Priority Queue Q:
H
(t,H
8), (x, 13)
H
t
x
1
8
13
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
6.4: Single-Source Shortest Paths
Frank Stajano
Lent 2015
Thomas Sauerwald
20
Outline
Bellman-Ford
Dijkstra’s Algorithm
All-Pairs Shortest Path
APSP via Matrix Multiplication
Johnson’s Algorithm
6.4: Single-Source Shortest Paths
T.S.
2
The Bellman-Ford Algorithm
BELLMAN-FORD(G,w,s)
0: assert(s in G.vertices())
1: for v in G.vertices()
2:
v.predecessor = None
3:
v.d = Infinity
4: s.d = 0
5:
6: repeat |V|-1 times
7:
for e in G.edges()
8:
Relax edge e=(u,v): Check if u.d + w(u,v) < v.d
9:
if e.start.d + e.weight.d < e.end.d:
10:
e.end.d = e.start.d + e.weight
11:
e.end.predecessor = e.start
12:
13: for e in G.edges()
14:
if e.start.d + e.weight.d < e.end.d:
15:
return FALSE
16: return TRUE
6.4: Single-Source Shortest Paths
T.S.
17
The Bellman-Ford Algorithm
BELLMAN-FORD(G,w,s)
0: assert(s in G.vertices())
1: for v in G.vertices()
2:
v.predecessor = None
3:
v.d = Infinity
4: s.d = 0
5:
6: repeat |V|-1 times
7:
for e in G.edges()
8:
Relax edge e=(u,v): Check if u.d + w(u,v) < v.d
9:
if e.start.d + e.weight.d < e.end.d:
10:
e.end.d = e.start.d + e.weight
11:
e.end.predecessor = e.start
12:
13: for e in G.edges()
14:
if e.start.d + e.weight.d < e.end.d:
15:
return FALSE
16: return TRUE
Can we terminate earlier if there is a pass that keeps all .d variables?
6.4: Single-Source Shortest Paths
T.S.
17
The Bellman-Ford Algorithm
BELLMAN-FORD(G,w,s)
0: assert(s in G.vertices())
1: for v in G.vertices()
2:
v.predecessor = None
3:
v.d = Infinity
4: s.d = 0
5:
6: repeat |V|-1 times
7:
for e in G.edges()
8:
Relax edge e=(u,v): Check if u.d + w(u,v) < v.d
9:
if e.start.d + e.weight.d < e.end.d:
10:
e.end.d = e.start.d + e.weight
11:
e.end.predecessor = e.start
12:
13: for e in G.edges()
14:
if e.start.d + e.weight.d < e.end.d:
15:
return FALSE
16: return TRUE
Can we terminate earlier if there is a pass that keeps all .d variables?
Yes, because if pass i keeps all .d variables, then so does pass i + 1.
6.4: Single-Source Shortest Paths
T.S.
17
The Bellman-Ford Algorithm (modified)
BELLMAN-FORD-NEW(G,w,s)
0: assert(s in G.vertices())
1: for v in G.vertices()
2:
v.predecessor = None
3:
v.d = Infinity
4: s.d = 0
5:
6: repeat |V| times
7:
flag = 0
8:
for e in G.edges()
9:
Relax edge e=(u,v): Check if u.d + w(u,v) < v.d
10:
if e.start.d + e.weight.d < e.end.d:
11:
e.end.d = e.start.d + e.weight
12:
e.end.predecessor = e.start
13:
flag = 1
14:
if flag = 0 return TRUE
15:
16: return FALSE
Can we terminate earlier if there is a pass that keeps all .d variables?
Yes, because if pass i keeps all .d variables, then so does pass i + 1.
6.4: Single-Source Shortest Paths
T.S.
17
The Bellman-Ford Algorithm (modified)
BELLMAN-FORD-NEW(G,w,s)
0: assert(s in G.vertices())
1: for v in G.vertices()
2:
v.predecessor = None
3:
v.d = Infinity
4: s.d = 0
5:
6: repeat |V| times
7:
flag = 0
8:
for e in G.edges()
9:
Relax edge e=(u,v): Check if u.d + w(u,v) < v.d
10:
if e.start.d + e.weight.d < e.end.d:
11:
e.end.d = e.start.d + e.weight
12:
e.end.predecessor = e.start
13:
flag = 1
14:
if flag = 0 return TRUE
15:
16: return FALSE
Can we terminate earlier if there is a pass that keeps all .d variables?
Yes, because if pass i keeps all .d variables, then so does pass i + 1.
6.4: Single-Source Shortest Paths
T.S.
17
Outline
Bellman-Ford
Dijkstra’s Algorithm
All-Pairs Shortest Path
APSP via Matrix Multiplication
Johnson’s Algorithm
6.4: Single-Source Shortest Paths
T.S.
21
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
6.4: Single-Source Shortest Paths
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
6.4: Single-Source Shortest Paths
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
6.4: Single-Source Shortest Paths
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
1
5
1
5
s
3
∞
0
3
0
2
6
1
2
5
1
3
4
6.4: Single-Source Shortest Paths
3
∞
1
4
∞
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
1
5
1
5
s
3
∞
0
3
0
2
6
1
2
5
1
3
4
6.4: Single-Source Shortest Paths
3
∞
1
4
∞
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
1
5
1
5
s
3
∞
0
3
0
2
6
1
2
5
1
3
4
6.4: Single-Source Shortest Paths
3
∞
1
4
∞
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
1
5
1
5
s
3
∞
0
3
0
2
6
1
2
5
1
3
4
6.4: Single-Source Shortest Paths
3
∞
1
4
∞
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
1
5
1
5
s
3
∞
0
3
0
2
6
1
2
5
1
3
4
6.4: Single-Source Shortest Paths
3
∞
1
4
∞
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
1
5
1
5
s
3
∞
0
3
0
2
6
1
2
5
1
3
4
