On Erd˝os-Gallai and Havel-Hakimi algorithms

26 A. Iványi, L. Lucz, T. F. Móri, P. Sótér
Corollary 13 If n ≥ 1, then
and
E(n + 1) E(n)
<
R(n + 1) R(n)
E(n)
lim
n→∞ R(n)
(35)
1
= . (36)
2
Proof. This assertion is a direct consequence of (25) and (27).
The expected value of the number of jumping elements has a substantial
influence on the running time of algorithms using the jumping elements. Therefore
the following two assertions are useful.
The number of different elements in an n-bounded sequence b is called the
rainbow number of the sequence, and it will be denoted by rn(b).
Lemma 14 Let b be a random n-bounded sequence. Then the expectation and
variance of its rainbow number are as follows.
E[rn(b)] = n 1 − 1 − 1
n
= n 1 − 1
+ O(1), (37)
e
1 − 1
n n
Var[rn(b)] = n 1 − 1
n
1 −
n
n
+ n(n − 1) 1 − 2
n
− 1 −
n
1
2n
n
= n
1 −
e
2
+ O(1). (38)
e
Proof. Let ξi denote the indicator of the event that number i is not contained
in a random n-bounded sequence. Then the rainbow number of a random
sequence is n − n−1 i=0 ξi, hence its expectation equals n − n−1 i=0 E[ξi]. Clearly,
E[ξi] = 1 − 1
n (39)
n
holds independently of i, thus
E[rn(b)] = n 1 − 1 − 1
n . (40)
n