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Incomputer science,cycle detection orcycle finding is thealgorithmic problem of finding a cycle in asequence ofiterated function values.

For anyfunctionf that maps afinite setS to itself, and any initial valuex₀ inS, the sequence of iterated function values

${\displaystyle x_0,x_1=f(x_0),x_2=f(x_1),\dots ,x_i=f(x_{i-1}),\dots }$

must eventually use the same value twice: there must be some pair of distinct indicesi andj such thatx_i =x_j. Once this happens, the sequence must continueperiodically, by repeating the same sequence of values fromx_i tox_{j − 1}. Cycle detection is the problem of findingi andj, givenf andx₀.

Several algorithms are known for finding cycles quickly and with little memory.Robert W. Floyd'stortoise and 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 ofexponential search. 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 ofpseudorandom number generators andcryptographic hash functions,computational number theory algorithms, detection ofinfinite loops in computer programs and periodic configurations incellular automata, automatedshape analysis oflinked list data structures, and detection ofdeadlocks fortransactions management inDBMS.

The figure shows a functionf that maps the setS = {0,1,2,3,4,5,6,7,8} to itself. If one starts fromx₀ = 2 and repeatedly appliesf, one sees the sequence of values

2, 0, 6, 3, 1, 6, 3, 1, 6, 3, 1, ....

The cycle in this value sequence is6, 3, 1.

LetS be any finite set,f be anyendofunction fromS to itself, andx₀ be any element ofS. For anyi > 0, letx_i =f(x_{i − 1}). Letμ be the smallest index such that the valuex_μ reappears infinitely often within the sequence of valuesx_i, and letλ (the loop length) be the smallest positive integer such thatx_μ =x_{λ +μ}. The cycle detection problem is the task of findingλ andμ.^[1]

One can view the same problemgraph-theoretically, by constructing afunctional graph (that is, adirected graph in which each vertex has a single outgoing edge) the vertices of which are the elements ofS and the edges of which map an element to the corresponding function value, as shown in the figure. The set of verticesreachable from starting vertexx₀ form a subgraph with a shape resembling theGreek letter rho (ρ): a path of lengthμ fromx₀ to acycle ofλ vertices.^[2]

Practical cycle-detection algorithms do not findλ andμ exactly.^[1] They usually find lower and upper boundsμ_l ≤μ ≤μ_h for the start of the cycle, and a more detailed search of the range must be performed if the exact value ofμ is needed. Also, most algorithms do not guarantee to findλ directly, but may find some multiplekλ <μ +λ. (Continuing the search for an additionalkλ/q steps, whereq is the smallest prime divisor ofkλ, will either find the trueλ or prove thatk = 1.)

Except in toy examples like the above,f will not be specified as a table of values. Such a table impliesO(|S|)space complexity, and if that is permissible, building a predecessor array (associative array mappingx_i toi) while iteratingf will detect the first repeated value when it is visited the second time, at which point the value in the predecessor array isμ and the current index isμ+λ. Rather, a cycle detection algorithm is given ablack box for generating the sequencex_i, and the task is to findλ andμ using very little memory.

The black box might consist of an implementation of the recurrence functionf, but it might also store additional internal state to make the computation more efficient. Althoughx_i =f(x_i−1) must be truein principle, this might be expensive to compute directly; the function could be defined in terms of thediscrete logarithm ofx_i−1 or some otherdifficult-to-compute property which can only be practically computed in terms of additional information. In such cases, the number of black boxes required becomes a figure of merit distinguishing the algorithms.

A second reason to use one of these algorithms is that they arepointer algorithms which do no operations on elements ofS other than testing for equality. An associative array implementation requires computing ahash function on the elements ofS, orordering them. But cycle detection can be applied in cases where neither of these are possible.

The classic example isPollard's rho algorithm forinteger factorization, which searches for a factorp of a given numbern by looking for valuesx_i andx_i+λ which are equal modulopwithout knowingp in advance. This is done by computing thegreatest common divisor of the differencex_i −x_i+λ with a known multiple ofp, namelyn. If the gcd is non-trivial (neither 1 norn), then the value is a proper factor ofn, as desired.^[2] Ifn is not prime, it must have at least one factorp ≤√n, and by thebirthday paradox, a random functionf has an expected cycle length (modulo p) of√p ≤⁴√n.

