gq(z) = −eb(z)e−z+ δ1eb(z/2)e−z− δ1e−z/2, and the Poisson generating function of the second moment
M˜q(z) =
(2 ˜Mq0(z/2) + 2 ˜Lq0(z/2)2+ 4(1 − e−z/2) ˜Lq0(z/2) + 1 + ˜gq(z), if q = (?, . . .);
M˜q0(z/2) + 2(1 − e−z/2) ˜Lq0(z/2) + 1 + ˜gq(z), if q = (S, . . .).
Moreover, we have for the poissonized variance
V˜q(z) = δqV˜q0(z/2) + ˜hq(z),
where ˜hq(z) was defined in the introduction. The remaining analysis now proceeds from these functional equational equations as in the previous section.
4 k-dimensional Bucket Digital Search Trees
Again, we start from a distributional recurrence for Xq,nwhich for the current situation reads as follows
Xq,n+b
=d
(Xq0,In+ Xq∗0,n−In+ 1, if q = (?, . . .);
Xq0,In+ 1, if q = (S, . . .), (n ≥ 0),
where the notation is as before and initial conditions are given by Xq,0= 0 and Xq,1= · · · = Xq,b−1= 1.
From here, we can in principle proceed as before. However, we will see that the equation satisfied by the Poisson generating function is more complicated. More precisely, we have to cope with a differential-functional equation compared with the differential-functional equation from the trie case. Here, we will first use Laplace transform to get rid of the differential operator. Then, after suitable normalization, we will be able to proceed as before. This combined use of Laplace and Mellin transform was introduced in [3] and we direct the interested reader to that paper for more details concerning technicalities.
Poissonization. We again define
P˜q(z, y) = e−zX
n≥0
E(eXq,ny)zn n!. Then,
b
X
j=0
b j
P˜q(z, y) = eyP˜q0(z/2, y)δ1.
Taking derivatives yields for the Poisson generating function of mean and second moment (denoted as before)
b
X
j=0
b j
L˜(j)q (z) = δ1L˜q0(z/2) + 1 (7)
12 Michael Fuchs and
b
X
j=0
b j
M˜q(j)(z) =
(2 ˜Mq0(z/2) + 4 ˜Lq0(z/2) + 2 ˜Lq0(z/2)2+ 1, if q = (?, . . .);
M˜q0(z/2) + 2 ˜Lq0(z/2) + 1, if q = (S, . . .).
The first step is again to show that ˜Lq(z) and ˜Mq(z) are JS-admissible. Therefore, we need the follow-ing result which is proved by a reduction to the trie case (see [3] for similar results).
Proposition 4 Assume that we have
b
X
j=0
b j
f˜q(j)(l)(z) = δl+1f˜q(l+1)(z/2) + ˜gq(l)(z), (0 ≤ l < k),
where all involved functions are entire and0 at z = 0. Moreover, assume that ˜gq(l)(z) is JS-admissible for0 ≤ l < k. Then, ˜fq(l)(z) is JS-admissible for 0 ≤ l < k.
From this it then follows as in the trie case that ˜Lq(z) and ˜Mq(z) are JS-admissible.
Next, we consider the poissonized variance ˜Vq(z) = ˜Mq(z) − ˜Lq(z)2. An easy computation proves that
b
X
j=0
b j
V˜(j)(z) = δ1V˜q0(z/2) + ˜hq(z),
where ˜hq(z) was defined in the introduction. Then, from the JS-admissibility of ˜Lq(z) and ˜Mq(z), we obtain as for tries the following result.
Proposition 5 As n → ∞,
Var(Xq,n) = ˜Vq(n) + O
n2u/k−1 .
Asymptotic Expansion of ˜Lq(z). Again, we first consider the mean value. Note that due to the differ-ential operator it is not possible to iterate (7). Therefore, we first have to get rid of the differdiffer-ential operator which is achieved by applying Laplace transform. This yields
(s + 1)bL [˜Lq(z); s] = 2δ1L [˜Lq0(z); 2s] + (s + 1)b−1/s. (8) Next, we normalize with Q(s) from the introduction. Therefore, set ¯Lq(s) =L [˜Lq(z); s]/Q(−s)band G(s) = (s + 1)¯ b−1/(Q(−2s)bs). Then,
L¯q(s) = 2δ1L¯q0(2s) + ¯G(s).
Now, we can iterate and obtain
L¯q(s) = 2k+uL¯q(2ks) +
k−1
X
l=0
2lδ1· · · δlG(2¯ ls).
Observe that this is a similar functional equation as in the trie case. Hence, we can proceed as before.
