Welcome to Lesson 16!
Learning ObjectivesBy the end of this class, you should know...
 How do you traverse a graph using the Breadth First Search (BFS) Algorithm?
 What is the SSSP problem?
 How does BFS solve this problem?
1. Breadth First Search Breadth First Search (BFS) is a Graph Traversal Algorithm that solves the Single Source Shortest Path (SSSP) problem.
 In other words, it finds the shortest path (fewest edges) between a source vertex and any other vertex in the graph.
 BFS traverses vertices and edges, moving across the graph rather than down each path, hence its name.
 Results in a level order traversal
 It also creates a BFS tree which is a subtree of the original graph that is rooted at the source vertex.
 The
algorithm uses a Queue, as well as a bookkeeping scheme wherein
vertices are assigned colors to keep track of which vertices have been
visited as the algorithm progresses.
1.1. BFS AlgorithmBFS(G, s) for all x in V(G) color[x] = white distance[x] = 1 parent[x] = Nil color[s] = grey distance[s] = 0 Enqueue(Q,s) while(Q is not empty) x = front of Q Dequeue(Q,x)
for all y in adj[x] if color[y] == white color[y] = grey distance[y] = distance[x] + 1 parent[y] = x Enqueue(Q, y) color[x] = black
1.2. BFS BigO Runtime: The BigO runtime of BFS is considered to be O(n + m)
 Where n = number of vertices in the Graph and m = the number of edges.
 Every vertex and every edge (or a subset there of) will be explored in best, worst and average cases.
1.3. BFS Example
 Let's trace through this algorithm with an example.
 Consider the following graph G1.
 Suppose that we wish to find the shortest distance from A to any other vertex in our graph.
 In other words, we consider A to be our source vertex.
 Recall that shortest distance means fewest number of edges.
 As we trace BFS on the above graph, we will need to do some book keeping.
 Thus, we should start by drawing the below chart to help organize us.
Vertex  Adjacency_List[]
 Color[]  Distance[]  Parent[]  A   G
 0  NIL  B
  W
 1  NIL  C   W  1  NIL  D
  W  1  NIL  E   W  1  NIL
 F   W  1  NIL 
Queue: A Colors
 Here the Color column for each vertex will be set to one of three possible values: white, grey or black.
 White indicates that a vertex has not been processed
 Grey indicates that a vertex is in process
 Black indicates that a vertex is finished being processed
Distance
 The distance column indicates distance to the source vertex
 The distance to the source from the source is 0, so we can immediately assign it a value.
 Other vertices are assigned the value of 1 to start. This value will be updated as the algorithm progresses.
Parent
 The
parent column indicates which vertex immediately proceeded the current
vertex in the path from the source to the current vertex.
 NIL indicates that a parent has not been assigned.
 However, in the case of the source vertex, the value of NIL will never be updated.
Queue
 The Queue will maintain an order in which the vertices need to be processed.
 The Queue is prefilled only with the value of the source.
 As vertices are processed, they are added to the back of the queue.
 When they are finished being processed, they are dequeued.
A Word on the Source Vertex
 The source vertex is indicated in purple.
 I initialize it differently than the other vertices as it is already in progress as the algorithm starts
 Note that I could apply the BFS algorithm using any vertex in my Graph as the source.
1.4. Review of Adjacency List Representation of a Graph
 Take a sheet of paper and recreate the table above.
 Also draw the corresponding graph on your paper
 Then, fill in the adjacency list for each of the vertices.
 Hold onto the paper when you are finished, as you will fill in the table as the BFS algorithm is applied.
1.5. BFS by Level
Image source.
2. BFS Activity2.1. First Graph
 Trace BFS on the following graph, using 8 as the source vertex
 Don't forget to use the following chart and to darken the lines of the BFS tree that results from tracing the graph
Vertex  Adjacency_List[]
 Color[]  Distance[]  Parent[]  0   W
 1  NIL  1
  W
 1  NIL  2
  W  1  NIL  3
  W  1  NIL  4
  W  1  NIL
 5   W  1  NIL  6   W  1  NIL  7   W  1  NIL
 8   W  1  NIL

Queue:
2.2. Second Graph
 Trace BFS on the following graph, using A as the source vertex
Vertex  Adjacency_List[]
 Color[]  Distance[]  Parent[]  A   W
 1  NIL  B
  W
 1  NIL  C
  W  1  NIL  D
  W  1  NIL  E
  W  1  NIL
 F   W  1  NIL  G   W  1  NIL  H   W  1  NIL

Queue:
Wrap Up Answer the review questions for this lesson on Canvas
Have a Great Day!
