Spanning Minimum Trees
2704 867 849 ORD 1846 SFO 337 LAX 1464 DFW 1235 1121 MIA 2342 802 1391 740 621 187 JFK 184 BWI 1090 946 144 1258 BOS PVD
Minimum Spanning Trees
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Outline and Reading
Minimum Spanning Trees (7.3)
The PrimJarnik Algorithm (7.3.2) Kruskal's Algorithm (7.3.1) Baruvka's Algorithm (7.3.3)
Minimum Spanning Trees 2
Definitions A crucial fact
Minimum Spanning Tree
Spanning subgraph
Spanning tree
Subgraph of a graph G containing all the vertices of G Spanning subgraph that is itself a (free) tree Spanning tree of a weighted graph with minimum total edge weight Communications networks Transportation networks
ORD
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10
PIT
9 6 7 3 5
DEN
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Minimum spanning tree (MST)
STL
8
DCA
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Applications
DFW
ATL
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Minimum Spanning Trees
Cycle Property
Cycle Property:
f 2 6 8
8 9 3 7 e
Let T be a minimum spanning tree of a weighted graph G Let e be an edge of G that is not in T and let C be the cycle formed by e with T For every edge f of C, weight(f) weight(e) Proof: By contradiction If weight(f) > weight(e) we can get a spanning tree of smaller weight by replacing e with f
C
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7
Replacing f with e yields a better spanning tree f 2 6 8 9 3 8 7 e 4
C
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Minimum Spanning Trees
Partition Property
Partition Property:
U
f
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V 4
Consider a partition of the vertices of 9 5 G into subsets U and V 2 8 Let e be an edge of minimum weight 3 8 across the partition e There is a minimum spanning tree of 7 G containing edge e Replacing f with e yields Proof: another MST Let T be an MST of G If T does not contain e, consider the U V 7 cycle C formed by e with T and let f f 4 be an edge of C across the partition By the cycle property, 9 5 2 weight(f) weight(e) 8 Thus, weight(f) = weight(e) 3 8 e We obtain another MST by replacing f with e 7
Minimum Spanning Trees 5
PrimJarnik's Algorithm
Similar to Dijkstra's algorithm (for a connected graph) We pick an arbitrary vertex s and we grow the MST as a cloud of vertices, starting from s We store with each vertex v a label d(v) = the smallest weight of an edge connecting v to a vertex in the cloud At each step:
We add to the cloud the vertex u outside the cloud with the smallest distance label We update the labels of the vertices adjacent to u
Minimum Spanning Trees
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PrimJarnik's Algorithm (cont.)
A priority queue stores the vertices outside the cloud
Locatorbased methods
Key: distance Element: vertex
We store three labels with each vertex:
insert(k,e) returns a locator replaceKey(l,k) changes the key of an item
Distance Parent edge in MST Locator in priority queue
Algorithm PrimJarnikMST(G) Q new heap-based priority queue s a vertex of G for all v G.vertices() if v = s setDistance(v, 0) else setDistance(v, ) setParent(v, ) l Q.insert(getDistance(v), v) setLocator(v,l) while Q.isEmpty() u Q.removeMin() for all e G.incidentEdges(u) z G.opposite(u,e) r weight(e) if r < getDistance(z) setDistance(z,r) setParent(z,e) Q.replaceKey(getLocator(z),r)
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Minimum Spanning Trees
Example
2 2 A B 5 8 7 2 2 A 7 5 8 7 5 C 9 8 E 7 7 8 C 9 8 E 7 7 4 F 3 0 2 2 A B 5 8 7 D 4 F 3 0 A 2 2 B 5 8 7 7 5 C 9 8 E 7
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7 5 C 9 8
D
7 4 F 3 7
0
E
B
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7 4 F 3 4
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Minimum Spanning Trees
Example (contd.)
