the one that is alphabetically first. Classify each edge as a tree edge or back edge, and give the
pre and post number of each vertex.
A
B
C
Basic Algorithms — Spring 2017 — Problem Set 3
Due:
F
E 19
D March
1. DFS mechanics. For each graph, show the “DFS forest” resulting from an execution of DFS. Whenever
there is a choice of vertices, choose the one that is alphabetically first. Identify the cross, forward, and back
H
edges, and label each vertex with its discovery Gand finishing
time.I
3.2. Perform depth-first search on each of the following graphs; whenever there’s a choice of vertices,
A is alphabetically first. Classify each edge
pick the one that
(i)
(ii)as a tree edge, forward edge, back
edge, or cross edge, and give the pre and post number of each vertex.
D
C
B (a)
E
A
G
A
(b)
B
E
C
H
F
D
J
I
A
B
B
C
H
D
E
FC
G
G
H
ID
In each case answer
questions.
H
G
F the following
F
E
2. DFS mechanics. Run the DFS-based topological sort algorithm on the following graph. Whenever there is
(a) In
order
are the
components
(SCCs)
a choice
of what
vertices,
choose
the strongly
one that connected
is alphabetically
first. Show
thefound?
“DFS forest”, including discovery
3.3.
Run
the
DFS-based
topological
ordering
algorithm
on
the
following
Which
are source
SCCs
and which
are sink
SCCs?of the vertices. graph. Whenever you have
and (b)
finishing
times.
Give the
resulting
topological
ordering
a choice of vertices to explore, always pick the one that is alphabetically first.
(c) Draw the “metagraph” (each meta-node is an SCC of G).
(d) What is the minimum Anumber of edges you
G graph to make it strongly
D must add to this
connected?
C
F
3.5. The reverse of a directed graph G = (V, E) is another directed graph GR = (V, E R ) on the same
H E}.
R
vertex set, but with all edgesBreversed; that is, E E
= {(v, u) : (u, v) ∈
Give a linear-time algorithm for computing the reverse of a graph in adjacency list format.
(a) Indicate the pre and post numbers of the nodes.
(i)
B
G
3.6.
In an
undirectedRun
graph,
the degree d(u) of a
vertex u isalgorithm
the number
of neighbors
u has,
equiv3.
DFS
mechanics.
the strongly-connected
components
on this
graph. Show
the or
“DFS
forest”,
(b) number
What areofthe
sources
and sinks
of the graph?
alently,discovery
the
edges
incident
Inofa the
directed
graph, we Draw
distinguish
between
the
including
and finishing
times,
forupon
both it.
runs
DFS algorithm.
the resulting
component
What
topological
is of
found
by the
indegree
d(c)
which
is with
theordering
number
edges
intoalgorithm?
u, and the outdegree dout (u), the number of
in (u),
graph.
As usual,
when
faced
a choice
A
(ii) among vertices, pick the one that is alphabetically first.
edges leaving
u. many topological orderings does this graph have?
(d) How
D
C
!
A
B = 2|E|.C
(a)3.4.
Show
an undirected
d(u)
u∈Valgorithm
Runthat
the in
strongly
connectedgraph,
components
on the following directed graphs G. When
R
(b)
Use
part
(a)
to
show
that
in
an
undirected
graph,
there
be an
evenpick
number
of that
vertices
doing DFS on G : whenever there is a choice of vertices
tomust
explore,
always
the one
is
E
alphabetically
first.
whose
degree
is
odd.
F
F
E
(c) Does a similar statement hold forDthe number
of vertices
with odd indegree in a directed
101
graph?
H
J
3.7.
A bipartite graph is a graph G = (V, E) whose
vertices
can be
H
G
I partitioned into two sets (V = V 1 ∪V2
I
and V1 ∩ V2 = ∅) such that there are no edges between vertices in the same set (for instance, if
u, v ∈ V1 , then there is no edge between u and v).
each case answer the following
questions.
