COP 3530: Computer Science III

1
COP 3530 Computer Science III Summer
2005 Graphs and Graph Algorithms Part 5
Instructor Mark Llewellyn
markl_at_cs.ucf.edu CSB 242, 823-2790 http//ww
w.cs.ucf.edu/courses/cop3530/summer05
School of Computer Science University of Central
Florida
2
All-Pairs Shortest Paths

• Dijkstras algorithm was a single source shortest
  path algorithm.
• Rather than defining one vertex as a starting
  vertex, suppose that we would like to find the
  distance between every pair of vertices in a
  weighted graph G. In other words, the shortest
  path from every vertex to every other vertex in
  the graph.
• One option would be to run Dijkstras algorithm
  in a loop considering each vertex once as the
  starting point. A better option would be to use
  the Floyd-Warshall dynamic programming algorithm
  (typically referred to as Floyds algorithm).

3

• Dijkstras Algorithm Single Source Shortest
  Path
• 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
4

• DAG based Algorithm - Single Source Shortest
  Path
• 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

-2
D
C
3
-5
8
E
5
6
9
B
2
A
5
All-Pairs Shortest Paths
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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

j?
i?
B
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

• 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

9
Partition Property

• 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

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
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MST Prim-Jarniks Algorithm
?
6
D
C
?
1
6
8
E
6
2
?
7
2
B
A
?
0
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MST Prim-Jarniks Algorithm
?
6
D
C
1
2,A
6
8
E
6
2
7,A
7
2
B
A
2,A
0
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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
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
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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
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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
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Example (cont.)
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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

The forest
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MST Kruskals Algorithm
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MST Kruskals Algorithm
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MST Kruskals Algorithm
E
D
C
B
A
27
MST Kruskals Algorithm
E
D
C
B
A
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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
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
MIA
2342
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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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

B
44
Baruvkas Algorithm

• Like Kruskals Algorithm, Baruvkas algorithm
  grows many clouds at once.
• Each iteration of the while-loop halves the
  number of connected components 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