6.4: Single-Source Shortest Paths
3
∞
1
4
∞
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
1
5
1
5
s
3
∞
0
3
0
2
5
1
2
5
1
3
4
6.4: Single-Source Shortest Paths
3
∞
1
4
∞
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
1
5
1
5
s
3
∞
0
3
0
2
5
1
2
5
1
3
4
6.4: Single-Source Shortest Paths
3
∞
1
4
∞
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
1
5
1
5
s
3
∞
0
3
0
2
5
1
2
5
1
3
3
∞
1
4
9
4
6.4: Single-Source Shortest Paths
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
1
5
1
5
s
3
∞
0
3
0
2
5
1
2
5
1
3
3
∞
1
4
9
4
6.4: Single-Source Shortest Paths
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
1
5
1
5
s
3
∞
0
3
0
2
5
1
2
5
1
3
3
∞
1
4
9
4
6.4: Single-Source Shortest Paths
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
1
5
1
5
s
3
∞
0
3
0
2
5
1
2
5
1
3
3
∞
1
4
9
4
6.4: Single-Source Shortest Paths
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
6.4: Single-Source Shortest Paths
T.S.
22
Dijkstra’s Algorithm
Overview of Dijkstra
Requires that all edges have non-negative weights
Use a special order for relaxing edges
The order follows a greedy-strategy (similar to Prim’s algorithm):
1. Maintain set S of vertices u with u.δ = u.d
2. At each step, add a vertex v ∈ V \ S with minimal v .d (= v .δ)
3. Relax all edges leaving v
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
can actually skip edges which go to a vertex in S
1: S = ∅
2: Q = V
(cf. implementation of Dijksta in the handout)
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
T.S.
22
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
(s, 0), (t, ∞), (x, ∞), (y , ∞), (z, ∞)
t
∞
10
x
1
s
∞
9
0
7
3
2
6
4
5
∞
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(s,
0), (t, ∞), (x, ∞), (y , ∞), (z, ∞)
H
H
t
∞
10
x
1
s
∞
9
0
7
3
2
6
4
5
∞
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(s,
0), (t, ∞), (x, ∞), (y , ∞), (z, ∞)
H
H
t
∞
10
x
1
s
∞
9
0
7
3
2
6
4
5
∞
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(s,
0), (t, 10), (x, ∞), (y , ∞), (z, ∞)
H
H
t
x
1
10
∞
10
s
9
0
7
3
2
6
4
5
∞
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(s,
0), (t, 10), (x, ∞), (y , ∞), (z, ∞)
H
H
t
x
1
10
∞
10
s
9
0
7
3
2
6
4
5
∞
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(s,
0), (t, 10), (x, ∞), (y , 5), (z, ∞)
H
H
t
x
1
10
∞
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(s,
0), (t, 10), (x, ∞), (y , 5), (z, ∞)
H
H
t
x
1
10
∞
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t, 10), (x, ∞), (y ,
5), (z, ∞)
H
H
t
x
1
10
∞
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t, 10), (x, ∞), (y ,
5), (z, ∞)
H
H
t
x
1
10
∞
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t, 8), (x, ∞), (y ,
5), (z, ∞)
H
H
t
x
1
8
∞
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t, 8), (x, ∞), (y ,
5), (z, ∞)
H
H
t
x
1
8
∞
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t, 8), (x, 14), (y ,
5), (z, ∞)
H
H
t
x
1
8
14
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t, 8), (x, 14), (y ,
5), (z, ∞)
H
H
t
x
1
8
14
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
2
∞
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t, 8), (x, 14), (y ,
5), (z, 7)
H
H
t
x
1
8
14
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t, 8), (x, 14), (y ,
5), (z, 7)
H
H
t
x
1
8
14
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
(t, 8), (x, 14), H
(z,
7)
H
H
t
x
1
8
14
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
(t, 8), (x, 14), H
(z,
7)
H
H
t
x
1
8
14
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
(t, 8), (x, 14), H
(z,
7)
H
H
t
x
1
8
14
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
(t, 8), (x, 14), H
(z,
7)
H
H
t
x
1
8
14
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
(t, 8), (x, 13), H
(z,
7)
H
H
t
x
1
8
13
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
(t, 8), (x, 13), H
(z,
7)
H
H
t
x
1
8
13
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t,
8), (x, 13)
H
H
t
x
1
8
13
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t,
8), (x, 13)
H
H
t
x
1
8
13
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t,
8), (x, 13)
H
H
t
x
1
8
13
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t,
8), (x, 13)
H
H
t
x
1
8
13
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t,
8), (x, 9)
H
H
t
x
1
8
9
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(t,
8), (x, 9)
H
H
t
x
1
8
9
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(x,
9)
H
H
t
x
1
8
9
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(x,
9)
H
H
t
x
1
8
9
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(x,
9)
H
H
t
x
