Concrete mathematics : a foundation for computer science

8.5 HASHING 405
(8.32), and we have Mean(S) = (n- 1)/2, Var(S) = (n2 - 1)/12. Hence
n2-1 (m-l)(n-1) =~ (n-1)(6m+n-5)
VP==+ -Jm2 12m2 ’
(8&v)
Once again we have gained the desired speedup factor of 1 /m. If m = n/inn
and n + 00, the average number of probes per successful search in this case
is about i Inn, and the standard deviation is asymptotically (Inn)/&!.
On the other hand, we might suppose that sk = (kH,))’ for 1 6 k 6 n;
this distribution is called “Zipf’s law!’ Then Mean(G) = n/H,, and Var( G) =
in(n + 1)/H,, - n’/Hi. The average number of probes for m = n/inn as
n + oo is approximately 2, with standard deviation asymptotic to G/d.
In both cases the analysis allows the cautious souls among us, who fear
the worst case, to rest easily: Chebyshev’s inequality tells us that the lists
will be nice and short, except in extremely rare cases.
Case 2, continued: Variants of the variance.
We have just computed the variance of the number of probes in a successful
search, by considering P to be a random variable over a probability space
with mn.n elements (h,, . . . , hn; k). But we could have adopted another point
OK, gang, time of view: Each pattern (h, , . . . , h,) of hash values defines a random variable
to put on your
P/h,... , h,), representing the probes we make in a successful search of a
skim suits again.
-Friendly TA
particular hash table on n given keys. The average value of PI (h, , . . . , h,),
A(h,, . . . ,&I = ~p.Pr(Pl(hl,...,h,)=p), (8.98)
p=l
can be said to represent the running time of a successful search. This quantity
A(h,, . . . , h,) is a random variable that depends only on (h, , . . . , h,), not on
the final component k; we can write it in the form
A(h,,... ,hn) = $kPb,,...,hn;k),
k=l
since P/(hl,... , h,) = p with probability
~~=, Pr(P(hl,... ,h,;k)=p) = xE=, m nsk[P(hl,... ,h,;k)=p]
~~=, Prh , . . . , hn; k) ~~=, m nSk
= fsk[P(h I,..., h,;k)=p].
k=l