Shortest Path Algorithms

1
Shortest Path Algorithms
2
Routing Algorithms

• Shortest path routing
• What is a shortest path?
• Minimum number of hops?
• Minimum distance?
• There is a weight associated with each link
• Weight can be a measure of congestion in the
  link, propagation delay etc.
• Weight of a path is the sum of weight of all
  links
• Shortest path is the minimum weight path

3
Weight of path 1 2.5
Weight of path 2 3.0
4
Computation of shortest paths

• Enumerate all paths?
• Exponential complexity
• Several polynomial complexity algorithms exist
• Dijkstras algorithm (greedy algorithm)
• Bellman-ford algorithm (distributed algorithm)
• Flloyd-Warshall algorithm (dynamic programming)
• Johnsons algorithm

5
Dijkstras algorithm

• Assumes a directed graph

• Given any node, finds the shortest path to every
  other node in the graph

• O(V log V E)

6

• Let source node be s
• Maintains shortest path estimate for every
  vertex v, (d(v))
• estimate is what it believes to be the
  shortest path from s
• and the list of vertices for whom the
  shortest path is known
• Initially the list of vertices for whom the
  shortest path is known is empty and
• the estimates are infinity for all vertices
  except the source vertex itself.

7

• It holds that whenever the estimate d(v) is
  finite for a vertex v, there exists a path from
  the source s to v with weight d(v)
• It turns out that the estimate is accurate for
  the vertex with the minimum value of this
  estimate
• Shortest path is known for this vertex (v)
• This vertex (v) is added to the list of vertices
  for whom shortest path is known
• Shortest path estimates are upgraded for every
  vertex which has an edge from v, and is not in
  this known list.

8
Estimate Upgrade Procedure

• Suppose vertex v is added to the list newly, and
  we are upgrading the estimate for vertex u
• d(v) is the shortest path estimate for v, d(u) is
  the estimate for u
• w(v, u) is the weight of the edge from v to u
• d(u) -gt min(d(u), d(v) w(v, u))

9
Intuition behind the upgrade procedure

• Assume that d(u) and d(v) are finite
• So there exists a path to v from s of weight
  d(v), (s, v1, v2,..v)
• Hence there exists a path from s to u (s, v1,
  v2,..v, u) of weight d(v) w(v, u)
• Also, there exists a path to u of weight d(u).
• So the shortest path to u can not have weight
  more than either d(u) or d(v) w(v, u).
• So we upgrade the estimate by the minimum of the
  two.

10
Notation

• Source vertex s
• Shortest path estimate of vertex v d(v)
• Weight of edge (u, v) w(u, v)
• Set of vertices whose shortest paths are known S
• Q V \ S

11
Algorithm
1. For every vertex v in V, d(v) ? 2. d(s)
0 3. Q V S ? 4. While Q is non-empty
(a) vExtract-min(Q)
12
(b) Add v to S (c ) Remove v from Q (d) For
each vertex u in Adj(v) d(u) -gt min(d(u), d(v))
w(v, u))
13
Example
14
Algorithm Complexity

• Statement 1 is executed V times
• Statements 2 and 3 are executed once
• Loop at statement 4 is executed V times
• Every extract-min operation can be done in at
  most V operations
• Statement 4(a) is executed total V2 times
• Statements 4(b) and 4(c ) are executed V times
  each (total)
• Observe that statement 4(d) is executed E times

15

• So overall complexity is O(V2 E) and this
  is same as O(V2)
• Using improved data structures complexity can be
  reduced
• O((V E)log V) using binary heaps
• O(V log V E) using fibonacci heaps

16
Proof of Correctness

• Exercise
• Verify that whenever d(v) is finite, there is a
  path from source s to vertex v of weight d(v)

17
Assumptions

• Assume that source s is connected to every vertex
  in the graph, and all edge weights are finite
  
  Also, assume that edge weights are positive.
• Let p(s, v) be the weight of the shortest path
  from s to v.
• Will show that the graph terminates with
  d(v)shortest path weights for every vertex

18

• Will first show that once a vertex v enters S,
  d(v) equals the shortest path weight from source
  s, at all subsequent times.
• Clearly this holds in step 1, as source enters S
  in step 1, and d(s) 0
• Let this not hold for the first time in step k gt
  1
• Thus the vertex u added has d(u) gt p(s, u)
• Consider the situation just before insertion of
  u.
• Consider the true shortest path, p, from s to u.
  