4. Dijkstra mechanics. Run Dijkstra’s algorithm on the following graph, starting from vertex A. Whenever
(a)
Give
aconnected
linear-time
algorithm
toone
determine
whether an undirected
is bipartite.
a) In what order are
the strongly
components
(SCCs)
found?
there
is a choice
of vertices,
choose the
that
is alphabetically
first. Show graph
the state
of the algorithm just
(b)
There
are
many
other
ways
to
formulate
this
property.
For
instance,
an
undirected
graph
after
each
vertex
is
removed
from
the
queue.
b) Which are source SCCs and which are sink SCCs?
is
bipartite
if
and
only
if
it
can
be
colored
with
just
two
colors.
c) Draw the “metagraph” (each meta-node is an SCC of G).
Prove the
following
an undirected
graph
bipartite if and only if it contains
d) What is the minimum number
of edges
you formulation:
must add to this
graph to make
it is
strongly
no cycles of odd length.
connected?
(c) At most how many colors are needed to color in an undirected graph with exactly one odde reverse of a directed graph G = (V, E) is another directed graph GR = (V, E R ) on the same
length cycle?
R
rtex set, but with all edges reversed; that is, E = {(v, u) : (u, v) ∈ E}.
ve a linear-time algorithm
for computing
thehave
reverse
of acontainers
graph in adjacency
list format.
3.8. Pouring
water. We
three
whose sizes
are 10 pints, 7 pints, and 4 pints, re-
Theof7-pint
andu 4-pint
containers
start outu full
an undirected graph, spectively.
the degree d(u)
a vertex
is the number
of neighbors
has,oforwater,
equiv- but the 10-pint container is
initially
empty.
We
are
allowed
one
type
of
operation:
pouring
thethe
contents of one container into
ently, the number of edges incident upon it. In a directed graph, we distinguish between
another,
stopping
only
when
the
source
container
is
empty
or
the
degree din (u), which is the number of edges into u, and the outdegree dout (u), the number ofdestination container is full.
We want to know if there is a sequence of pourings that leaves exactly 2 pints in the 7- or 4-pint
ges leaving u.
1
container.
!
a) Show that in an undirected graph, u∈V d(u) = 2|E|.
b) Use part (a) to show that in an undirected graph, there must be an even number of vertices
5. Carrying stones. You are given a directed acyclic graph G = (V, E) with n vertices and m edges, along
with two distinguished vertices s, t ∈ V . At each vertex v ∈ V , there are a number of stones q(v) (so q(v) is
a nonnegative integer). Your goal is to find a path from s to t, picking up stones along the way, and arrive
at t carrying as many stones as possible. At each node v along the path, you are allowed to pick up at most
q(v) stones (and for simplicity, assume that q(t) = 0). However, there is a complication: each edge e ∈ E has
a capacity c(e) (which is a positive integer), and you are not allowed to carry more than c(e) stones across
that edge. If necessary, you can drop stones before crossing an edge.
Show how to solve this problem in time O(n + m). The output should be an optimal path from s to t, and for
every vertex along the path, your output should include the number of stones to be picked up or dropped at
that vertex. Assume the the graph is given in adjacency list form, and that you can fetch the q and c values
in constant time.
6. Gotham City. The mayor of Gotham City has made all the streets of the city one way. We can model
the street layout as a directed graph G = (V, E), where the vertices V of the graph correspond to street
intersections (locations), and the edges E correspond the the roads connecting intersections. So for every
pair of distinct vertices u, v ∈ V , if there is an edge from u to v, then there is no edge from v to u. Assume
a sparse representation for G.
(a) The mayor claims that one can legally drive from any one location in the city to any other. Show how
to verify the mayor’s claim using a linear-time algorithm that takes as input the graph G.
(b) Suppose the mayor’s claim is not necessarily true: that it may not be possible to legally drive from any
one location to any other.
Let us call a location v safe if wherever you legally drive starting at v, you can always legally drive back
to v. Said another way, v is safe if the following property holds: for every location w, if one can legally
drive from v to w, then one can legally drive back from w to v.
Give a linear-time algorithm that outputs all of the safe locations, given the graph G as input.
