1 GRAPHS - ADVANCED APPLICATIONS Minimim Spanning Trees Shortest Path Transitive Closure.

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GRAPHS - ADVANCED APPLICATIONS

Minimim Spanning Trees

Shortest Path

Transitive Closure

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Graph applicationsGraph applications

Graphs can be used for:

Finding a route to drive from one city to another

Finding connecting flights from one city to another

Determining least-cost highway connections

Designing optimal connections on a computer chip

Implementing compilers

Doing garbage collection

Representing family histories

Doing similarity testing (e.g. for a dating service)

Playing games

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Elementary Graph OperationsElementary Graph Operations

Graph traversals provide the basis for many elementary graph operations:

Spanning trees on graphs

Graph cycles

Connected components of a graph

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Spanning Tree

Connected sub graph that includes all vertices of the original connected graph

The subgraph is a tree

If original graph has n vertices, the spanning tree has

n vertices and n-1 edges.

No cycle allowed in this sub graph

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Minimum-cost spanning tree

A minimum-cost spanning tree of a

connected weighted graph is :

a collection of edges connecting all vertices such that

The sum of the weights of the edges is the smallest possible.

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Spanning trees Suppose you have a connected undirected graph

ConnectedConnected: every node is reachable from every other node

UndirectedUndirected: edges do not have an associated direction ... then a spanning tree is a connected sub graph in which

there are no cycles

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A connected,undirected graph

Four of the spanning trees of the graph

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Finding a spanning tree – We did this in class

To find a spanning tree of a graph,1. pick an initial node and call it part of the spanning

tree 2. do a search from the initial node: Each time you find a node that is not in the spanning

tree, add to the spanning tree both the new node and the edge you followed to get to it

7An undirected graph Result of a BFS

starting from topResult of a DFSstarting from top

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Spanning Tree - Minimum Edges

Minimum number of edges to keep the graph connected

If have N vertices, spanning tree has N-1N-1 edges

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Minimizing costs

Suppose you want to supply a set of houses (say, in a new subdivision) with:

electric power

Water

sewage lines

telephone lines

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Minimum cost spanning tree

To keep costs down:

Connect these houses with a spanning tree (of, for example, power lines)

However, the houses are not all equal distances apartall equal distances apart

To reduce costs even further, you could connect the

houses with a minimum-cost spanning tree

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Minimum Spanning Tree (MST)

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Spanning tree with minimum weight

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Algorithm For Finding MST

1. All nodes are unselected,

2. Mark node v selected to get started – can be any node

3. For each node of graph,

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Find the edge with minimum weight minimum weight that connects an unselected node with a selected node

Mark this unselected node as selected

}

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Demos For Finding MST

Step 1: mark vertex AA as selected

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Step 2: find the minimum weighted edge connected to vertex A, and mark the other vertex on this edge as selected.

We choose {A to D}

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Step 3: find the minimum weighted edge connected to vertices set { A, D } ,

and mark the other vertex on this edge as selected.

We choose { A, B }

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Step 4: find the minimum weighted edge

connected to vertices set

{ A, D, B} , and mark the other vertex on this edge as selected.

We choose {B,C}

vertices set { A, D, B, C} { A, D, B, C}

No more nodes!No more nodes!

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Step 5: All vertices are marked as selected, So we find the minimum spanning tree

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Complexity For Finding MST

N nodes, E edges

Analysis: N * E = O(N*E) N * E = O(N*E)

Number of nodes * the number of Edges

Better implementation:

Heap O(logE) where is E is number of edges(weights)

Initialization of heap: O(E log E)O(E log E)

Number of edges * log of E

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Cost - Definition

Routing Protocol Cost = congestion, bandwidth, delay …

cost Umass

Router

cost

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Minimum-Cost Spanning Trees

A minimum-cost spanning tree of a connected weighted graph is a collection of edges connecting all vertices collection of edges connecting all vertices such that the sum of the weights of the edges is the smallest possible.

Prim’s algorithmPrim’s algorithm: always pick the edge with the smallest always pick the edge with the smallest weight to weight to any nodeany node. It is a . It is a greedygreedy algorithm. algorithm.

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Kruskal’s algorithm

T = empty spanning tree;

E = set of edges;

N = number of nodes in graph;

while T has fewer than N - 1 edges

{ remove an edge (v, w) of lowest cost from E

if adding (v, w) to T would create a cycle

then discard (v, w)

else add (v, w) to T }

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MST Algorithms

Finding an edge of lowest cost can be done just by sorting the edges –

so could use a heap

Efficient testing for a cycle requires a fairly complex algorithm (UNION-FIND)

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MCST --- Another Example

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MCST --- Example (cont’d)

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All nodes visited!

