Graphs
Two ingredients
1. vertices (nodes) v
2. edges(undirected or directed)
Examples: road networks, the web, social networks
The minimum Cut problem
Input: undirected graph G = (V, E) (parallel edges allowed)
Goal: compute a cut with fewest number of Crossing edges (a min cut)
Sparse vs. Dense Graphs
let n = # of vertices, m = # of edges
In most applications, m is Omega(n) and O(n^2)
In a "sparse graph", m is O(n) or close to it
In a "dense graph", m is closer to Theta(n^2)
Two ways to represent a Graph
1. The Adjacency Matrix
2. The Adjacency List
Which one is better? Depends on graph density and operation needed.
Random Contraction Algorithm
while there are more than 2 vertices:
-pick a remaining edge(u, v) uniformly at random
-merge(or "contract") u and v into a single vertex
-remove self-loops
return cut represented by final 2 vertices
Karger's Min-Cut Algorithm -------Random Contraction Algorithm(Python code):
import random
import copy
import time
def contract(ver, e):
while len(ver) > 2: #create a new graph every time (not efficient)
ind = random.randrange(0, len(e))
[u, v] = e.pop(ind) #pick a edge randomly
ver.remove(v) #remove v from vertices
newEdge = list()
for i in range(len(e)):
if e[i][0] == v: e[i][0] = u
elif e[i][1] == v: e[i][1] = u
if e[i][0] != e[i][1]: newEdge.append(e[i]) # remove self-loops
e = newEdge
return(len(e)) #return the number of the remained edges
if __name__ == '__main__':
f = open('kargerMinCut.txt')
_f = list(f)
edges = list() #initialize vertices and edges
vertices = list()
for i in range(len(_f)): #got 2517 different edges
s = _f[i].split()
vertices.append(int(s[0]))
for j in range(1, len(s)):
if [int(s[j]), int(s[0])] not in edges:
edges.append([int(s[0]), int(s[j])])
result = list()
starttime = time.clock()
for i in range(2000): #we take n^2logn times so that the Pr(allfail) <= 1/n where n is the number of vertics
v = copy.deepcopy(vertices) #notice: deepcopy
e = copy.deepcopy(edges)
r = contract(v, e)
result.append(r)
endtime = time.clock()
#print(result)
print(min(result))
print(endtime - starttime)