Chapter 5 Shortest Path LabelCorrecting

1
Chapter 5Shortest Path Label-Correcting

• FIFO Label-Correcting Algorithm
• All Pairs Shortest Paths
• Detecting Negative Cycles

2
Introduction

• With negative arc lengths, Dijkstras algorithm
  may fail why? may need to revise some node
  labels
• Single source goal
• detect a negative cycle if it exists,
• otherwise, find shortest paths from s to all
  other nodes
• Label-correcting algorithms maintain a distance
  label d(j) for every node j
• At intermediate nodes, d(j) is an upper bound on
  the length of a shortest path from s to j
• When the algorithm terminates (w/o a negative
  cycle), d(j) is the shortest path length
• No label is permanent until the algorithm stops

3
Optimality Conditions

• What are necessary and sufficient conditions for
  a set of labels d(j) to be shortest path lengths?
• Necessary
• (otherwise, we could find a shorter path with i
  as pred(j) )
• These are also sufficient conditions! (proof in
  text)
• But if the network contains a negative cycle,
  then no set of distance labels will satisfy the
  NS conditions for at least some nodes, can
  always decrease d(j) by going around the negative
  cycle once more
• For now, assume no negative cycles find d(j)
  that satisfy conditions.

4
FIFO Label-Correcting Algorithm

• algorithm FIFO label-correcting
• begin
• d(s) 0 and pred(s) 0 d(j) ? for each
  node j ? s
• LIST s
• while LIST ? ? do
• begin
• remove the first element from LIST
• for each arc (i, j) ?A(i) do
• if d(j) gt d(i) cij then
• begin
• d(j) d(i) cij pred(j) i
• if j ?LIST then append j to the end of LIST
• end
• end
• end

5
Analysis

• How do we know the algorithm will stop?
• Assume arc lengths are integers. Say C is the
  largest arc length. Each d(j) is bounded above
  by nC and from below by nC. Since each update of
  d(j) decreases it by at least 1, d(j) will be
  updated at most 2nC times. Since each iteration
  updates a distance label, there will be at most
  2n2C iterations, or O(n2C)
• When it stops, the distance labels will satisfy
  the optimality conditions
• FIFO algorithm actually solves the problem in
  O(nm) time

6
Detecting a Negative Cycle

• Keep track of how many times the algorithm
  examines each node.
• If it examines any node more than n-1 times, then
  it must be considering paths containing more than
  n-1 arcs, i.e., paths with cycles.
• The only way for a path with more arcs to have
  shorter total arc length is that it is not a path
  but a walk containing a negative cycle
• Therefore, if algorithm examines a node more than
  n-1 times, stop and conclude a negative cycle
  exists

7
Dequeue Version of Label-Correcting

• A dequeue is a list to (from) which elements can
  be added (deleted) at either end
• In the dequeue implementation of the
  label-correcting algorithm,
• nodes are selected from the front of the list
• nodes are added to the front of the list if they
  have been examined earlier
• nodes are added to the rear of the list otherwise
• This implementation is more efficient because it
  updates distances sooner and reexamines fewer
  nodes.

8
Detecting Negative Cycles II

• Every so often, check to see if the predecessor
  graph (which is to become the shortest path tree)
  contains a cycle
• such a cycle would occur if a negative cycle were
  driving the path lengths down
• How to detect a cycle in the predecessor graph?
• Call the source node labeled and all other nodes
  unlabeled
• Examine each unlabeled node k
• Assign a label k to node k
• Trace predecessor indices back from k and assign
  the label k to all nodes until a previously
  labeled node, say l, is reached
• If nodes k and l have the same label, then the
  pred graph is cyclic
• Requires O(n) time
• Do this after every an distance updates

9
All Pairs Shortest Paths

• Assume the network is strongly connected and does
  not contain a negative cycle
• Problem find a shortest path from every node to
  every other node. For each pair of nodes i and
  j, we seek di, j the length of a shortest
  path from i to j.
• For sparse networks, can simply apply the single
  source shortest path algorithm n times, with a
  different node as source node each time
• For dense networks, an all-pairs label-correcting
  algorithm is more efficient

10
Floyd-Warshall Algorithm

• Optimality conditions A set of node labels
  di, j represent shortest path lengths if and
  only if they satisfy the triangle inequality
• Floyd-Warshall updates distances in a clever way
  until they satisfy the optimality conditions
• Let be the length of a shortest path from node
  i to node j if only nodes 1,, k-1 are used as
  intermediate nodes. Compute recursively as