Chapter 4 Shortest Path LabelSetting Algorithms

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Chapter 4Shortest Path Label-Setting Algorithms

• Introduction Assumptions
• Applications
• Dijkstras Algorithm

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Problem Definition Assumptions

• Problem Given a network G (N, A) in which
  each arc (i, j) has an associated length or cost
  cij, let node s be the source. The length of a
  directed path is the sum of the lengths of the
  arcs in the path. For every node i ? s, find a
  shortest length directed path from s to i.
• Assumptions
• All arc lengths are integers (if rational but not
  integer, multiply them by a suitably large
  number)
• The network contains a directed path from node s
  to every node i ? s (if not, add an arc (s, i)
  with very large cost)
• The network does not contain a negative cycle
  (otherwise see Ch. 5)
• The network is directed (transform undirected
  arcs w/positive costs as in Ch. 2 undirected
  arcs w/negative costs will create neg. cycles)

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Types of Shortest Path Problems

• Single-source shortest path
• One node to all others with nonnegative arc
  lengths Chapter 4
• Variations maximum capacity path, maximum
  reliabiltiy path
• One node to all others with arbitrary arc lengths
  Chapter 5
• All-pairs shortest path every node to every
  other node Chapter 5
• String model for shortest path from s to t
• Arcs strings, knots nodes hold s and t and
  pull tight.
• Shortest paths will be taut for i and j on a
  shortest path connected by arc (i, j), distance
  s-i plus cij ? distance s-j
• Associated dual maximization problem pulling
  s and t as far apart as possible

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Single-Source Shortest Paths

• Solution is a shortest-path tree rooted at s.
• Property 4.1. If the path s i1 i2 ih
  k is a shortest path from s to k, then for every
  q 2, 3, , h-1, the subpath s i1 i2
  iq is a shortest path from the source node to iq.
• Property 4.2. Let the vector d represent the
  shortest path distances. Then a directed path P
  from s to k is a shortest path if and only if for
• Store the shortest path tree as a vector of n-1
  predecessor nodes pred(j) is the node i that
  satisfies above equality.

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Acyclic Networks Reaching

• Examine the nodes in topological order perform a
  breadth-first search to find a shortest-path
  tree.
• Reaching Algorithm
• 0. d(s) ? 0, d(j) ?? for j ? s, i ? s
• 1. If A(i) is empty, then stop. Otherwise, to
  examine node i, scan the arcs in A(i). If for
  any arc (i, j), d(j) ? d(i) cij , then set d(j)
  d(i) cij .
• 2. Set i to the next node in topological order
  and return to 1.
• Solves shortest path problem on acyclic networks
  in O(m) time.

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Dijkstras Algorithm

• Shortest paths from source node to all other
  nodes with nonnegative arc lengths (cycles
  permitted)
• Output
• d(i) is the distance from s to i along a
  shortest path
• pred(i) is the predecessor of i along a shortest
  path
• Intermediate
• S set of permanently labeled nodes (L in
  GIDEN)
• set of temporarily labeled nodes (P in GIDEN)
• GIDEN also has set U for unlabeled nodes
• At each iteration, one node moves to S from

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Dijkstras Algorithm in GIDEN
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Complexity

• Node selection requires time proportional to
• Distance updates are performed for each arc
  emanating from node i total of m updates in the
  entire algorithm
• Since
• For dense networks,
• Complexity can be improved for sparse networks by
  cleverness and special data structures