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| {{about|iterated functions|another use|Cycle detection (graph theory)}}
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| In [[computer science]], '''cycle detection''' is the [[algorithm]]ic problem of finding a cycle in a [[sequence]] of [[iterated function]] values.
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| For any [[function (mathematics)|function]] ''ƒ'' that maps a [[finite set]] ''S'' to itself, and any initial value ''x''<sub>0</sub> in ''S'', the sequence of iterated function values
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| :<math> x_0,\ x_1=f(x_0),\ x_2=f(x_1),\ \dots,\ x_i=f(x_{i-1}),\ \dots</math>
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| must eventually use the same value twice: there must be some ''i'' ≠ ''j'' such that ''x<sub>i</sub>'' = ''x<sub>j</sub>''. Once this happens, the sequence must continue by repeating the [[cycle (mathematics)|cycle]] of values from ''x<sub>i</sub>'' to ''x<sub>j</sub>''<sub>−1</sub>. Cycle detection is the problem of finding ''i'' and ''j'', given ''ƒ'' and ''x''<sub>0</sub>.
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| ==Example==
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| [[Image:Functional graph.svg|thumb|240px|A function from and to the set {0,1,2,3,4,5,6,7,8} and the corresponding functional graph]]
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| The figure shows a function ƒ that maps the set ''S'' = {0,1,2,3,4,5,6,7,8} to itself. If one starts from ''x''<sub>0</sub> = 2 and repeatedly applies ƒ, one sees the sequence of values
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| :2, 0, 6, 3, 1, 6, 3, 1, 6, 3, 1, ....
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| The cycle in this value sequence is 6, 3, 1.
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| ==Definitions==
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| Let ''S'' be any finite set, ƒ be any function from ''S'' to itself, and ''x''<sub>0</sub> be any element of ''S''. For any ''i'' > 0, let ''x<sub>i</sub>'' = ƒ(''x<sub>i</sub>''<sub>−1</sub>). Let μ be the smallest index such that the value ''x''<sub>μ</sub> reappears infinitely often within the sequence of values ''x<sub>i</sub>'', and let λ (the loop length) be the smallest positive integer such that ''x''<sub>μ</sub> = ''x''<sub>λ+μ</sub>. The cycle detection problem is the task of finding λ and μ.
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| One can view the same problem [[graph theory|graph-theoretically]], by constructing a [[functional graph]] (that is, a [[directed graph]] in which each vertex has a single outgoing edge) the vertices of which are the elements of ''S'' and the edges of which map an element to the corresponding function value, as shown in the figure. The set of vertices [[Reachability|reachable]] from any starting vertex ''x''<sub>0</sub> form a [[Glossary of graph theory#Subgraphs|subgraph]] with a shape resembling the [[Rho (letter)|Greek letter rho]] (ρ): a path of length μ from ''x''<sub>0</sub> to a [[Cycle graph|cycle]] of λ vertices.
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| ==Computer representation==
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| Generally, ƒ will not be specified as a table of values, as we have given it in the figure above. Rather, we may be given access either to the sequence of values ''x<sub>i</sub>'', or to a subroutine for calculating ƒ. The task is to find λ and μ while examining as few values from the sequence or performing as few subroutine calls as possible. 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.
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| 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 ƒ. 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. We may view a cycle detection algorithm for this application as having the following capabilities: it initially has in its memory an object representing a pointer to the starting value ''x''<sub>0</sub>. 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 ƒ 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: in Pollard's rho algorithm, it is implemented by testing whether the difference between two stored values has a nontrivial [[Greatest common divisor|gcd]] with the number to be factored. In this context, we will call an algorithm that only uses pointer copying, advancement within the sequence, and equality tests a ''pointer algorithm''.
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| ==Algorithms==
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| If the input is given as a subroutine for calculating ƒ, the cycle detection problem may be trivially solved using only λ+μ function applications, simply by computing the sequence of values ''x<sub>i</sub>'' and using a [[data structure]] such as a [[hash table]] to store these values and test whether each subsequent value has already been stored. However, the space complexity of this algorithm is λ+μ, unnecessarily large. 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.