1
8
9
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Complete Run of Dijkstra (Figure 24.6)
Priority Queue Q:
H
(x,
9)
H
H
t
x
1
8
9
10
s
9
0
7
3
2
6
4
5
5
y
6.4: Single-Source Shortest Paths
7
2
z
T.S.
23
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
T.S.
24
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
Runtime (using Fibonacci Heaps)
T.S.
24
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
Runtime (using Fibonacci Heaps)
T.S.
24
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
Runtime (using Fibonacci Heaps)
Initialization (l. 0-2): O(V )
T.S.
24
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
Runtime (using Fibonacci Heaps)
Initialization (l. 0-2): O(V )
T.S.
24
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
Runtime (using Fibonacci Heaps)
Initialization (l. 0-2): O(V )
ExtractMin (l. 4): O(V · log V )
T.S.
24
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
Runtime (using Fibonacci Heaps)
Initialization (l. 0-2): O(V )
ExtractMin (l. 4): O(V · log V )
T.S.
24
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
Runtime (using Fibonacci Heaps)
Initialization (l. 0-2): O(V )
ExtractMin (l. 4): O(V · log V )
DecreaseKey (l. 7): O(E · 1)
T.S.
24
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
Runtime (using Fibonacci Heaps)
Initialization (l. 0-2): O(V )
ExtractMin (l. 4): O(V · log V )
DecreaseKey (l. 7): O(E · 1)
T.S.
24
Runtime of Dijkstra’s Algorithm
DIJKSTRA(G,w,s)
0: INITIALIZE(G,s)
1: S = ∅
2: Q = V
3: while Q 6= ∅ do
4:
u = Extract-Min(Q)
5:
S = S ∪ {u}
6:
for each v ∈ G.Adj[u] do
7:
RELAX(u, v , w)
8:
end for
9: end while
6.4: Single-Source Shortest Paths
Runtime (using Fibonacci Heaps)
Initialization (l. 0-2): O(V )
ExtractMin (l. 4): O(V · log V )
DecreaseKey (l. 7): O(E · 1)
⇒ Overall: O(V log V + E)
T.S.
24
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
S
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
6.4: Single-Source Shortest Paths
x
y
S
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
⇒
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
x
y
S
u.d
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
⇒
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
x
y
S
u.d ≤ y .d
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
⇒
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
x
y
S
u.d ≤ y .d
u is extracted before y
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
⇒
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
x
y
S
u.d ≤ y .d = y .δ
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
⇒
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
x
y
S
u.d ≤ y .d = y .δ
since x.d = x.δ when x is extracted, and then
(x, y ) is relaxed ⇒ Convergence Property
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
⇒
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
x
y
S
u.δ < u.d ≤ y .d = y .δ
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
⇒
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
x
y
S
u.δ < u.d ≤ y .d = y .δ
This contradicts that y is on a shortest path
from s to u.
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
⇒
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
x
y
S
u.δ < u.d ≤ y .d = y .δ
This contradicts that y is on a shortest path
from s to u.
This step requires non-negative weights!
6.4: Single-Source Shortest Paths
T.S.
25
Correctness of Dijkstra’s Algorithm
Theorem 24.6
For any directed graph G = (V , E) with non-negative edge weights w :
E → R+ and source s, Dijkstra terminates with u.d = u.δ for all u ∈ V .
Proof: Invariant: If v is extracted, v .d = v .δ
There are edge cases like
s = x and/or y = u!
Suppose there is u ∈ V , when extracted,
u.d > u.δ
u
s
Let u be the first vertex with this property
⇒
Take a shortest path from s to u and let
(x, y ) be the first edge from S to V \ S
x
y
S
u.δ < u.d ≤ y .d = y .δ
This contradicts that y is on a shortest path
from s to u.
6.4: Single-Source Shortest Paths
T.S.
25
2) Remove vertex s and its incident edges
e
3. For every vertex v 2 V , run Dijkstra on (G, E, w)
0
0
s
0
4
8
0
0
1
3
2
4
7
4
0
1
6
6.5: All-Pairs Shortest Paths
0
4
13
5
2
5
0
0
0
10
2
T.S.
6.5: All-Pairs Shortest Paths
Frank Stajano
Lent 2015
0
Thomas Sauerwald
11
Outline
Bellman-Ford
Dijkstra’s Algorithm
All-Pairs Shortest Path
APSP via Matrix Multiplication
Johnson’s Algorithm
6.5: All-Pairs Shortest Paths
T.S.
2
Formalizing the Problem
All-Pairs Shortest Path Problem
Given: directed graph G = (V , E), V = {1, 2, . . . , n}, with edge
weights represented by a matrix W :


