Probabilistic Methods Algorithmic Aspects
Probabilistic Methods Algorithmic Aspects
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Theorem 3. 4 For a finite set V = {vi, . . . , Vn} Q 3?'' with n 5^i;. -0 and \\v, \\ 1. \ 0}. Then we can find in 0(n(P) steps another nontrivial linear dependence a such that for E = {v : a{v) > ^} u* G f: and \E\ n.
Further calculation shows that R{k, k) > ^(1 + o(l))A;2''/^, yielding only a small constant factor improvement. The dependencies being rare, things are brighter when we apply the lemma to the off-diagonal Ramsey numbers R(k, t), with one of the parameters small. We can prove, usi...ng the lemma, that R{3, t)>{^-o{i))ty\ogH and that There is no known better bound for i?(4, t) without using the local lemma. And this still leaves a gap, since the upper bound is i?(4, .
We conclude this section with yet another application of the local lemma. (Strictly speaking, this is a special case of Theorem 4. 1. ) Consider n points and n sets made out of these n points. Suppose, further, that each set has exactly 10 points and that each point is in exactly 10 sets; i. E. The degree of each set and the degree of each point equal 10.
What to read after Probabilistic Methods Algorithmic Aspects?
You can find similar books in the "Read Also" column, or choose other free books by Prasad Tetali to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Theorem 3. 4 For a finite set V = {vi, . . . , Vn} Q 3?'' with n 5^i;. -0 and \\v, \\ 1. \ 0}. Then we can find in 0(n(P) steps another nontrivial linear dependence a such that for E = {v : a{v) > ^} u* G f: and \E\ n.
Further calculation shows that R{k, k) > ^(1 + o(l))A;2''/^, yielding only a small constant factor improvement. The dependencies being rare, things are brighter when we apply the lemma to the off-diagonal Ramsey numbers R(k, t), with one of the parameters small. We can prove, usi...ng the lemma, that R{3, t)>{^-o{i))ty\ogH and that There is no known better bound for i?(4, t) without using the local lemma. And this still leaves a gap, since the upper bound is i?(4, .
We conclude this section with yet another application of the local lemma. (Strictly speaking, this is a special case of Theorem 4. 1. ) Consider n points and n sets made out of these n points. Suppose, further, that each set has exactly 10 points and that each point is in exactly 10 sets; i. E. The degree of each set and the degree of each point equal 10.
What to read after Probabilistic Methods Algorithmic Aspects?
You can find similar books in the "Read Also" column, or choose other free books by Prasad Tetali to read onlineMoreLess
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