Thus, we again apply Mellin transform. First, note that the Mellin transform of ¯Lq(s) exists in a non-trivial strip. Moreover, due to the rapid growth of Q(s) at infinity (see [3]), the Mellin transform of ¯G(s)
Partial match retrievals in digital trees 13 exists in the strip h1, ∞i. Applying Mellin transform yields
M [¯Lq(s); ω] = M [ ¯G(s); ω]
1 − 2k−ωk+u
k−1
X
l=0
δ1· · · δl2l−ωl, <(ω) ∈ h1 + u/k, ∞i.
Next, by inverse Mellin transform and shifting the line of integration to the left, we obtain
L¯q(z) ∼
∞
X
r=−∞
crs−1−u/k−2πir/(kL), (s → 0).
Since, Q(−s)b = 1 + O(|s|) as s → 0, the same asymptotic expansion holds forL [˜Lq(z); s] as well.
Finally, by formal inverse Laplace transform (see [3] for technical details justifying this step), we have L˜q(z) ∼ zu/kP1(log2z1/k), (z → ∞),
where P1is a computable, 1-periodic function. A more careful analysis shows that the above asymptotic expansion holds uniformly for |z| → ∞ and | arg(z)| ≤ π/2 − .
Asymptotic Expansion of ˜Vq(z). Here, we proceed as above and obtain
V¯q(s) = 2k+uV¯q(2ks) +
k−1
X
l=0
δ1· · · δl2lH¯q(l)(2ls),
where ¯Vq(s) =L [ ˜Vq(z); s]/Q(−s)band ¯Hq(l)(s) = (L [˜hq(l)(z); s] + p(s))/Q(−2s)bwith
p(s) =(1 + s)b−1+ (−1)b
s + 2 .
Now, observe that
˜hq(z) =
(O(z2u/k−2), if z → ∞;
O(1), if z → 0+,
where the first bound follows from the bound of the previous paragraph (which we are allowed to dif-ferentiate due to Ritt’s theorem; see [9]) and the second bound is trivial. This together with the growth properties of Q(s) then in turn yields
H¯q(s) =
(O(1/s), if s → ∞;
O(s−β), if s → 0+,
where β > 0 is an arbitrary constant. Consequently, the Mellin transform of ¯Hq exists in the strip h1, ∞i.
The remaining proof of Theorem 2 proceeds then as in the previous paragraph.
14 Michael Fuchs Simplification of the Fourier Coefficients for b = 1. First, by iteration of (8),
L [˜Lq(z); s] =1 s
X
j≥0
δ∗q,j
(s + 1) · · · (2js + 1),
where δ∗q,j =Qj
l=1δl. Next, by partial fraction expansion, L [˜Lq(z); s] =1
s X
j≥0 j
X
l=0
(−1)j−l2−(j−l+12 )δq,j∗ (2ls + 1)QlQj−l =1
s X
l≥0
δ¯∗q,l (2ls + 1)Ql, where
δ¯q,l=X
j≥0
(−1)j2−(j+12 ) Qj
δq,j+l.
Consequently, by inverse Laplace transform L˜q(z) =X
l≥0
δ¯q,l Ql
(1 − e−z/2l).
This implies
L˜0q(z) =X
l≥0
δ¯q,l
2lQl
e−z/2l, L˜0q(z)2= X
l,h≥0
δ¯q,l¯δq,h
2l+hQlQh
e−z/2l−z/2h.
Plugging this into (3) (note that for b = 1, we have ˜hq(z) = ˜L0q(z)2) and using 1
Q(−2s)= 1 Q(1)
X
j≥0
(−1)j2−(j2) Qj(s + 2−j) together with some standard computations proves the claim.
5 Conclusion
In this paper, we gave a new and simpler approach to the variance of partial match queries in k-dimensional bucket digital trees. Our method used standard tools from the analysis of algorithm such as poissonization-depoissonization and Mellin transform. The main simplification comes from the poissonized variance which incorporates cancellations at a much earlier stage compared to previous derivations.
Our approach allowed us to derive asymptotic expansions of the variance in k-dimensional bucket tries, k-dimensional bucket PATRICIA tries and k-dimensional bucket digital search trees. In all cases, the variance is asymptotic to nu/kP (log2n1/k) where P is a 1-periodic function. Since the mean has the same order, our results show that the cost of partial match retrievals is concentrated around the mean.
We conclude by pointing out that even though we only derived the main term in the asymptotic expan-sions, our approach can be straightforwardly applied to derive longer asymptotic expanexpan-sions, too.
Partial match retrievals in digital trees 15
Acknowledgements
We are indebted to the anonymous referees for many helpful comments. Financial support of the National Science Counsel is acknowledged as well.
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