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Chapter 0: Q2E (page 18)

Show that, if c is a positive real number, then g(n) = 1 + c + c2 + 路 路 路 + cn is:

(a) 螛(1) if c < 1.

(b) 螛(n) if c = 1.

(c) 螛(cn) if c > 1.

The moral: in big-螛 terms, the sum of a geometric series is simply the first term if the series is strictly decreasing, the last term if the series is strictly increasing, or the number of terms if the series is unchanging.

Short Answer

Expert verified

The sum of the series can be calculated by using the following formula:

sn=a(rn-1)r-1

Where a is the first term of the GP and r is the common ratio.

Step by step solution

01

Simplifying the Geometric Progression

For the given series,

a=1,r=c

Applying the sum of the GP formula, we get:

sn=1cn-1c-1=cn-1c-1

02

Proving the result for c < 1 using the hit and trial method.

let ,c =0 which is less than 1, and put the value of c in sn, we get:

sn=0n-10-1=0-1-1鈭�on=0=1

Therefore, we can say thatg(n) is 螛(1)whenc<1.

03

for c = 1,

Use the limits to prove this,

limc鈫�1cn-1c-1=n.1n-1鈭�limx鈫�axn-anx-anan-1=n.1=n

So,

sn=n鈯�sn=鈯�n

04

for c > 1

For any number c > 1, the last term of the series can be used to find the theta notation of the entire series because that term will be largest. So, the last term of the series is cnwhose theta notation will be螛(cn)

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Most popular questions from this chapter

Here鈥檚 a problem that occurs in automatic program analysis. For a set of variablesx1,......,xn, you are given some equality constraints, of the form 鈥� xi=xj鈥� and some disequality constraints, of the form 鈥� xi鈮�xj.鈥� Is it possible to satisfy all of them?

For instance, the constraints.

x1=x2,x2=x3,x3=x4,x1鈮�x4

cannot be satisfied. Give an efficient algorithm that takes as input m constraints over n variables and decides whether the constraints can be satisfied.

There are many variants of Rudrata鈥檚 problem, depending on whether the graph is undirected or directed, and whether a cycle or path is sought. Reduce the DIRECTED RUDRATA PATH problem to each of the following.(a)The (undirected) RUDRATA PATH problem.(b) The undirected RUDRATA PATH problem, which is just like RUDRATA PATH except that the endpoints of the path are specified in the input.

The tramp steamer problem. You are the owner of a steamship that can apply between a group of port cities V . You make money at each port: a visit to city i earns you a profit of pi dollars. Meanwhile, the transportation cost from port i to port j is cij>0 .You want to find a cyclic route in which the ratio of profit to cost is maximized.

To this end, consider a directed graph G=(V,E) whose nodes are ports, and which has edges between each pair of ports. For any cycle C in this graph, the profit-to-cost ratio is

role="math" localid="1658920675878" r(c)=鈭�i,jicPij鈭�i,jicCij

Let r' be the maximum ratio achievable by a simple cycle. One way to determine r' is by binary search: by first guessing some ratio r , and then testing whether it is too large or too small. Consider any positive r>0 . Give each edge (i,j) a weight of wij=rcij-pj .

  1. Show that if there is a cycle of negative weight, then .
  2. Show that if all cycles in the graph have strictly positive weight, then r<r*.
  3. Give an efficient algorithm that takes as input a desired accuracy 鈭�>0 and returns a simple cycle c for which r(C)3r*-鈭� Justify the correctness of your algorithm and analyze its running time in terms of |V|,鈭� and R=max(i,j)iE(PJCIJ) .

Alice wants to throw a party and is deciding whom to call. She has n people to choose from, and she has made up a list of which pairs of these people know each other. She wants to pick as many people as possible, subject to two constraints: at the party, each person should have at least five other people whom they know and five other people whom they don鈥檛 know. Give an efficient algorithm that takes as input the list of n people and the list of pairs who know each other and outputs the best choice of party invitees. Give the running time in terms of n

The k鈭�SPANNING TREE problem is the following.Input: An undirected graph G=(V,E) Output: A spanning tree of G in which each node has degree 鈮�k, if such a tree exists.Show that for any k鈮�2:

  1. k鈭� SPANNING TREE is a search problem.
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