19

• Since s is in S, and u is in Q, path p must jump
  from S to Q at some point.

S
Q
u
s
x
y
Path p
Let the jump have end point x in S, and y in Q
(possibly s x, and u y)
We will argue that y and u are different vertices
Since path p is the shortest path from s to u,
the segment of path p between s and x, is the
shortest path from s to x, and that between s and
y is the shortest from s to y
20
S
Q
u
s
w(x,y)
y
x
Path p

• Weight of the segment between s and x is d(x)
• since x is in S, d(x) is the weight of the
  shortest path to x

Weight of the segment between s and y is d(x)
w(x, y)
Thus, p(s, y) d(x) w(x, y)
Also, d(y) lt d(x) w(x, y) p(s, y)
Follows that d(y) p(s, y)
However, d(u) gt p(s, u). So, u and y are different
21
Since, y appears somewhere along the shortest
path between s and u, but y and u are different,
p(s, y) lt p(s, u) Using the fact that all edges
have positive weight
Hence, d(y) p(s, y) lt p(s, u) lt d(u)
Both y and u are in Q. So, u should not be chosen
in this step
So, whenever a vertex u is inducted in S, d(y)
p(s, y).
Once d(u) equals p(s, u) for any vertex it can
not change any further (d(u) can only decrease
or remain same, and d(u) can not fall below p(s,
u).
Since the algorithm terminates only when S V, we
are done!
22

• We have proved only for edges with positive
  weight
• One can prove for edges with nonnegative weight
• Shortcoming
• Does not hold for edges with nonnegative weight
• Centralized algorithm

23
Exercise This computation gives shortest path
weights only. Modify this algorithm to generate
shortest paths as well!
24
Bellman-ford Algorithm

• Applies as long as there are no nonpositive
• weight cycles
• If there are circles of weight 0 or less, then
  the shortest paths are not well defined
• Capable of full distributed operation
• O(VE) complexity
• slower than Dijkstra

25
Algorithm description

• Every node v maintains a shortest path weight
  estimate, d(v)
• These estimates are initialized to infinity, for
  all vertices except source, s, d(s)0
• Every node repeatedly updates its shortest path
  estimate as follows

26
(No Transcript)
27
Example
28
Complexity Analysis

• Initialization step takes V 1 steps
• The loop in statement 3 is executed V times
• Each execution takes E steps
• Overall, there are V 1 VE steps
• O(VE)

29
Proof that it works

• Assume that all vertices are reachable from
  source, s.
• Thus there is a shortest path to any vertex v
  from s.
• Assume that the graph has no cycles of weight 0
  or less
• So the shortest paths can not have more than
  V-1 edges.
• We will prove that at the termination of
  Bellman-Ford algorithm, d(v)p(s,v) for every
  vertex v.
• We will show that if there is one shortest path
  to a vertex of k hops, then after the kth
  execution of the loop in statement 3, d(v)
  freezes at p(s, v)

30
We know the above holds for k 0, as d(s) p(s,
s) 0 at all times.
Let the above hold for 1,.,k. We will show that
this holds for k 1
So, by induction hypothesis, d(y) p(s, y) after
the kth iteration and at all subsequent times .
So by the estimate update procedure, d(u) lt d(y)
w(y, u) p(s, y) w(y, u) weight of path p
p(s, u) after the k1 th iteration and all
subsequent times.
31
We have just shown that d(u) lt p(s, u) after the
k1 th iteration
Again verify that as long as d(v) is finite, d(v)
is length of some path to vertex v.
Hence d(u) gt p(s, u) always
Thus, d(u) p(s, u), always after the k1th
iteration.
32
Features of this algorithm

• Note that a node needs information about its
  neighbors only!
• So we do not need a global processor.