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| ===Tortoise and hare===
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| [[Image:Tortoise and hare algorithm.svg|thumb|300px|Floyd's "tortoise and hare" cycle detection algorithm, applied to the sequence 2, 0, 6, 3, 1, 6, 3, 1, ...]]
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| '''Floyd's cycle-finding algorithm''', also called the "[[The Tortoise and the Hare|tortoise and the hare]]" algorithm, is a pointer algorithm that uses only two pointers, which move through the sequence at different speeds. The algorithm is named for [[Robert W. Floyd]], who invented it in the late 1960s.<ref>Floyd describes algorithms for listing all simple cycles in a [[directed graph]] in a 1967 paper: {{citation|last=Floyd|first=R.W.|authorlink=Robert W. Floyd|title=Non-deterministic Algorithms|journal=J. ACM|pages=636–644|volume=14|issue=4|year=1967|url=http://doi.acm.org/10.1145/321420.321422|doi=10.1145/321420.321422}}. However this paper does not describe the cycle-finding problem in functional graphs that is the subject of this article. An early description of the tortoise and hare algorithm appears in {{citation|first=Donald E.|last=Knuth|authorlink=Donald Knuth|title=The Art of Computer Programming, vol. II: Seminumerical Algorithms|publisher=Addison-Wesley|year=1969}}, exercises 6 and 7, page 7. Knuth (p.4) credits Floyd for the algorithm, without citation.</ref>
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| The key insight in the algorithm is that, for any integers {{nowrap|''i'' ≥ μ}} and {{nowrap|''k'' ≥ 0}}, {{nowrap|1=''x<sub>i</sub>'' = ''x''<sub>''i'' + ''k''λ</sub>,}} where λ is the length of the loop to be found. In particular, whenever {{nowrap|1=''i'' = ''m''λ ≥ μ}}, it follows that {{nowrap|1=''x<sub>i</sub>'' = ''x''<sub>2''i''</sub>}}.
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| 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 ν of a repetition that is a multiple of λ.
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| Once ν is found, the algorithm retraces the sequence from its start to find the first repeated value ''x''<sub>μ</sub> in the sequence, using the fact that λ divides ν and therefore that {{nowrap|1=''x''<sub>μ</sub> = ''x''<sub>ν + μ</sub>}}. Finally, once the value of μ is known it is trivial to find the length λ of the shortest repeating cycle, by searching for the first position {{nowrap|μ + λ}} for which {{nowrap|1=''x''<sub>μ + λ</sub> = ''x''<sub>μ</sub>}}.
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| The algorithm thus maintains two pointers into the given sequence, one (the tortoise) at ''x<sub>i</sub>'', and the other (the hare) at ''x''<sub>2''i''</sub>. 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 ν.
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| The following [[Python (programming language)|Python]] code shows how this idea may be implemented as an algorithm.
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| <source lang="python">
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| def floyd(f, x0):
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| # The main phase of the algorithm, finding a repetition x_mu = x_2mu
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| # The hare moves twice as quickly as the tortoise
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| # Eventually they will both be inside the cycle
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| # and the distance between them will increase by 1 until
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| # it is divisible by the length of the cycle.
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| tortoise = f(x0) # f(x0) is the element/node next to x0.
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| hare = f(f(x0))
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| while tortoise != hare:
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| tortoise = f(tortoise)
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| hare = f(f(hare))
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|
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| # at this point the position of tortoise which is the distance between
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| # hare and tortoise is divisible by the length of the cycle.
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| # so hare moving in circle and tortoise (set to x0) moving towards
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| # the circle will intersect at the beginning of the circle.
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| # Find the position of the first repetition of length mu
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| # The hare and tortoise move at the same speeds
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| mu = 0
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| tortoise = x0
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| while tortoise != hare:
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| tortoise = f(tortoise)
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| hare = f(hare)
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| mu += 1
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|
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| # Find the length of the shortest cycle starting from x_mu
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| # The hare moves while the tortoise stays still
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| lam = 1
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| hare = f(tortoise)
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| while tortoise != hare:
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| hare = f(hare)
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| lam += 1
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|
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| return lam, mu
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| </source> | |
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| This code only accesses the sequence by storing and copying pointers, function evaluations, and equality tests; therefore, it qualifies as a pointer algorithm. The algorithm uses {{nowrap|O(λ + μ)}} operations of these types, and O(1) storage space.
