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Chapter 4 Shortest Path Label-Setting Algorithms by pptfiles


									               Chapter 4
Shortest Path Label-Setting Algorithms
         Introduction & Assumptions
             Dijkstra’s Algorithm
    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.
   – 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)
      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
         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  i, j   P, d  j   d i   cij
Store the shortest path tree as a vector of n-1 predecessor
   nodes: pred(j) is the node i that satisfies above equality.

         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.

               Dijkstra’s Algorithm

Shortest paths from source node to all other nodes with
   nonnegative arc lengths (cycles permitted)
   d(i) is the distance from s to i along a shortest path
   pred(i) is the predecessor of i along a shortest path
   S = set of permanently labeled nodes (L in GIDEN)
   S  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 S
Dijkstra’s Algorithm in GIDEN


• Node selection requires time proportional to
          n   n  1   n  2   ...  1  O  n2 
• Distance updates are performed for each arc emanating
  from node i; total of m updates in the entire algorithm
• Since n2  m, the complexity is O  n2 
• For dense networks, n 2  m
   – Complexity can be improved for sparse networks by cleverness
     and special data structures


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