473: CS Fundamental Algorithms, Spring 2012
Homework 1 (due Tuesday, 23:55:00, January 31, 2012)
This homework contains three problems. Read the instructions for submitting the homework on the course webpage. Read the course policies before starting the homework.
Collaboration Policy: For this homework, Problems 13 can be worked in groups of up to three
students.
Each student individually have to also do quiz 1 online.
Submission guidelines:
For every group of students, one and only one student have to upload the solutions to the
homework to the class moodle page. Together with the solutions to the three questions, the
submitting student of the group also have to upload a text le that contains the netid of the
submitting students. Specically, this text le should be named group.txt. Each student netid
(and nothing else) would be written as is on each line of this text le. As such, an example such
le might look like:
eric2
jeff7
cheng27
Version: 2.11
1. (45 pts.) Briges and cut vertices.
Given a connected undirected graph G = (V, E ), an edge e = (u, v )
c
d
b
e
is called a bridge , or a cut-edge , if removing e disconnects the
graph into two pieces, one containing u and the other containing
i
f
v . A vertex u is called a separating vertex , or cut-vertex , if
g
h
removing u leaves the graph into two or more disconnected pieces;
k
j
note that u does not count as one of the pieces in this denition.
p
l
Your goal in this problem is to develop a linear time algorithm to
nd all the bridges and cut-vertices of a given graph using DFS.
q
rs
t
Let T be a DFS tree of G (note that it is rooted at the rst node
from which DFS is called). For a node v we will use the notation Tv to denote the sub-tree
of T hanging at v (Tv includes v ).
(A) (4 pts.) In the graph shown above, identify all the bridges and cut-vertices (i.e., list all
the bridge edges and cut vertices).
(B) (5 pts.) Prove that any bridge of G has to be a tree edge in the tree generated by every
DFS(G). Recall the property of DFS in undirected graphs. Why does this show that
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the maximum number of bridges is n 1? What is a graph that achieves this bound?
(C) (8 pts.) Suppose e = (u, v ) is a tree-edge in DFS(G) with pre(u) < pre(v ) (that means
u is the ancestor of v ). Prove that e is a bridge if and only if (need to prove both
directions) there is no edge from any node in Tv to either u or any of its ancestors.
(D) pts.) (5 For each node u dene:
low(u) = min
(E)
(F)
(G)
(H)
pre(u)
pre(w) where (v, w) is a back edge for some descendant v of u.
Give a linear time algorithm that computes the low value for all nodes by adapting
DFS(G). Give the altered pseudo-code of DFS(G) to do this. There is no need to prove
that your code is correct.
(5 pts.) Give a linear time algorithm that identies all the bridges of G using the low
values and the steps above. Specically, provide pseudo-code for a linear time algorithm
to do so. There is no need to prove that your code is correct.
(5 pts.) Prove that the root of the DFS tree is a cut-vertex if and only if it has two or
more children.
(8 pts.) Prove that a non-root vertex u of the DFS tree T is a cut-vertex if and only if
it has a child v such that no node in Tv has a backedge to a proper ancestor of u (that
is, an ancestor of u which is not u itself).
(5 pts.) The above two properties can be used to nd all the cut-vertices in linear time.
Give the pseudo-code for a linear time algorithm to do so. There is no need to prove
that your code is correct.
(It is instructive to run DFS(G) on the example graph and compute the pre values and the
lowvalues for each node.)
2. (25 pts.) Semi-strongly connected?
Let G = (V, E ) be a directed graph. Dene a relation R on the nodes V as follows: uRv i
u can reach v or v can reach u.
(a) (5 pts.) Is R an equivalence relation? If yes, give a proof, otherwise give an example
to show it is false.
(b) (20 pts.) Call G semi-strongly-connected if for every pair of nodes (u, v ), uRv . Give a
linear time algorithm to determine if G is semi-strongly-connected. By linear time we
mean an algorithm that runs in time O(m + n) where m = |E | and n = |V |.
3. (30 pts.) Diameter and low-degree vertex
(a) (22 pts.) Let G = (V, E ) be an undirected connected graph. Suppose G has a pair
of nodes s, t that are distance d apart. Show that there is a vertex v G such that
the degree of v is at most 6n/d where n = |V |. Hint: Consider BFS(s) and show the
existence of three adjacent layers with total of at most 6n/d vertices.
(b) (8 pts.) Show that for directed graphs the above is not true. More precisely, give an
example of a directed graph on n nodes such that there is a pair of nodes s, t such that
distance from s to t is (n) and for each node v in the graph, both the in-degree and
out-degree are (n).
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