weight of edge (i, j) for an edge (i, j) ∈ E,
wi,j = ∞
if there is no edge from i to j,


0
if i = j.
6.5: All-Pairs Shortest Paths
T.S.
3
Formalizing the Problem
All-Pairs Shortest Path Problem
Given: directed graph G = (V , E), V = {1, 2, . . . , n}, with edge
weights represented by a matrix W :


weight of edge (i, j) for an edge (i, j) ∈ E,
wi,j = ∞
if there is no edge from i to j,


0
if i = j.
Goal: Obtain a matrix of shortest path weights L, that is
(
weight of a shortest path from i to j, if j is reachable from i
`i,j =
∞
otherwise.
6.5: All-Pairs Shortest Paths
T.S.
3
Formalizing the Problem
All-Pairs Shortest Path Problem
Given: directed graph G = (V , E), V = {1, 2, . . . , n}, with edge
weights represented by a matrix W :


weight of edge (i, j) for an edge (i, j) ∈ E,
wi,j = ∞
if there is no edge from i to j,


0
if i = j.
Goal: Obtain a matrix of shortest path weights L, that is
(
weight of a shortest path from i to j, if j is reachable from i
`i,j =
∞
otherwise.
Here we will only compute the weight of the shortest
path without keeping track of the edges of the path!
6.5: All-Pairs Shortest Paths
T.S.
3
Outline
Bellman-Ford
Dijkstra’s Algorithm
All-Pairs Shortest Path
APSP via Matrix Multiplication
Johnson’s Algorithm
6.5: All-Pairs Shortest Paths
T.S.
4
A Recursive Approach
i
k
j
Basic Idea
Any shortest path from i to j of length k ≥ 2 is the concatenation of
a shortest path of length k − 1 and an edge
6.5: All-Pairs Shortest Paths
T.S.
5
A Recursive Approach
i
k
j
Basic Idea
Any shortest path from i to j of length k ≥ 2 is the concatenation of
a shortest path of length k − 1 and an edge
(m)
Let `i,j be min. weight of any path from i to j with at most m edges
6.5: All-Pairs Shortest Paths
T.S.
5
A Recursive Approach
i
k
j
Basic Idea
Any shortest path from i to j of length k ≥ 2 is the concatenation of
a shortest path of length k − 1 and an edge
(m)
Let `i,j be min. weight of any path from i to j with at most m edges
(1)
Then `i,j = wi,j , so L(1) = W
6.5: All-Pairs Shortest Paths
T.S.
5
A Recursive Approach
i
k
j
Basic Idea
Any shortest path from i to j of length k ≥ 2 is the concatenation of
a shortest path of length k − 1 and an edge
(m)
Let `i,j be min. weight of any path from i to j with at most m edges
(1)
Then `i,j = wi,j , so L(1) = W
How can we obtain L(2) from L(1) ?
6.5: All-Pairs Shortest Paths
T.S.
5
A Recursive Approach
i
k
j
Basic Idea
Any shortest path from i to j of length k ≥ 2 is the concatenation of
a shortest path of length k − 1 and an edge
(m)
Let `i,j be min. weight of any path from i to j with at most m edges
(1)
Then `i,j = wi,j , so L(1) = W
How can we obtain L(2) from L(1) ?
(2)
(1)
(1)
`i,j = min `i,j , min `i,k + wk ,j
1≤k ≤n
6.5: All-Pairs Shortest Paths
T.S.
5
A Recursive Approach
i
k
j
Basic Idea
Any shortest path from i to j of length k ≥ 2 is the concatenation of
a shortest path of length k − 1 and an edge
(m)
Let `i,j be min. weight of any path from i to j with at most m edges
(1)
Then `i,j = wi,j , so L(1) = W
How can we obtain L(2) from L(1) ?
(2)
(1)
(1)
`i,j = min `i,j , min `i,k + wk ,j
1≤k ≤n
(m)
`i,j =
6.5: All-Pairs Shortest Paths
T.S.
5
A Recursive Approach
i
k
j
Basic Idea
Any shortest path from i to j of length k ≥ 2 is the concatenation of
a shortest path of length k − 1 and an edge
(m)
Let `i,j be min. weight of any path from i to j with at most m edges
(1)
Then `i,j = wi,j , so L(1) = W
How can we obtain L(2) from L(1) ?
(2)
(1)
(1)
`i,j = min `i,j , min `i,k + wk ,j
1≤k ≤n
(m)
(m−1)
(m−1)
`i,j = min `i,j
, min `i,k
+ wk ,j
1≤k ≤n
6.5: All-Pairs Shortest Paths
T.S.
5
A Recursive Approach
i
j
k
Basic Idea
Any shortest path from i to j of length k ≥ 2 is the concatenation of
a shortest path of length k − 1 and an edge
(m)
Let `i,j be min. weight of any path from i to j with at most m edges
(1)
Then `i,j = wi,j , so L(1) = W
How can we obtain L(2) from L(1) ?
(2)
(1)
(1)
`i,j = min `i,j , min `i,k + wk ,j
1≤k ≤n
Recall that wj,j = 0!
(m)
(m−1)
(m−1)
`i,j = min `i,j
, min `i,k
+ wk ,j
1≤k ≤n
6.5: All-Pairs Shortest Paths
T.S.
5
A Recursive Approach
i
k
j
Basic Idea
Any shortest path from i to j of length k ≥ 2 is the concatenation of
a shortest path of length k − 1 and an edge
(m)
Let `i,j be min. weight of any path from i to j with at most m edges
(1)
Then `i,j = wi,j , so L(1) = W
How can we obtain L(2) from L(1) ?
(2)
(1)
(1)
`i,j = min `i,j , min `i,k + wk ,j
1≤k ≤n
Recall that wj,j = 0!
(m)
(m−1)
(m−1)
(m−1)
`i,j = min `i,j
, min `i,k
+ wk ,j = min `i,k
+ wk ,j
1≤k ≤n
6.5: All-Pairs Shortest Paths
1≤k ≤n
T.S.
5
Example of Shortest Path via Matrix Multiplication (Figure 25.1)
2
3
1
−4
7
5
2
6
4
8
3
1
4
−5
6.5: All-Pairs Shortest Paths
T.S.
6
Example of Shortest Path via Matrix Multiplication (Figure 25.1)
2
3
1
−4
7
5



