Shortest path algorithms

1
Shortest path algorithms

• Dijkstra (aka Shortest Path First (SPF))
• Bellman-Ford (aka Ford-Fulkson)
• Floyd-Warshall

2
Shortest path problem

• For a directed graph G(N,A), each arc (x,y) has
  associated with it a number d(x,y) that
  represents the length (previously known as
  weight) of the arc.
• The length of a path is defined as the sum of the
  lengths of the individual arcs comprising the
  path.
• The shortest path problem is to determine the
  path with the minimum path length from s to t.

3
Dijkstra algorithm

• This algorithm is applicable to graphs with
  positive arc lengths.
• Given a source node, Dijkstra algorithm
  determines the shortest path to a specific node,
  or it can be run until the shortest paths to all
  nodes other than the source node has been found.
• Let D(x) denote the the path length from node x
  to the source node.

4
Dijkstra algorithm (cont.)

• The basic principle is illustrated as follows.

• Denote those nodes to which the shortest path
  from the source has been found as permanent node,
  the other nodes tentative node. As the algorithm
  proceeds, each node is assigned either a
  tentative label or a permanent label
• For those tentative nodes, the shortest path must
  be via some permanent node. Based on this
  principle, the shortest path from S to one of the
  tentative nodes can be determined.

shortest paths have been found for these nodes
shortest paths have NOT yet been found for these
nodes
D(y)
D(x)
y
x
d(y,x)
S
z
D(x)D(y)d(y,x) or d(s,x)
5
Dijkstra algorithm stated

• Dijkstra algorithm can be stated as follows.
• Step 0 The source node s is given a permanent
  label of (0,s) Assign tentative (t) labels (
  D(x)?, u(x) - ) to all nodes other than the
  source node where D(x) is the distance from
  source to this node over the (tentative) shortest
  path and u(x) is the upstream node in the
  shortest path. Let y denote the previous node
  that has been assigned a permanent label set
  ys.
• Step 1 For each tentative node x, redefine d(x)
  as D(x) min D(x), D(y)d(y,x)
• Step 2 Search over all the tentative labels and
  change the label of a node ( D(x), u(x) ) to
  permanent (p) if the D(x) is the smallest among
  all the tentative labels update u(x) as well.
• Step 3 Set yx (update y) If the specific node
  has not yet been given a permanent label, go to
  step 2 otherwise stop (the shortest path from s
  to the specific node has been determined).

6
Dijkstra algorithm an example
3
A
B
2
Steps 2,3 Find p, y
2
4
7
S
T
3
2
3
D
C
Step 1 Update D(x)
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Dijkstra algorithm an example (cont.)
Another example in flash movie
8
Bellman-Ford algorithm

• Dijkstra algorithm cannot accommodate graphs with
  negative arc lengths.
• Bellman-Ford algorithm includes some changes to
  Dijkstra algorithm.
• In the i-th iteration, a set of nodes Y denotes
  the shortest paths with at most i arcs (known as
  the shortest ? i path).
• In Step 1, compute ??D(x) minD(x),D(y)d(y,x)
  ?? for all nodes for y in Y.
• Terminate the algorithm only after Step 1 fails
  to lower D(x) values of the nodes.

9
Bellman-Ford algorithm stated

• Bellman-Ford algorithm can be stated as follows.
• Step 0 The source node s is given a label of
  (0,s) Assign labels ( D(x)?, u(x) - ) to all
  nodes other than the source node where D(x) is
  the distance from source to this node over the
  shortest path and u(x) is the upstream node
  in the shortest path. Let Y be the set containing
  s.
• Step 1 For each node x that are neighbor nodes
  (nodes that are adjacent to x) of a node in Y,
  redefine d(x) as D(x) min D(x), D(y)d(y,x)
  for y in Y.
• Step 2 Stop if Step 1 does not lower the value
  of any D(X).
• Step 3 Add x to the set Y. Go to Step 1.

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Bellman-Ford algorithm an example
3
A
B
2
Steps 2,3 update Y
2
6
7
S
T
3
2
3
D
C
Step 1 Update D(x)
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Bellman-Ford algorithm another example
Note that Bellman-Ford algorithm may fail if
there exits a cycle with negative total weights.
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Floyd-Warshall algorithm

• The Floyd-Warshall algorithm determines the
  shortest path between all pairs of nodes.
• The arc distance can be negative.
• The Floyd-Warshall algorithm iterates on the set
  of nodes that are allowed as intermediate nodes
  on the path.