1. Single-Source Shortest Path
Jeff Chastine

2. How does Google Maps work?
Jeff Chastine

3. Single-Source Shortest Path
How many ways can I go from SPSU to KSU?
How can I represent the map?
What are vertices?
What are edges?
What are weights?
Jeff Chastine

4. Single-Source Shortest Path
For this problem
Disallow cycles
Still have weight function 𝑤:𝐸→𝑅
Calculate path p and shortest path 𝜕 where
Note: Breadth works for non-weighted edges
𝑤 𝑝 = 𝑖=1 𝑘 𝑤( 𝑣 𝑖−1 , 𝑣 𝑖 )
𝜕 𝑢,𝑣 = min 𝑤 𝑝 :𝑢 𝑝 𝑣 𝑖𝑓 ∃ 𝑝 ∞
Jeff Chastine

5. Single-Source Shortest Path
Has optimal sub-structure. Why?
Path is 𝑢 𝑝 𝑢𝑤 𝑥 𝑝 𝑥𝑤 𝑤 𝑝 𝑤𝑣 𝑣 is fastest
Assume faster way between x and w
Cut and paste faster way and you get optimal!
Graph can’t have a negative cycle. Why?
Can the graph have a positive cycle?
Can the shortest path contain a positive cycle?
Jeff Chastine

6. How it’s Done in General
A node v has a predecessor node 𝜋(𝑣)
Tells us how we got there
NIL for first node
End with a directed graph rooted at start node s
Relaxation (pay attention!)
Maintain an upper-bound cost for each node d[v]
Initially, all nodes are marked as ∞
If we find a cheaper path to node v, relax (update) the cost 𝑑[𝑣] and predecessor 𝜋(𝑣)
Jeff Chastine

7. Relaxation Example (using a breadth-first traversal)∞ 2 5 a 11 d ∞ 99 3 c ∞
Original Graph
Predecessor and Cost
INITIALIZE-SINGLE-SOURCE(G, s)
Jeff Chastine

8. Start with source node 𝑎
Relaxation Example b ∞ 2 5 a 11 d ∞ 99 3 c ∞
Original Graph
Predecessor and Cost
Start with source node 𝑎
Jeff Chastine

9. Update connected nodes
Relaxation Example b 2 2 5 a 11 d ∞ 99 3 c 99
Original Graph
Predecessor and Cost
Update connected nodes
Jeff Chastine

10. Relaxation Example b 2 a d ∞ c 99 Continue with 𝑏 Original Graph5 a 11 d ∞ 99 3 c 99
Original Graph
Predecessor and Cost
Continue with 𝑏
Jeff Chastine

11. Relaxation Example b 2 a d 7 c 99 Original Graph Predecessor and Cost5 a 11 d 7 99 3 c 99
Original Graph
Predecessor and Cost
Jeff Chastine

12. Relaxation Example b 2 a d 7 c 99 This guy can chill out… RELAX!5 a 11 d 7 99 3 c 99
Original Graph
Predecessor and Cost
This guy can chill out… RELAX!
Jeff Chastine

13. Relaxation Example b 2 a d 7 c 13 Original Graph Predecessor and Cost5 a 11 d 7 99 3 c 13
Original Graph
Predecessor and Cost
Jeff Chastine

14. Relaxation Example But wait! We’re not done! b 2 a d 7 c 135 a 11 d 7 99 3 c 13
Original Graph
Predecessor and Cost
But wait! We’re not done!
Jeff Chastine

15. Relaxation Example No relaxation b 2 a d 7 c 13 Original Graph5 a 11 d 7 99 3 c 13
Original Graph
Predecessor and Cost
No relaxation
Jeff Chastine

16. Relaxation Example b 2 a d 7 c 13 Original Graph Predecessor and Cost5 a 11 d 7 99 3 c 13
Original Graph
Predecessor and Cost
Jeff Chastine

17. Relaxation Example Needs to relax again! b 2 a d 7 c 13 Original Graph5 a 11 d 7 99 3 c 13
Original Graph
Predecessor and Cost
Needs to relax again!
Jeff Chastine

18. Relaxation Example Needs to relax again! b 2 a d 7 c 10 Original Graph5 a 11 d 7 99 3 c 10
Original Graph
Predecessor and Cost
Needs to relax again!
Jeff Chastine

19. Final Predecessor and Cost
Relaxation Example b 2 2 5 a 11 d 7 99 3 c 10
Original Graph
Predecessor and Cost
Final Predecessor and Cost
Jeff Chastine

20. How does Google Maps work?
PWND!
Jeff Chastine

21. Bellman-Ford Algorithm
Works with negatively weighted edges
Detects if negative cycle exists
Consistently uses relaxation
Runs in Θ(𝑉𝐸)
Jeff Chastine

22. Bellman-Ford Algorithm
BELLMAN-FORD(G, w, s)
INITIALIZE-SINGLE-SOURCE(G, s)
for i ← 1 to |V[G]| -1
foreach edge (u, v) ∈ E[G]
RELAX(u, v, w)
if d[v] > d[u] + w(u, v)
then return FALSE
return TRUE
Jeff Chastine

23. Dijkstra’s Algorithm Is more efficient than Bellman-Ford
Doesn’t work with negative edges
Has a set S of vertices that it has already traversed. Heapifies V.
In general
Picks vertex u from V - S with minimum estimate
Relaxes everything connected to u
Adds u to S
Repeats until V - S is the empty set
Jeff Chastine

24. Dijkstra’s Algorithm DIJKSTRA (G, w, s) INITIALIZE-SINGLE-SOURCE(G, s)
Q ← V[G]
while Q ≠∅
do u ← EXTRACT-MIN(Q)
S←S ∪ {u}
foreach vertex v ∈ Adj[u]
do RELAX(u, v, w)
Jeff Chastine

25. Summary Both algorithms use Bellman-Ford (𝑂(𝐸 𝑉)) Dijkstra (𝑂(𝐸 𝑙𝑔𝑉))
A predecessor graph
Costs to each node
Relaxation
Bellman-Ford (𝑂(𝐸 𝑉))
Works with negative weights
Detects negative cycles
Dijkstra (𝑂(𝐸 𝑙𝑔𝑉))
More efficient
Doesn’t work with negative weights
Jeff Chastine