Dijkstras Algorithm for Finding Shortest Paths

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Dijkstras Algorithm for Finding Shortest Paths

• Irena Pevac

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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.

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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.

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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.

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Example
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Start at A
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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))

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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).