## Description

CS 341: Algorithms

Homework 2

You are allowed to discuss with others but are not allowed to use any references other than the course

notes and the three reference books. Please list your collaborators for each question. You must write your

own solutions. See the course outline for the homework policy.

There are totally 52 marks. The full mark is 50. This homework is counted 10% of the course grade.

1. Programming Problem: A Game with Two Tokens1

(20 marks)

Let’s consider the following game on an n × n square grid. Some squares of the grid are marked as

obstacles, and one grid square is marked as the target. There are two tokens – token A and token B.

In the beginning, each of them is placed in a distinct square that is neither an obstacle nor the target.

In each turn, the player must move one of the tokens from its current position as far as possible upward,

downward, left, or right. That is, the player chooses a token and a direction, and moves the token

along that direction, stopping just before it hits (1) the edge of the board, (2) an obstacle square, or

(3) the other token. The goal of the game is to move either of the tokens to the target square. When

the goal is attained, the game finishes. Note that moving past the target square doesn’t count.

Your task is to return a shortest sequence of moves to finish the game, or report that it is impossible.

Input: The first line of input has a positive integer n, the size of the grid. Each of the next n lines

contains n characters, representing the grid. Here’s a table of all possible characters and their meaning:

character meaning

A token A

B token B

T the target

# an obstacle

. an empty square

It is guaranteed that 2 ≤ n ≤ 20, and that the characters A, B, T appear exactly once in the input.

Output: If there is no solution, output −1 on a single line. Otherwise, on the first line output k, the

minimum number of moves to finish the game, followed by k lines representing the moves in order.

Each line must be in the form {A|B} {up|down|left|right}. The sample test case should give you a

better idea of the output format.

Sample Input 1:

5

…A.

..B..

..T..

.#…

….#

1This problem is from Chapter 5 of the textbook “Algorithms” by Jeff Erickson, available online and under the CC BY 4.0

license. Wording is slightly modified.

1

Sample Output 1:

4

A right

A down

A left

B down

Explanation:

Sample Input 2:

5

..#A.

..B##

..T..

.#…

….#

Sample Output 2:

-1

2

2. Written Problem: Red-Blue Paths2

(10 marks)

Suppose we are given an undirected graph G = (V, E) where some edges are red and the remaining

edges are blue. Describe an algorithm to find a shortest walk in G from one vertex s to another vertex

t in which no three consecutive edges have the same color, or report that no such walks exist. That is,

if the walk contains two red edges in a row, the next edge must be blue, and if the walk contains two

blue edges in a row, the next edge must be red. You will get full marks if the time complexity of the

algorithm is O(|V | + |E|) and the proofs are correct.

For example, given the following graph as input, your algorithm should return (s, a, d, e, f, e, g, c, t),

which is the shortest legal walk from s to t.

3. Written Problem: Reachability (10 marks)

Given a directed graph G = (V, E), design an algorithm to find a vertex s ∈ V from which all other

vertices are reachable (i.e. there is a directed path from s to v for all v ∈ V ) or report that no such

vertices exist. You will get full marks if the time complexity is O(|V |+|E|) and the proofs are correct.

(a) Vertex 1 can reach all other vertices. (b) No vertex can reach all other vertices.

2This problem is also from Chapter 5 of the textbook “Algorithms” by Jeff Erickson.

3

4. Written Problem: One-Way Streets (12 marks)

Imagine in a city where all the streets are two-way streets. Perhaps because the streets are too narrow,

the government would like to see if it is possible to make all streets one way, so that there is still a way

to drive from any street to any other street in the city.

Show how to model this problem as a graph problem. Then design an algorithm to determine if it is

possible, and when it is possible to output one solution to assign the directions.

You will get full marks if the time complexity of the algorithm is O(n + m) and the proofs are correct,

where m is the number of streets in the city and n is the number of intersections in the city.

Figure 2: There is a solution in this example.

?

Figure 3: There is no solution in this example.

4