Expected Parallel Time And Sequential Space Complexity of Graph And Digraph Prob

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Expected Parallel Time And Sequential Space Complexity of Graph And Digraph Prob
John Reif
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1. At the end of (a) of phase i of stage 2, each large super- vertex has size at least s(i)/(6 log n) .
We now will show LEMMA 3. 1. If a supervertex has size ^s(i), then it is declared large idth -(S-1) probability at least 1-n -15- Proof. Let p(i) be the number of cells of the table of the supervertex, which stay at value after (a) of stage 2. We know that s(i) is increasing with 4 i and s (1) is 21og n. The probability that the particular supervertex fails to be declared large is prob{There
...is a particular memory cell staying at value 0} = (l-l/(s(i)/(B log n)))"^^' ^ e"^' ^°^ " so prob{p(i) = O} ^ n-n = n "" o -2 We now inductively assume that for iJJl, with probability >l-n, "At least — r nodes are in large supervertices" . The basis, i = l, was proved previously.
2^ Let N. Be the number of large supervertices at end of (a) of phase i.
Fact 1 . For any n, p, 3 with n>0, l^p>0 and

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