|
 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| • |
We
know each node pays an
|
|
amount
£ the number of
|
|
|
|
ranks
in its rank group
|
|
|
| • |
How
many ranks in a rank
|
|
|
group?
|
|
|
|
– |
A rank
r is in rank-group g iff
|
|
|
|
log*r =
g
|
|
|
|
– |
This
is equivalent to saying
|
|
|
tower(g-1)
< r £ tower(g)
|
|
|
|
– |
Thus
the number of ranks in
|
|
|
|
rank-group
g is
|
|
|
tower(g)
- tower(g-1)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| • |
Thus
a node in rank-group g
|
|
|
|
pays
at most tower(g)-
|
|
|
|
tower(g-1)
credits
|
|
|
| • |
How
many nodes in rank-
|
|
|
group
g?
|
|
|
|
– |
There are n/2r nodes of rank
|
|
|
r (see
the earlier lemma)
|
|
|
|
– |
# nodes
in rank-group g
|
|
|
|
= n/2tower(g-1)+1
+ n/2tower(g-1)+2
|
|
|
+... + n/2tower(g)
|
|
|
<
n/2tower(g-1) (1/2 + 1/4 +...)
|
|
|
= n/2tower(g-1)
|
|
|
=
n/tower(g)
|
|
|
|