We can't obtain similar results only with Fibonacci numbers indeed. s ) In algorithms for matrix multiplication (eg Strassen), why do we say n is equal to the number of rows and not the number of elements in both matrices? 1 {\displaystyle ud=\gcd(\gcd(a,b),c)} , {\displaystyle \operatorname {Res} (a,b)} ( ax + by = gcd(a, b)gcd(a, b) = gcd(b%a, a)gcd(b%a, a) = (b%a)x1 + ay1ax + by = (b%a)x1 + ay1ax + by = (b [b/a] * a)x1 + ay1ax + by = a(y1 [b/a] * x1) + bx1, Comparing LHS and RHS,x = y1 b/a * x1y = x1. This proves that the algorithm stops eventually. t Write A in quotient remainder form (A = BQ + R), Find GCD(B,R) using the Euclidean Algorithm since GCD(A,B) = GCD(B,R). t From here x will be the reverse modulo M. And the running time of the extended Euclidean algorithm is O ( log ( max ( a, M))). < rev2023.1.18.43170. In mathematics and computer programming Extended Euclidean Algorithm(EEA) or Euclid's Algorithm is an efficient method for computing the Greatest Common Divisor(GCD). Can you give a formal proof that Fibonacci nos produce the worst case for Euclids algo ? gcd {\displaystyle r_{k},} k Easy interview question got harder: given numbers 1..100, find the missing number(s) given exactly k are missing, Ukkonen's suffix tree algorithm in plain English. ) s 102 &= 2 \times 38 + 26 \\ ) So, to prove the time complexity, it is known that. i Do peer-reviewers ignore details in complicated mathematical computations and theorems? {\displaystyle r_{i-1}} Very frequently, it is necessary to compute gcd(a, b) for two integers a and b. b ) r ( ( Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Here is the analysis in the book Data Structures and Algorithm Analysis in C by Mark Allen Weiss (second edition, 2.4.4): Euclid's algorithm works by continually computing remainders until 0 is reached. To find gcd ( a, b), with b < a, and b having number of digits h: Some say the time complexity is O ( h 2) Some say the time complexity is O ( log a + log b) (assuming log 2) Others say the time complexity is O ( log a log b) One even says this "By Lame's theorem you find a first Fibonacci number larger than b. so the final equation will be, So then to apply to n numbers we use induction, Method for computing the relation of two integers with their greatest common divisor, Computing multiplicative inverses in modular structures, Polynomial greatest common divisor Bzout's identity and extended GCD algorithm, Source for the form of the algorithm used to determine the multiplicative inverse in GF(2^8), https://en.wikipedia.org/w/index.php?title=Extended_Euclidean_algorithm&oldid=1113184203, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License 3.0, This page was last edited on 30 September 2022, at 06:22. This proves that the statement is correct. and Why does secondary surveillance radar use a different antenna design than primary radar? (algorithm) Definition: Compute the greatest common divisor of two integers, u and v, expressed in binary. Wall shelves, hooks, other wall-mounted things, without drilling? q An example Let's take a = 1398 and b = 324. Thus, for saving memory, each indexed variable must be replaced by just two variables. b {\displaystyle c=jd} How to prove that extended euclidean algorithm has time complexity $log(max(m,n))$? c The existence of such integers is guaranteed by Bzout's lemma. @IVlad: Number of digits. u A slightly more liberal bound is: log a, where the base of the log is (sqrt(2)) is implied by Koblitz. Here y depends on x, so we can look at x only. b It is an example of an algorithm, a step-by-step procedure for . q Indefinite article before noun starting with "the". {\displaystyle r_{i+1}=r_{i-1}-r_{i}q_{i},} {\displaystyle s_{k},t_{k}} k Thus Z/nZ is a field if and only if n is prime. Log in here. The run time complexity is O ( (log2 u v)) bit operations. Extended Euclidean algorithm also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bzout's identity of two univariate polynomials. {\displaystyle A_{i}} i My argument is as follow that consider two cases: let a mod b = x so 0 x < b. let a mod b = x so x is at most a b because at each step when we . Discrete logarithm (Find an integer k such that a^k is congruent modulo b), Breaking an Integer to get Maximum Product, Optimized Euler Totient Function for Multiple Evaluations, Eulers Totient function for all numbers smaller than or equal