What is the running time of Karger algorithm to find the minimum cut in a graph?
The runtime of the algorithm is O(n2) since each merge operation takes O(n) time (going through at most O(n) edges and vertices), and there are n − 2 merges until there are 2 supernodes left.
What is the probability of finding a min cut?
The probability that that cut is the minimum cut is P = 1 / (n^2/2 – n/2) , which is much better than just picking a cut completely randomly. If you run the algorithm once, your probability of getting the min cut is P , but your probability of not getting it is 1 – P .
How do you find the minimal cut set on a graph?
The minimum cut of a weighted graph is defined as the minimum sum of weights of edges that, when removed from the graph, divide the graph into two sets. , and the sum of weights of these two edges are minimum among all other cuts in this graph.
What is meant by randomized algorithm can you describe about how global minimum cut can be found using this?
In computer science and graph theory, Karger’s algorithm is a randomized algorithm to compute a minimum cut of a connected graph. It was invented by David Karger and first published in 1993. are “reattached” to the merged node, effectively producing a multigraph.
What is min-cut problem?
The minimum cut problem (abbreviated as “min cut”), is defined as follows: Input: Undirected graph G = (V,E) Output: A minimum cut S, that is, a partition of the nodes of G into S and V \ S that minimizes the number of edges going across the partition.
What is a cut in a graph?
In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition.
What is min cut problem?
What is the advantage of randomized min-cut?
The first advantage is performance; randomized algo- rithms run faster than the best-known deterministic algorithms for many problems. The second advantage is that many randomized algorithms are simpler to describe and implement than deterministic algorithms of comparable performance.
What is the cut produced by Karger’s algorithm?
The number of edges in the resultant graph is the cut produced by Karger’s algorithm. Karger’s algorithm is a Monte Carlo algorithm and cut produced by it may not be minimum. For example, the following diagram shows that a different order of picking random edges produces a min-cut of size 3.
How to find the smallest cut of an undirected graph?
Given an undirected and unweighted graph, find the smallest cut (smallest number of edges that disconnects the graph into two components). The input graph may have parallel edges. For example consider the following example, the smallest cut has 2 edges. A Simple Solution use Max-Flow based s-t cut algorithm to find minimum cut.
How to find minimum s-t cut in a simple solution?
A Simple Solution use Max-Flow based s-t cut algorithm to find minimum cut. Consider every pair of vertices as source ‘s’ and sink ‘t’, and call minimum s-t cut algorithm to find the s-t cut. Return minimum of all s-t cuts. Best possible time complexity of this algorithm is O (V 5) for a graph.