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Chapter 3: Q16E (page 109)

Suppose a CS curriculum consists of n courses, all of them mandatory. The prerequisite graph G has a node for each course, and an edge from course v to course w if and only if v is a prerequisite for w. Find an algorithm that works directly with this graph representation, and computes the minimum number of semesters necessary to complete the curriculum (assume that a student can take any number of courses in one semester). The running time of your algorithm should be linear.

Short Answer

Expert verified

The linear algorithm that computes the minimum number of semesters necessary to complete the curriculum in linear time is as follows:

Input: Graph G

Output: number of required semesters.

Add n vertices that has din=0to the first queue

n=din-1

if din-1=0

add nto the second queue

process second queue

if semester coincides with class schedule

return n

Step by step solution

01

Explain the given information

Consider the CS curriculum consists of n courses. The prerequisite graph G has node for each course and an edge from course v to course w, if and only if v is a prerequisite forw

02

Step 2: Give the linear time algorithm that computes the minimum number of semesters

The linear algorithm that computes the minimum number of semesters necessary to complete the curriculum in linear time is as follows:

Input: Graph G

Output: number of required semesters.

Add n vertices that has dn=0to the first queue

n=dn-1

if dn-1=0

add n to the second queue

process second queue

if semester coincides with class schedule

return n

The above algorithm runs in the linear-time.

Therefore, The linear algorithm that computes the minimum number of semesters necessary to complete the curriculum in linear time has been provided.

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Most popular questions from this chapter

Rewrite the explore procedure (Figure 3.3) so that it is non-recursive (that is, explicitly use a stack). The calls to pre visit and post visit should be positioned so that they have the same effect as in the recursive procedure.

Biconnected componentsLet G=(V,E) be an undirected graph. For any two edgese,e'∈E,, we’ll saye:e'if eithere=e'or there is a (simple) cycle containing both e and e'.

a. Show that : is an equivalence relation (recall Exercise 3.29) on the edges.

The equivalence classes into which this relation partitions the edges are called the biconnected components of G . A bridge is an edge which is in a biconnected component all by itself.

A separating vertexis a vertex whose removal disconnects the graph.

b. Partition the edges of the graph below into biconnected components, and identify the bridges and separating vertices.

Not only do biconnected components partition the edges of the graph, they almost partition the vertices in the following sense.

c. Associate with each biconnected component all the vertices that are endpoints of its edges. Show that the vertices corresponding to two different biconnected components are either disjoint or intersect in a single separating vertex.

d. Collapse each biconnected component into a single meta-node, and retain individual nodes for each separating vertex. (So there are edges between each component node and its separating vertices.) Show that the resulting graph is a tree.

DFS can be used to identify the biconnected components, bridges, and separating vertices of a graph in linear time.

e. Show that the root of the DFS tree is a separating vertex if and only if it has more than one child in the tree.

f. Show that a non-root vertex v of the DFS tree is a separating vertex if and only if it has a child v' none of whose descendants (including itself) has a back edge to a proper ancestor of v .

g. For each vertex u define:

Iowu=minpreuprew

Where (v,w) is a back edge for some descendant v of u.

(h) Show how to compute all separating vertices, bridges, and biconnected components of a graph in linear time.

Run the strongly connected components algorithm on the following directed graphs G. When doing DFS on GR: whenever there is a choice of vertices to explore, always pick the one that is alphabetically first.

In each case answer the following questions.

(a) In what order are the strongly connected components (SCCs) found?

(b) Which are source SCCs and which are sink SCCs?

(c) Draw the “metagraph” (each meta-node is an SCC of G).

(d) What is the minimum number of edges you must add to this graph to make it strongly connected

You are given tree T=(V,E) along with a designated root node r∈V. The parent of any node V≠r, denoted p(V), is defined to be the node adjacent to v in the path from r to v . By convention, p(r)=r. For k>1, define pk(v)pk-1(pv)andp1(v)=p(v)(so pk(v)is the k th ancestor of v ). Each vertex v of the tree has an associated non-negative integer label l(v). Given a linear-time algorithm to update the labels of all the vertices T according to the following rule: lnew(v)=l(plvv).

On page 102, we defined the binary relation “connected” on the set of vertices of a directedgraph. Show that this is an equivalence relation(see Exercise 3.29), and conclude that it partitions the vertices into disjoint strongly connected components.
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