L(1) = W = 

2
4
8
4
6
0
∞
∞
2
∞
3
0
4
∞
∞
3
1
8
∞
0
−5
∞
−5
∞
1
∞
0
6
6.5: All-Pairs Shortest Paths
−4
7
∞
∞
0





T.S.
6
Example of Shortest Path via Matrix Multiplication (Figure 25.1)
2
3
1
−4
7
5



L(1) = W = 

2
4
8
4
6
0
∞
∞
2
∞
3
0
4
∞
∞
3
1
8
∞
0
−5
∞
−5
∞
1
∞
0
6
6.5: All-Pairs Shortest Paths
−4
7
∞
∞
0








L(2) = 

T.S.
0
3
∞
2
8
3
0
4
−1
∞
8
−4
0
−5
1
?
1
5
0
6
−4
7
11
−2
0
6





Example of Shortest Path via Matrix Multiplication (Figure 25.1)
2
3
1
−4
7
5



L(1) = W = 

2
4
8
4
6
0
∞
∞
2
∞
3
0
4
∞
∞
3
1
8
∞
0
−5
∞
−5
∞
1
∞
0
6
6.5: All-Pairs Shortest Paths
−4
7
∞
∞
0
(2)
`1,4 = min{0 + ∞, 3 + 1, 8 + ∞, ∞ + 0, − 4 + 6}








L(2) = 

T.S.
0
3
∞
2
8
3
0
4
−1
∞
8
−4
0
−5
1
?
1
5
0
6
−4
7
11
−2
0
6





Example of Shortest Path via Matrix Multiplication (Figure 25.1)
2
3
1
−4
7
5



L(1) = W = 

2
4
8
4
6
0
∞
∞
2
∞
3
0
4
∞
∞
3
1
8
∞
0
−5
∞
−5
∞
1
∞
0
6
6.5: All-Pairs Shortest Paths
−4
7
∞
∞
0
(2)
`1,4 = min{0 + ∞, 3 + 1, 8 + ∞, ∞ + 0, − 4 + 6}








L(2) = 

T.S.
0
3
∞
2
8
3
0
4
−1
∞
8
−4
0
−5
1
2
1
5
0
6
−4
7
11
−2
0
6





Example of Shortest Path via Matrix Multiplication (Figure 25.1)
2
3
1
−4
7
5
2
4
8
3
1
4
6

0
∞
∞
2
∞
3
0
4
∞
∞
8
∞
0
−5
∞
∞
1
∞
0
6

0
3
7
2
8
3
0
4
−1
5
−3
−4
0
−5
1
2
1
5
0
6


L(1) = W = 



L(3) = 

−5
−4
7
∞
∞
0
−4
−1
11
−2
0
6.5: All-Pairs Shortest Paths
(2)
`1,4 = min{0 + ∞, 3 + 1, 8 + ∞, ∞ + 0, − 4 + 6}








L(2) = 

0
3
∞
2
8
3
0
4
−1
∞
8
−4
0
−5
1
2
1
5
0
6
−4
7
11
−2
0





T.S.
6





Example of Shortest Path via Matrix Multiplication (Figure 25.1)
2
3
1
−4
7
5
2
4
8
3
1
4
6

0
∞
∞
2
∞
3
0
4
∞
∞
8
∞
0
−5
∞
∞
1
∞
0
6

0
3
7
2
8
3
0
4
−1
5
−3
−4
0
−5
1
2
1
5
0
6


L(1) = W = 



L(3) = 

−5
−4
7
∞
∞
0
−4
−1
11
−2
0
6.5: All-Pairs Shortest Paths
(2)
`1,4 = min{0 + ∞, 3 + 1, 8 + ∞, ∞ + 0, − 4 + 6}











0
3
∞
2
8
3
0
4
−1
∞
8
−4
0
−5
1
 0
 3

= 7

2
8
1
0
4
−1
5
−3
−4
0
−5
1


L(2) = 

L(4)
T.S.
2
1
5
0
6
2
1
5
0
6
−4
7
11
−2
0
−4
−1
?
−2
0
6










Example of Shortest Path via Matrix Multiplication (Figure 25.1)
2
3
1
−4
7
5
2
4
8
3
1
4
6

0
∞
∞
2
∞
3
0
4
∞
∞
8
∞
0
−5
∞
∞
1
∞
0
6

0
3
7
2
8
3
0
4
−1
5
−3
−4
0
−5
1
2
1
5
0
6


L(1) = W = 



L(3) = 

−5
−4
7
∞
∞
0
−4
−1
11
−2
0
(2)
`1,4 = min{0 + ∞, 3 + 1, 8 + ∞, ∞ + 0, − 4 + 6}











0
3
∞
2
8
3
0
4
−1
∞
8
−4
0
−5
1
 0
 3

= 7

2
8
1
0
4
−1
5
−3
−4
0
−5
1


L(2) = 

L(4)
2
1
5
0
6
2
1
5
0
6
−4
7
11
−2
0
−4
−1
?
−2
0
(4)










`3,5 = min{7 − 4, 4 + 7, 0 + ∞, 5 + ∞, 11 + 0}
6.5: All-Pairs Shortest Paths
T.S.
6
Example of Shortest Path via Matrix Multiplication (Figure 25.1)
2
3
1
−4
7
5
2
4
8
3
1
4
6

