CSC 2300 Data Structures

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CSC 2300Data Structures Algorithms

• April 3, 2007
• Chapter 9. Graph Algorithms

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Today

• Shortest-Path Algorithms
• Unweighted Shortest Paths
• Dijkstras Algorithm

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Single-Source Shortest Path

• Problem Given as input a weighted graph G
  (V,E) and a distinguished vertex s, find the
  shortest weighted path from s to every other
  vertex in G.
• Example

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Edges of Negative Costs

• Edges of negative costs can cause problem.
• Example
• Consider a path from v5 to v4.
• What happens to shortest paths when a
  negative-cost cycle is present in a graph?
• In the absence of negative-cost cycles, the
  shortest path from s to s is zero.

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Unweighted Shortest Paths

• We may say that all edges have weight 1.

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Illustration

• Say initial vertex is v3
• After finding all vertices whose path length from
  s is 1

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Illustration

• After finding all vertices whose path length from
  s is 2
• Final shortest paths

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Breadth-first Search

• This strategy for searching a graph is known as
  breadth-first search.
• It operates by processing the vertices in layers
  the vertices closest to the start are evaluated
  first, and the most distant vertices are
  evaluated last.
• What is the equivalent traversal scheme for trees?

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Initial Configuration of Table
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Pseudocode

• Running time of this algorithm?

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Inefficiency

• Here is an example
• How to remove this efficiency?
• Use a queue.

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Improved Pseudocode
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Example
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Weighted Shortest Paths

• Dijkstras algorithm
• Greedy algorithm solves a problem in stages by
  doing what appears to be the best choice at each
  stage.

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Dijkstras Algorithm

• Initial configuration After v1 is declared
  known

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Dijkstras Algorithm

• After v4 is declared known After v2 is declared
  known

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Dijkstras Algorithm

• After v5 and v3 are known After v7 is declared
  known

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Dijkstras Algorithm

• After v6 is declared known and algorithm
  terminates

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Stages of Dijkstras Algorithm
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Pseudocode of Dijkstras Algorithm

• Running time?