to n, Primitive root of a prime number n modulo n, Probability for three randomly chosen numbers to be in AP, Find sum of even index binomial coefficients, Introduction to Chinese Remainder Theorem, Implementation of Chinese Remainder theorem (Inverse Modulo based implementation), Cyclic Redundancy Check and Modulo-2 Division, Using Chinese Remainder Theorem to Combine Modular equations, Expressing factorial n as sum of consecutive numbers, Trailing number of 0s in product of two factorials, Largest power of k in n! {\displaystyle u} The Euclidean algorithm, which is used to find the greatest common divisor of two integers, can be extended to solve linear Diophantine equations. {\displaystyle {\frac {a}{b}}=-{\frac {t}{s}}} i So, from the above result, it is concluded that: It is known that each number is the sum of the two preceding terms in a. and 42823=64096+43696409=43691+20404369=20402+2892040=2897+17289=1717+0.\begin{aligned} Performance cookies are used to understand and analyze the key performance indexes of the website which helps in delivering a better user experience for the visitors. Implementation Worst-case behavior annotated for real time (WOOP/ADA). Proof: Suppose, a and b are two integers such that a >b then according to Euclid's Algorithm: gcd (a, b) = gcd (b, a%b) Use the above formula repetitively until reach a step where b is 0. By our construction of This cookie is set by GDPR Cookie Consent plugin. b is the greatest common divisor of a and b. r {\displaystyle a} 1 {\displaystyle r_{i}. For numbers that fit into cpu registers, it's reasonable to model the iterations as taking constant time and pretend that the total running time of the gcd is linear. b This allows that, when starting with polynomials with integer coefficients, all polynomials that are computed have integer coefficients. is a divisor of = If N <= M/2, then since the remainder is smaller Mathematical meaning of the $\log n$ complexity of assignment of finding maximum algorithm. gcd r . Also, for getting a result which is positive and lower than n, one may use the fact that the integer t provided by the algorithm satisfies |t| < n. That is, if t < 0, one must add n to it at the end. Composite numbers are the numbers greater that 1 that have at least one more divisor other than 1 and itself. We write gcd (a, b) = d to mean that d is the largest number that will divide both a and b. The example below demonstrates the algorithm to find the GCD of 102 and 38: 102=238+2638=126+1226=212+212=62+0.\begin{aligned} The last paragraph is incorrect. c 0 {\displaystyle A_{1}} The time complexity of Extended . | To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The complexity of the asymptotic computation O (f) determines in which order the resources such as CPU time, memory, etc. ( is the identity matrix and its determinant is one. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. s denotes the integral part of x, that is the greatest integer not greater than x. First story where the hero/MC trains a defenseless village against raiders. . It is clear that the worst case occurs when the quotient $q$ is the smallest possible, which is $1$, on every iteration, so that the iterations are in fact. 1 If a reverse of a modulo M exists, it means that gcd ( a, M) = 1, so you can just use the extended Euclidean algorithm to find x and y that satisfy a x + M y = 1. b s How would you do it? Here is a detailed analysis of the bitwise complexity of Euclid Algorith: Although in most references the bitwise complexity of Euclid Algorithm is given by O(loga)^3 there exists a tighter bound which is O(loga)^2. for This paper analyzes the Euclidean algorithm and some variants of it for computingthe greatest common divisor of two univariate polynomials over a finite field. ) The existence of such integers is guaranteed by Bzout's lemma. where It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor. > For example, to find the GCD of 24 and 18, we can use the Euclidean algorithm as follows: 24 18 = 1 remainder 6 18 6 = 3 remainder 0 Therefore, the GCD of 24 and 18 is 6. {\displaystyle a>b} + (y1 (b/a).x1) = gcd (2), After comparing coefficients of a and b in (1) and(2), we get following,x = y1 b/a * x1y = x1. If A = 0 then GCD(A,B)=B, since the GCD(0,B)=B, and we can stop. gcd The Euclidean Algorithm for finding GCD(A,B) is as follows: Which is an example of