulo some value d might be stored. Woodruff with the suggestion of storing a random sample of previously seen values, making an appropriate random choice at each step so that the sample remains random. Nivasch. An item is stored in the memory table, with this technique, when no later item has a smaller value. As Nivasch shows, the items with this technique can be maintained using a stack data structure, and each successive sequence value need be compared only to the top of the stack. The algorithm terminates when the repeated sequence element with smallest value is found. Running the same algorithm with multiple stacks, using random permutations of the values to reorder the values within each stack, allows a time. However, even the version of this algorithm with a single stack is not a pointer algorithm, due to the comparisons needed to determine which of two values is smaller. Any cycle detection algorithm that stores at most M values from the input sequence must perform at least (. This is the application cited by Knuth in describing Floyd's method. For more complex generators, the sequence of values in which the cycle is to be found may not represent the output of the generator, but rather its internal state. Several number- theoretic algorithms are based on cycle detection, including Pollard's rho algorithm for integer factorization. For instance, Quisquater and Delescaille. The technique may also be used to find a collision in a cryptographic hash function. If a node in the list incorrectly points to an earlier node in the same list, the structure will form a cycle that can be detected by these algorithms. The cryptographic algorithms of Kaliski et al. In this application, cycle detection in the phase space of an orbital system may be used to determine whether the system is periodic to within the accuracy of the simulation. ISBN 9. 78. 14. 20. Joux (2. 00. 9), p. Knuth, Donald E. II: Seminumerical Algorithms, Addison- Wesley, p. Handbook of Applied Cryptography, by Alfred J. Floyd, R. W. Phan, Luca Henzen (2. Joux (2. 00. 9), Section 7. Floyd's cycle- finding algorithm, pp. ACM Symposium on Theory of Computing, pp.
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