0
∞
∞
2
∞
3
0
4
∞
∞
8
∞
0
−5
∞
∞
1
∞
0
6

0
3
7
2
8
3
0
4
−1
5
−3
−4
0
−5
1
2
1
5
0
6


L(1) = W = 



L(3) = 

−5
−4
7
∞
∞
0
−4
−1
11
−2
0
(2)
`1,4 = min{0 + ∞, 3 + 1, 8 + ∞, ∞ + 0, − 4 + 6}











0
3
∞
2
8
3
0
4
−1
∞
8
−4
0
−5
1
 0
 3

= 7

2
8
1
0
4
−1
5
−3
−4
0
−5
1


L(2) = 

L(4)
2
1
5
0
6
2
1
5
0
6
−4
7
11
−2
0
−4
−1
3
−2
0
(4)










`3,5 = min{7 − 4, 4 + 7, 0 + ∞, 5 + ∞, 11 + 0}
6.5: All-Pairs Shortest Paths
T.S.
6
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
Computing L(m) :
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
Computing L(m) :
(m)
(m−1)
`i,j = min `i,k
+ wk ,j
1≤k ≤n
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
Computing L(m) :
(m)
(m−1)
`i,j = min `i,k
+ wk ,j
1≤k ≤n
X (m−1)
(m−1)
(L
· W )i,j =
`i,k
× wk ,j
1≤k ≤n
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
Computing L(m) :
(m)
(m−1)
`i,j = min `i,k
+ wk ,j
1≤k ≤n
X (m−1)
(m−1)
(L
· W )i,j =
`i,k
× wk ,j
1≤k ≤n
The correspondence is as follows:
min ⇔
X
+ ⇔ ×
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
Computing L(m) :
(m)
(m−1)
`i,j = min `i,k
+ wk ,j
1≤k ≤n
X (m−1)
(m−1)
(L
· W )i,j =
`i,k
× wk ,j
1≤k ≤n
The correspondence is as follows:
min ⇔
X
+ ⇔ ×
∞ ⇔ ?
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
Computing L(m) :
(m)
(m−1)
`i,j = min `i,k
+ wk ,j
1≤k ≤n
X (m−1)
(m−1)
(L
· W )i,j =
`i,k
× wk ,j
1≤k ≤n
The correspondence is as follows:
min ⇔
X
+ ⇔ ×
∞ ⇔ 0
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
Computing L(m) :
(m)
(m−1)
`i,j = min `i,k
+ wk ,j
1≤k ≤n
X (m−1)
(m−1)
(L
· W )i,j =
`i,k
× wk ,j
1≤k ≤n
The correspondence is as follows:
min ⇔
X
+ ⇔ ×
∞ ⇔ 0
0 ⇔ ?
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
Computing L(m) :
(m)
(m−1)
`i,j = min `i,k
+ wk ,j
1≤k ≤n
X (m−1)
(m−1)
(L
· W )i,j =
`i,k
× wk ,j
1≤k ≤n
The correspondence is as follows:
min ⇔
X
+ ⇔ ×
∞ ⇔ 0
0 ⇔ 1
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(m)
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
L(n−1) = L(n) = L(n+1) = . . . = L, since every shortest path uses at most
n − 1 = |V | − 1 edges (assuming absence of negative-weight cycles)
Computing L(m) :
L(m) can be
(m)
(m−1)
`i,j = min `i,k
+ wk ,j
computed in O(n3 )
1≤k ≤n
X (m−1)
(L(m−1) · W )i,j =
`i,k
× wk ,j
1≤k ≤n
The correspondence is as follows:
min ⇔
X
+ ⇔ ×
∞ ⇔ 0
0 ⇔ 1
6.5: All-Pairs Shortest Paths
T.S.
7
Computing L(n−1) efficiently
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
For, say, n = 738, we subsequently compute
L(1) , L(2) , L(3) , L(4) , . . . , L(737) = L
6.5: All-Pairs Shortest Paths
T.S.
8
Computing L(n−1) efficiently
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
Takes O(n · n3 ) = O(n4 )
For, say, n = 738, we subsequently compute
L(1) , L(2) , L(3) , L(4) , . . . , L(737) = L
6.5: All-Pairs Shortest Paths
T.S.
8
Computing L(n−1) efficiently
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
Takes O(n · n3 ) = O(n4 )
For, say, n = 738, we subsequently compute
L(1) , L(2) , L(3) , L(4) , . . . , L(737) = L
Since we don’t need the intermediate matrices, a more efficient way is
L(1) , L(2) , L(4) , . . . , L(512) , L(1024) = L
6.5: All-Pairs Shortest Paths
T.S.
8
Computing L(n−1) efficiently
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
Takes O(n · n3 ) = O(n4 )
For, say, n = 738, we subsequently compute
L(1) , L(2) , L(3) , L(4) , . . . , L(737) = L
Since we don’t need the intermediate matrices, a more efficient way is
L(1) , L(2) , L(4) , . . . , L(512) , L(1024) = L
Takes O(log n · n3 ).
6.5: All-Pairs Shortest Paths
T.S.
8
Computing L(n−1) efficiently
(m)
`i,j = min
1≤k ≤n
(m−1)
`i,k
+ wk ,j
Takes O(n · n3 ) = O(n4 )
For, say, n = 738, we subsequently compute
L(1) , L(2) , L(3) , L(4) , . . . , L(737) = L
Since we don’t need the intermediate matrices, a more efficient way is
L(1) , L(2) , L(4) , . . . , L(512) , L(1024) = L
Takes O(log n · n3 ).
We need L(4) = L(2) ·L(2) = L(3) ·L(1) ! (see Ex. 25.1-4)
6.5: All-Pairs Shortest Paths
T.S.
8
Outline
Bellman-Ford
Dijkstra’s Algorithm
All-Pairs Shortest Path
APSP via Matrix Multiplication