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| ===Brent's algorithm===
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| [[Richard Brent (scientist)|Richard P. Brent]] described an alternative cycle detection algorithm that, like the tortoise and hare algorithm, requires only two pointers into the sequence.<ref name="brent">{{citation|first=R. P.|last=Brent|authorlink=Richard Brent (scientist)|title=An improved Monte Carlo factorization algorithm|journal=BIT|volume=20|year=1980|pages=176–184|url=http://wwwmaths.anu.edu.au/~brent/pd/rpb051i.pdf|doi=10.1007/BF01933190|issue=2}}.</ref> However, it is based on a different principle: searching for the smallest [[power of two]] 2<sup>''i''</sup> that is larger than both λ and μ. For ''i'' = 0, 1, 2, etc., the algorithm compares ''x''<sub>2<sup>''i''</sup>−1</sub> with each subsequent sequence value up to the next power of two, stopping when it finds a match. It has two advantages compared to the tortoise and hare algorithm: it finds the correct length λ of the cycle directly, rather than needing to search for it in a subsequent stage, and its steps involve only one evaluation of ƒ rather than three.
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| The following Python code shows how this technique works in more detail. | |
| <source lang="python">
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| def brent(f, x0):
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| # main phase: search successive powers of two
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| power = lam = 1
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| tortoise = x0
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| hare = f(x0) # f(x0) is the element/node next to x0.
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| while tortoise != hare:
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| if power == lam: # time to start a new power of two?
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| tortoise = hare
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| power *= 2
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| lam = 0
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| hare = f(hare)
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| lam += 1
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| # Find the position of the first repetition of length lambda
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| mu = 0
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| tortoise = hare = x0
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| for i in range(lam):
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| # range(lam) produces a list with the values 0, 1, ... , lam-1
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| hare = f(hare)
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| while tortoise != hare:
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| tortoise = f(tortoise)
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| hare = f(hare)
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| mu += 1
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|
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| return lam, mu
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| </source>
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| Like the tortoise and hare algorithm, this is a pointer algorithm that uses {{nowrap|O(λ + μ)}} tests and function evaluations and O(1) storage space.
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| It is not difficult to show that the number of function evaluations can never be higher than for Floyd's algorithm.
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| Brent claims that, on average, his cycle finding algorithm runs around 36% more quickly than Floyd's and that it speeds up the Pollard rho algorithm by around 24%. 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.
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| ===Time–space tradeoffs===
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| A number of authors have studied techniques for cycle detection that use more memory than Floyd's and Brent's methods, but detect cycles more quickly. 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. Following Nivasch,<ref name="nivasch">{{citation|first=Gabriel|last=Nivasch|title=Cycle detection using a stack|journal=Information Processing Letters|volume=90|year=2004|pages=135–140|doi=10.1016/j.ipl.2004.01.016|issue=3}}.</ref> we survey these techniques briefly.
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| *Brent<ref name="brent"/> already describes variations of his technique in which the indices of saved sequence values are powers of a number ''R'' other than two. 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 λ+μ.<ref>{{citation|first1=Claus P.|last1=Schnorr|authorlink1=Claus P. Schnorr|first2=Hendrik W.|last2=Lenstra|authorlink2=Hendrik Lenstra|title=A Monte Carlo Factoring Algorithm With Linear Storage|journal=Mathematics of Computation|volume=43|issue=167|year=1984|pages=289–311|doi=10.2307/2007414|publisher=American Mathematical Society|jstor=2007414}}.</ref><ref name="teske">{{citation|first=Edlyn|last=Teske|title=A space-efficient algorithm for group structure computation|journal=Mathematics of Computation|volume=67|issue=224|year=1998|pages=1637–1663|doi=10.1090/S0025-5718-98-00968-5}}.</ref>
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| *Sedgewick, Szymanski, and Yao<ref>{{citation|first1=Robert|last1=Sedgewick|authorlink1=Robert Sedgewick (computer scientist)|first2=Thomas G.|last2=Szymanski|first3=Andrew C.-C.|last3=Yao|authorlink3=Andrew Yao|title=The complexity of finding cycles in periodic functions|journal=SIAM Journal on Computing|volume=11|issue=2|year=1982|pages=376–390|doi=10.1137/0211030}}.</ref> provide a method that uses ''M'' memory cells and requires in the worst case only <math>(\lambda+\mu)(1+cM^{-1/2})</math> function evaluations, for some constant ''c'', which they show to be optimal. 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.
