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lecture 18

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sajinsc

The document discusses the algorithms of breadth-first search (BFS) and depth-first search (DFS) on graphs. It provides pseudocode for BFS and DFS, and examples of running the algorithms on sample graphs. Key points include: - BFS uses a queue to explore all neighbors of a vertex before moving to the next level. It finds the shortest paths from the source. - DFS uses recursion to explore as deep as possible before backtracking. It identifies tree edges, back edges, and forward edges. - Both BFS and DFS run in O(V+E) time on a graph with V vertices and E edges.

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CS 332: Algorithms Graph Algorithms
Administrative ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Review: Graphs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Review: Representing Graphs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Review: Graph Searching ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Review: Breadth-First Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Review: Breadth-First Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Review: Breadth-First Search ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What does v->p represent? What does v->d represent?
Breadth-First Search: Example         r s t u v w x y
Breadth-First Search: Example   0      r s t u v w x y s Q:
Breadth-First Search: Example 1  0 1     r s t u v w x y w Q: r
Breadth-First Search: Example 1  0 1 2 2   r s t u v w x y r Q: t x
Breadth-First Search: Example 1 2 0 1 2 2   r s t u v w x y Q: t x v
Breadth-First Search: Example 1 2 0 1 2 2 3  r s t u v w x y Q: x v u
Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: v u y
Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: u y
Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: y
Breadth-First Search: Example 1 2 0 1 2 2 3 3 r s t u v w x y Q: Ø
BFS: The Code Again ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What will be the running time? Total running time: O(V+E) Touch every vertex: O(V) u = every vertex, but only once ( Why? ) So v = every vertex that appears in some other vert’s adjacency list
BFS: The Code Again ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What will be the storage cost in addition to storing the graph? Total space used: O(max(degree(v))) = O(E)
Breadth-First Search: Properties ,[object Object],[object Object],[object Object],[object Object],[object Object]
Depth-First Search ,[object Object],[object Object],[object Object],[object Object]
Depth-First Search ,[object Object],[object Object],[object Object]
Depth-First Search: The Code ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Depth-First Search: The Code ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What does u->d represent?
Depth-First Search: The Code ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What does u->f represent?
Depth-First Search: The Code ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Will all vertices eventually be colored black?
Depth-First Search: The Code ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What will be the running time?
Depth-First Search: The Code ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Running time: O(n 2 ) because call DFS_Visit on each vertex, and the loop over Adj[] can run as many as |V| times
Depth-First Search: The Code ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],BUT, there is actually a tighter bound. How many times will DFS_Visit() actually be called?
Depth-First Search: The Code ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],So, running time of DFS = O(V+E)
Depth-First Sort Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
DFS Example source vertex
DFS Example 1 | | | | | | | | source vertex d f
DFS Example 1 | | | | | | 2 | | source vertex d f
DFS Example 1 | | | | | 3 | 2 | | source vertex d f
DFS Example 1 | | | | | 3 | 4 2 | | source vertex d f
DFS Example 1 | | | | 5 | 3 | 4 2 | | source vertex d f
DFS Example 1 | | | | 5 | 6 3 | 4 2 | | source vertex d f
DFS Example 1 | 8 | | | 5 | 6 3 | 4 2 | 7 | source vertex d f
DFS Example 1 | 8 | | | 5 | 6 3 | 4 2 | 7 | source vertex d f
DFS Example 1 | 8 | | | 5 | 6 3 | 4 2 | 7 9 | source vertex d f What is the structure of the grey vertices? What do they represent?
DFS Example 1 | 8 | | | 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
DFS Example 1 | 8 |11 | | 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
DFS Example 1 |12 8 |11 | | 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
DFS Example 1 |12 8 |11 13| | 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
DFS Example 1 |12 8 |11 13| 14| 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
DFS Example 1 |12 8 |11 13| 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f
DFS: Kinds of edges ,[object Object],[object Object],[object Object],[object Object]
DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f Tree edges
DFS: Kinds of edges ,[object Object],[object Object],[object Object],[object Object]
DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f Tree edges Back edges
DFS: Kinds of edges ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f Tree edges Back edges Forward edges
DFS: Kinds of edges ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
DFS Example 1 |12 8 |11 13|16 14|15 5 | 6 3 | 4 2 | 7 9 |10 source vertex d f Tree edges Back edges Forward edges Cross edges
DFS: Kinds of edges ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
DFS: Kinds Of Edges ,[object Object],[object Object],[object Object],[object Object],source F?
DFS: Kinds Of Edges ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],source C?
DFS And Graph Cycles ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
DFS And Cycles ,[object Object],DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; }
DFS And Cycles ,[object Object],DFS(G) { for each vertex u  G->V { u->color = WHITE; } time = 0; for each vertex u  G->V { if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v  u->Adj[] { if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; }
DFS And Cycles ,[object Object],[object Object],[object Object],[object Object],[object Object]

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lecture 18