If the input is given as a subroutine for calculatingf, the cycle detection problem may be trivially solved using onlyλ +μ function applications, simply by computing the sequence of valuesx_i and using adata structure such as ahash table to store these values and test whether each subsequent value has already been stored. However, the space complexity of this algorithm is proportional toλ +μ, unnecessarily large. Additionally, to implement this method as apointer 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.

Floyd's cycle-finding algorithm is a pointer algorithm that uses only two pointers, which move through the sequence at different speeds. It is also called the "tortoise and the hare algorithm", alluding to Aesop's fable ofThe Tortoise and the Hare.

The algorithm is named afterRobert W. Floyd, who was credited with its invention byDonald Knuth.^[3][4] However, the algorithm does not appear in Floyd's published work, and this may be a misattribution: Floyd describes algorithms for listing all simple cycles in adirected graph in a 1967 paper,^[5] but this paper does not describe the cycle-finding problem in functional graphs that is the subject of this article. In fact, Knuth's statement (in 1969), attributing it to Floyd, without citation, is the first known appearance in print, and it thus may be afolk theorem, not attributable to a single individual.^[6]

The key insight in the algorithm is as follows. If there is a cycle, then, for any integersi ≥μ andk ≥ 0,x_i =x_{i +kλ}, whereλ is the length of the loop to be found,μ is the index of the first element of the cycle, andk is a whole integer representing the number of loops. Based on this, it can then be shown thati =kλ ≥μ for somek if and only ifx_i =x_2i (ifx_i =x_2i in the cycle, then there exists somek such that2i =i +kλ, which implies thati =kλ; and if there are somei andk such thati =kλ, then2i =i +kλ andx_2i =x_{i +kλ}). 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 periodν of a repetition that is a multiple ofλ. Onceν is found, the algorithm retraces the sequence from its start to find the first repeated valuex_μ in the sequence, using the fact thatλ dividesν and therefore thatx_μ =x_{μ +v}. Finally, once the value ofμ is known it is trivial to find the lengthλ of the shortest repeating cycle, by searching for the first positionμ +λ for whichx_{μ +λ} =x_μ.

The algorithm thus maintains two pointers into the given sequence, one (the tortoise) atx_i, and the other (the hare) atx_2i. At each step of the algorithm, it increasesi 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 ofi > 0 for which the tortoise and hare point to equal values is the desired valueν.

The followingPython code shows how this idea may be implemented as an algorithm.

deffloyd(f,x0)->(int,int):"""Floyd's cycle detection algorithm."""# Main phase of algorithm: finding a repetition x_i = x_2i.# The hare moves twice as quickly as the tortoise and# the distance between them increases by 1 at each step.# Eventually they will both be inside the cycle and then,# at some point, the distance between them will be# divisible by the period λ.tortoise=f(x0)# f(x0) is the element/node next to x0.hare=f(f(x0))whiletortoise!=hare:tortoise=f(tortoise)hare=f(f(hare))# At this point the tortoise position, ν, which is also equal# to the distance between hare and tortoise, is divisible by# the period λ. So hare moving in cycle one step at a time,# and tortoise (reset to x0) moving towards the cycle, will# intersect at the beginning of the cycle. Because the# distance between them is constant at 2ν, a multiple of λ,# they will agree as soon as the tortoise reaches index μ.# Find the position μ of first repetition.mu=0tortoise=x0whiletortoise!=hare:tortoise=f(tortoise)hare=f(hare)# Hare and tortoise move at same speedmu+=1# Find the length of the shortest cycle starting from x_μ# The hare moves one step at a time while tortoise is still.# lam is incremented until λ is found.lam=1hare=f(tortoise)whiletortoise!=hare:hare=f(hare)lam+=1returnlam,mu

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 usesO(λ +μ) operations of these types, andO(1) storage space.^[7]

Richard P. Brent described an alternative cycle detection algorithm that, like the tortoise and hare algorithm, requires only two pointers into the sequence.^[8] However, it is based on a different principle: searching for the smallestpower of two2ⁱ that is larger than bothλ andμ. Fori = 0, 1, 2, ..., the algorithm comparesx_2ⁱ−1 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 lengthλ of the cycle directly, rather than needing to search for it in a subsequent stage, and its steps involve only one evaluation of the functionf rather than three.^[9]

