On Erd˝os-Gallai and Havel-Hakimi algorithms

More documents

Recommendations

Info

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
On the other hand, Here and Var[rn(b)] = Var On Erdős-Gallai and Havel-Hakimi algorithms 27 n−1 i=0 ξi Var[ξi] = n−1 = Var[ξi] + 2 i=0 cov[ξi, ξj] = E[ξiξj] − E[ξi]E[ξj] = 0≤i
Page 1 and 2: Acta Univ. Sapientiae, Informatica,
Page 3 and 4: On Erdős-Gallai and Havel-Hakimi a
Page 7 and 8: 04 if Hn odd 05 L = False 06 return
Page 25: On Erdős-Gallai and Havel-Hakimi a
Page 39: On Erdős-Gallai and Havel-Hakimi a

On Erd˝os-Gallai and Havel-Hakimi algorithms

Create successful ePaper yourself

Delete template?

Save as template?