Algorithms and Data Structures for External Memory

Notations and Acronyms 149
PDM parallel disk model (Chapter 2).
SRM simple randomized merge sort (Section 5.2.1).
RCD randomized
(Section 5.1.3).
cycling distribution sort
TPIE Transparent
(Chapter 17).
Parallel I/O Environment
f(k) ≈ g(k) f(k) is approximately equal to g(k).
f(k) ∼ g(k) f(k) is asymptotically equal to g(k), as k →∞:
f(k)
lim
k→∞ g(k) =1.
f(k)=O � g(k) �
f(k) is big-oh of g(k), as k →∞: there exist
constants c>0 and K>0 such that � �
�f(k) � ≤
c � f(k)=Ω
�
�g(k) �, for all k ≥ K.
� g(k) �
f(k) is big-omega of g(k), as k →∞: g(k) =
O � f(k) � f(k)=Θ
.
� g(k) �
f(k) is big-theta of g(k), as k →∞: f(k) =
O � g(k) � and f(k) =Ω � g(k) � f(k)=o
.
� g(k) �
f(k) is little-oh of g(k), as k →∞:
f(k)
lim
k→∞ g(k) =0.
f(k)=ω � g(k) �
f(k) is little-omega of g(k), as k →∞: g(k) =
o � f(k) � .
⌈x⌉ ceiling of x: the smallest integer k satisfying
k ≥ x.
⌊x⌋ floor of x: the largest integer k satisfying k ≤ x.
min{x,y} minimum of x and y.
max{x,y} maximum of x and y.
Prob{R} probability that relation R is true.
logb x base-b logarithm of x; ifbis not specified, we use
b =2.
lnx natural logarithm of x: loge x.