The path length of random skip lists - Institut für Analysis und ...

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With a = Em>1 Q ,"- 1 . In order to get the asymptotic behaviour of [z'- ' ]H(z) for n oo we extend the contour of integration to a large rectangle and shift the left side to the line Re z = 1 + E, E > 0. Then [z" -1 ]H(z) equals the sum of residues at the additional poles included in the contour with an error term of order O(n 1+ E) (compare (3] for technical details of the estimate of the error term) . In our instance the additional poles originate from the second integral, and they are located at z = 2 (triple) resp . z = 2 + Xk, k E Z \ {0} (simple) . The computation of the residues can be performed by hand, or, more preferably, e .g . by Maple, and we find for the residue at z = 2 2(Q - 1)2 + Hn2) 1 n + 1 ~H,2,_i Hn 1 (1 + 2 2 L 2 L2 L ) + 3 + L2 - 2(a +,3) + 2L + (Q 1 I )L }' (2 .31) where L = log Q, H,,, resp . H(,,,2) are the harmonic numbers of first resp . second order, and ,0= E 1 2 . m>1 (Qm- 1 ) The asymptotic equivalent of (2 .31) is 2 g 2 2(Q - 1) 2 n12 Q n + n 2 logQ n C L 2 L l 2 1 1 'Y 'Y 'Y2 ~2 1 1 l +n (6+L2-a-~-2L L2+ 2L2+ 12L2+ 4L + 2(Q - IL) ) +O (n log e n) . (2.32) I The first order poles at z = 2 + Xk, k E Z \ {0}, give a contribution of the form n 2 68 (log Q n) + 0(n), (2 .33) where 58(x) is a continuous periodic function of period 1 and mean zero . The Fourier coefficients of 68(x) can be obtained explicitly from the residues at the above . mentioned poles . They can be used to show that the amplitude of 68(x) is very small (because of the exponential decrease of the F-function along . vertical lines) . Since from the practical .point of view 6 8 (x) has almost no influence on the variance, we omit these calculations here . Combining (2 .32), (2 .33) and Proposition 1 we get the asymptotic equivalent of Var(Xn,) according to (2 .11) . The reader should take notice of the fact that the term of order n2 in E(X n )2 contains the square 6 2 (logQ n) of the fluctuating term 6 1 (logQ n), where 61(x) has mean zero . Therefore we have 2 Var(Xn,) = (Q - - 2(a +)(3) + [ 1)2n2 2L + 12 2L + 6L2 (Q _ 1 1)L 2L 6 1 0 +n 2 64(logQ n) + 0(n 1 +6 ), E > 0, (2 .34) where [ 62 ] o is the mean of 62 (x), and 64 (x) is a small periodic function of period 1 and mean zero .
where The quantity a- +,3 = 7r 2 a + Q = li(L) = 6L 2 can be rewritten as 1 1 47x2 47r2 2L + 24 - L2 It ( L )' (2 .35) ekx h(x) _ k (ekx - 1)2 (2 .36) Equation (2 .35) follows from the functional equation for the Dedekind r7-function and is proved e .g . in [6] . The alternative representation for a +,3 given by (2 .35) is extremely useful for numerical purposes, since the term i2 h( 4L2 ) is very small for "reasonable" values of q resp . Q = q -1 . The term [6fl 0 can be computed explicitly from the Fourier expansion in Proposition 1 . Since (compare e .g . [18]) JF(zy)I = Y 7 sinh iry (2 .37) we find 1 2 L2 [bl ] o = al (2 .38) with a, defined in the theorem . Therefore the proof of Theorem I is complete . 3. The total cost As already mentioned in the Introduction, the total cost Cn is the sum of the horizontal path length X,,, and the vertical path length Y,-, where Y,, is n times the height of the skiplist . The variance of the latter parameter was already studied in [5] resp . [17] and we have the asymptotic equivalents for E(Yn ) resp . Var(Yn ) as presented in the Introduction . It is an easy consequence that E(C n ) = E(X n ) + E(Yn) fulfills the asymptotic relation from Proposition 1 (ii) . In order to compute Var(Cn ) = Var(Xn + Yn ) we use Var(Cn ) = Var(Xn ) + Var(Yn ) + 2Cov(Xn , Yn ), (3 .1) where the covariance Cov(X n , Yn ) = E(X.Yn ) - E(X n )E(Yn ) will be studied in the sequel . We start with the term E(XnYn ) . The generating function of n+ l E(X n+ 1 Yn+l) is E F=m (z) (mProb(G < m) + 1: kProb(G = k)) , (3 .2) M>1 k>m where G stands for the random variable producing the last elment of the skip list (a,, . . .) . (Observe that the random variables in question were assumed to be i .i .d .) From (4 .2) we get immediately the expression M> 1 m F=m(z) m + q = (F(z) - (1 - qm) F
Page 1 and 2: Acta Informatica 31, 775-792 (1994)
Page 3 and 4: The path length of random skip list
Page 5 and 6: q 0 .1 0 .2 0 .3 0.31 . . . 0 .4 0
Page 7 and 8: The path length of random skip list
Page 9 and 10: Lemma 1 . Let A(z) be a formal powe
Page 11: -1 w S< 1(w)S
Page 15 and 16: 4. Alternative representations for
Page 17 and 18: then Hence and or b(p) = b(Qmrr) =

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