A158466 Numerators of EH(n), the expected value of the height of a probabilistic skip list with n elements and p=1/2.

0, 2, 8, 22, 368, 2470, 7880, 150266, 13315424, 2350261538, 1777792792, 340013628538, 203832594062416, 131294440969788022, 822860039794822168, 177175812995012739374, 231553634961214157747264, 1813465925343969651214825522, 14983458468103810854318443432

Offset

0,2

Comments

A probabilistic skip list is a data structure for sorted elements with O(log n) average time complexity for most operations. The probability p is a fixed internal parameter of the skip list.

n fair coins are flipped in a single toss. Those that show tails are collected and reflipped in another single toss. The process is repeated until all the coins show heads. H(n) is the discrete random variable that denotes the number of tosses required. P(H(n)<= k) = (1-(1/2)^k)^n. - Geoffrey Critzer, Dec 13 2009

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

P. V. Poblete, J. I. Munro and T. Papadakis, The binomial transform and the analysis of skip lists, Theor. Comput. Sci. 352, 1 (Mar. 2006), 136-158.

William Pugh, Skip lists: a probabilistic alternative to balanced trees, Communications of the ACM, v.33 n.6, 668-676, June 1990

Wikipedia, Skip list

Formula

EH(n) = Sum_{k>0} k * ((1-(1/2)^k)^n - (1-(1/2)^(k-1))^n).

EH(n) = -Sum_{k=1..n} (-1)^k * C(n,k) / (1-(1/2)^k).

Example

0, 2, 8/3, 22/7, 368/105, 2470/651, 7880/1953, 150266/35433, 13315424/3011805, 2350261538/513010785, 1777792792/376207909 ... = A158466/A158467

Maple

EH:= n-> -add((-1)^k *binomial(n, k) /(1-(1/2)^k), k=1..n):
seq(numer(EH(n)), n=0..20);

Mathematica

Table[Sum[x*((1-2^(-x))^n-(1-2^-(x-1))^n), {x, 1, Infinity}], {n, 0, 20}] (* Geoffrey Critzer, Dec 13 2009 *)

Crossrefs

Denominators of EH(n): A158467.

Cf. A278327.

Sequence in context: A238530 A303287 A321068 * A065694 A178129 A203298

Adjacent sequences: A158463 A158464 A158465 * A158467 A158468 A158469

Keyword

frac,nonn

Author

Alois P. Heinz, Mar 19 2009

Status

approved