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Shortest Path Problem

Weight: cost, distance, travel time, hop …

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Prim’s Algorithm: Heap Implementation

Consider using a heap to hold the keys D[0…n-1].

We use a ( Min Heap):

To find the minimum in heap array using tree formulas, To find the minimum in heap array using tree formulas, we call we call Extract_Min(heap) Extract_Min(heap)

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ExtractMin

Extract_Min removes the minimum element from the heap

which is in the root

by swapping in the last element. O(1)

Then sift down last element(LgV)

We get O(log V) using the usual removal heap operation

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PRIMS ALGORITHM- Heap implementation

Extract_Min removes the minimum element from the heap which is in the root.

Heap Implementation:

O(E log V) or

Number of Edges times the log of # of Vertices

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Single-Source Shortest Paths – Prim’s

To find the shortest path from one given node to all the others

Given a source x. Find the path from x to v

such that v is in V-x and the path is minimum.

If the weights in the edges are all one we already have a solution: breadth first search

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Single Source Shortest Paths

Given a weighted connected graph G=(V,E), and given a pair of vertices vs (source) and vd (destination) Î V what is the shortest path from vs to vd? That is, what is the path that has the smallest sum of edge weights?

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Source vertex vs is vertex 1Bold line is shortest path from 1(vs) to 2 (vd)

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SP from A to H = SP from A to E + SP from E to H.

SP from A to H = SP from A to B + SP from B to H.

SP from A to H = SP from A to C + SP from C to H.

In general: SP from A to H = SP from A to vi + SP from vi to H; vi.

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• Use Dijkstra’s (greedy) algorithm if graph has only positive weights

• Use Bellman-Ford’s algorithm if graph has positive and

negative weights (but not negative cycles…)

• Both algorithms can run on directed or undirected graphs

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Single-Source Shortest Paths The solution when we have different weights is a variation of

Prim's algorithm

Before we chose the node that was closest to the partial minimum tree.

Now we want the node which is closest to the source node

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Single-Source Shortest Paths

Algorithm:

1.Start from the source node

2.make this node part of the a shortest path tree A=(VA,E

A)

3. Find an edge (x,y) such that x in the VA and y is not,

so that the path from the initial node to y is minimum

4. add y and the edge (x,y) to the tree A.

Algorithm:

1.Start from the source node

2.make this node part of the a shortest path tree A=(VA,E

A)

3. Find an edge (x,y) such that x in the VA and y is not,

so that the path from the initial node to y is minimum

4. add y and the edge (x,y) to the tree A.

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Single-Source Shortest Paths

The "question" here is: how do we find the next edge to be added in the tree?

We again can make use of existing data structures As the case with Prim's algorithms for the MST we can

use a priority queue

The algorithms described in the following slides is normally attributed to E. Dijkstra

Let's understand the algorithm graphically first.

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An Example(adapted from Sedgewick)

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The Algorithm

The algorithm here is the almost the same as the one for the minimum spanning tree

The distance to the initial node is considered instead of the absolute value of the edge

Inserting the node in the priority queue is done by considering all the edges that leave the node plus the value (the priority) assigned to the node that is being removed.

Delete is done based on the priority queue strategy

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ANOTHER EXAMPLE

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Dijkstra’s Algorithm (Cont’d)

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S = {A,C,B,E}

S = {A,C,B}

Cost of SP from A to vi through S

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S = {A,C,B,E,D}


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Dijkstra’s Algorithm for Shortest Paths

S = {0} /* Current MST */

for i = 0 to n

D[i] = M[0][i] /* Shortest path length from 0 to i */

for i = 1 to n-1

find the smallest D[v] such that v S

S = S {v}

for all vertices u S

if (D[u] > D[v] + M[v][u]) then

D[u] = D[v] + M[v][u]

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Dijkstra’s Algorithm --- Example

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Example – Dijkstra Algorithm

Greedy Algorithm Assume all weight of edge >0

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best

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Animation Online

Dijkstra’s Shortest Path Algorithm - at my website

1 GRAPHS - ADVANCED APPLICATIONS Minimim Spanning Trees Shortest Path Transitive Closure.

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Transcript of 1 GRAPHS - ADVANCED APPLICATIONS Minimim Spanning Trees Shortest Path Transitive Closure.