Graphs (Contd.)

Overview

  • Depth First Search
  • Topological Sort using DFS
  • Dijkstra Shortest Path Algorithm (single source multiple destination)
  • Bellman - Ford Algorithm

DFS

In DFS we take a node and visit its child until we find a dead end in graph.

BFS

In BFS we take node and visit its all adjacent childrens and then move forward

Difference

Similarity

In DFS we traverse through whole graph.

In BFS we traverse through whole graph.

Motivation for DFS

  • Edge Classification
  • Cycle Detection
  • Topological Sorting

Edge Classifiction

Edges are of three type :

  1. Back Edge : Visiting a parent vertex from a decendent
  2. Cross Edge : From one tree to the other. 
  3. Forward Edge : visiting a visited vertex from a parent to child
  4. Tree Edge : An edge vising a new vertex via an edge

Let's Keep it simple.

 

you all might have solved maze in past.

 

Try this one.

Your Entry is from 0 and Exit is at 8

Maze

Steps

  • Select a vertex to begin with
  • From that vertex consider its neighbor and continue forward  
  • Again consider the neighbor of the vertex recently discovered
  • March forward till you find the dead end
  • On dead end turn back and find another possible path
  • Do this till all the vertex of a graph are visited.

pseudo-code for selecting a vertex to begin with

DFS()
for vertex in V:
    if vertex is not visited
        visited[vertex]=true
        DFS-Visit(vertex)


DFS-Visit( vertex )
    
    Phase - I

    for nodes adjacent to vertex
        if( visited[node]==false )
            DFS-Visit( node )

    Phase - II

Let Dissect It

Step 1: 

            Select the vertex

Entry : 0

Let Dissect It

Step 2: 

            Consider its Neighbour

Entry : 0

Let Dissect It

Step 3: 

            Consider Neighbor's Neighbor

Entry : 0

Let Dissect It

Step 4: 

            Carry on till all are visited

Entry : 0

Steps for Topological Sort:

  • Run DFS on a graph for all vertexes.
  • Insert the Vertexes when they are processed in other words during phase - II.
  • Present the list in reversed order this a topologically sorted graph.

Problems You can Attempt:

1. UVa 124 - Following Orders

2. UVa 200 - Rare Order

3. UVa 784 - Maze Exploration (Flood Fill) (Special one)

Dijkstra's Algorithm

Task :

          To find shortest path from one vertex to another.

What is Shortest Path?

Try yourself:

function Dijkstra(Graph, source):

	for each vertex v in Graph:	// Initialization

	    dist[v] := infinity	// initial distance from source to vertex v is set to infinite

	    previous[v] := undefined	// Previous node in optimal path from source


	dist[source] := 0	// Distance from source to source

	Q := the set of all nodes in Graph	// all nodes in the graph are un-optimized - 
                                                // thus are in Q

	while Q is not empty:	// main loop

	    u := node in Q with smallest dist[ ]

	    remove u from Q


	    for each neighbor v of u:	// where v has not yet been removed from Q.

	        alt := dist[u] + dist_between(u, v)

	        if alt < dist[v]	// Relax (u,v)
	            dist[v] := alt
	            previous[v] := u

	return previous[ ]

Again Implementing

Try the algorithm on the graph given here.

Source : 0

Assumptions:

  • G is the set of all the vertexes of the graph that are unprocessed.
  • dist[] is array storing distance of any node from a source vertex.
  • No negative cycles.
  • Infinity is a very large value :P

Step 1 : 

Stand at zero and consider all the edges from it to its adjacent vertices.

Step 2 : 

Push all the vertex of a graph into a list with distances undefined or infinity.

Step 3 : 

Stand at vertex with minimum distance value that is source.

Step 4 : 

Get the minimum distance for all the vertex adjacent to it initialized.

Step 5 : 

Now get the vertex with minimum value of distance from source from the set G.

Step 6 : 

Again, get the minimum distance for all the vertex adjacent to it initialized or updated.

Step 7 : 

We get an updated distance array with required output.

Scope of Improvements:

  • Using a better means to find minimum from distance array
  • Using adjacency list in place of Adjacency Matrix
  •  

Problems you can attempt

1. UVa 341 - Non-Stop Travel

2. UVa 929 - Number Maze (on a 2-D maze graph)

3. UVa 10278 - Fire Station

Bellman Ford's Algorithm

Dijkstra 

View : 

          Consider only those vertices with minimum distance or in other words adjacent to the vertex being processed.

Bellman Ford

View : 

          Consider all the vertices no matter which vertex is being processed.

Benefit :

  • Efficient in terms of runtime

Benefit :

  • Detect Negative Cycle which dijkstra cannot.
function BellmanFord(list vertices, list edges, vertex source)

   // This implementation takes in a graph, represented as
   // lists of vertices and edges, and fills two arrays
   // (distance and predecessor) with shortest-path
   // (less cost/distance/metric) information


   // Step 1: initialize graph
   for each vertex v in vertices:
       if v is source then distance[v] := 0
       else distance[v] := inf
       predecessor[v] := null


   // Step 2: update value of dist array with closer ones
   for i in V:
       for each edge (u, v) in Graph with weight w in edges:
           if distance[u] + w < distance[v]:
               distance[v] := distance[u] + w
               predecessor[v] := u


   // Step 3: check for negative-weight cycles
   for each edge (u, v) in Graph with weight w in edges:
       if distance[u] + w < distance[v]:
           error "Graph contains a negative-weight cycle"
   return distance[], predecessor[]

Problems you can attempt

1. UVa 558 - Wormholes (checking the existence of negative cycle)

2. UVa 10557 - XYZZY

3. UVa 11280 - Flying to Fredericton (modified Bellman Ford’s)

Thank You

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