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Graphs-Spanning Trees

November 17, 2016

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Topics

Spanning Trees in Unweighted Graphs

Minimum Spanning Trees in Weighted Graphs -

Prim’s Algorithm

Minimum Spanning Trees in Weighted Graphs - Kruskal’s Algorithm

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Definitions

Spanning tree: a tree that contains all vertices in the graph.Number of nodes: |V|Number of edges: |V|-1

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Spanning trees for unweighted graphs - data structures

A table (an array) Tsize = number of vertices,Tv = parent of vertex v

Adjacency lists

A queue of vertices to be processed

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Algorithm - initialization

1 Choose a vertex S and store it in a queue, seta counter = 0 (counts the processed nodes), andTs = 0 (to indicate the root),Ti = −1 ,i 6= s (to indicate vertex not processed)

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Algorithm - basic loop

2 While queue not empty and counter < |V | − 1 :Read a vertex V from the queueFor all adjacent vertices U :If Tu = −1 (not processed)Tu ← Vcounter ← counter + 1store U in the queue

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Algorithm – results and complexity

Result:Root: S, such that Ts = 0Edges in the tree: (Tv , V)

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

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Example

Table

0 1 is the root

1 edge (1,2)

2 edge (2,3)

1 edge (1,4)

2 edge (2,5)

Edge format: (Tv,V)

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Definitions

Minimum Spanning tree: a treethat contains all vertices in the graph,is connected,is acyclic, andhas minimum total edge weight.

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Applications of Minimum Spanning Trees

Communication networks

VLSI design

Transportation systems

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Minimum Spanning Tree - Prim’s algorithm

Given: Weighted graph.

Find a spanning tree with the minimal sum of the weights.Similar to shortest paths in weighted graphs.Difference: we record the weight of the current edge, not the length ofthe path .

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Data Structures

Three arrays:cost[v] = the weight of the shortest edge connecting v to a known vertexpath[v] = the last vertex to cause a change in cost[v]known[v] = True, if vertex v is fixed in the tree, False otherwise

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Data Structures (cont.)

Adjacency listsA priority queue of vertices to be processed.

Priority of each vertex is determined by the weight of edge that linksthe vertex to its parent.The priority may change if we change the parent, provided the vertex isnot yet fixed in the tree.

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Prim’s Algorithm

For all vcost[v ] =∞; known[v ] = false; path[v ] = −

cost[s] = 0Insert s onto a priority queue that percolates vertices

with lowest cost[] to the top.While priority queue not empty

DeleteMin v from priority queueIf not known[v ]

known[v ] = trueFor all unknown successors w of v

If weight[v ,w ] < cost[w ]cost[w ] = weight[v ,w ]path[w ] = vinsert w onto priority queue

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Results and Complexity

At the end of the algorithm, the tree would be represented with itsedges

{(v , path[v ])|v = 1, 2, . . . , |V |}

Complexity: O(|E |log(|V |))

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Example

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Kruskal’s Algorithm

The algorithm works with :set of edges,tree forests,each vertex belongs to only one tree in the forest.

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The Algorithm

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

do nothingif 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

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Operations on Disjoint Sets

Comparison: Are the vertices in the same tree?

Union: Combine two disjoint sets to form one set - the new treeImplementation

Union/find operations: the unions are represented as trees - nlogncomplexity.The set of trees is implemented by an array.

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Complexity of Kruskal’s Algorithm

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

Union/find operations on disjoint sets represented as trees: treesearch operations, with complexity O(log|V |)DeleteMin in Binary heaps with N elements is O(logN),

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

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Complexity of Kruskal’s Algorithm (Cont.)

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)

Thus for the binary heap we haveO(|E |log(|E |)) = O(|E |log(|V |2) = O(2|E |log(|V |)= O(|E |log(|V |))Since for each edge we perform DeleteMin operation and union/findoperation, the overall complexity is:

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

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Discussion

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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