1 graphs and graph algorithms. 2 graph traversals zdepth first traversal. ystarting from a source...

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Graphs and Graph Algorithms

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Graph Traversals

Depth First Traversal. Starting from a source node, mark the

node as visited Choose an ordering of the edges from

this node. Do a Depth First Traversal of the vertex

the first edge is connected to, provided the vertex has not yet been visited.

Repeat the previous step for all edges.

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Depth First Traversal From Vertex A

Assume ordering is alphabetical.From A, go to B.

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Depth First Traversal From Vertex A

From B, go to D.

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Depth First Traversal From Vertex A

From D, go to C.

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Depth First Traversal From Vertex A

From C, go to E.

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Depth First Traversal From Vertex A

From E, go to F.

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Depth First Traversal From Vertex A

Done.

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Breadth First Traversal

Mark the source vertex as visitedVisit all the children of the source

(provided they have not already been visited)

Perform a Breadth First Traversal of all of the children.

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Breadth First Traversal From Vertex A

Assume ordering is alphabetical.From A, go to B.

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Breadth First Traversal From Vertex A

From A, go to C.

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Breadth First Traversal From Vertex A

From B, go to D.

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Breadth First Traversal From Vertex A

From B, go to E.

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Breadth First Traversal From Vertex A

From B, go to F.

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Breadth First Traversal From Vertex A

Done.

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Definitions of Connectivity

An undirected graph is connected if there is a path from v to w for all vertices v and w in the vertex set.

A directed graph is strongly connected if there is a path from v to w for all vertices v and w in the vertex set.

A directed graph is weakly connected if its underlying undirected graph is connected.

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Underlying Undirected Graphs

Every directed graph has an underlying undirected graph.

This graph is found as follows: It has the same vertices as the directed graph If the directed graph has an edge (v,w), then

the underlying undirected graph has edges (v,w) and (w,v).

Simply put, to find the underlying undirected graph, remove the arrows.

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Testing for Connectivity

For undirected graphs: Apply a DFT or a BFT from any vertex. If all

vertices are visited, the graph is connected.For weakly connected directed graphs:

Create the underlying undirected graph and test it for connectivity as above.

For strongly connected directed graphs: Apply a DFT or a BFT from every vertex. If all

vertices are reachable every time, the graph is strongly connected.

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Dijkstra’s Algorithm

Start with the Finished Set of Nodes containing only the source.

Find all nodes not in the Finished Set of Nodes, but connected to one of these nodes. Call this set Next Nodes.

Find the node in the set of Next Nodes which has minimum cost to link to the Finished Set of Nodes.

Add this node to the Finished Set of Nodes

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Dijkstra’s Algorithm II

Repeat the previous 3 steps until all nodes are added to the Finished Set of Nodes.

Note: To determine the minimum cost to link a node: Let PrevNode be the one currently in the

Finished set of Nodes; Let ThisNextNode be the one it is linked to. The cost of ThisNextNode = cost of PrevNode

+ cost of the link between the two.

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Dijkstra’s Algorithm Example

Suppose A wishes to send to F. What is the cheapest path?

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Dijkstra’s Algorithm Example

Tan nodes are in the Finished Set.Blue nodes are in the Next Set.

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Dijkstra’s Algorithm Example

What happens next?

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Dijkstra’s Algorithm Example

C is least cost among Next Node set, so it is added to Finished set.

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How do I update the Next Node Set?

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Dijkstra’s Algorithm Example

D, E, F added to Next Node Set. What are their costs?

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What is next for Finished Set of nodes?

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Dijkstra’s Algorithm Example

B is added to Finished set of nodes.

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How do I update the Next Set of Nodes?

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Dijkstra’s Algorithm Example

E’s minimum changes to 6.

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Next for Finished Set?

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Dijkstra’s Algorithm Example

Next, update Next Set costs:

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D has minimum cost.

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Dijkstra’s Algorithm Example

Next for Finished Set?

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Dijkstra’s Algorithm Example

No cost updates for Next Nodes here; finish it up:

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Either E or F; I chose E.

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Dijkstra’s Algorithm Example

All nodes in finished set...done

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Dijkstra Discussion

This example is centralized; can we make it distributed? Each node would need the entire graph

before starting. Each node would need to send its

connection costs to everyone else.Can we make it adaptive?

When costs change, they would need to be sent to everyone and the algorithm rerun.

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Bellman-Ford Algorithm

Basic idea: Keep track of current minimum cost to

all other nodes . When the minimum cost to another

node from one of your neighbors changes, maybe you can change your cost to that node too.

Initially, you only know the cost to your neighbors.

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Bellman-Ford Explanation

Thus, the minimum cost from A to F is the minimum of: The cost from A to B plus the cost from B to F; The cost from A to C plus the cost from C to F.

Thus, each source node can fill in its destination minimums by looking at its neighbor’s minimums.

This process iterates until no changes are made to the list of minimums.

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Bellman-Ford Example I

This is the same graph as for Dijkstra’s Algorithm and the initial table.

