Single-Source Shortest Path Jeff Chastine

How does Google Maps work? Jeff Chastine

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

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

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

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

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

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

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

Relaxation Example b 2 a d ∞ c 99 Continue with 𝑏 Original Graph 5 a 11 d ∞ 99 3 c 99 Original Graph Predecessor and Cost Continue with 𝑏 Jeff Chastine

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

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

Relaxation Example b 2 a d 7 c 13 Original Graph Predecessor and Cost 5 a 11 d 7 99 3 c 13 Original Graph Predecessor and Cost Jeff Chastine

Relaxation Example But wait! We’re not done! b 2 a d 7 c 13 5 a 11 d 7 99 3 c 13 Original Graph Predecessor and Cost But wait! We’re not done! Jeff Chastine

Relaxation Example No relaxation b 2 a d 7 c 13 Original Graph 5 a 11 d 7 99 3 c 13 Original Graph Predecessor and Cost No relaxation Jeff Chastine

Relaxation Example b 2 a d 7 c 13 Original Graph Predecessor and Cost 5 a 11 d 7 99 3 c 13 Original Graph Predecessor and Cost Jeff Chastine

Relaxation Example Needs to relax again! b 2 a d 7 c 13 Original Graph 5 a 11 d 7 99 3 c 13 Original Graph Predecessor and Cost Needs to relax again! Jeff Chastine

Relaxation Example Needs to relax again! b 2 a d 7 c 10 Original Graph 5 a 11 d 7 99 3 c 10 Original Graph Predecessor and Cost Needs to relax again! Jeff Chastine

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

How does Google Maps work? PWND! Jeff Chastine

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

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

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

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

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