One of the assignments in my algorithms class is to design an exhaustive search algorithm to solve the clique problem. That is, given a graph of size n, the algorithm is supposed to determine if there is a complete sub-graph of size k. I think I've gotten the answer, but I can't help but think it could be improved. Here's what I have:
input: A graph represented by an array A[0,...n-1], the size k of the subgraph to find.
output: True if a subgraph exists, False otherwise
Algorithm (in python-like pseudocode):
def clique(A, k):
P = A x A x A //Cartesian product
for tuple in P:
if connected(tuple):
return true
return false
def connected(tuple):
unconnected = tuple
for vertex in tuple:
for test_vertex in unconnected:
if vertex is linked to test_vertex:
remove test_vertex from unconnected
if unconnected is empty:
return true
else:
return false
input: An adjacency matrix of size n by n, and k the size of the subgraph to find
output: All complete subgraphs in A of size k.
Algorithm (this time in functional/Python pseudocode):
//Base case: return all vertices in a list since each
//one is a 1-clique
def clique(A, 1):
S = new list
for i in range(0 to n-1):
add i to S
return S
//Get a tuple representing all the cliques where
//k = k - 1, then find any cliques for k
def clique(A,k):
C = clique(A, k-1)
S = new list
for tuple in C:
for i in range(0 to n-1):
//make sure the ith vertex is linked to each
//vertex in tuple
for j in tuple:
if A[i,j] != 1:
break
//This means that vertex i makes a clique
if j is the last element:
newtuple = (i | tuple) //make a new tuple with i added
add newtuple to S
//Return the list of k-cliques
return S
Does anybody have any thoughts, comments, or suggestions? This includes bugs I might have missed as well as ways to make this more readable (I'm not used to using much pseudocode).
Fortunately, I talked to my professor before submitting the assignment. When I showed him the pseudo-code I had written, he smiled and told me that I did way too much work. For one, I didn't have to submit pseudo-code; I just had to demonstrate that I understood the problem. And two, he was wanting the brute force solution. So what I turned in looked something like this:
input: A graph G = (V,E), the size of the clique to find k
output: True if a clique does exist, false otherwise
Algorithm:
UPDATE: Added second version. I think this is getting better although I haven't added any fancy dynamic programming (that I know of).
UPDATE 2: Added some more commenting and documentation to make version 2 more readable. This will probably be the version I turn in today. Thanks for everyone's help! I wish I could accept more than one answer, but I accepted the answer by the person that's helped me out the most. I'll let you guys know what my professor thinks.
Some comments:
connected(tuple)
doesn't look right. Don't you need to reset unconnected
inside the loop?