Graphs

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Graphs Paths

• Presentation Part II

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Graph representation

• Given graph G (V, E).
• May be either directed or undirected.
• Two common ways to represent for algorithms
• Adjacency lists.
• Adjacency matrix.
• When expressing the running time of an algorithm,
  its often in terms of both V and E.

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Adjacency lists

• Array Adj of V lists, one per vertex.
• Vertex us list has all vertices v such that (u,
  v) ? E. (Works for both directed and undirected
  graphs.)

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Example

• For an undirected graph

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Example

• For a directed graph

Same asymptotic space and time.
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Adjacency matrix
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Adjacency matrix
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Breadth-first search

• Input Graph G (V, E), either directed or
  undirected, and source vertex s ? V.
• Output dv distance (smallest of edges)
  from s to v, for all v ? V.

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Idea

• Idea Send a wave out from s.
• First hits all vertices 1 edge from s.
• From there, hits all vertices 2 edges from s.
• Etc.
• Use FIFO queue Q to maintain wavefront.
• v ? Q if and only if wave has hit v but has not
  come out of v yet.

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Breadth-First Search

• Run-time O(VE)
• Shortest-path tree
• the final weight is the minimum distance
• keep predecessors and get the shortest path
• all BFS shortest paths make a tree

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Breadth-First SearchExample
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Breadth-First Search

• Since each vertex gets a finite d value at most
  once, values assigned to vertices are
  monotonically increasing over time.
• Time O(V E).
• O(V) because every vertex enqueued at most once.
• O(E) because every vertex dequeued at most once
  and we examine (u, v) is in E only when u is
  dequeued. Therefore, every edge examined at most
  once if directed, at most twice if undirected.

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Depth-first search

• Input G (V, E), directed or undirected. No
  source vertex given!
• Output 2 timestamps on each vertex
• dv discovery time
• f v finishing time

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Depth-first search

• Will methodically explore every edge.
• Start over from different vertices as necessary.
• As soon as we discover a vertex, explore from it.
• Unlike BFS, which puts a vertex on a queue so
  that we explore from it later.
• As DFS progresses, every vertex has a color
• WHITE undiscovered
• GRAY discovered, but not finished (not done
  exploring from it)
• BLACK finished (have found everything reachable
  from it)
• Discovery and finish times Unique integers from
  1 to 2V.
• For all v, dv lt f v.
• In other words, 1 dv lt f v 2V.

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Depth-First Search

• Methodically explore all vertices and edges
• All vertices are White, t 0
• For each vertex u do if u is White then
  Visit(u)
• Procedure Visit(u)
• color u Gray
• du ? t ? t 1
• for each v adjacent to u do
• if v is White then Visit(v)
• color u Black
• f u ? t ? t 1
• Gray vertices stack of recursive calls

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Review Depth-First Search

• Depth-first search is another strategy for
  exploring a graph
• Explore deeper in the graph whenever possible.
• Edges are explored out of the most recently
  discovered vertex v that still has unexplored
  edges.
• When all of vs edges have been explored,
  backtrack to the vertex from which v was
  discovered

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DFS Applications.

• Testing whether graph is connected.
• Computing a spanning forest of graph.
• Computing a path between two vertices of graph or
  equivalently reporting that no such path exists.
• Computing a cycle in graph or equivalently
  reporting that no such cycle exists.

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DFS vs. BFS

• DFS seems cleaner than BFS recursive, it doesn't
  make use of an external data structure (a queue),
  and is shorter too.
• But it is important to realize these two
  algorithms address very different needs.
• An example of BFS usage is calculating shortest
  paths.
• A Depth First Search performed on the World Wide
  Web would quickly overload any given web server
  with an ever growing number of requests.
• Therefore a Breadth First Search is preferred,
  since it dispatches requests to many servers at a
  time.

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DFS vs. BFS
Applications DFS BFS
Spanning forest, connected components, paths, cycles ? ?
Shortest paths ?
Biconnected components ?
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Example 2
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Example 2 animated
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DFS Example
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DFS Example
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DFS Example
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DFS Example
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DFS Example
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DFS Example
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DFS Example
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DFS Example
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DFS Example
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What is the structure of the yellow vertices?
What do they represent?
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DFS Example
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DFS Example
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DFS Example
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DFS Example
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DFS Example
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Snapshot