COSC 2007Data Structures II

Chapter 14Graphs III

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Topics

Spanning trees DFS Spanning tree BFS Spanning tree

Minimum spanning trees Algorithms Prim Dijkstra

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Spanning Trees A special kind of undirected graph

Contains all the vertex Is a tree There may be several spanning trees for a given graph

If we have a connected undirected graph with cycle, we obtain a spanning tree by removing edges until we have no cycles

We can determine whether a connected graph contains a cycle simply by counting its nodes & edges

If we add an edge to a spanning tree, then the resulting graph is no longer a tree, because it will produce a cycle containing the new edge

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Spanning Trees

Examplea

fg

eh

d

c

b

i

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Spanning Trees Example

Original graph

3 different spanning trees

a

dc

ba

dc

ba

dc

b

a

dc

b

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Spanning Trees Observations about undirected graphs:

A connected undirected graph that has N nodes must have at least N-1 edges

A connected undirected graph that has N nodes and exactly N-1 edges contains no cycles

A connected undirected graph that has N nodes & more than N-1 edges must contain at least one cycle

Two algorithms for determining a spanning tree of a graph, based on traversal algorithms

DFS BFS

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Spanning Trees

Spanning tree recursive (DFS) algorithm

dfs_Tree_R(v)

// Recursive version

Mark v as visited

for (each unvisited node u adjacent to v)

{ Mark the edge from u to v

dfs_Tree_R(u)

} // end for

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Spanning Trees

Spanning tree recursive (DFS) algorithm example Order of nodes visited

a b c d g e f h i

a

f g

eh

d

c

b

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Root

6

75

4

3

218

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Spanning Trees Spanning tree iterative (BFS) algorithm

Uses queuebfs_Tree_I( v)// Iterative versionq.createQueue( )// Add v to queue amd mark itq.enqueue(v)Mark v as visitedwhile (!q.isEmpty ( ) ){ q.dequeue(w)

for (each unvisited node u adjacent to w){ Mark u as visited

Mark edge between w & uq.enqueue(u)

} // end for} // end while

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Spanning Trees

Minimum Spanning Tree (MST) Weighted connected, undirected graph

a

f g

eh

d

c

b

i

8

4

97

2

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1

25

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Spanning Trees

Minimum Spanning Tree (MST) If a graph has weighted edges, it is called a Network The weight can be thought of as a cost or Distance Minimum spanning tree (MST):

A subgraph of a given connected graph that is connected and that minimizes the sum of the weights of its edges

If we have a graph G with weighted edges, then any spanning tree has associated weight which is equal to the sum of the weights of its edges

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Spanning Trees

Minimum Spanning Tree (MST) Weighted connected, undirected graph

a

f g

eh

d

c

b

i

8

4

97

2

4

6

1

25

3

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Prim's Algorithm

Building a MST of an undirected graph Idea:

Choose the root for the MST Start with the MST having the root node Repeatedly add to MST the graph node that is closest, in the

sense of having least weight, to MST Continue until all nodes have been added

 At the end of each pass, the marked tree is a MST for that subgraph of G consisting of the nodes chosen so far, along with all edges of G connecting these nodes.

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Prim's Algorithm

Prim”s Algorithm(v)// Determines a MST for a weighted (non-negative),

//connected, undirected graph, beginning with node v

Mark node v as visited & include it in the MST

while ( there are unvisited nodes)

{ Find the least cost edge (v, u) to the MST

Mark u as visited

Add node u & the edge (v, u) to the MST

} // end while

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Prim's Algorithm

Minimum Spanning Tree (MST)

a

f g

eh

d

c

b

i

8

4

97

2

4

6

1

25

3

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Dijkstra's Algorithm One of the greedy algorithms

At each step the best "local" solution to the problem is the one chosen

Finds the shortest (cheapest) path between two nodes in a weighted directed graph Known as “single-source least-cost” problem

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Dijkstra's Algorithm

A

B

CD

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Example: Cheapest (shortest-whatever) path from A to E?

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Dijkstra's Algorithm For each vertex keep track of

Cost Cheapest known cost to get this vertex from the source

Path The path with cheapest known cost to get this vertex from

the source

Done Is the path and cost guaranteed to be optimum?

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Dijkstra's Algorithm

A

B

CD

E30 60

10010

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What is the Minimum Spanning Tree?

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Dijkstra's Algorithm Set cost of source vertex to 0 While destination vertex no done do

Find cheapest not Done vertex (X) Mark it as done

It is guaranteed to be so

For each vertex X do Calculate total cost of getting to vertex If total cost < current cost

Replace current cost with lowest cost Mark route (path) as X

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Dijkstra's Algorithm

Do this

2010

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