Selmer, Ernst S.

English

University of Bergen

ON SHELLSORT AND THE FROBENIUS PROBLEMERNST S. SELMERFebruary 1987Report No 48' iDepartment of MathematicsUniversity o£ BergenISSN - 0332-54075007 BergenNORWAYON SHELLSORT AND THE FROBENIUS PROBLEMERNST S. SELMERReport No 48 February 1987ON SHELLSORT AND THE FROBENIUS PROBLEMERNST S. SELMERAbst raet.A bound 0(N^ +^k ) for the running time o£ Shellsort, with0 (l°g passes, is proved very simply by application of a Frobeniusbasis with k elements.1 . Shellsort theorFor a description of Shellsort and the number-theoreticalproblem of Frobenius, we refer to two recent papers by Sedgewick [6]and by Incerpi and Sedgewick [4]. In the former paper, Sedgewickimproves the Shellsort bound 0(N to 0(N 4^3 ), using a "resultof Selmer" [7] on a Frobenius basis with three elements. It is ofcourse nice for a number-theoretician to see that his "useless"mathematics can really be applied. In all fairness, however, itshould be made clear (as stated in [7]) that "my" result is reallydue to Hofmeister [3], as a special case of a general and rathercomplicated theorem. What I did in [7] was to give a direct, simpleproof for this special case.Later (but published before [6]), Incerpi and Sedgewick [4]have improved the bound 0(N 4//3 ) to 0(N^ +e ), and further to0(N I+e/y/log N ) . In both cases, they circumvent the standardapproach of Frobenius bases. Their proof of the latter bound isvery nice, and I cannot in any way improve on it. Their proof of1 “f* øthe bound 0(N ) does, however, result in an unnecessarilycomplicated increment sequence. The purpose of the present paperis to describe a simpler method, using a classical result inFrobenius theory.In [4] p. 217, an increasing "base sequence" = , a^,of natural numbers is used to produce the increments hj of aShellsort. A number c of different produet sequences are interleaved, each such sequence consisting of certain produets of celements . We shall see that one produet sequence will suffice.In fact, we can define the increments by = 1 andDepartment of Mathematics, University of Bergen,N-5000 Bergen, Norway2h a . la.J-1 J j > 1a. 0j +c-2 *JFrom thøse, wg form a Frobenius basis with c + 1 elements(1)(say). Und.Gr thG condition (5) below for thG base sequence {a.} ,wg can thcn dGtGrminG Gxplicitly thG Frobenius number(2)This expression is clearly 0(hj +1/c )(i£ each term a i is withina constant factor o£ thG prcvious onc). Just as in Thcorcm 2 o£ [4]wg thcn g g t thG running time of Shellsort bounded by 0 (N^^ + ).2. Frobonius theory.Wg opcratG with a Frobenius basisAlrcady Frobenius rGalizGd that a dGtGrmination of g(Bp) in thegGncral casG was extremely difficult. Hg therefore invited hisaudicncGS to look for good uppGr bounds for g(Bp)ThG first such bound was given alrcady in the 1942 paper byA. Brauer [l] (indeed the first "sGrious" paper to be written on theproblem of Frobenius). Letwith equality if the following condition is satisfiedFor all i = 2, 3, k-1 , + + 1 as a linear* combination of b.,/d. , bVd. , .... b./d. with nonli 2 i * i inegative integer coefficients.