Johnson’s Algorithm
6.5: All-Pairs Shortest Paths
T.S.
9
Johnson’s Algorithm
Overview
6.5: All-Pairs Shortest Paths
T.S.
10
Johnson’s Algorithm
Overview
allow negative-weight edges and negative-weight cycles
6.5: All-Pairs Shortest Paths
T.S.
10
Johnson’s Algorithm
Overview
allow negative-weight edges and negative-weight cycles
one pass of Bellman-Ford and |V | passes of Dijkstra
6.5: All-Pairs Shortest Paths
T.S.
10
Johnson’s Algorithm
Overview
allow negative-weight edges and negative-weight cycles
one pass of Bellman-Ford and |V | passes of Dijkstra
after Bellman-Ford, edges are reweighted s.t.
6.5: All-Pairs Shortest Paths
T.S.
10
Johnson’s Algorithm
Overview
allow negative-weight edges and negative-weight cycles
one pass of Bellman-Ford and |V | passes of Dijkstra
after Bellman-Ford, edges are reweighted s.t.
all edge weights are non-negative
6.5: All-Pairs Shortest Paths
T.S.
10
Johnson’s Algorithm
Overview
allow negative-weight edges and negative-weight cycles
one pass of Bellman-Ford and |V | passes of Dijkstra
after Bellman-Ford, edges are reweighted s.t.
all edge weights are non-negative
shortest paths are maintained
6.5: All-Pairs Shortest Paths
T.S.
10
Johnson’s Algorithm
Overview
allow negative-weight edges and negative-weight cycles
one pass of Bellman-Ford and |V | passes of Dijkstra
after Bellman-Ford, edges are reweighted s.t.
all edge weights are non-negative
shortest paths are maintained
Adding a constant to every edge doesn’t work!
6.5: All-Pairs Shortest Paths
T.S.
10
Johnson’s Algorithm
Overview
allow negative-weight edges and negative-weight cycles
one pass of Bellman-Ford and |V | passes of Dijkstra
after Bellman-Ford, edges are reweighted s.t.
all edge weights are non-negative
shortest paths are maintained
Adding a constant to every edge doesn’t work!
3
4
1004
1003
8
1008
2
−4
1002
1
7
−5
1001 995
996
1007
6
6.5: All-Pairs Shortest Paths
1006
T.S.
10
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
3
4
8
2
−4
1
7
−5
6
6.5: All-Pairs Shortest Paths
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
0
0
s
3
4
8
0
0
2
−4
0
1
7
−5
6
6.5: All-Pairs Shortest Paths
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
0
0
s
3
4
8
0
0
2
−4
0
1
7
−5
6
6.5: All-Pairs Shortest Paths
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
0
0
s
0
4
8
0
0
−1
3
2
−4
0
1
7
−4
6
6.5: All-Pairs Shortest Paths
−5
−5
0
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
0
0
s
0
4
8
0
0
−1
3
2
−4
0
1
7
−4
6
6.5: All-Pairs Shortest Paths
−5
−5
0
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
0
0
s
0
4
8
0
0
−1
3
2
−4
0
1
7
−4
6
6.5: All-Pairs Shortest Paths
−5
−5
0
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
0
0
s
0
4
8
0
0
−1
3
2
−4
0
1
7
−4
6
6.5: All-Pairs Shortest Paths
0
4
13
−5
2
0
−5
0
0
10
0
2
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
e
3. For every vertex v ∈ V , run Dijkstra on (G, E, w)
0
0
s
0
4
8
0
0
−1
3
2
−4
0
1
7
−4
6
6.5: All-Pairs Shortest Paths
0
4
13
−5
2
0
−5
0
0
10
0
2
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
e
3. For every vertex v ∈ V , run Dijkstra on (G, E, w)
0
0
s
0
4
8
0
0
−1
3
2
−4
0
1
7
−4
6
6.5: All-Pairs Shortest Paths
0
4
13
−5
2
0
−5
0
0
10
0
2
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
e
3. For every vertex v ∈ V , run Dijkstra on (G, E, w)
0
0
s
0
4
8
0
0
−1
3
2
−4
0
1
7
−4
6
6.5: All-Pairs Shortest Paths
0
4
13
−5
2
0
−5
0
0
10
0
2
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
e
3. For every vertex v ∈ V , run Dijkstra on (G, E, w)
0
0
s
0
4
8
0
0
−1
3
2
−4
7
−4
6
0
Direct: 7, Detour: −1
6.5: All-Pairs Shortest Paths
1
0
4
13
−5
2
0
−5
0
0
10
0
2
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
e
3. For every vertex v ∈ V , run Dijkstra on (G, E, w)
0
0
s
0
4
8
0
0
−1
3
2
−4
7
−4
6
0
Direct: 7, Detour: −1
6.5: All-Pairs Shortest Paths
1
0
4
13
−5
2
0
−5
0
0
10
0
2
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
e
3. For every vertex v ∈ V , run Dijkstra on (G, E, w)
0
0
s
0
4
8
0
0
−1
3
2
−4
7
−4
6
0
Direct: 7, Detour: −1
6.5: All-Pairs Shortest Paths
1
0