(c) Suppose again that the mayor’s claim is not necessarily true. Nevertheless, an emergency arises, and
you must drive from location s to location t, even though there may not be a legal route from s to t.
You want to plan a route that traverses some edges in the graph in the reverse (i.e., illegal) direction. In
order to minimize the chance of getting caught by the police, your planned route should minimize the
number of illegal edges. The number of legal edges in the route is irrelevant.
Give a linear-time algorithm that computes a route from s to t that minimizes the number of illegal
edges (or reports that no route exists), given the graph G and locations s and t as input.
7. Help! You are given a directed graph G = (V, E), along with vertices s, t ∈ V . Some edges are labeled with
a letter ‘a’–‘z’, while other edges have no label. Your goal is to find a path from s to t that spells out the
word “help”. So the path should consist of edges labeled ‘h’, ‘e’, ‘l’, ‘p’, in that order, along with any number
of other edges. These other edges may be either labeled or unlabeled, but unlabeled edges are preferred. So
among all such paths, you should find one with the minimum number of labeled edges.
Design a linear time algorithm to solve this problem.
8. Invasion. Two armies simultaneously invade a country. Let’s call them the “red army,” and the “blue army.”
The red army starts out occupying city a, and the blue army starts out occupying city b. Both armies fan
out simultaneously in all directions, and whichever army arrives at a city first, occupies that city, and blocks
the other army from either occupying or transiting through that city. The occupying army leaves a small
occupation force at that city, but the remainder of the army continues to fan out to all neighboring cities.
In case of a tie, the city is occupied by neither army, and neither army may transit the city. The army that
occupies the most cities wins the war. The question is: which army wins? Let’s model this problem as a
directed graph with positive edge weights. The nodes in the graph represent the cities, and the edges represent
roads between cities. The weight of an edge (u, v) represents the amount of time required for either army to
travel from city u to city v. Design and analyze an efficient algorithm to solve this problem. The input is a
directed, weighted graph, along with distinct nodes a and b. The output is “red wins,” “blue wins,” or “tie.”
Observations and hints: You might think that you could just run Dijkstra twice, once starting at a and once
starting at b. However, the tie-breaking rule means that this won’t work. Why? You might want to come up
with an example graph that shows why this does not work.
2
To solve this, you will have to modify Dijkstra’s algorithm. The idea is to associate with each city two
pieces of information: a running estimate for red’s best time to reach that city, a running estimate for blue’s
best time to reach that city. In every step of the algorithm, we greedily choose one city whose status is
“undecided” and move it into the “decided” category, assigning it to either red, blue, or neither, and then
update the estimates for the other undecided cities accordingly.
Fill in the details of the above idea, and try to carefully prove the correctness of it using a loop invariant
similar to what was done in class for Dijkstra. In your proof, you should identify where we used the fact that
the edge weights are positive. You can use a priority queue in your algorithm, without worrying about how
that is implemented.
9. Subway madness. The city of Brombus has n subway stations v1 , . . . vn and k subway systems B1 , . . . Bk .
Each system Bs has the following structure: there is a fixed entrance fee ps to enter Bs (from any station
vi ), and there is a set of fares wts (vi , vj ) to go from station vi to station vj on system Bs . We assume that
each entrance fee ps is a positive number, and that each fare wts (vi , vj ) is a positive number or ∞.
When traveling from vi to vj from the street, one can enter any subway system Bs , pay admission ps , travel to
some intermediate city v` by paying the fare wts (vi , v` ), and then (if necessary) repeat the same process, using
either a different subway system (and paying its entrance fee), or using the same subway system (without
needing to pay the entrance fee again), until one reaches the desired destination vj .
Design an algorithm that takes as input the entrance fee and fare data as above, and outputs a table of values
Mij for i, j = 1, . . . , n, where Mij is the minimum cost required to get from vi to vj . Your algorithm does
not need to compute the corresponding routes. State its complexity as a function of n and k. For full credit,
your algorithm should run in time O(n3 k).
Hint: First, run an all-pairs shotest path algorithm on each subway system.
3