The following Python code shows how this technique works in more detail.

defbrent(f,x0)->(int,int):"""Brent's cycle detection algorithm."""# main phase: search successive powers of twopower=lam=1tortoise=x0hare=f(x0)# f(x0) is the element/node next to x0.# this assumes there is a cycle; otherwise this loop won't terminatewhiletortoise!=hare:ifpower==lam:# time to start a new power of two?tortoise=harepower*=2lam=0hare=f(hare)lam+=1# Find the position of the first repetition of length λtortoise=hare=x0foriinrange(lam):hare=f(hare)# The distance between the hare and tortoise is now λ.# Next, the hare and tortoise move at same speed until they agreemu=0whiletortoise!=hare:tortoise=f(tortoise)hare=f(hare)mu+=1returnlam,mu

Like the tortoise and hare algorithm, this is a pointer algorithm that usesO(λ +μ) tests and function evaluations andO(1) storage space. 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 36% more quickly than Floyd's and that it speeds up thePollard rho algorithm by around 24%. He also performs anaverage 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.^[8]

R. W. Gosper's algorithm^[10][11] finds the period${\displaystyle \lambda }$, and the lower and upper bound of the starting point,${\displaystyle {\mu }_l}$ and${\displaystyle {\mu }_u}$, of the first cycle. The difference between the lower and upper bound is of the same order as the period, i.e.${\displaystyle {\mu }_l+\lambda \approx {\mu }_h}$.

The algorithm maintains an array of tortoises${\displaystyle T_j}$. For each${\displaystyle x_i}$:

• For each${\displaystyle 0\le j\le {\log }_{2}i,}$ compare${\displaystyle x_i}$ to${\displaystyle T_j}$.
• If${\displaystyle x_i=T_j}$, a cycle has been detected, of length${\displaystyle \lambda =(i-2^j){mod2}^{j+1}+1.}$
• If no match is found, set${\displaystyle T_k\leftarrow x_i}$, where${\displaystyle k}$ is thenumber of trailing zeros in the binary representation of${\displaystyle i+1}$. I.e. the greatest power of 2 which divides${\displaystyle i+1}$.

If it is inconvenient to vary the number of comparisons as${\displaystyle i}$ increases, you may initialize all of the${\displaystyle T_j=x_0}$, but must then return${\displaystyle \lambda =i}$ if${\displaystyle x_i=T_j}$ while.

The main features of Gosper's algorithm are that it is economical in space, very economical in evaluations of the generator function, and always finds the exact cycle length (never a multiple). The cost is a large number of equality comparisons. It could be roughly described as a concurrent version of Brent's algorithm. While Brent's algorithm uses a single tortoise, repositioned every time the hare passes a power of two, Gosper's algorithm uses several tortoises (several previous values are saved), which are roughly exponentially spaced. According to the note inHAKMEM item 132,^[11] this algorithm will detect repetition before the third occurrence of any value, i.e. the cycle will be iterated at most twice. HAKMEM also states that it is sufficient to store${\displaystyle \lceil {\log }_2\lambda \rceil }$ previous values; however, this only offers a saving if we knowa priori that${\displaystyle \lambda }$ is significantly smaller than${\displaystyle \mu }$. The standard implementations^[10] store${\displaystyle \lceil {\log }_2(\mu +2\lambda )\rceil }$ values. For example, assume the function values are 32-bit integers, so${\displaystyle \mu +\lambda \le 2^{32}}$ and${\displaystyle \mu +2\lambda \le 2^{33}.}$ Then Gosper's algorithm will find the cycle after less than${\displaystyle \mu +2\lambda }$ function evaluations (in fact, the most possible is${\displaystyle 3\cdot 2^{31}-1}$), while consuming the space of 33 values (each value being a 32-bit integer).

