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Advanced Algorithms Lab 2

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CPS688 – Advanced Algorithms
Lab 2
Lab Instructions:
Problem 1 – N-Queens
The N Queen is the problem of placing N chess queens on an N×N chessboard so that no two
queens attack each other.
The idea is to place queens one by one in different columns, starting from the leftmost column.
When placing a queen in a column, you should check for clashes with already placed queens (no
more than one Q in the row and column).
See the example below.
Input: your input should take the number of matrix size N. your matrix is a zeros matrix
Output: your output is a binary matrix which has 1s for the blocks where queens are placed.
Input Output
4 0 0 1 0
1 0 0 0
0 0 0 1
0 1 0 0
CPS688 – DR. OMAR FALOU 2
Problem 2 – Acyclic Graph:
Given an undirected graph, you are required to check if the graph has a cycle.
Hint: consider using BFS/DFS
Input
Your program will be tested against multiple test cases. Each test case begins with two integers
n and e, representing the number of vertices and edges. The next e lines represent the vertices
that are connected by an edge.
Output
For each test case, print “no” if the graph contains a cycle; else print “yes”.
Sample Input Sample Output
6 6
0 1
0 3
1 2
2 4
3 4
3 5
no
6 5
0 1
0 2
0 3
1 4
2 5
yes
Problem 3 – Minimum Spanning Tree
Given a weighted undirected graph, you are required to find the minimum spanning tree in the
graph using Prim’s algorithm.
Input: Your program will be tested against multiple test cases. Each test case begins with two
integers n and e, representing the number of vertices and edges. The next e lines represent the
vertices that are connected by an edge and the weight of that edge.
Output: For each test case, print the sum of weights on the edges of the minimum spanning tree
and the corresponding edges with their weights.
CPS688 – DR. OMAR FALOU 3
Sample Input Sample Output
4 5
0 1 10
0 2 6
0 3 5
1 3 15
2 3 4
Edge 2-3 has a weight of 4
Edge 0-3 has a weight of 5
Edge 0-1 has a weight of 10
MST = 19
Problem 4
John enters a candy shop and in his hand is a very light-thin bag. He is given 20 mins to collect
as much candy as possible, but he needs to make sure that the bag doesn’t tear apart. There’s a
fixed number of candies in the shop – each with its own weight and sentimental value. Of course,
his goal is to put as many candies in the bag as possible while also maximizing the number of
best-quality candies so that he can enjoy the treat during the weekend. Plus, he doesn’t want to
share the candy with anyone since the candy placed in the bag is of the most value with respect
to him.
But John needs your help! So, can you be kind and help him choose the candy he’s required to
put in the bag and the ones he’s supposed to keep at the store?
Input:
• The first line contains one integer N; the number of candies present in the shop. Then 3
lines follow.
• The second line contains N integers where Ni represents the sentimental value of the
candy w.r.t John
• The third line contains N integers where Ni represents the weight of each candy.
• The last line contains one integer W representing the maximum weight that can be
carried by the bag
Output: Output the highest sentimental aggregated value
Sample Input Sample Output
3
6 10 12
1 2 3
5
22
CPS688 – DR. OMAR FALOU 4
Problem 5 – Longest Increasing Subsequence
The Longest Increasing Subsequence (LIS) problem is to find the length of the longest
subsequence of a given sequence such that all elements of the subsequence are sorted
in increasing order. For example, the length of LIS for array A = {10, 22, 9, 33, 21, 50, 41,
60, 80} is 6 and LIS of A is {10, 22, 33, 50, 60, 80}.
Write a Dynamic programming solution for LIS. Print the LIS and the elements that formed
this LIS.
Hint: this problem may be converted to a Longest Common Subsequence problem. Create
another array (Array B) of unique elements of the original array (array A) and sort it. The
Longest Increasing Subsequence of A must be present as a subsequence in B.
Sample Input Sample Output
10 22 9 33 21 50 41 60 LIS = 5
LIS is: 10 22 33 50 60

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