IT용어위키

Generic Greedy Minimum Spanning Tree Algorithm is a fundamental approach for constructing a Minimum Spanning Tree (MST) by iteratively selecting the smallest available edge that does not form a cycle. It is the basis for well-known MST algorithms such as Kruskal’s and Prim’s algorithms.

Concept

The generic greedy MST algorithm follows a greedy strategy:

Initialize an empty set to store the MST edges.
Sort all edges by weight (if not already sorted).
Iterate through edges, selecting the smallest edge that does not form a cycle.
Repeat until the MST contains exactly (N - 1) edges, where N is the number of vertices.

This algorithm ensures that the MST is constructed efficiently by always choosing the lowest-cost option available at each step.

Algorithm

The generic greedy MST algorithm can be described as:

Initialize an empty set T for the MST.
Sort all edges in non-decreasing order of weight.
For each edge (u, v) in sorted order:
- If adding (u, v) to T does not form a cycle, include it in the MST.
Repeat until the MST contains (N - 1) edges.

Example

Consider the following weighted graph:

Vertex Pair	Edge Weight
A - B	4
A - C	3
B - C	1
B - D	2
C - D	5

Applying the generic greedy MST algorithm:

Select B - C (1) (smallest edge).
Select B - D (2) (next smallest).
Select A - C (3) (next smallest, does not form a cycle).
The MST is complete with edges B - C, B - D, A - C.

Total MST weight: 1 + 2 + 3 = 6

Implementation

A simple implementation of the generic greedy MST algorithm using Kruskal's approach:

class UnionFind:
    def __init__(self, n):
        self.parent = list(range(n))

    def find(self, u):
        if self.parent[u] != u:
            self.parent[u] = self.find(self.parent[u])
        return self.parent[u]

    def union(self, u, v):
        root_u = self.find(u)
        root_v = self.find(v)
        if root_u != root_v:
            self.parent[root_u] = root_v

def generic_greedy_mst(edges, n):
    edges.sort(key=lambda x: x[2])
    uf = UnionFind(n)
    mst = []
    
    for u, v, weight in edges:
        if uf.find(u) != uf.find(v):
            uf.union(u, v)
            mst.append((u, v, weight))
        if len(mst) == n - 1:
            break

    return mst

edges = [(0, 1, 4), (0, 2, 3), (1, 2, 1), (1, 3, 2), (2, 3, 5)]
mst = generic_greedy_mst(edges, 4)
print("Minimum Spanning Tree:", mst)

Properties

Greedy Approach
- Always selects the smallest available edge at each step.
Cycle Avoidance
- Ensures no cycles are formed, maintaining the tree structure.
Efficiency
- Runs in O(E log E) time complexity when implemented with a priority queue or sorting.

Applications

Network Design
- Used in designing efficient communication, power, and transportation networks.
Approximation Algorithms
- Forms the basis for solving hard problems like the Traveling Salesman Problem (TSP).
Cluster Analysis
- Used in hierarchical clustering techniques.