1. EMIS 8374 Shortest Path Problems: Introduction Updated 9 February 2008

2. Assumptions All arc costs (cij) are integers.
The network contains a directed path from the source node to every other node in the network.
No negative (directed) cycles

3. A Network with Negative Cycles3 2 4 1 1 2 1 6 3 -3 -3 3 3 5 1 7

4. Shortest Path Problems
Acyclic Networks

5. Topological Ordering
Renumber the nodes so that for every arc (i,j), i < j
Let k := 1
Find a node i with in-degree = 0
If no such node then quit (no T.O.)
Else let order(i) := k, k := k+1
Remove i and all incident arcs (i,j)
Goto Step 2

6. A Network with no Topological Ordering1 2 3

7. Theorem
A network is acyclic if and only if it has a topological ordering of its nodes.
Complexity of T.O. Algorithm:
O(m)

8. T.O. Algorithm 2 4 1 6 1 3 5

9. T.O. Algorithm 2 4 3 5 6 2

10. T.O. Algorithm 2 4 6 5 3

11. T.O. Algorithm 4 5 2 4 6 6

12. Topological Ordering 4 5 2 4 1 6 1 6 3 5 2 3

13. The Reaching Algorithm: Initialization
Number the nodes by a topological ordering.
Let d(s) := 0; d(i) :=  for all other nodes.
d(i) indicates the length of the shortest path found so far from s to i.
When d(i) = , it means we have not yet found a path from s to i.
Pred(s) := 0;

14. The Reaching Algorithm: Main Loop
for i = 1 to n do
for (i,j) in A(i) do
if d(j) > d(i) + cij then
begin
let d(j) := d(i) + cij
pred(j) := i
end

15. Reaching Algorithm Example  2 4 5 7 1 1 2 6  1 6 5 2 3 -2  

16. Examine A(1) 7   2 4 5 7 1 1 2 6  1 6 5 2 3 -2   5

17. Examine A(2) 7 6  2 4 5 7 1 1 2 6  1 6 5 2 3 -2  5 3

18. Examine A(3) 6 5  2 4 5 7 1 1 2 6  9 1 6 5 2 3 -2 5 3

19. Examine A(4) 5 7  2 4 5 7 1 1 2 6 9 1 6 5 2 3 -2 5 3

20. Examine A(5) 5 7 2 4 5 7 1 1 2 6 8 9 1 6 5 2 3 -2 5 3

21. Examine A(5) 5 7 2 4 5 7 1 1 2 6 8 1 6 5 2 3 -2 5 3

22. Complexity of the Reaching Algorithm
for i = 1 to n do
for (i,j) in A(i) do
if d(j) > d(i) + cij then
begin
let d(j) := d(i) + cij
pred(j) := i;
end
Outer loop (for i := i+1) executes n times.
Examining an arc is O(1).
Comparison, updates of d(j) and pred(j) are O(1).
Each arc is examined once.
O(n+m) = O(m)

23. Shortest Path Tree
We can store a set of shortest paths from s to all other nodes in a tree
Use the pred(j) label to keep track of the last arc (i,j) that sets d(j) = d(i) + cij

24. Shortest Path Tree 7 6 7 5  2 4 5 1 1 2 6 8  5 2 3 -2  5 3 