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Posted 2021-01-14 helpingmips

CIS 675 (Fall 2018) Disclosure Sheet
Name:
HW #
Yes No Did you consult with anyone on parts of this assignment, including other students, TAs, or
the instructor?
Yes No Did you consult an outside source (such as an Internet forum or a book other than the course
textbook) on parts of this assignment?
If you answered Yes to one or more questions, please give the details here:
By submitting this sheet through my Blackboard account, I assert that the information on this sheet is true.
This disclosure sheet was based on one originally designed by Profs. Royer and Older.
Homework 3:
Due October 15 at midnight Eastern time. Submit your solutions typed and
in a pdf document. To receive full credit, explain your answers.
If you collaborate with another student or use outside sources, please list
those students’ names and the URL/title/etc. of the sources that you referred
to. Collaboration is permitted, but you must write up your own solutions.
1. Suppose we have a directed graph with non-negative edge weights. It is
possible for this graph to have multiple shortest paths between two nodes. In
class, we saw how to find the length of the shortest path from a node s to all
other nodes. Suppose that in addition to finding the length of these shortest
paths, we also want to know how many shortest paths there are. Describe
an algorithm to do this.
2. There is a series of lectures that you would like to attend. Each lecture
i has a start time si and an end time ei. To attend a lecture, you must
attend the whole thing, and you can only attend one lecture at a time. Your
goal is to maximize the number of lectures that you attend. Design a greedy
algorithm to find the number of lectures that you can attend.
Hint: Consider the ending times of each lecture. If you pick a lecture that
ends at time t, which other lectures are you still able to attend? If you can
choose a lecture that ends sooner or a lecture that ends later, which gives
you more freedom for later selections? The algorithm that I am looking for
here follows the simple rule of at each step, selecting the lecture that has the
earliest ending time (and of course, don’t pick anything that interferes with
your past choices).
3. In the Grand Tour problem, you are given a list of n cities along with their
pairwise distances (i.e., you know the distance between every two cities). All
distances are positive and symmetric (the distance between A and B is equal
to the distance between B and A). You begin at a specified city C0, and
must perform a ‘tour’, in which you visit every other city exactly once, and
then return to C0. The goal is to minimize the total distance traveled.
Design a reasonable greedy algorithm for solving this problem. Does it al-
1
ways find the correct answer (i.e., the shortest tour)? If yes, prove that it is
always correct, and if no, provide a counterexample.
4. Suppose that you have a weighted, undirected graph, and are trying to
find the length of the shortest path from a node s to some other node t. You
decide to use Dijkstra’s algorithm for this. However, your graph is gigantic,
and running Dijkstra’s on the whole thing will take a very long time. So
instead, you perform two simultaneous searches in parallel. In one search,
you begin at node s, and gradually discover shortest paths from s to other
nodes. In the other search, you begin at node t, and gradually discover
shortest paths from t to other nodes (and because the graph is undirected,
this process also tells you the shortest paths from those nodes to t). In each
iteration, suppose the first search has discovered shortest paths from s to
some set of nodes Rs, and the second search has discovered paths from t to
some set of nodes Rt.
Initially, Rs only contains s and Rt only contains t. This process continues
until Rs and Rt have some overlap; that is, you find some node u that is
present in both Rs and Rt. Then if the distance from s to u is d1 and the distance
from u to t is d2, you report that the distance between s and t is d1+d2.
Will this procedure give you the correct shortest path from s to t? If yes,
prove that it will work. If no, give an example showing where it can fail.
5. Let G be a directed graph. Let s be a ‘start vertex’. An infinite path
of G is an infinite sequence v0, v1, v2, ... of vertices such that v0 = s and for
all i > 0, there is an edge from vi to vi+1. In other words, this is a path of
infinite length. Because G has a finite number of vertices, some vertices in
an infinite path are visited infinitely often.
1. If p is an infinite path, let Inf(p) be the set of vertices that occur
infinitely many times in p. Prove that Inf(p) is a subset of a single
strongly connected component of G.
2. Describe an algorithm to determine if G has an infinite path. Prove
that it is correct.
Extra Credit: A perfect matching in a graph G is a set of edges S such that
every node from G is adjacent to (i.e., appears in) exactly one edge from S
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(no more and no less). The graph might represent students in a classroom:
each student is willing to work with some other students on a partner project,
and we want to figure out how to pair them so that everyone has one partner.
For example, in the graph below, the bolded edges represent a perfect
matching.
Suppose you are given a binary tree T with vertices V and edges E. This
tree is not necessarily a complete binary tree: i.e., every node has at most
2 children, but might only have 0 or 1 child. Describe an ecient
greedy
algorithm to determine if T has a perfect matching. Your algorithm does not
have to work for non-tree graphs.

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