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

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It is easily verified that both remaining integrals coincide with the negative sum of the corresponding residues at the poles with real part greater than 0 . Collecting the residues at z = I (double pole) and z = k E N, k > 2 (simple poles) we finally obtain L 2 .2 QL 2 3 QL [62] '0 - 12 + 6 +1- (Q -1) 2 2Q-1 + 2L - 2L log :2 - 2L E (- I)k k - (4 .6) (k + 1)k(k - 1)(Q 1) Inserting (4 .6) in expression (2 .34) gives for the variance of the horizontal path length the following alternative formula . Corollary 1 . Var(Xn ) _ (Q - 1)2n22 2 log 2 - 1 + L k 5 Q11 1: + 2 (k + 1)k(k _-1 1)(Q k - 1) ) k>2 +(Q Q 1)' k l Q 1)2 + 64(logQ n) } + O(n'+E) - 2 k k>I (Q e > 0 . In a similar manner we obtain Corollary 2 . G) Var(Y,,,) _ n2 _ k I log 2 2 k ( Q1)-1 k> .) + 6 5 (logQ n) + O(n' +E ), (ii) Cov(Xn , Yn ) = (Q - 1)n2 { L (2log2- 2 + 2 1 + 1 1 k>2 Q 1 + 2E (-I)k k>2 (k,+ 1)k(k - 1)(Qk - 1)) Q (Q - 1) 2 k> I Q k - (k + I )Q k , . + b6(log Q n)) + O(n") (Qk - 1 )- Again, Var(CC ) is easily computed from the terms above using (3 .1) . 5. Variations Here we first discuss the analogous notion of the path length where only strictly greater elements count as right-to-left maxima . This is quite natural from a combinatorial point of view . However, since there is no relevance with respect to a computer science problem, we confine ourselves to a sketch and, to ease the presentation, use the analogous notations without explicitly stating them . We write p' in a unique way as p'=amrr, where or E {1, . . .,m}*, -r E {1, . . .,m-
then Hence and or b(p) = b(Qmrr) = b(o) + b(rr) + I-rj + I , b(e) = 0 . P=m(z, y) = pqm-' zyP< m (z, y)P- (z)UImB 2 + Ul L! pqm- I z = pq,j-' z ' am-111 z qJ .V (Z) = p (1 - ) ;>o 91i ID j )mffj -1~ To get the coefficient of z" -I is now very simple . Apart from exponentially small terms, the expectation is with the constants a = En ti p ( 2 ) + pna - pa4, Q 1- 1 and a4 = = a + ,Q . The other variant (equality does not count) leads to
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 and 12: -1 w S< 1(w)S
Page 13 and 14: where The quantity a- +,3 = 7r 2 a
Page 15: 4. Alternative representations for

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