4
13
−5
2
0
−5
0
0
10
0
2
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
e
3. For every vertex v ∈ V , run Dijkstra on (G, E, w)
0
0
s
0
4
8
0
0
−1
3
2
−4
7
−4
6
0
Direct: 7, Detour: −1
6.5: All-Pairs Shortest Paths
1
0
4
13
−5
2
0
−5
0
0
10
0
2
Direct: 10, Detour: 2
T.S.
11
How Johnson’s Algorithm works
Johnson’s Algorithm
1. Add a new vertex s and directed edges (s, v ), v ∈ V , with weight 0
2. Run Bellman-Ford on this augmented graph with source s
If there are negative weight cycles, abort
Otherwise:
e
1) Reweight every edge (u, v ) by w(u,
v ) = w(u, v ) + u.δ − v .δ
2) Remove vertex s and its incident edges
e
3. For every vertex v ∈ V , run Dijkstra on (G, E, w)
Runtime: O(V ·E +V ·(V log V +E))
0
0
s
0
4
8
0
0
−1
3
2
−4
7
−4
6
0
Direct: 7, Detour: −1
6.5: All-Pairs Shortest Paths
1
0
4
13
−5
2
0
−5
0
0
10
0
2
Direct: 10, Detour: 2
T.S.
11
Correctness of Johnson’s Algorithm
Theorem
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
6.5: All-Pairs Shortest Paths
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 1.
6.5: All-Pairs Shortest Paths
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 1.
Let u.δ and v .δ be the distances from the fake source s
6.5: All-Pairs Shortest Paths
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 1.
Let u.δ and v .δ be the distances from the fake source s
u.δ + w(u, v ) ≥ v .δ
6.5: All-Pairs Shortest Paths
(triangle inequality)
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 1.
⇒
Let u.δ and v .δ be the distances from the fake source s
u.δ + w(u, v ) ≥ v .δ
(triangle inequality)
e
w(u,
v ) + u.δ + w(u, v ) ≥ w(u, v ) + u.δ − v .δ + v .δ
6.5: All-Pairs Shortest Paths
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 1.
⇒
Let u.δ and v .δ be the distances from the fake source s
u.δ + w(u, v ) ≥ v .δ
(triangle inequality)
e
w(u,
v ) + u.δ + w(u, v ) ≥ w(u, v ) + u.δ − v .δ + v .δ
e
⇒
w(u,
v) ≥ 0
6.5: All-Pairs Shortest Paths
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 2.
6.5: All-Pairs Shortest Paths
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 2.
Let p = (v0 , v1 , . . . , vk ) be any path
6.5: All-Pairs Shortest Paths
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 2.
Let p = (v0 , v1 , . . . , vk ) be any path
In the original graph, the weight is
6.5: All-Pairs Shortest Paths
Pk
i=1
w(vi−1 , vi ) = w(p).
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 2.
Let p = (v0 , v1 , . . . , vk ) be any path
In the original graph, the weight is
Pk
i=1
w(vi−1 , vi ) = w(p).
In the reweighted graph, the weight is
k
X
i=1
e i−1 , vi )
w(v
6.5: All-Pairs Shortest Paths
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 2.
Let p = (v0 , v1 , . . . , vk ) be any path
In the original graph, the weight is
Pk
i=1
w(vi−1 , vi ) = w(p).
In the reweighted graph, the weight is
k
X
i=1
e i−1 , vi ) =
w(v
k
X
i=1
(w(vi−1 , vi ) + vi−1 .δ − vi .δ)
6.5: All-Pairs Shortest Paths
T.S.
12
Correctness of Johnson’s Algorithm
Theorem
e
w(u,
v ) = w(u, v ) + u.δ − v .δ
For any graph G = (V , E, w) without negative-weight cycles:
1. After reweighting, all edges are non-negative
2. Shortest Paths are preserved
Proof of 2.
Let p = (v0 , v1 , . . . , vk ) be any path
In the original graph, the weight is
Pk
i=1
w(vi−1 , vi ) = w(p).
In the reweighted graph, the weight is
k
X
i=1
e i−1 , vi ) =
w(v
k
X
i=1
(w(vi−1 , vi ) + vi−1 .δ − vi .δ) = w(p) + v0 .δ − vk .δ
6.5: All-Pairs Shortest Paths
T.S.
12
Comparison of all Shortest-Path Algorithms
SSSP
Algorithm
sparse
2
APSP
dense
dense
X
Dijkstra
V log V
V2
V 2 log V
V3
X
Matrix Mult.
–
–
V 3 log V
V 3 log V
(X)
6.5: All-Pairs Shortest Paths
–
V
weights
4
V
–
V
3
Bellman-Ford
Johnson
V
sparse
3
negative
2
V log V
T.S.
V
3
X
13
Comparison of all Shortest-Path Algorithms
SSSP
Algorithm
sparse
2
APSP
dense
dense
X
Dijkstra
V log V
V2
V 2 log V
V3
X
Matrix Mult.
–
–
V 3 log V
V 3 log V
(X)
–
V
weights
4
V
–
V
3
Bellman-Ford
Johnson
V
sparse
3
negative
2
V log V
V
3
X
can handle negative weight edges,
but not negative-weight cycles
6.5: All-Pairs Shortest Paths
T.S.
13