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| *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.<ref>{{citation|first1=Paul C.|last1=van Oorschot|first2=Michael J.|last2=Wiener|title=Parallel collision search with cryptanalytic applications|journal=Journal of Cryptology|volume=12|issue=1|year=1999|pages=1–28|doi=10.1007/PL00003816}}.</ref><ref name="qd">{{citation|first1=J.-J.|last1=Quisquater|first2=J.-P.|last2=Delescaille|contribution=How easy is collision search? Application to DES|title=Advances in Cryptology – EUROCRYPT '89, Workshop on the Theory and Application of Cryptographic Techniques|series=Lecture Notes in Computer Science|volume=434|publisher=Springer-Verlag|pages=429–434|url=http://www.springerlink.com/content/0wnq56dava5nnm17/}}.</ref> More simply, Nivasch<ref name="nivasch"/> credits D. P. 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.
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| *Nivasch<ref name="nivasch"/> describes an algorithm that does not use a fixed amount of memory, but for which the expected amount of memory used (under the assumption that the input function is random) is logarithmic in the sequence length. 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)|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–space tradeoff similar to the previous algorithms. 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.
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| Any cycle detection algorithm that stores at most ''M'' values from the input sequence must perform at least <math>\scriptstyle (\lambda+\mu)(1+\frac{1}{M-1})</math> function evaluations.<ref name="fich">{{citation|first=Faith Ellen|last=Fich|contribution=Lower bounds for the cycle detection problem|title=Proc. 13th ACM Symp. Theory of Computation|year=1981|doi=10.1145/800076.802462|pages=96–105}}.</ref><ref>{{citation|first1=Eric W.|last1=Allender|authorlink1=Eric Allender|first2=Maria M.|last2=Klawe|authorlink2=Maria Klawe|title=Improved lower bounds for the cycle detection problem|journal=Theoretical Computer Science|volume=36|issue=2–3|pages=231–237|year=1985|doi=10.1016/0304-3975(85)90044-1}}.</ref>
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| ==Applications==
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| Cycle detection has been used in many applications.
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| *Determining the cycle length of a [[pseudorandom number generator]] is one measure of its strength. This is the application cited by Knuth in describing Floyd's method. Brent<ref name="brent"/> describes the results of testing a [[linear congruential generator]] in this fashion; its period turned out to be significantly smaller than advertised. 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.
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| *Several [[Number theory|number-theoretic]] algorithms are based on cycle detection, including [[Pollard's rho algorithm]] for integer factorization<ref>{{citation|first=J. M.|last=Pollard|title=A Monte Carlo method for factorization|journal=BIT|volume=15|year=1975|pages=331–334|doi=10.1007/BF01933667|issue=3}}.</ref> and his related [[Pollard's kangaroo algorithm|kangaroo algorithm]] for the [[discrete logarithm]] problem.<ref>{{citation|first=J. M.|last=Pollard|title=Monte Carlo methods for index computation (mod ''p'')|journal=Math. Comp.|volume=32|issue=143|year=1978|pages=918–924|doi=10.2307/2006496|publisher=American Mathematical Society|jstor=2006496}}.</ref>
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| *In [[Cryptography|cryptographic]] applications, the ability to find two distinct values ''x''<sub>μ−-1</sub> and ''x''<sub>λ+μ−-1</sub> mapped by some cryptographic function ƒ to the same value ''x''<sub>μ</sub> may indicate a weakness in ƒ. For instance, Quisquater and Delescaille<ref name="qd"/> apply cycle detection algorithms in the search for a message and a pair of [[Data Encryption Standard]] keys that map that message to the same encrypted value; [[Burt Kaliski|Kaliski]], [[Ron Rivest|Rivest]], and [[Alan Sherman|Sherman]]<ref name="krs">{{citation|first1=Burton S., Jr.|last1=Kaliski|first2=Ronald L.|last2=Rivest|authorlink2=Ron Rivest|first3=Alan T.|last3=Sherman|title=Is the Data Encryption Standard a group? (Results of cycling experiments on DES)|journal=Journal of Cryptology|volume=1|issue=1|year=1988|pages=3–36|doi=10.1007/BF00206323}}.</ref> also use cycle detection algorithms to attack DES. The technique may also be used to find a [[hash collision|collision]] in a [[cryptographic hash function]].