(4)Further, Brauer and Seelbinder [2] showed that this condition is alsonecessary for equality in (3).B c + 1 {h j +1 ’ h j+2 ’••• hj +C + 1 }  {b i >b 2 >•••> b c + 1 }r-n C + 1 c + 1g(Bc +l) = i-2 a j +i ' 2 h j +i ' i-1 h j +i, , • • • , , , , • • • , - 1d 0 =O, d 1 =b 1 ; d i = gcd (b 1 ,b 2 ,...,b_L ), 2 i< kThen Brauer showed thatk d. 1(3) g(Bk ) S l b i (“j— '1) ,i= 1 i3The proofs in the two papers quoted are rather complicated.Later, a ceiy simple proof of the above results has been given byRddseth [s].We now apply this to the basis (1), and illustrate in the casec = 3 , hence k = 4 . I£ we write gcd(m,n) = (m,n) , thenWe will thereforeassume that the base sequence satisfies( a i , +r ) - 1 for r-1, 2 , 3 and all ik 1 . In the generalcase, the corresponding condition isThe conditions (4) are trivially satisfied, sinceTn the general case, we similarly haveIt remains to find base sequences satisfying (5). Oneobvious possibility is suggested in [4]: Choose a > 1 , and a ras the smallest prime k a .An interesting alternative stems from Sedgewick f s first paper[6], where his Theorem 6 in fact corresponds to c = 2 above (buthe does not give explicitly) , withV) .< ~ 3- -3. . 1 3. . ~ , fø = a a a1 J J+l J+2 ’ 2 j+l j+2 j+3b 3 = a j +2 a j +3 a j+4 ’ b 4 = a j +3 a j +4 a j +5d 2 = a j+l a j+2 lf a j ’a j +3"* = 1d 3 aj+2 lf also *- a j+l ’aj “ (a j+l >a j +4-* " 1d 4 =1 if also (a jt2 ,a j+3 ) = Ca J+2 ,a j+4 ) = (a j+2 ,a. +5 )(5) gcd(ai , a i+r ) =l, r=l, 2, c, i = 1,2, ....b 3 _ b 2 b 4 ba j +3 a j +4 “ a j+4 HJ ’ =aj +3 a j +4 a j +5 = a j+s 3^b - i b .r 1? = a j +c+ i-i zr > 1= 2 ’ 3 > •••> cI+l J 1We can thus use (3) with equality. Since = 0 andd i-1 /d i = a j+i-2 ’ 1= 2, 3, 4, we getc 'j 4 4g (B 4 3 = .1. a j +i-2b i - .1. b i >I=2 J i=lwhere . The generalization to (2) is immediate.42 i + 1(6) a 3iThe conditions (5) for r = 1, 2 are clearly satisfied, since? i +1 i + ia i+l a i " Z » a i+2 a i = 3 * 2 • In fact, the choice (6) ispossible also for c= 3 , since now a. +3 -a. = 7-2 i+l , and 7|a.This stems from the fact that 2 is not a primitive root of 7 ,2 = 1 , and 3 (mod 7) for all t .For c-4 , we similarly form a^ + 4 -a. = (2^-1)2 =3«5»2Since 2 is a primitive root both of 3 and of 5 , we must now tryto make = 2 m- n , where n (odd) is divisible by 15 . Thesmallest choice of n also possible modulo 7 is n = 45 , so wecan putQuite surprisingly, this choice is possible for all c < 9sinceHere 2 is not a primitive root modulo any of the primes p = 17,31, 73, 127 , and it turns out that always 2 Z % 45 (mod p) forthese four primes.1 0For c=lo , however, we have 2 -1 = 3*11«31 , where 2 isa primitive root of 11 , so we must have 11|n . Trying to combinewith the earlier primes considered, we end up with quite a large n ,For all sorting purposes, it is hardly practical to choosec> 9 . We can then use (7), or (6) for c=2, 3 . I leave it tothe sorting specialists (like Sedgewick) to test whether my aboveprocedure for Shellsort can compete with procedures describedearlier.(7) a. = 2 i+s - 452 5 -l =3l , 2 6 -1 = 3 2 -7, 2 7 -1 =127 , 2 8 -1 =3-5- 1 7 , 2 9 -1 =7- 735REFERENCES3. G. Hofmeister, Zu einem Problem von Frobenius , Kgl. Norske Vid.Selsk. Skrifter 1966 Nr. 5, 1-37.4. J. Incerpi and R. Sedgewick, Improved upper bounds on Shellsort,J. Comput. System Sci. 31 (1985), 210-224.6. R. Sedgc vvick , A new upper bound for Shellsort, J. of Algorithms 7(1 986) , 1 59-1 73.7. E. S. Selmei, On the linear diophantine problem of Frobenius,J. reine angew. Math. 293/294 (1977), 1-17.1* Brauer, On a problem of partitions, Amer. J. Math. 64 (1942),299-312.2. A. Brauer and B. M. Seelbinder, On a problem of partitions. 11,Amer. J. Math. 76 (1954), 343-346.5. 0. J. Rddseth, Two reniarks on. linsar forma in non-nogativsintegers. Math. Scand. 51 (1 982), 1 93-198 .Depotbiblioteket78sd 20 220

On Shellsort and the Frobenius problem

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On Shellsort and the Frobenius problem

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