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Chapter 0: 11 E (page 11)

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.

Short Answer

Expert verified

2. Enumerating the source and destination solves the given problem.

Step by step solution

01

Explain the RUDTATA PATH problem

Consider a directed or undirected graph, In which there exists a path that visits each vertex exactly once. The path is known as RUDRATA PATH.

02

Reduce undirected RUDTATA PATH problem

Consider the directed graph G, in which each vertex v is categorized into new and old vertices i+o. In-degree vertices are represented as i and the out-degree vertices are represented as o. Consider that every directed edge is connected to the vertex that is transformed intoi+onew undirected edge of the vertex.

Assume that the vertex connected to the incoming edges is called the incoming vertex and the vertex connected to the outgoing edges is called the outgoing vertex. Connect each incoming vertex to undirected edges.

Thus, the graph G is transformed into G'. Consider the following example,

Therefore, the undirected graph is reduced to RUDATA PATH.

03

The undirected RUDRATA (s,t)− PATH problem.

Consider the directed graph G, in which each vertex is categorized into new and old vertices . In-degree vertices are represented as and the out-degree vertices are represented as . Consider that every directed edge is connected to the vertex that is transformed into new undirected edge of the vertex.

Based on the above graph information, enumerate the path as . Consider the vertex source as , and the vertex that ends the cycle is t.

Therefore, enumerating the source and destination solves the given problem.

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Most popular questions from this chapter

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.
  2. k鈭� SPANNING TREE is NP-complete. (Hint: Start with k=2 and consider the relation between this problem and RUDRATA PATH.)

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

Consider the following game. A 鈥渄ealer鈥� produces a sequence s1路路路sn of 鈥渃ards,鈥� face up, where each card si has a value vi. Then two players take turns picking a card from the sequence, but can only pick the first or the last card of the (remaining) sequence. The goal is to collect cards of largest total value. (For example, you can think of the cards as bills of different denominations.) Assume n is even. (a) Show a sequence of cards such that it is not optimal for the first player to start by picking up the available card of larger value. That is, the natural greedy strategy is suboptimal. (b) Give an O(n2) algorithm to compute an optimal strategy for the first player. Given the initial sequence, your algorithm should precompute in O(n2) time some information, and then the first player should be able to make each move optimally in O(1) time by looking up the precomputed information.

Question: 0.1. In each of the following situations, indicate whether 蹿=翱(驳),辞谤蹿=惟(驳),or both (in which case f=鈯�(g))

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.
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