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| *Cycle detection may be helpful as a way of discovering [[infinite loop]]s in certain types of [[computer program]]s.<ref>{{citation|first=Allen|last=Van Gelder|title=Efficient loop detection in Prolog using the tortoise-and-hare technique|journal=Journal of Logic Programming|volume=4|issue=1|pages=23–31|year=1987|doi=10.1016/0743-1066(87)90020-3}}.</ref>
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| *[[Oscillator (cellular automaton)|Periodic configurations]] in [[cellular automaton]] simulations may be found by applying cycle detection algorithms to the sequence of automaton states.<ref name="nivasch"/>
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| *[[Shape analysis (software)|Shape analysis]] of [[linked list]] data structures is a technique for verifying the correctness of an algorithm using those structures. 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.<ref>{{citation|first1=Mikhail|last1=Auguston|first2=Miu Har|last2=Hon|contribution=Assertions for Dynamic Shape Analysis of List Data Structures|pages=37–42|title=AADEBUG '97, Proceedings of the Third International Workshop on Automatic Debugging|series=Linköping Electronic Articles in Computer and Information Science|publisher=[[Linköping University]]|url=http://www.ep.liu.se/ea/cis/1997/009/04/|year=1997}}.</ref>
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| *Teske<ref name="teske"/> describes applications in [[computational group theory]]: determining the structure of an [[Abelian group]] from a set of its generators. The cryptographic algorithms of Kaliski et al.<ref name="krs"/> may also be viewed as attempting to infer the structure of an unknown group.
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| *Fich<ref name="fich"/> briefly mentions an application to [[computer simulation]] of [[celestial mechanics]], which she attributes to [[William Kahan]]. 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.
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| * In [[Common Lisp]], the [[S-expression]] printer, under control of the <code>*print-circle*</code> variable, detects circular list structure and prints it compactly.
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| ==References==
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| {{reflist|30em}}
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| ==External links==
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| *Gabriel Nivasch, [http://www.gabrielnivasch.org/fun/cycle-detection The Cycle Detection Problem and the Stack Algorithm]
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| *[http://c2.com/cgi/wiki/Curry?TortoiseAndHare Tortoise and Hare], Portland Pattern Repository
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| {{DEFAULTSORT:Cycle Detection}}
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Combinatorial algorithms]]
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| [[Category:Articles with example Python code]]
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Before you resolve whether or not stainless-steel cookware is value buying, lets first focus on what chrome steel cookware is. Chrome steel is fabricated from an alloy, or a mix of metals. Most commonly, primary iron with chromium, nickel or some other minor metals. The chromium offers rust safety and provides your cookware durability. The nickel offers rust safety as nicely, and adds a elegant look. Most properly made stainless-steel cookware has copper or aluminum added to the underside of the pan or pot. That is accomplished to will increase the power of the pot or pan to conduct heat.
The most effective chrome steel cookware is the principle class, however still it is divided into several subcategories primarily based on the standard and the worth vary. It may be confusing to decide on the best stainless-steel cookware out of the categories that will meet your necessities. That is the place we took a step ahead to elucidate you all the data that will be helpful for you to understand how to decide on the best stainless-steel cookware. The most effective stainless-steel cookware set is manufactured from low cost to costly and high quality constructed pots and pans.
