Minimum Spanning Tree

Minimum Spanning Tree

Only used for undirected graph

1. Kruskal's Algorithm

1.1 Description

1.2 Procedure

Suppose:

• \(G\) : Graph
• \(|N|\) : Node set of \(G\)
• \(|A|\) : Edge set of \(G\)

Step 0 : Initialization.

Create a forest \(T_0\) (a set of trees), where each node in the graph \(G\) is a separate tree, (i.e, \(T_0\) only contain all nodes of the graph \(G\) )

Create a edge set \(E_0\) containing all the edges in the graph \(G\)

Step 1: Stopping criterion. If \(A_{k} \neq \varnothing\) and \(T_{k}\) is not yet spanning (i.e., the edge number of \(T_{k}\) is less than the node number minus 1), continue to Step 2, otherwise stop.

Step 2: Select the edge \(e_k\) with minimum weight from \(A_k\)

• (1) Remove \(e_k\) from \(A_k\):

  \[A_{k+1} = A_k - \{e_k\} \]

• (2) If \(e_k\) connects two different trees, which can combining two trees into a single tree, then add it to the forest \(T_k\)

  \[T_{k+1} = T_k \ \text{ add a edge } \ \{e_k \} \]

  Otherwise, that is, add \(e_k\) will create a loop for a tree, do nothing.

Then set \(k = k+1\), go to Step 1.

At the termination of the algorithm, the forest forms a minimum spanning forest of the graph. If the graph is connected, the forest has a single component and forms a minimum spanning tree.

1.3 Algorithm

2. Prim's Algorithm

2.1 Description

2.2 Procedure

Suppose

• \(N = \{ 1,2,\cdots, n\}\) : the set of nodes of the network
• \(C_k\) : set of nodes that have been permanently connected at iteration \(k\),
• \(\bar{C}_k\) : set of nodes as yet to be connected permanently after iteration \(k\).

The following steps describe the minimal spanning tree algorithm :

Step 0: Initiation. Set \(C_0 = \varnothing\), \(\bar{C}_0=N\), \(k=0\)

Start with any node \(i\) in the unconnected set \(\bar{C}_0\) and set

\[C_1=\{i\}, \ \bar{C}_1=N-\{i\}, \ k=1 \]

Step 1: Check the stopping criterion. If \(\bar{C}_k=\varnothing\), stop. Otherwise, go to Step 2.

Step 2: During each loop \(k\), perform the following two sub-steps:

Step 2.1: Select a node \(j^*\) from the unconnected set \(\bar{C}_k\) that yields the shortest link to someone node in the connected set \(C_n\)

Step 2.2: Adding node \(j^*\) to set \(C_n\) will not contain a cycle:

\[C_{k+1} = C_{k} + \{j^*\}, \ \bar{C}_{k+1} = \bar{C}_{k} - \{j^*\}, \ k=k+1 \]

2.3 Algorithm

Suppose:

• \(\mathbf{D}\) : distance matrix, if there are not edge between two node, the distance will be set as positive infinite.
• \(\mathbf{T}\) : Output, the distance matrix of minimal spanning tree
• \(t_{i^*j^*}, \, t_{j^*i^*}\) : the distance between node \(i^*\) and \(j^*\) of \(\mathbf{T}\).

\[\begin{align*} \hline & \text{Minimum Spanning Tree} \ (\mathbf{D}, \mathbf{T}=\inf) \\ \hline & C_1=\{i\},\ \bar{C}_1=N-\{i\}, \ k=1\\ & \text{for } k = 2, \cdots, n \ ( \, \text{equal to : While }\bar{C}_k \neq \varnothing ) \\ & \qquad d^*=\inf \\ & \qquad \text{for } i \in C_k :\\ & \qquad \qquad \text{for } j \in \bar{C}_k :\\ & \qquad \qquad \qquad \text{if } d^* > d_{ij}: \\ & \qquad \qquad \qquad \qquad d^*,i^*,j^* \leftarrow d_{ij},i,j \\ & \qquad t_{i^*j^*}, \, t_{j^*i^*} \leftarrow d^* \\ & \qquad C_k \leftarrow C_k + \{j^*\} \\ & \qquad \bar{C}_k \leftarrow \bar{C}_k - \{j^*\} \\ \hline \end{align*} \]

2.4 Implement by Python

2.4.1 Using network structure

2.4.2 Using constructed functions in NetworkX

# return: type of Graph
MT = nx.minimum_spanning_tree(G, weight='weight', algorithm='kruskal', ignore_nan=False)
# return: edges of MT, type of iterator
edges = nx.minimum_spanning_edges(G, algorithm='kruskal', weight='weight', keys=True, data=True, ignore_nan=False)

• Parameters:
  • algorithm: {'kruskal', 'prim', 'boruvka'}

3. Boruvka's Algorithm

References

[1] H. A. Taha, "Section 6.2 Minimal Spanning Tree Algorithm" in Operations research: an introduction, Tenth edition, Global edition. Harlow, England London New York Boston Amsterdam Munich: Pearson Education, 2017, p.p. 250-255.

[2] Mathematical Programming, website.

[3] Kruskal's algorithm, Wikipedia, website.