Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 594 — #602
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Chapter 14
Sums and Asymptotics
f g, the “asymptotically equal” relation.
f D o.g/, the “little Oh” relation.
f D O.g/, the “big Oh” relation.
f D ‚.g/, the “Theta” relation.
f D O.g/ AND NOT.g D O.f //.
(b) Indicate the implications among the assertions in part (a). For example,
f D o.g/ IMPLIES f D O.g/:
Problem 14.40.
Recall that if f and g are nonnegative real-valued functions on Z C , then f D O.g/
iff there exist c; n 0 2 Z C such that
8n n 0 : f .n/ cg.n/:
For each pair of functions f and g below, indicate the smallest c 2 Z C , and
for that smallest c, the smallest corresponding n 0 2 Z C , that would establish
f D O.g/ by the definition given above. If there is no such c, write 1.
(a) f .n/ D 1 2 ln n2 ; g.n/ D n. c D , n 0 =
(b) f .n/ D n; g.n/ D n ln n. c D , n 0 =
(c) f .n/ D 2 n ; g.n/ D n 4 ln n c D , n 0 =
.n 1/
(d) f .n/ D 3 sin
C 2; g.n/ D 0:2.
100
c D , n 0 =
Problem 14.41.
Let f; g be positive real-valued functions on finite, connected, simple graphs. We
will extend the O./ notation to such graph functions as follows: f D O.g/ iff
there is a constant c > 0 such that
f .G/ c g.G/ for all connected simple graphs G with more than one vertex:
For each of the following assertions, state whether it is True or False and briefly
explain your answer. You are not expected to offer a careful proof or detailed
counterexample.
Reminder: V .G/ is the set of vertices and E.G/ is the set of edges of G, and G
is connected.