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Advanced Algorithms Lab 3

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CPS688 – Advanced Algorithms
Lab 3
Lab Instructions:
Problem 1 – Rod Cutting Problem: Note: this problem was asked during a Microsoft interview
Given a rod of steel, you need to cut it into pieces in a way to optimize the total revenue. The
decision is where to cut the rod given that different piece lengths have different revenue.
Formally,
Given a rod of length n inches and a table of prices p[i], i=1,2,…,n, find the maximum revenue
r[n] obtainable by cutting up the rod and selling the pieces.
Rod lengths are integers
For i=1, 2,…,n we know the price(revenue) p[i] of a rod of length i inches.
Length i 1 2 3 4 5 6 7 8 9 10
Price p[i] 1 5 8 9 10 17 17 20 24 30
Cut possibilities for a rod of length 4:
1+1+1+1 for a total price of 1+1+1+1=4
2+1+1 for a total price of: 5+1+1+1=8
2+2 for a total price of 5+5=10
3+1 for a total price of 8+1=9
For a rod of length 4: 2+2 is optimal (p[2]+p[2]=10)
CPS688 – DR. OMAR FALOU 2
One approach to solve this problem is the dynamic programming bottom-up method. This
approach typically depends on some natural notion of the “size” of a subproblem, such that
solving any particular subproblem depends only on solving “smaller” subproblems. We sort the
subproblems by size and solve them in size order, smallest first. When solving a particular
subproblem, we have already solved all of the smaller subproblems its solution depends upon,
and we have saved their solutions. We solve each subproblem only once, and when we first see
it, we have already solved all of its prerequisite subproblems.
Here we proactively compute the solutions for smaller rods first, knowing that they will later be
used to compute the solutions for larger rods. The answer will once again be stored in r[n].
Input
Your program will be tested against multiple test cases. Each test case begins with an integer n
representing the length of the rod followed by n elements representing the prices.
Output
For each test case, determine the maximum value obtainable by cutting up the rod and selling
the pieces.
Sample Input Sample Output
8
1 5 8 9 10 17 17 20
22
CPS688 – DR. OMAR FALOU 3
Problem 2 – Strongly Connected Components
Given a directed graph, write a program that checks if the given graph is strongly connected
or not. A directed graph is said to be strongly connected if every vertex is reachable from
every other vertex.
Following is Kosaraju’s DFS-based solution that does two DFS traversals of graph:
1. Initialize all vertices as not visited.
2. Do a DFS traversal of graph starting from any arbitrary vertex v. If DFS traversal doesn’t
visit all vertices, then return false.
3. Reverse all arcs (or find transpose or reverse of graph)
4. Mark all vertices as not-visited in reversed graph.
5. Do a DFS traversal of reversed graph starting from same vertex v (Same as step 2). If
DFS traversal doesn’t visit all vertices, then return false. Otherwise return true.
The idea is, if every node can be reached from a vertex v, and every node can reach v, then the
graph is strongly connected. In step 2, we check if all vertices are reachable from v. In step 4, we
check if all vertices can reach v (In reversed graph, if all vertices are reachable from v, then all
vertices can reach v in original graph).
Input
Your program will be tested against multiple test cases. Each test case begins with two
integers n and e, representing the number of antennas and cables. The next e lines represent
the antennas that are connected by a cable.
Output
For each test case, print “yes” if all the antennas can communicate with each other; else print
“no”.
Sample Input Sample Output
4 4
0 1
1 2
2 3
3 0
yes
6 7
0 1
0 2
2 4
3 1
3 5
4 5
5 0
no

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