|
b
n = (log
a
n)/(log
a
b). First, can you see why this is true? Second, why does this mean that
we usually don’t worry about bases?
2-5. Show that any increasing exponential (Q(k
n
) for k > 1) asymptotically dominates any polynomial
(Q(n
j
) for j > 0).
21
http://wikipedia.org
Chapter 2
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40
2-6. Show that any polynomial (that is, Q(n
k
), for any constant k > 0) asymptotically dominates any
logarithm (that is, Q(lg n)). (Note that the polynomials here include, for example, the square root,
for k = 0.5.)
2-7. Research or conjecture the asymptotic complexity of various operations on Python lists, such as
indexing, item assignment, reversing, appending, and inserting (the latter two discussed in the “Black
Box” sidebar on list). How would these be different in a linked list implementation? What about,
for example, list.extend?
2-8. Show that the expressions Q(f) + Q(g) = Q(f + g) and Q(f) · Q(g) = Q(f · g) are correct. Also, try your
hand at max(Q(f), Q(g)) = Q(max(f, g)) = Q(f + g).
2-9. In Appendix C, you’ll find a numbered list of statements about trees. Show that they are equivalent.
2-10. Let T be an arbitrary rooted tree with at least three nodes, where each internal node has exactly
two children. If T has n leaves, how many internal nodes does it have?
2-11. Show that a directed acyclic graph (DAG) can have any underlying structure whatsoever. Put
differently, any undirected graph can be the underlying graph for a DAG, or, given a graph, you can
always orient its edges so that the resulting digraph is a DAG.
2-12. Consider the following graph representation: You use a dictionary and let each key be a pair
(tuple) of two nodes, with the corresponding value set to the edge weight. For example W[u, v] = 42.
What would be the advantages and disadvantages of this representation? Could you supplement it to
mitigate the downsides?
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