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

Author

Alois P. Heinz, Mar 19 2009

Keywords

• frac
• nonn

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

Examples

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

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

Crossrefs

Denominators of EH(n): A158467.

Cf. A278327.

Programs

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 *)

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).