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A B C D E FA NA B,2 C,1 ? ? ?B A,2 NA ? D,3 E,4 F,7C A,1 ? NA D,3 E,6 F,7D ? B,3 C,3 NA ? F,2E ? B,4 C,6 ? NA F,6F ? B,7 C,7 D,2 E,6 NA

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Bellman-Ford Example II

A B C D E FA NA B,2 C,1 ? ? ?B A,2 NA ? D,3 E,4 F,7C A,1 ? NA D,3 E,6 F,7D ? B,3 C,3 NA ? F,2E ? B,4 C,6 ? NA F,6F ? B,7 C,7 D,2 E,6 NA

A B C D E FA NA B,2 C,1 C,4 B,6 C,8B A,2 NA A,3 D,3 E,4 D,5C A,1 A,3 NA D,3 E,6 D,5D C,4 B,3 C,3 NA B,7 F,2E B,6 B,4 C,6 B,7 NA F,6F C,8 D,5 D,5 D,2 E,6 NA

First Iteration (Repeated):

Second Iteration:

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A B C D E FA NA B,2 C,1 C,4 B,6 C,8B A,2 NA A,3 D,3 E,4 D,5C A,1 A,3 NA D,3 E,6 D,5D C,4 B,3 C,3 NA B,7 F,2E B,6 B,4 C,6 B,7 NA F,6F C,8 D,5 D,5 D,2 E,6 NA

Bellman-Ford Example III

Second Iteration (Repeated):

Third Iteration:A B C D E F

A NA B,2 C,1 C,4 B,6 C,6B A,2 NA A,3 D,3 E,4 D,5C A,1 A,3 NA D,3 E,6 D,5D C,4 B,3 C,3 NA B,7 F,2E B,6 B,4 C,6 B,7 NA F,6F D,6 D,5 D,5 D,2 E,6 NA

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Bellman-Ford Discussion

This example is centralized; can we make it distributed? Each node needs to send its costs to its

neighbors only. In our example, A only needs the rows

for B and C to decide on its cost to all destinations.

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Bellman-Ford Discussion II

Can we make it adaptive? Each node keeps the costs from its

neighbors; when these costs change, it only needs to pick the minimum among the old costs and the new one.

NOTE: when a cost changes, a node must notify its neighbors.

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Minimum Cost Spanning Trees

Prim’s Algorithm May be viewed as a “vertex-centric”

approach. Basically the same as Dijkstra’s except that

the cost of a node is its minimum cost connection to the finished set, rather than minimum connection to source vertex.

Proceed with adding the minimum vertex to the finished set until all vertices are added.

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Prim’s Algorithm Example

Same graph as for Dijkstra’s Algorithm, but let’s start at D.

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Prim’s Algorithm Example

Next for Finished Set?

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Prim’s Algorithm Example

Insert F.Next for Finished Set?

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Prim’s Algorithm Example

Insert B. Note change of E’s weight.Next for Finished Set?

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Prim’s Algorithm Example

Insert A. Note change of C’s weight.Next for Finished Set?

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Prim’s Algorithm Example

Insert C. Next for Finished Set?

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Prim’s Algorithm Example

Insert E. Done.

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Prim’s Discussion

This is not the same minimum spanning tree as produced by Dijkstra’s Algorithm.

Starting from node A, we added (C,D) rather than (B,D).

Both edges have the same weight, so the overall cost is the same.

This means that there may be more than one minimum spanning tree!

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Kruskal’s Algorithm

More of an “edge-centric” approach.Idea: sort the edges; add edges from

least weight to maximum weight with the restriction that the newly added edge does not create a cycle.

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Kruskal’s Algorithm Example

Which edge has minimum weight?

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Kruskal’s Algorithm Example

Edge (A,C) is minimum. What’s next?

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Kruskal’s Algorithm Example

Edge (A,B) is minimum. What’s next?

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Kruskal’s Algorithm Example

Edge (D,F) is minimum. What’s next?

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Kruskal’s Algorithm Example

Edge (B,D) is minimum. What’s next?

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Kruskal’s Algorithm Example

Edge (C,D) is minimum, but creates a cycle. Here, we choose (B,E). What’s next?

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Kruskal’s Algorithm Example

All vertices are now in the tree, so we are done.

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Kruskal’s Algorithm -- Detecting Cycles

Neat algorithm, but how are cycles detected?

Answer: Start by considering each vertex in its

own distinct set. Whenever an edge (v,w) is considered, v

and w must be in different vertex sets. When adding edge (v,w), merge the sets

containing v and w into one set.

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Kruskal’s Algorithm Example with edge sets

Vertex Sets: ({A},{B},{C},{D},{E},{F}}

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Kruskal’s Algorithm Example

Vertex Sets: ({A,C},{B},{D},{E},{F}}

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Kruskal’s Algorithm Example

Vertex Sets: ({A,B,C},{D},{E},{F}}

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Kruskal’s Algorithm Example

Vertex Sets: ({A,B,C},{D,F},{E}}

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Kruskal’s Algorithm Example

Vertex Sets: ({A,B,C,D,F},{E}}

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Kruskal’s Algorithm Example

Note that (C,D) is minimum, but C and D are in the same vertex set.

Vertex Sets: ({A,B,C,D,F},{E}}

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Kruskal’s Algorithm Example

Vertex Sets: ({A,B,C,D,E,F}}All vertices are now in one set, so

we are done.

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The End Slide