Chapter 9: Graphs

Spanning Trees

Mark Allen Weiss: Data Structures and Algorithm Analysis in Java

Lydia Sinapova, Simpson College

Spanning Trees

• Spanning Trees in Unweighted Graphs
• Minimal Spanning Trees in Weighted Graphs - Prim’s Algorithm
• Minimal Spanning Trees in Weighted Graphs - Kruskal’s Algorithm
• Animation

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Definitions

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Spanning tree: a tree that contains all vertices in the graph.

  Number of nodes: |V|

Number of edges: |V|-1

Spanning trees for unweighted graphs - data structures

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A table (an array) T

size = number of vertices,

Tv= parent of vertex v

Adjacency lists

A queue of vertices to be processed

Algorithm - initialization

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1. Choose a vertex S and store it in a queue, set

a counter = 0 (counts the processed nodes), and Ts = 0 (to indicate the root),

Ti = -1 ,i ≠ s (to indicate vertex not processed)

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Algorithm – basic loop

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2. While queue not empty and counter < |V| -1 :

Read a vertex V from the queue

    For all adjacent vertices U :

If Tu = -1 (not processed)

Tu ← V

counter ← counter + 1

store U in the queue

Algorithm �– results and complexity

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Result:

Root: S, such that Ts = 0

Edges in the tree: (Tv , V)

Complexity: O(|E| + |V|) - we process all edges and all nodes

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Example

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Table

0 // 1 is the root

1 // edge (1,2)

2 // edge (2,3)

1 // edge (1,4)

2. // edge (2,5)

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Edge format: (Tv,V)

Minimum Spanning Tree - �Prim's algorithm

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Given: Weighted graph.

Find a spanning tree with the minimal sum of the weights.

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Similar to shortest paths in a weighted graphs.

Difference: we record the weight of the current edge, not the length of the path .

Data Structures

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A table : rows = number of vertices,

three columns:

T(v,1) = weight of the edge from

parent to vertex V

T(v,2) = parent of vertex V

T(v,3) = True, if vertex V is fixed in the tree,

False otherwise

Data Structures (cont)

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Adjacency lists

A priority queue of vertices to be processed.

Priority of each vertex is determined by the weight of edge that links the vertex to its parent.

The priority may change if we change the parent, provided the vertex is not yet fixed in the tree.

Prim’s Algorithm

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Initialization

For all V, set

first column to –1 (not included in the tree)

second column to 0 (no parent)

third column to False (not fixed in the tree)

Select a vertex S, set T(s,1) = 0 (root: no parent)

Store S in a priority queue with priority 0.

While (queue not empty) loop

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1. DeleteMin V from the queue, set T(v,3) = True

2. For all adjacent to V vertices W:

If T(w,3)= True do nothing – the vertex is fixed

If T(w,1) = -1 (i.e. vertex not included)

T(w,1) ← weight of edge (v,w)

T(w,2) ← v (the parent of w)

insert w in the queue with

priority = weight of (v,w)

While (queue not empty) loop (cont)

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If T(w,1) > weight of (v,w)

T(w,1) ← weight of edge (v,w)

T(w,2) ← v (the parent of w)

update priority of w in the queue

remove, and insert with new priority = weight of (v,w)

Results and Complexity

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At the end of the algorithm, the tree would be represented in the table with its edges

{(v, T(v,2) ) | v = 1, 2, …|V|}

 Complexity: O(|E|log(|V|))

Kruskal's Algorithm

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The algorithm works with :

set of edges,

tree forests,

each vertex belongs to only one tree in the forest.

The Algorithm

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1. Build |V| trees of one vertex only - each vertex is a tree of its own. Store edges in priority queue

2. Choose an edge (u,v) with minimum weight

if u and v belong to one and the same tree,

do nothing

if u and v belong to different trees,

link the trees by the edge (u,v)

3. Perform step 2 until all trees are combined into one tree only

Example

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Edges in

Priority Queue:

(1,2) 1

(3,5) 2

(2,4) 3

(1,3) 4

(4,5) 4

(2,5) 5

(1,5) 6

Example (cont)

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Forest of 5 trees

Edges in

Priority Queue:

(1,2) 1

(3,5) 2

(2,4) 3

(1,3) 4

(4,5) 4

(2,5) 5

(1,5) 6

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Process edge (1,2)

Process edge (3,5)

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Example (cont)

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Edges in

Priority

Queue:

(2,4) 3

(1,3) 4

(4,5) 4

(2,5) 5

(1,5) 6

Process edge (2,4)

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Process edge (1,3)

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All trees are combined in one, the algorithm stops

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Operations on Disjoint Sets

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Comparison: Are the vertices in the same tree

Union: combine two disjoint sets to form one set - the new tree

Implementation

Union/find operations: the unions are represented as trees - nlogn complexity.

The set of trees is implemented by an array.

Complexity of Kruskal’s Algorithm

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O(|E|log(|V|)

Explanation:

The complexity is determined by the heap operations and the union/find operations

Union/find operations on disjoint sets represented as trees: tree search operations, with complexity O(log|V|)

DeleteMin in Binary heaps with N elements is O(logN),

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When performed for N elements, it is O(NlogN).

Complexity (cont)

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Kruskal’s algorithm works with a binary heap that contains all edges: O(|E|log(|E|))

However, a complete graph has

|E| = |V|*(|V|-1)/2, i.e. |E| = O(|V|²)

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Thus for the binary heap we have

O(|E|log(|E|)) = O(|E|log (|V|²) = O(2|E|log (|V|)

= O(|E|log(|V|))

Complexity (cont)

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Since for each edge we perform DeleteMin operation and union/find operation, the overall complexity is:

O(|E| (log(|V|) + log(|V|)) = O(|E|log(|V|))

Discussion

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Sparse trees - Kruskal's algorithm :

Guided by edges

Dense trees - Prim's algorithm :

The process is limited by the number of the processed vertices

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