You will see that magnetic stainless-steel in the layer on the surface of some quality items of stainless steel. This is to make it suitable with induction stovetops, which contain the use of a quickly charging electromagnetic subject to heat cookware. Excessive-high quality chrome steel, like All-Clad , makes use of three layers of steel—the austenite layer of steel on the inside, ferrite steel on the skin, and a layer of aluminum sandwiched between the 2 for optimal warmth conductivity (metal alone doesn't conduct warmth evenly). Lesser-high quality stainless steel is usually just one layer of austenitic stainless steel.
Aesthetically speaking, chrome steel is a clever selection should you prefer to display or hold pots or pans. The clear, crisp look of all chrome steel kitchenware can rework a mishmash of cookware into a classy décor statement. Chrome steel kettles, such because the Cuisinart Tea Kettle will combine particular person kitchenware into a cohesive and nice entity. Consider buying stainless-steel utensils as properly. Already acquired a gorgeous stainless-steel cookware collection? The Cuisinart Chef’s Collection stainless pot rack may be the of entirety for a kitchen, liberating up space and making these pots and pans readily accessible. Get the chrome steel cookware of your culinary goals at Macy’s!
Onerous-anodized aluminum cookware is one of the most popular forms of material, regardless that many people don't fairly perceive the development. Arduous-anodized aluminum is plain aluminum that has been processed in a sequence of chemical baths charged with an electrical current. The result is a cloth that has the identical superior heat conductivity as aluminum but is non-reactive with acidic foods, equivalent to tomatoes, and twice as onerous as stainless steel. Two drawbacks to onerous-anodized cookware are that it's not dishwasher-safe and, as a result of it's not magnetic, it is not going to work with induction range tops.
The enamel over metal method creates a bit that has the warmth distribution of carbon steel and a non-reactive, low-stick floor. Such pots are a lot lighter than most other pots of similar dimension, are cheaper to make than stainless steel pots, and do not have the rust and reactivity issues of forged iron or carbon steel. citation needed Enamel over metal is good for large stockpots and for other massive pans used principally for water-based mostly cooking. Because of its mild weight and simple cleanup, enamel over metal can also be fashionable for cookware used while tenting. Clad aluminium or copper edit
Unique specialty cookware pieces served a la carte to go with any cookware set are constructed of a durable Stainless Metal with a brushed exterior finish. Designed with an influence bonded, aluminum disk encapsulated base which distributes warmth rapidly and evenly to allow exact temperature control. Handles are riveted for sturdiness and efficiency. The New Specialty Cookware is suitable for all vary sorts including induction. In addition to the multi use function, one other unique characteristic is bottom to high inside volume markings in both quarts and metric measurement; and each bit comes with a tempered glass lid, oven safe to 350°F.
Whether or not you are a cooking fanatics, knowledgeable chef or just cooking for your family you know the importance of getting a completely stocked kitchen. Not solely do you want the best substances, however you also want the precise tools to get the job executed. In any sort of fundamental cooking coaching lesson, you'll be taught that stainless-steel is your new best friend in terms of kitchen cookware. What you will also study is that quality cooking gear does not normally come at a discounted value. For this reason, it is important to take good care of your cookware! Listed here are some fundamentals for stainless-steel care.
To fight the uneven heating drawback, most stainless steel pans are laminations of aluminum or copper on the underside to spread the heat around, and stainless steel inside the pan to offer a cooking floor that's impervious to no matter you might put inside. In my experience, this stainless steel surface is still too sticky to fry on, and in case you ever burn it you get a permanent bother spot. Should you loved this informative article and you would want to receive more information relating to stainless steel cookware reviews generously visit our own webpage. But, sometimes a stainless steel cooking surface turns out to be useful when you may't use aluminum (see below) so I maintain some around. Select one thing with a fairly thick aluminum layer on the underside.
Effectively, except you’re a metals expert and go inspect the manufacturing facility where the metal is made to see whether or not or not their manufacturing course of creates a pure austenite without corrosive supplies fashioned, you’re not going to know for certain whether or not the craftsmanship of your stainless is of the highest high quality. I believe your finest guess is to simply buy high-high quality stainless-steel from the beginning, from a brand with a status for good high quality. However, I believe I have found out one way that you would be able to determine if the stainless cookware you already have is doubtlessly reactive.