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
𝑘
𝑤 𝑝 =
𝑤(𝑣𝑖−1 , 𝑣𝑖 )
𝑖=1
𝑝
𝜕 𝑢, 𝑣 = min 𝑤 𝑝 : 𝑢 → 𝑣 𝑖𝑓 ∃ 𝑝
∞
• Note: Breadth works for non-weighted edges
Jeff Chastine
Single-Source Shortest Path
• Has optimal sub-structure. Why?
𝑝𝑢𝑤
𝑝𝑥𝑤
𝑝𝑤𝑣
– Path is 𝑢
𝑥
𝑤
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
b
5
11
a
∞
d
3
99
∞
0
c
∞
Original Graph
Predecessor and Cost
INITIALIZE-SINGLE-SOURCE(G, s)
Jeff Chastine
Relaxation Example
2
b
5
11
a
∞
d
3
99
∞
0
c
∞
Original Graph
Predecessor and Cost
Start with source node 𝑎
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
∞
0
3
99
c
99
Original Graph
Predecessor and Cost
Update connected nodes
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
∞
0
3
99
c
99
Original Graph
Predecessor and Cost
Continue with 𝑏
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
0
7
3
99
c
99
Original Graph
Predecessor and Cost
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
0
7
3
99
c
99
Original Graph
Predecessor and Cost
This guy can chill out… RELAX!
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
0
7
3
99
c
13
Original Graph
Predecessor and Cost
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
0
7
3
99
c
13
Original Graph
Predecessor and Cost
But wait! We’re not done!
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
0
7
3
99
c
13
Original Graph
Predecessor and Cost
No relaxation
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
0
7
3
99
c
13
Original Graph
Predecessor and Cost
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
0
7
3
99
c
13
Original Graph
Predecessor and Cost
Needs to relax again!
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
0
7
3
99
c
10
Original Graph
Predecessor and Cost
Needs to relax again!
Jeff Chastine
Relaxation Example
2
b
5
11
a
2
d
0
7
3
99
c
10
Original Graph
Predecessor and Cost
Final Predecessor and Cost
Jeff Chastine
How does Google Maps work?
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)
1 INITIALIZE-SINGLE-SOURCE(G, s)
2 for i ← 1 to |V[G]| -1
3
foreach edge (u, v) ∈ E[G]
4
RELAX(u, v, w)
5 foreach edge (u, v) ∈ E[G]
6
if d[v] > d[u] + w(u, v)
7
then return FALSE
8 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)
1 INITIALIZE-SINGLE-SOURCE(G, s)
2 S←∅
3 Q ← V[G]
4 while Q ≠ ∅
5
do u ← EXTRACT-MIN(Q)
6
S←S ∪ {u}
7
foreach vertex v ∈ Adj[u]
8
do RELAX(u, v, w)
Jeff Chastine
Summary
• Both algorithms use
– 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