2 2 A B 5 8 7 7 5 C 9 8 E 3 D 7 4 F 3 2 2 A 7 5 8 7 5 C 9 8 E 3
9
4
0
B
D
7 4 F 3 4
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Minimum Spanning Trees
Analysis
Graph operations
Label operations
Method incidentEdges is called once for each vertex We set/get the distance, parent and locator labels of vertex z O(deg(z)) times Setting/getting a label takes O(1) time Each vertex is inserted once into and removed once from the priority queue, where each insertion or removal takes O(log n) time The key of a vertex w in the priority queue is modified at most deg(w) times, where each key change takes O(log n) time
Priority queue operations
PrimJarnik's algorithm runs in O((n + m) log n) time provided the graph is represented by the adjacency list structure
The running time is O(m log n) since the graph is connected
Minimum Spanning Trees 10
Recall that v deg(v) = 2m
Kruskal's Algorithm
A priority queue stores the edges outside the cloud
At the end of the algorithm
Key: weight Element: edge
We are left with one cloud that encompasses the MST A tree T which is our MST
Algorithm KruskalMST(G) for each vertex V in G do define a Cloud(v) of {v} let Q be a priority queue. Insert all edges into Q using their weights as the key T while T has fewer than n-1 edges do edge e = T.removeMin() Let u, v be the endpoints of e if Cloud(v) Cloud(u) then Add edge e to T Merge Cloud(v) and Cloud(u) return T
Minimum Spanning Trees
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Data Structure for Kruskal Algortihm
The algorithm a maintains forest of trees An edge is accepted it if connects distinct trees We need a data structure that maintains a partition, i.e., a collection of disjoint sets, with the operations: find(u): return the set storing u union(u,v): replace the sets storing u and v with their union
Minimum Spanning Trees
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Representation of a Partition
Each set is stored in a sequence Each element has a reference back to the set
operation find(u) takes O(1) time, and returns the set of which u is a member. in operation union(u,v), we move the elements of the smaller set to the sequence of the larger set and update their references the time for operation union(u,v) is min(nu,nv), where nu and nv are the sizes of the sets storing u and v
Whenever an element is processed, it goes into a set of size at least double, hence each element is processed at most log n times
Minimum Spanning Trees 13
PartitionBased Implementation
A partitionbased version of Kruskal's Algorithm performs cloud merges as unions and tests as finds.
Algorithm Kruskal(G): Input: A weighted graph G. Output: An MST T for G. Let P be a partition of the vertices of G, where each vertex forms a separate set. Let Q be a priority queue storing the edges of G, sorted by their weights Let T be an initially-empty tree while Q is not empty do (u,v) Q.removeMinElement() if P.find(u) != P.find(v) then Running time: Add (u,v) to T O((n+m)log n) P.union(u,v) return T
Minimum Spanning Trees 14
Kruskal Example
1846 SFO 337 LAX 1464
2704
867 849 ORD 740 621 187 JFK 184 1391 BWI
BOS PVD 144 1258
802
1090 DFW 1235 1121 MIA 2342
Minimum Spanning Trees 15
946
Example
1846 SFO 337 LAX 1464
2704
867 849 ORD 740 621 187 JFK 184 1391 BWI
BOS PVD 144 1258
802
1090 DFW 1235 1121 MIA 2342
Minimum Spanning Trees 16
946
Example
1846 SFO 337 LAX 1464
2704
867 849 ORD 740 621 187 JFK 184 1391 BWI
BOS PVD 144 1258
802
1090 DFW 1235 1121 MIA 2342
Minimum Spanning Trees 17
946
Example
1846 SFO 337 LAX 1464
2704
867 849 ORD 740 621 187 JFK 184 1391 BWI
BOS PVD 144 1258
802
1090 DFW 1235 1121 MIA 2342
Minimum Spanning Trees 18
946
Example
1846 SFO 337 LAX 1464
2704
867 849 ORD 740 621 187 JFK 184 1391 BWI
BOS PVD 144 1258
802
1090 DFW 1235 1121 MIA 2342
Minimum Spanning Trees 19
946
Example
1846 SFO 337 LAX 1464
2704
867 849 ORD 740 621 187 JFK 184 1391 BWI
BOS PVD 144 1258
802
1090 DFW 1235 1121 MIA 2342
Minimum Spanning Trees 20
946
Example
1846 SFO 337 LAX 1464
2704
867 849 ORD 740 621 187 JFK 184 1391 BWI
BOS PVD 144 1258
802
1090 DFW 1235 1121 MIA 2342
Minimum Spanning Trees 21
946
Example
1846 SFO 337 LAX 1464
2704
867 849 ORD 740 621 187 JFK 184 1391 BWI
BOS PVD 144 1258
802
1090 DFW 1235 1121 MIA 2342
Minimum Spanning Trees 22
946
Example
1846 SFO 337 LAX 1464
2704
867 849 ORD 740 621 187 JFK 184...