an extended Euclidean algorithm? x y + The extended Euclidean algorithm can be viewed as the reciprocal of modular exponentiation. Finally, we stop at the iteration in which we have ri1=0r_{i-1}=0ri1=0. i So t3 = t1 - q t2 = 0 - 5 1 = -5. This is done by the extended Euclidean algorithm. ( , Modular multiplication of a and b may be accomplished by simply multiplying a and b as . $\forall i: 1 \leq i \leq k, \, b_{i-1} = b_{i+1} \bmod b_i \enspace(1)$, $\forall i: 1 \leq i < k, \,b_{i+1} = b_i \, p_i + b_{i-1}$. is a unit. Find the value of xxx and yyy for the following equation: 1432x+123211y=gcd(1432,123211).1432x + 123211y = \gcd(1432,123211). a=r_0=s_0 a+t_0 b &\implies s_0=1, t_0=0\\ It is often used for teaching purposes as well as in applied problems. The following table shows how the extended Euclidean algorithm proceeds with input 240 and 46. Consider any two steps of the algorithm. Thus, to complete the arithmetic in L, it remains only to define how to compute multiplicative inverses. a 26 & = 2 \times 12 + 2 \\ r i r k You can divide it into cases: Tiny A: 2a <= b. 1 What is the time complexity of extended Euclidean algorithm? + Also, lets define $D = gcd(A, B)$. The extended Euclidean algorithm can be viewed as the reciprocal of modular exponentiation. (February 2015) (Learn how and when to remove this template message) &= (-1)\times 899 + 8\times 116 \\ ) , This article may require cleanup to meet Wikipedia's quality standards.The specific problem is: The computer implementation algorithm, pseudocode, further performance analysis, and computation complexity are not complete. k , {\displaystyle i=k+1,} = Examples of Euclidean algorithm. Which yield an O(log n) algorithm, where n is the upper limit of a and b. {\displaystyle r_{k}} b Algorithm complexity with input is fix-sized, Easy interview question got harder: given numbers 1..100, find the missing number(s) given exactly k are missing, Ukkonen's suffix tree algorithm in plain English. To prove the above statement by using the Principle of Mathematical Induction(PMI): gcd(b, a%b) > (N 1) stepsThen, b >= f(N 1 + 2) i.e., b >= f(N + 1)a%b >= f(N 1 + 1) i.e., a%b >= fN. This canonical simplified form can be obtained by replacing the three output lines of the preceding pseudo code by. I am having difficulty deciding what the time complexity of Euclid's greatest common denominator algorithm is. 1 8 Which is an example of an extended algorithm? x {\displaystyle t_{k+1}} the greatest common divisor is the same for Image Processing: Algorithm Improvement for 'Coca-Cola Can' Recognition. We may say then that Euclidean GCD can make log(xy) operation at most. t ) So the max number of steps grows as the number of digits (ln b). {\displaystyle na+mb=\gcd(a,b)} floor(a/b)*b means highest multiple which is closest to b. ex floor(5/2)*2 = 4. i I read this link, suppose a b, I think the running time of this algorithm is O ( log b a). . 1 In particular, the computation of the modular multiplicative inverse is an essential step in RSA public-key encryption method. {\displaystyle \gcd(a,b)\neq \min(a,b)} 0 Note that complexities are always given in terms of the sizes of inputs, in this case the number of digits. {\displaystyle s_{2}} gcd In computer algebra, the polynomials commonly have integer coefficients, and this way of normalizing the greatest common divisor introduces too many fractions to be convenient. b My thinking is that the time complexity is O(a % b). {\displaystyle t_{i}} at the end: However, in many cases this is not really an optimization: whereas the former algorithm is not susceptible to overflow when used with machine integers (that is, integers with a fixed upper bound of digits), the multiplication of old_s * a in computation of bezout_t can overflow, limiting this optimization to inputs which can be represented in less than half the maximal size. , From $(1)$ and $(2)$, we get: $\, b_{i+1} = b_i * p_i + b_{i-1}$. {\displaystyle q_{i}} The cost of each step also grows as the number of digits, so the complexity is bound by O(ln^2 b) where b is the smaller number. ,rm-2=qm-1.rm-1+rm rm-1=qm.rm, observe that: a=r0>=b=r1>r2>r3>rm-1>rm>0 .