1. SINGLE-SOURCE SHORTEST PATHS

2. Shortest-Path Variants
Single-Source Single-Destination (1-1)
No good solution that beats 1-M variant
Thus, this problem is mapped to the 1-M variant
Single-Source All-Destination (1-M)
Need to be solved (several algorithms)
We will study this one
All-Sources Single-Destination (M-1)
Reverse all edges in the graph
Thus, it is mapped to the (1-M) variant
All-Sources All-Destinations (M-M)
We will study it (if time permits)

3. Shortest Path Shortest Path = Path of minimum weight d(u,v)=
min{ω(p) : u v}; if there is a path from u to v,
 otherwise.

4. Algorithms to Cover
Dijkstra’s Algorithm
Shortest Path is DAGs

5. Initialization Maintain d[v] for each v in V
d[v] is called shortest-path weight estimate
and it is upper bound on δ(s,v)
INIT(G, s)
for each v  V do
d[v] ← ∞
π[v] ← NIL
d[s] ← 0
Upper bound
Parent node

6. Relaxation RELAX(u, v) if d[v] > d[u]+w(u,v) then
π[v] ← u
When you find an edge (u,v) then check this condition and relax d[v] if possible u v u v 2 2 5 9 5 6
Change d[v]
No change 5 7 5 6 2 2 u v u v

7. Algorithms to Cover
Dijkstra’s Algorithm
Shortest Path is DAGs

8. Dijkstra’s Algorithm For Shortest Paths
Non-negative edge weight
Like BFS: If all edge weights are equal, then use BFS, otherwise use this algorithm
Use Q = min-priority queue keyed on d[v] values

9. Dijkstra’s Algorithm For Shortest Paths
DIJKSTRA(G, s)
INIT(G, s)
S←Ø > set of discovered nodes
Q←V[G] > put all vertices in a queue
while Q ≠Ø do
u←EXTRACT-MIN(Q)
S←S U {u}
for each v in Adj[u] do
RELAX(u, v) > May cause
> DECREASE-KEY(Q, v, d[v])

10. Example: Initialization Step

11. Example 10 5 s u v y x 1 3 9 4 6 7 2 2

12. Example 8 14 5 7 s y x 10 1 3 9 4 6 2 u v 2

13. Example 8 13 5 7 s u v y x 1 2 3 9 4 6 10

14. Example 8 u v 9 5 7 y x 10 1 2 3 4 6 s

15. Example u 8 9 5 7 v y x 10 1 2 3 4 6 s 5

16. Dijkstra’s Algorithm Analysis
O(V)
O(V Log V)
Total in the loop: O(V Log V)
Total in the loop: O(E Log V)
O(Log V) for each call
Time Complexity: O (E Log V)

17. Algorithms to Cover
Dijkstra’s Algorithm
Shortest Path is DAGs

18. Key Property in DAGs If there are no cycles  it is called a DAG
(Directed Acyclic Graph)
In DAGs, nodes can be sorted in a linear order such that all edges are forward edges
Topological sort

19. Single-Source Shortest Paths in DAGs
Shortest paths are always well-defined in DAGs
no cycles => no negative-weight cycles even if there are negative-weight edges
Idea:
No backward edges
Process nodes left to right. When reach node X its cost will not change any more because there are no backward edges from the coming nodes
We will be done in 1 pass

20. Single-Source Shortest Paths in DAGs
DAG-SHORTEST PATHS(G, s)
TOPOLOGICALLY-SORT the vertices of G
INIT(G, s)
for each vertex u taken in topologically sorted order do
for each v in Adj[u] do
RELAX(u, v)

21. Example 6 1 u r s t v w 5 2 7 –1 –2      4 3 2

22. Example 6 1 u r s t v w 5 2 7 –1 –2      4 3 2    4 3 2
Any nodes left from the source s will have cost ∞ (not reachable from s)

23. Example 6 1 u r s t v w 5 2 7 –1 –2  2 6   4 3 2

24. Example 6 1 u r s t v w 5 2 7 –1 –2  2 6 6 4 4 3 2

25. Example 6 1 u r s t v w 5 2 7 –1 –2  2 6 5 4 4 3 2

26. Example 6 1 u r s t v w 5 2 7 –1 –2  2 6 5 3 4 3 2

27. Example 6 1 u r s t v w 5 2 7 –1 –2  2 6 5 3 4 3 2

28. Single-Source Shortest Paths in DAGs: Analysis
O(V+E)
DAG-SHORTEST PATHS(G, s)
TOPOLOGICALLY-SORT the vertices of G
INIT(G, s)
for each vertex u taken in topologically sorted order do
for each v in Adj[u] do
RELAX(u, v)
O(V)
Total O(E)
Time Complexity: O (V + E)

29. Shortest-Path Variants
Single-Source Single-Destination (1-1)
No good solution that beats 1-M variant
Thus, this problem is mapped to the 1-M variant
Single-Source All-Destination (1-M)
Need to be solved (several algorithms)
We will study this one
All-Sources Single-Destination (M-1)
Reverse all edges in the graph
Thus, it is mapped to the (1-M) variant
All-Sources All-Destinations (M-M)
We will study it (if time permits)