CSC 2300 Data Structures

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CSC 2300Data Structures Algorithms

• April 17, 2007
• Chapter 9. Graph Algorithms

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Today

• Network Flow
• Minimum Spanning Tree
• Prims Algorithm

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Network Flow

• Given a directed graph G(V,E) with edge
  capacities cvw.
• Examples amount of water that flow through a
  pipe, or amount of traffic on a street between
  two intersections.
• Also given two vertices source s and sink t.
• Goal determine the maximum amount of flow that
  can pass from s to t.

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Example

• A graph and its maximum flow

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Procedure

• Start with our graph G.
• Construct a flow graph Gf, which gives the flow
  that has been attained at the current stage.
• Initially, all edges in Gf have no flow.
• Also construct a graph Gr, called the residual
  graph.
• Each edge of Gr tells us how much more flow can
  be added.

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Initial Stage

• Graphs G, Gf, and Gr

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Stage 2

• Two units of flow are added along s, b, d, t

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Stage 3

• Two units of flow are added along s, a, c, t

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Stage 4

• One unit of flow added along s, a, d, t
• Algorithm terminates with correct solution.

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Greedy Algorithm

• We have described a greedy algorithm, which does
  not always work.
• Here is an example
• Algorithm terminates with suboptimal solution.

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Improve Algorithm

• How to make the algorithm work?
• Allow it to change its mind!
• For every edge (v,w) with flow fvw in the flow
  graph, we will add an edge in the residual graph
  (w,v) of capacity fvw.
• What are we doing?
• We are allowing the algorithm to undo its
  decisions by sending flow back in the opposite
  direction.

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Augmenting Path

• Old
• New

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Improved Algorithm

• Two units of flow are added along s, b, d, a, c,
  t
• Algorithm terminates with correct solution.

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Algorithm Always Works?

• Theorem. If the edge capacities are rational
  numbers, then the improved algorithm always
  terminates with a maximal flow.
• What are rational numbers?

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Running Time

• Say that the capacities are all integers and the
  maximal flow is f.
• Each augmenting flow increases the flow value by
  at least one.
• How many stages?
• Augmenting path can be found by which algorithm?
• Unweighted shortest path.
• Total time?
• O( f E ).

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Bad Case

• This is a classic bad case for augmenting
• The maximum flow is easily seen by inspection.
• Random augmentations could go along a path that
  includes (a,b).
• What is the worst case for number of
  augmentations?
• 2,000,000.

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Remedy

• How to get around problem on previous slide?
• Choose the augmenting path that gives the largest
  increase in flow.
• Which algorithm?
• Modify the Dijkstras algorithm that solves a
  weighted shortest-path problem.

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Minimum Spanning Tree
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Example
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Second Example

• A graph and its minimum spanning tree

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Prims Algorithm

• Outline

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Prims Algorithm
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Prims Algorithm Tables
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Another Algorithm

• Can you name another algorithm that is very
  similar to Prims algorithm?
• Dijkstras algorithm.
• What is the major difference?

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Running Time

• Without heaps?
• O( V2 ).
• Optimal for dense graphs.
• With heaps?
• O( E logV ).
• Good for sparse graphs.