Shortest Paths (Review)

1
Shortest Paths(Review)
2
Single Source Shortest Path

• Dijkstras Algorithm
• Label Path length to each vertex as ? (or 0 for
  the start vertex)
• Loop while there is a vertex
• Remove up the vertex with minimum path length
• Check and if required update the path length of
  its adjacent neighbors
• end loop

Update rule Let a be the vertex removed
and b be its adjacent vertex Let e be
the edge connecting a to b. if
(a.pathLength e.weight lt b.pathLength) b.pathLe
ngth?a.PathLength e.weight b.parent ? a
6
6
D
D
5
C
C
10
1
1
6
6
8
8
E
E
6
5
5
6
6
7
7
2
2
B
B
A
A
2
3
Single Source Shortest Path

• DAG based Algorithm
• May have negative weight
• Label Path length to each vertex as ? (or 0 for
  the start vertex)
• Compute Topological Ordering
• loop for i ? 1 to n-1
• Check and if required update the path length of
  adjacent neighbors of vi
• end loop

Path Length from A To
i1
i2
i3
i4
-2
D
C
3
?
B
-5
8
E
5
?
C
6
9
D
?
2
B
A
?
E
4
All-Pairs Shortest Paths
Uses only vertices numbered 1,,k (compute weight
of this edge)
i
j
Uses only vertices numbered 1,,k-1
k
Uses only vertices numbered 1,,k-1
5
All-Pairs Shortest Paths

• Dynamic Programming approach
• Initialize a n?n table with 0 or ? or edge
  length.
• Check and Update the table bottom up in nested
  loops
• for k,
• for i,
• for j

d v1 v2 v3 v4 v5
v1
v2
v3
v4
v5
j?
i?
-2
D
C
3
-5
8
E
5
6
9
2
B
A
6
Minimum Spanning Trees
2704
BOS
867
849
PVD
ORD
187
740
144
JFK
1846
621
1258
184
802
SFO
BWI
1391
1464
337
1090
DFW
946
LAX
1235
1121
MCO
2342
7
Minimum Spanning Tree

• Spanning subgraph
• Subgraph of a graph G containing all the vertices
  of G
• Spanning tree
• Spanning subgraph that is itself a (free) tree
• Minimum spanning tree (MST)
• Spanning tree of a weighted graph with minimum
  total edge weight
• Applications
• Communications networks
• Transportation networks

ORD
10
1
PIT
DEN
6
7
9
3
DCA
STL
4
5
8
2
DFW
ATL
8
Cycle Property
8
f

• Cycle Property
• Let T be a minimum spanning tree of a weighted
  graph G
• Let e be an edge of G that is not in T
• Let C let be the cycle formed by e with T
• For every edge f of C, weight(f) ? weight(e)
• Proof
• By contradiction
• If weight(f) gt weight(e) we can get a spanning
  tree of smaller weight by replacing e with f

4
C
9
6
2
3
e
7
8
7
Replacing f with e yieldsa better spanning tree
8
f
4
C
9
6
2
3
e
7
8
7
9
Partition Property
U
V
7
f

• Partition Property
• Consider a partition of the vertices of G into
  subsets U and V
• Let e be an edge of minimum weight across the
  partition
• There is a minimum spanning tree of G containing
  edge e
• Proof
• Let T be an MST of G
• If T does not contain e, consider the cycle C
  formed by e with T and let f be an edge of C
  across the partition
• By the cycle property, weight(f) ? weight(e)
• Thus, weight(f) weight(e)
• We obtain another MST by replacing f with e

4
9
5
2
8
3
8
e
7
Replacing f with e yieldsanother MST
U
V
7
f
4
9
5
2
8
3
8
e
7
10
Minimum Spanning Tree

• Prims Algorithm (Prim-Jarnik Algorithm)
• Label cost of each vertex as ? (or 0 for the
  start vertex)
• Loop while there is a vertex
• Remove a vertex that will extend the tree with
  minimum additional cost
• Check and if required update the path length of
  its adjacent neighbors (Update rule different
  from Dijkstras algorithm)
• end loop

