Shortest Path

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Shortest Path
Consider the following weighted undirected graph
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Such that cost(5?v1) cost(v1? v2) cost (
?20) is minimum
2
How About Using the MST Algorithm?
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1. Compute the MST
2. Take the path between the two nodes that goes
   through the MST

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The Fringe
Given a Graph G (V,E) and T a subset of V, the
fringe of T, is defined as Fringe(T)
(w,x) in E w ? T and x ?V - T
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Djikstra Algorithm Idea

• We are computing a shortest path from node u to a
  node v
• Start the algorithm with T u, construct
  Fringe(T)
• At each iteration, select the (z,w) in Fringe(T)
  such that the path from u to w has the lowest
  cost
• Stop when v is part of T (not of the fringe!)

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Example
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66
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The Dijkstras Algorithm
// Input G (G,V), nodes u and v in V //output
Shortest path from u to v
T ? u F ? Fringe(u) ET ? While v is not
in the T do Choose (z,w) in F with the
shortest path (u, u1),
(u1,u2),, (un,z) (z,w) with (u, u1),
(u1,u2),, (un,z) in ET T ? T ? w
ET ? ET ? (z,w) for each (w,x)
(w,x) in E - ET do If no (w,x) in
F then F ? F ? (w,x) else

If the path u ? x going through (w,x) has less
weight than the path u?x going through
(w,x) then F ? ( F - (w,x)) ? (w,x)
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Properties
Is Dijkstras Algorithm greedy?
Yes
Why Dijkstras Algorithm works?
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Complexity

• Actual complexity is O(Elog2 E)

• It is easy to see that it is at most EV
  (i.e., E2)
• the outer while is performed V times
• the inner for is performed at most E times

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Homework

• 010
• Write the pseudo-code for the algorithm inserting
  an element in a min-heap
• Do the same for a Heap
• Write the pseudo-code for isLeaf(i) (Slide 15 of
  class about Heaps)
• Change Dijkstras Algorithm (Slide 6) to compute
  the minimum distance from u to every node in the
  graph
• True or False the subgraph (T,ET) resulting from
  4 is an MST of the input graph
• In the graph of 2.b (Page 323). Follow Dijkstras
  Algorithm to compute the path from c to l
• 011
• Write the pseudo-code for the algorithm deleting
  the element with the minimum priority value in a
  min-heap
• Do the same but deleting the one with the highest
  priority for a Heap
• Write the pseudo-code for Parent(i) (Slide 15 of
  class about Heaps)
• Change Dijkstras Algorithm (Slide 6) to compute
  the minimum distance from u to every node in the
  graph
• True or False the subgraph (T,ET) resulting from
  4 is a tree connecting all nodes of the input
  graph
• In the graph of 2.b (Page 323). Follow Dijkstras
  Algorithm to compute the path from l to a