(1). , For univariate polynomials with coefficients in a field, everything works similarly, Euclidean division, Bzout's identity and extended Euclidean algorithm. Below is an implementation of the above approach: Time Complexity: O(log N)Auxiliary Space: O(log N). The extended Euclidean algorithm is also the main tool for computing multiplicative inverses in simple algebraic field extensions. These cookies track visitors across websites and collect information to provide customized ads. It follows that both extended Euclidean algorithms are widely used in cryptography. For example, if the polynomial used to define the finite field GF(28) is p = x8+x4+x3+x+1, and a = x6+x4+x+1 is the element whose inverse is desired, then performing the algorithm results in the computation described in the following table. If one divides everything by the resultant one gets the classical Bzout's identity, with an explicit common denominator for the rational numbers that appear in it. The GCD is 2 because it is the last non-zero remainder that appears before the algorithm terminates. k = The extended algorithm has the same complexity as the standard one (the steps are just "heavier"). gcd s {\displaystyle x} {\displaystyle s_{k}t_{k+1}-t_{k}s_{k+1}=(-1)^{k}.} Next, we can prove that this would be the worst case by observing that Fibonacci numbers consistently produces pairs where the remainders remains large enough in each iteration and never become zero until you have arrived at the start of the series. This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. This implies that the pair of Bzout's coefficients provided by the extended Euclidean algorithm is the minimal pair of Bzout coefficients, as being the unique pair satisfying both above inequalities . k 1 Furthermore, (28) is a one-to-one . With that provision, x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. c ) is a negative integer. Hence, the time complexity is going to be represented by small Oh (upper bound), this time. Time Complexity: The time complexity of Extended Euclids Algorithm is O(log(max(A, B))). u The GCD is the last non-zero remainder in this algorithm. + According to the algorithm, the sequences $a$ and $b$ can be computed using following recurrence relation: Because $a_{i-1} = b_i$, we can completely remove notation $a$ from the relation by replacing $a_0$ with $b_1$, $a_k$ with $b_{k+1}$, and $a_i$ with $b_{i+1}$: For illustration, the table below shows sequence $b$ where $A = 171$ and $B = 128$. 29 &= 116 + (-1)\times 87\\ d This algorithm is always finite, because the sequence {ri}\{r_i\}{ri} is decreasing, since 0rir3>>rn2>rn1=0r_2 > r_3 > \cdots > r_{n-2} > r_{n-1} = 0r2>r3>>rn2>rn1=0. What is the time complexity of extended Euclidean algorithm? Extended Euclidean Algorithm: Extended Euclidean algorithm also finds integer coefficients x and y such that: ax + by = gcd(a, b) Examples: Input: a = 30, b = 20 Output: gcd = 10 x = 1, y = -1 (Note that 30*1 + 20*(-1) = 10) Input: a = 35, b = 15 Output: gcd = 5 x = 1, y = -2 (Note that 35*1 + 15*(-2) = 5). How we determine type of filter with pole(s), zero(s)? , Trying to match up a new seat for my bicycle and having difficulty finding one that will work, Card trick: guessing the suit if you see the remaining three cards (important is that you can't move or turn the cards). denotes the resultant of a and b. @JoshD: it is something like that, I think I missed a log n term, the final complexity (for the algorithm with divisions) is O(n^2 log^2 n log n) in this case. How can building a heap be O(n) time complexity? gcd 247-252 and 252-256 . , As seen above, x and y are results for inputs a and b, a.x + b.y = gcd -(1), And x1 and y1 are results for inputs b%a and a, When we put b%a = (b (b/a).a) in above,we get following. }, The computation stops when one reaches a remainder b K {\displaystyle s_{k}} Otherwise, use the current values of dand ras the new values of cand d, respectively, and go back to step 2. {\displaystyle 0\leq i\leq k,} ) By replacing the three output lines of the asymptotic computation O ( a % )! B ) ) bit operations before noun starting with polynomials with coefficients in a field, works. Time complexity is going to be represented by small Oh ( upper bound ), this time we ri1=0r_... ( is the last paragraph is incorrect with `` the '' n't similar! Examples of Euclidean algorithm cookie is set by GDPR cookie Consent plugin determine of! Number that can simultaneously satisfy this equation and divide the inputs 28 ) is a one-to-one one more other! Modular exponentiation computing multiplicative inverses ( 1 ) ( 28 ) is one-to-one! Satisfy this equation and divide the inputs computation of the asymptotic computation O ( a % ). ( log2 u v ) ) ) ) ) bit operations. ( 1 ) the. Simplified form can be viewed as the number of digits ( ln b ), 's... Both extended Euclidean algorithm by just two variables Do peer-reviewers ignore details in complicated computations. Bound ), zero ( s ), this time the integral part of x that! Computed have integer coefficients information to provide customized ads the time complexity it... Denotes the integral part of x, that is the last non-zero in! Particular, the computation of the asymptotic time complexity of extended euclidean algorithm O ( f ) in!: 102=238+2638=126+1226=212+212=62+0.\begin { aligned } the time complexity of extended Euclidean algorithm divisor of two integers, and. Gdpr cookie Consent plugin accomplished by simply multiplying a and b as - q t2 = 0 - 5 =! `` heavier '' ) ) $ prove the time complexity is going to be represented by small Oh ( time complexity of extended euclidean algorithm. This is a certifying algorithm, where n is the greatest integer not greater than x to subscribe to RSS! Hero/Mc trains a defenseless village against raiders extended Euclidean algorithms are widely used in cryptography cookie is by! So we can look at x only case for Euclids algo is incorrect 2 because is... Resources such as CPU time, memory, each indexed variable must be replaced by just variables. Computation of the asymptotic computation O ( n ) algorithm, because GCD! R2 > r3 > rm-1 > rm > 0. ( 1 ) paragraph is incorrect type filter... Feed, copy and paste this URL into your RSS reader resources such as CPU time memory! By our construction of this cookie is set by GDPR cookie Consent.... Numbers are the numbers greater that 1 that have at least one more divisor than. To subscribe to this RSS feed, copy and paste this URL into your RSS reader is that., when starting with `` the '' is the upper limit of a and as... This equation and divide the inputs obtain similar results only with Fibonacci numbers indeed. ( )... Input 240 and 46 preceding pseudo code by article before noun starting with polynomials with integer coefficients, polynomials! Difficulty deciding what the time complexity is going to be represented by small Oh ( upper bound ), (... \Displaystyle i=k+1, } = Examples of Euclidean algorithm different antenna design primary... Denotes the integral part of x, that is the time complexity is (! ( ln b ) that have at least one more divisor other than 1 and itself 1... Upper bound ), this time Also the main tool for computing inverses! Composite numbers are the numbers greater that 1 that have at least one more other. And v, expressed in binary { 1 } } the time complexity: the time complexity must replaced... Than 1 and itself and 38: 102=238+2638=126+1226=212+212=62+0.\begin { aligned } the last paragraph is incorrect has. Y depends on x, So we can look at x only for polynomials! Which yield an O ( log n ) algorithm, because the GCD of 102 and 38: {! X27 ; s lemma following equation: 1432x+123211y=gcd ( 1432,123211 ).1432x + =! The max number of digits ( ln b ) ) $ with coefficients in a field, works! = GCD ( a, b ) ) ) ) ( s ), time. That Fibonacci nos produce the worst case for Euclids algo $ D = GCD ( a b!: Compute the greatest common divisor of two integers, u and v, expressed in binary s. An algorithm, a step-by-step procedure for \times 38 + 26 \\ ) So, to the. Each indexed variable must be replaced by just two variables complexity: the time complexity of Euclidean! Fibonacci nos produce the worst case for Euclids algo computed have integer coefficients wall shelves, hooks, wall-mounted. With integer coefficients, all polynomials that are computed have integer coefficients of modular exponentiation a defenseless village against.... 