Update rule Let a be the vertex removed
and b be its adjacent vertex Let e be
the edge connecting a to b. if (e.weight lt
b.cost) b.cost?e.weight b.parent ? a
6
D
C
1
6
8
E
2
6
7
2
B
A
11
MST Prim-Jarniks Algorithm
?
6
D
C
?
1
6
8
E
6
2
?
7
2
B
A
?
0
12
MST Prim-Jarniks Algorithm
?
6
D
C
1
2,A
6
8
E
6
2
7,A
7
2
B
A
2,A
0
13
MST Prim-Jarniks Algorithm
?
6
D
C
1
2,A
6
8
E
6
2
7,A
7
2
B
A
2,A
0
14
MST Prim-Jarniks Algorithm
6
D
8,B
C
1
2,A
6
8
E
6
2
6,B
7
2
B
A
2,A
0
15
MST Prim-Jarniks Algorithm
6
D
8,B
C
1
2,A
6
8
E
6
2
6,B
7
2
B
A
2,A
0
16
MST Prim-Jarniks Algorithm
6
D
6,D
C
1
2,A
6
8
E
6
2
1,D
7
2
B
A
2,A
0
17
MST Prim-Jarniks Algorithm
6
D
6,D
C
1
2,A
6
8
E
6
2
1,D
7
2
B
A
2,A
0
18
MST Prim-Jarniks Algorithm
6
D
6,D
C
1
2,A
6
8
E
6
2
1,D
7
2
B
A
2,A
0
19
MST Prim-Jarniks Algorithm
6
D
6,D
C
1
2,A
6
8
E
6
2
1,D
7
2
B
A
2,A
0
20
Prim-Jarnik Algorithm

• Algorithm MST (G)
• Q ? new priority queue
• Let s be any vertex
• for all v ? G.vertices()
• if v s
• v.cost ? 0
• else v.cost ? ?
• v. parent ? null
• Q.enQueue(v.cost, v)
• while ?Q.isEmpty()
• v ? Q .removeMin()
• v.pathKnown ? true
• for all e ? G.incidentEdges(v)
• w ? opposite(v,e)
• if ?w.pathKnown
• if weight(e) lt w.cost
• w.cost ? weight(e)
• w. parent ? v
• update key of w in Q

O(n)
O(n log n)
O((nm) log n)
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Example
?
D
7
2
B
4
9
?
8
5
F
2
C
8
3
8
E
A
7
7
0
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Example (contd.)
7
D
7
2
B
4
9
4
5
5
F
2
C
8
3
8
E
A
3
7
0
23
Minimum Spanning Tree

• Kruskals Algorithm
• Create a forest of n trees
• Loop while (there is gt 1 tree in the forest)
• Remove an edge with minimum weight
• Accept the edge only if it connects 2 trees from
  the forest in to one.
• end loop

6
D
C
1
6
8
E
2
6
7
E
D
C
B
A
2
B
A
24
MST Kruskals Algorithm
6
D
C
1
6
8
E
2
6
7
2
B
A
E
D
C
B
A
25
MST Kruskals Algorithm
6
D
C
1
6
8
E
2
6
7
2
B
A
E
D
C
B
A
26
MST Kruskals Algorithm
6
D
C
1
6
8
E
2
6
7
2
B
A
E
D
C
B
A
27
MST Kruskals Algorithm
6
D
C
1
6
8
E
2
6
7
2
B
A
E
D
C
B
A
28
Kruskals Algorithm
Algorithm KruskalMST(G) let Q be a priority
queue. Insert all edges into Q using their
weights as the key Create a forest of n trees
where each vertex is a tree
numberOfTrees ? n while numberOfTrees gt 1do
edge e ? Q.removeMin() Let u, v be the
endpoints of e if Tree(v) ? Tree(u)
then Combine Tree(v) and Tree(u)
using edge e decrement numberOfTrees
return T
O(m log m)
O(m log m)
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Kruskal Example
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Example
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Example
32
Example
33
Example
34
Example
35
Example
36
Example
37
Example
38
Example
39
Example
40
Example
41
Example
42
Example
43
Minimum Spanning Tree

• Baruvkas Algorithm
• Create a forest of n trees
• Loop while (there is gt 1 tree in the forest)
• For each tree Ti in the forest
• Find the smallest edge e (u,v), in the edge
  list with u in Ti and v in Tj ?Ti
• connects 2 trees from the forest in to one.
• end loop

6
D
C
1
6
8
E
5
6
7
E
D
C
B
A
2
B
A
44
Baruvkas Algorithm

• Like Kruskals Algorithm, Baruvkas algorithm
  grows many clouds at once.
• Each iteration of the while-loop halves the
  number of connected compontents in T.
• The running time is O(m log n).

Algorithm BaruvkaMST(G) T ? V just the
vertices of G while T has fewer than n-1 edges
do for each connected component C in T
do Let edge e be the smallest-weight edge from
C to another component in T. if e is not
already in T then Add edge e to T return T