Upon the${\displaystyle i}$-th evaluation of the generator function, the algorithm compares the generated value with${\displaystyle {\log }_{2}i}$ previous values; observe that${\displaystyle i}$ goes up to at least${\displaystyle \mu +\lambda }$ and at most${\displaystyle \mu +2\lambda }$. Therefore, the time complexity of this algorithm is${\displaystyle O((\mu +\lambda )\cdot \log (\mu +\lambda ))}$. Since it stores${\displaystyle {\log }_2(\mu +2\lambda )}$ values, its space complexity is${\displaystyle \mathrm{\Theta }(\log (\mu +\lambda ))}$. This is under the usualtransdichotomous model, assumed throughout this article, in which the size of the function values is constant. Without this assumption, we know it requires${\displaystyle \mathrm{\Omega }(\log (\mu +\lambda ))}$ space to store${\displaystyle \mu +\lambda }$ distinct values, so the overall space complexity is${\displaystyle \mathrm{\Omega }({\log }^2(\mu +\lambda )).}$

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 ahash 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,^[12] we survey these techniques briefly.

• Brent^[8] already describes variations of his technique in which the indices of saved sequence values are powers of a numberR other than two. By choosingR to be a number close to one, and storing the sequence values at indices that are near a sequence of consecutive powers ofR, a cycle detection algorithm can use a number of function evaluations that is within an arbitrarily small factor of the optimumλ +μ.^[13][14]
• Sedgewick, Szymanski, and Yao^[15] provide a method that usesM memory cells and requires in the worst case only${\displaystyle (\lambda +\mu )(1+c{M}^{-1/2})}$ function evaluations, for some constantc, which they show to be optimal. The technique involves maintaining a numerical parameterd, storing in a table only those positions in the sequence that are multiples ofd, and clearing the table and doublingd whenever too many values have been stored.
• Several authors have describeddistinguished 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 valued might be stored.^[16][17] More simply, Nivasch^[12] 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.
• Nivasch^[12] 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 astack 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.

Any cycle detection algorithm that stores at mostM values from the input sequence must perform at least${\displaystyle (\lambda +\mu )\left(1+\frac{1}{M-1}\right)}$ function evaluations.^[18][19]

Cycle detection has been used in many applications.

• Determining the cycle length of apseudorandom number generator is one measure of its strength. This is the application cited by Knuth in describing Floyd's method.^[3] Brent^[8] describes the results of testing alinear 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.
• Severalnumber-theoretic algorithms are based on cycle detection, includingPollard's rho algorithm for integer factorization^[20] and his relatedkangaroo algorithm for thediscrete logarithm problem.^[21]
• Incryptographic applications, the ability to find two distinct valuesx_μ−1 andx_λ+μ−1 mapped by some cryptographic functionƒ to the same valuex_μ may indicate a weakness inƒ. For instance, Quisquater and Delescaille^[17] apply cycle detection algorithms in the search for a message and a pair ofData Encryption Standard keys that map that message to the same encrypted value;Kaliski,Rivest, andSherman^[22] also use cycle detection algorithms to attack DES. The technique may also be used to find acollision in acryptographic hash function.^[23]
• Cycle detection may be helpful as a way of discoveringinfinite loops in certain types ofcomputer programs.^[24]
• Periodic configurations incellular automaton simulations may be found by applying cycle detection algorithms to the sequence of automaton states.^[12]
• Shape analysis oflinked 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.^[25] InCommon Lisp, theS-expression printer, under control of the*print-circle* variable, detects circular list structure and prints it compactly.
• Teske^[14] describes applications incomputational group theory: determining the structure of anAbelian group from a set of its generators. The cryptographic algorithms of Kaliski et al.^[22] may also be viewed as attempting to infer the structure of an unknown group.
• Fich (1981) briefly mentions an application tocomputer simulation ofcelestial mechanics, which she attributes toWilliam Kahan. In this application, cycle detection in thephase space of an orbital system may be used to determine whether the system is periodic to within the accuracy of the simulation.^[18]
• InMandelbrot Set fractal generation some performance techniques are used to speed up the image generation. One of them is called "period checking", which consists of finding the cycles in a point orbit. This article describes the "period checking" technique. You can find another explanationhere. Some cycle detection algorithms have to be implemented in order to implement this technique.

• Gabriel Nivasch,The Cycle Detection Problem and the Stack Algorithm
• Tortoise and Hare, Portland Pattern Repository
• Floyd's Cycle Detection Algorithm (The Tortoise and the Hare)
• Brent's Cycle Detection Algorithm (The Teleporting Turtle)

• Fixed points (mathematics)
• Combinatorial algorithms
• The Tortoise and the Hare

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