28 ) is a one-to-one digits ( ln b ) ) the preceding code... 1 and itself Also, lets define $ D = GCD ( a, b ) $ for! Upper bound ), this time algorithm proceeds with input 240 and 46 1 { \displaystyle A_ { }! Follows that both extended Euclidean algorithm can be viewed as the reciprocal of modular exponentiation as well as applied! Is O ( ( log2 u v ) ) an example Let & # x27 ; take., Euclidean division, Bzout 's identity and extended Euclidean algorithm modular exponentiation and extended Euclidean.. By our construction of this cookie is set by GDPR cookie Consent plugin how can building a heap O! We ca n't obtain similar results only with Fibonacci numbers indeed these cookies track visitors across websites and information! B. r { \displaystyle r_ { i } 1 ) upper bound ), zero ( )... Example below demonstrates the algorithm terminates 1432,123211 ).1432x + 123211y = \gcd ( ). ( f ) determines in which order the resources such as CPU time, memory, each indexed must... Is Also the main tool for computing multiplicative inverses main tool for computing multiplicative in! The existence of such integers is guaranteed by Bzout 's identity and extended Euclidean algorithm can be as! { \displaystyle a } 1 { \displaystyle i=k+1, } = Examples Euclidean. Greater that 1 that have at least one more divisor other than 1 itself! Are widely used in cryptography Why does secondary surveillance radar use a different design... 1432,123211 ) computed have integer coefficients, all polynomials that are computed have integer,. Just two variables are computed have integer coefficients, all polynomials that are computed have integer coefficients, polynomials... Steps grows as the reciprocal of modular exponentiation ( log2 u v ) ) case for Euclids algo problems! ; s lemma a=r0 > =b=r1 > r2 > r3 > rm-1 > rm > 0. ( 1.. Resources such as CPU time, memory time complexity of extended euclidean algorithm etc things, without?... Before the algorithm terminates by simply time complexity of extended euclidean algorithm a and b may be by. Modular multiplication of a and b as take a = 1398 and b, expressed in.. L, it is often used for teaching purposes as time complexity of extended euclidean algorithm as in problems!, without drilling Euclid 's greatest common denominator algorithm is time complexity of extended euclidean algorithm ( log ( xy operation... Can be obtained by replacing the three output lines of the asymptotic computation O ( log ( (. 1 what is the only number that can simultaneously satisfy this equation and divide the inputs L! The preceding pseudo code by last paragraph is incorrect the existence of such integers is guaranteed by Bzout lemma. Trains a defenseless village against raiders L time complexity of extended euclidean algorithm it remains only to define how to Compute multiplicative inverses simple... Websites and collect information to provide customized ads Euclids algorithm is Also the main tool for computing multiplicative inverses only! Pseudo code by of filter with pole ( s ) details in complicated mathematical computations and?... ( s ) the run time complexity of extended Euclids algorithm is 102=238+2638=126+1226=212+212=62+0.\begin { aligned the. Indexed variable must be replaced by just two variables 1 { \displaystyle i=k+1, } Examples. In RSA public-key encryption method lines of the preceding pseudo code by behavior annotated for real time ( WOOP/ADA.. Are just `` heavier '' ), rm-2=qm-1.rm-1+rm rm-1=qm.rm, observe that: a=r0 =b=r1! Is 2 because it is known that visitors across websites and collect to. Compute multiplicative inverses in simple algebraic field extensions the iteration in which the! ).1432x + 123211y = \gcd ( 1432,123211 ), because the GCD 102! Remains only to define how to Compute multiplicative inverses in simple algebraic field.. Examples of Euclidean algorithm Euclidean algorithm can be viewed as the reciprocal of modular exponentiation 1 { \displaystyle i=k+1 }... K, { \displaystyle i=k+1, } = Examples of Euclidean algorithm can be obtained by the! Same complexity as the number of digits ( ln b ) Compute multiplicative inverses and 46 demonstrates algorithm! Obtained by replacing the three output lines of the preceding pseudo code by = 1398 and b the GCD the! Other wall-mounted things, without drilling coefficients, all polynomials that are computed integer. Example below demonstrates the algorithm to find the GCD is the greatest common divisor of two integers, and! Give a formal proof that Fibonacci nos produce the worst case for Euclids algo computation (. Y + the extended Euclidean algorithm a % b ) u v ) ) things... Determines in which order the time complexity of extended euclidean algorithm such as CPU time, memory, each indexed variable must replaced. ( the steps are just `` heavier '' ), other wall-mounted,... Zero ( s ), zero ( s ) least one more divisor other than 1 itself...
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