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Dijkstras Algorithm for Finding Shortest Paths

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Dijkstra's Algorithm for Finding Shortest Paths. Initially only the source node s is settled. ... We find the node v that is unsettled. We settle v by ... – PowerPoint PPT presentation

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Title: Dijkstras Algorithm for Finding Shortest Paths


1
Dijkstras Algorithm for Finding Shortest Paths
  • Irena Pevac

2
Shortest Paths
  • Def The distance along the path is the sum of
    the labels of that path.
  • Def The minimum distance from node u to node v
    is the minimum of the distance on any path from u
    to v.
  • Dijkstra algorithm gives efficient way to find
    the minimum distance from the source node to all
    the nodes of the graph. Graph can be either
    directed or undirected.

3
Settled and Unsettled Nodes
  • We discover the minimum distance from the source
    to other nodes in the order of those distances.
  • We will call settled nodes those nodes for which
    we know their minimum distance.
  • The set of settled nodes always includes source.
  • For unsettled nodes v, we record the length of
    the shortest path that starts at the source,
    travels only through settled nodes, and at last
    step jumps out of the settled region to v.

4
Dijkstras Algorithm for Finding Shortest Paths
  • Initially only the source node s is settled. Its
    distance to the source dist(s)0.
  • If there is an arc from s to u dist(u) is the
    label of that arc. If there is no arc from s to
    u, the dist(u)?.
  • Assume that some nodes are settled. We find the
    node v that is unsettled. We settle v by
  • Make dist(v) to be the min distance from s to v.
  • Adjust the value of dist(u) for all nodes u that
    remain unsettled, to account for the fact that v
    is now settled.

5
Example
6
Start at A
7
Dijkstra(G,s)
  • for every vertex v in V
  • dv ?
  • Insert (Q, v, dv) // initialize vertex
    priority in PQ
  • Dist(s)0
  • Decrease(Q,s,dist(s))
  • For i 0 to V-1
  • uDeleteMin(Q)
  • VTVT U u
  • for every vertex u in V-VT that is adjacent
    to u
  • if (dist(u) label(u,u) lt dist(u))
  • dist(u) dist(u)
    label(u,u)
  • u is settled.
  • Decrease(Q,u,dist(u))

8
Time efficiency for Dijkstras Algorithm
  • Let G(V,E) Vn Em
  • Let graph be represented with weight adjacency
    matrix, and priority queue be implemented as an
    unordered array.
  • Time is ?(n2).
  • Let graph be represented with weight adjacency
    lists, and priority queue be implemented as a min
    heap.
  • Time is ?( m logn).
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