AoA Presentation MST

7/28/2019 AoA Presentation MST

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BY: GHULAM ABBAS

MS-III(CCN)

SUKKUR IBA


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TopicsDefinitions & Networking Basics

MST Algorithms

KruskalReverse Delete

Prim

Brief History

Importance


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Definitions & Networking

BasicsNetwork models are created from two major

building blocks:arcs (sometimes called edges), which are

connecting lines, and

nodes, which are the connecting points for the

arcs.Agraph is a structure that is built byinterconnecting nodes and arcs.

Adirected graph (often called a digraph) is agraph in which the arcs have specified

directions, as shown by arrowheads.Finally, a network is a graph (or more commonly

a digraph) in which the arcs have an associatedflow.

Practical Optimization: A Gentle Introduction,John W. Chinneck, 2010http: / /www.sce.car leton .ca/faculty/chinneck/po.htm l


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Networking Basics

Practical Optimization: A Gentle Introduction,John W. Chinneck, 2010http: / /www.sce.car leton.ca/faculty/chinneck/po.htm l

A networkis an arrangement of paths

connected at various points, through whichitems move.

Introduction to Management Science, 9th ed, Bernard W. Taylor III, 2006


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Definitions & Networking BasicsSome further definitions associated with graphs and networks:

chain: a sequence of arcs connecting two nodes i and j.For example, in Figure 8.1(a), we might connect nodes

A and E via the chains ABCE or ADCE.

path: a sequence of directed arcs connecting two

nodes. In this case, you must respect the arc

directions. For example, in Figure 8.1(b), a path from A

to E might be ABDE, but the chain ABCE is not a path

because it traverses arc BC in the wrong direction.

cycle: a chain that connects a node to itself without any

retracing.



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Definitions & Networking

Basicsconnected graph (or connected network): has

just one part. In other words you can reach anynode in the graph or network via a chain from

any other node.



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tree: a connected graph having no cycles.

Some examples are shown in Figure 8.2(a).


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Definitionsspanning tree: normally a tree selected from among thearcs in a graph or network so that all of the nodes in thetree are connected.



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Definitionsflow capacity: an upper (and sometimes lower) limit on

the amount of flow in an arc in a network. For example,the maximum flow rate of water in a pipe, or themaximum simultaneous number of calls on atelephone connection.

source (or source node): a node which introduces flow

into a network. This happens at the boundary betweenthe network under study and the external world.

sink (or sink node): a node which removes flow from anetwork. This happens at the boundary between thenetwork under study and the external world.



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The Minimum Spanning Tree

ProblemGiven a graph in which the arcs are labeled with the

distances between the nodes that they connect,find a spanning tree which has the minimum total

length.

NO CYCLES ALLOWED!!ALGORITHMS a problem solving procedure

following a set of rulesCycle- the nodes are connected by more than

one edge.



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Steps of the minimal spanning tree solutionmethod:

1. Select any starting node (conventionally, node 1 isselected).

2. Select the node closest to the starting node to jointhe spanning tree.

3. Select the closest node not presently in the spanningtree.

4. Repeat step 3 until all nodes have joined the

spanning tree.

Introduction to Management Science, 9th ed, Bernard W. Taylor III, 2006


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What is an MST AlgorithmAn MST algorithm is a way to show theshortest distance

There are many different algorithms:

KruskalReverse Delete

Prim

Introduction to Management Science, 9th

ed, Bernard W. Taylor III, 2006


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KRUSKALS ALGORITHM1. Pick out the smallest edges

2. Repeat Step 1 as long as the edges selected does not

create a cycle3. When all nodes have been connected, you are done

http://www.slideshare.net/zhaokatherine/minimum-spanning-tree
http://www.slideshare.net/zhaokatherine/minimum-spanning-treehttp://www.slideshare.net/zhaokatherine/minimum-spanning-treehttp://www.slideshare.net/zhaokatherine/minimum-spanning-treehttp://www.slideshare.net/zhaokatherine/minimum-spanning-treehttp://www.slideshare.net/zhaokatherine/minimum-spanning-treehttp://www.slideshare.net/zhaokatherine/minimum-spanning-tree


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REVERSE-DELETE ALGORITHMThis is the opposite of Kruskals algorithm

1. Start with all edges

2. Delete the longest edge3. Continue deleting longest edges as long as all nodes

are connected and no cycles are formed



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PRIMS ALGORITHM1. Pick out a node

2. Pick out the shortest edge that is connected to your

tree so far, as long as it doesnt create a cycle3. Continue this until all nodes are covered



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The execution of Kruskal's algorithm

The edges are considered by the algorithm insorted order by weight.

The edge under consideration at each step isshown with a red weight number.


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Grow the minimum spanning tree from theroot vertex r.

Q is a priority queue, holding all verticesthat are not in the tree now.

key[v] is the minimum weight of any edgeconnecting v to a vertex in the tree.

parent[v] names the parent of v in the tree.When the algorithm terminates, Q is empty;

the minimum spanning tree A for G is thus

A={(v,parent[v]):v

V-{r}}.Running time: O(||E||lg |V|).

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The execution of Prim's algorithm(moderatepart)

the rootvertex


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Bottleneck spanning tree: A spanning tree of Gwhose largest edge weight is minimum over allspanning trees of G. The value of the bottleneckspanning tree is the weight of the maximum-weightedge in T.

Theorem: A minimum spanning tree is also abottleneck spanning tree. (Challenge problem)


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MSTsKruskal (1956) and Prim (1957)

MSTs makes our life easier and we save money byusing short paths

MSTs was first seen in Poland, France, the CzechRepublic Slovakia, and Czechoslovakia


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IMPORTANCE OF MSTS1. We need it in computer science because it preventloops in a switched network with redundant paths

2. It is one of the oldest and most basic graphs intheoritical computer science



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Why we like MSTs1. MSTs are fun to work with2. It helps us find the shortest route

Theshortestdistance between two people is asMILE.


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THANKS FOR PATIENCE

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