Minimum Spanning Trees 2704 BOS 867 849 PVD
Minimum Spanning Trees 2704 BOS 867 849 PVD ORD 740 621 1846 LAX 1391 1464 1235 144 JFK 1258 184 802 SFO 337 187 BWI 1090 DFW 946 1121 MIA 2342 © 2010 Goodrich, Tamassia Minimum Spanning Trees 1
Minimum Spanning Trees Spanning subgraph n ORD Subgraph of a graph G containing all the vertices of G 1 Spanning tree n Spanning subgraph that is itself a (free) tree DEN Minimum spanning tree (MST) n q 10 Spanning tree of a weighted graph with minimum total edge weight PIT 9 6 STL 4 8 7 3 DCA 5 2 Applications n n Communications networks Transportation networks © 2010 Goodrich, Tamassia DFW Minimum Spanning Trees ATL 2
Cycle Property: Let T be a minimum spanning tree of a weighted graph G n Let e be an edge of G that is not in T and C let be the cycle formed by e with T n For every edge f of C, weight(f) weight(e) Proof: n By contradiction n If weight(f) > weight(e) we can get a spanning tree of smaller weight by replacing e with f Minimum Spanning Trees 2 4 C n © 2010 Goodrich, Tamassia 8 f 6 9 3 e 8 7 7 Replacing f with e yields a better spanning tree f 2 6 8 4 C 9 3 8 e 7 7 3
Partition Property U f Partition Property: Consider a partition of the vertices of G into subsets U and V n Let e be an edge of minimum weight across the partition n There is a minimum spanning tree of G containing edge e Proof: n Let T be an MST of G n 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 n By the cycle property, weight(f) weight(e) n Thus, weight(f) = weight(e) n We obtain another MST by replacing f with e n © 2010 Goodrich, Tamassia Minimum Spanning Trees V 7 4 9 5 2 8 8 3 e 7 Replacing f with e yields another MST U 2 f V 7 4 9 5 8 8 3 e 7 4
Kruskal’s Algorithm q q q Maintain a partition of the vertices into clusters Algorithm Kruskal. MST(G) for each vertex v in G do Create a cluster consisting of v n Initially, single-vertex let Q be a priority queue. clusters Insert all edges into Q n Keep an MST for each T cluster {T is the union of the MSTs of the clusters} n Merge “closest” clusters while T has fewer than n - 1 edges do and their MSTs e Q. remove. Min(). get. Value() [u, v] G. end. Vertices(e) A priority queue stores the A get. Cluster(u) edges outside clusters B get. Cluster(v) n Key: weight if A B then n Element: edge Add edge e to T merge. Clusters(A, B) At the end of the algorithm return T n One cluster and one MST © 2010 Goodrich, Tamassia Minimum Spanning Trees 5
Example 8 B 5 1 6 3 H D 9 5 1 2 C 11 7 A 9 C 11 7 10 © 2010 Goodrich, Tamassia E 6 F 3 D H 1 2 A Campus Tour E 6 5 H C 11 10 2 G 9 7 F 3 8 B 4 4 D 10 G 8 5 F G 8 B 4 E C 11 10 B A 9 7 A 1 G 4 E 6 F 3 D H 2 6
Example (contd. ) B 5 1 G 8 9 6 H D 9 5 1 2 C 11 7 A 4 E 6 H D 10 A 5 9 C 11 7 10 © 2010 Goodrich, Tamassia E ep 6 F 3 D H G st o 4 8 B 1 2 Campus Tour 2 four steps tw 8 B 1 G F 3 s 10 F 3 C 11 7 A E 8 B 4 G A 5 9 C 11 7 10 4 E 6 F 3 D H 2 7
Data Structure for Kruskal’s Algorithm q q The algorithm maintains a forest of trees A priority queue extracts the edges by increasing weight 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 operations: n n n make. Set(u): create a set consisting of u find(u): return the set storing u union(A, B): replace sets A and B with their union © 2010 Goodrich, Tamassia Minimum Spanning Trees 8
Recall of List-based Partition q q Each set is stored in a sequence Each element has a reference back to the set n n n q operation find(u) takes O(1) time, and returns the set of which u is a member. in operation union(A, B), we move the elements of the smaller set to the sequence of the larger set and update their references the time for operation union(A, B) is min(|A|, |B|) 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 © 2010 Goodrich, Tamassia Minimum Spanning Trees 9
Partition-Based Implementation q Partition-based version of Kruskal’s Algorithm n n q Cluster merges as unions Cluster locations as finds Running time O((n + m) log n) n n PQ operations O(m log n) UF operations O(n log n) © 2010 Goodrich, Tamassia Algorithm Kruskal. MST(G) Initialize a partition P for each vertex v in G do P. make. Set(v) let Q be a priority queue. Insert all edges into Q T {T is the union of the MSTs of the clusters} while T has fewer than n - 1 edges do e Q. remove. Min(). get. Value() [u, v] G. end. Vertices(e) A P. find(u) B P. find(v) if A B then Add edge e to T P. union(A, B) return T Minimum Spanning Trees 10
Prim-Jarnik’s Algorithm q q Similar to Dijkstra’s algorithm 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 label d(v) representing the smallest weight of an edge connecting v to a vertex in the cloud At each step: n n 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 © 2010 Goodrich, Tamassia Minimum Spanning Trees 11
Prim-Jarnik’s Algorithm (cont. ) q A heap-based adaptable priority queue with location-aware entries stores the vertices outside the cloud n n n q Key: distance Value: vertex Recall that method replace. Key(l, k) changes the key of entry l We store three labels with each vertex: n n n Distance Parent edge in MST Entry in priority queue © 2010 Goodrich, Tamassia Algorithm Prim. Jarnik. MST(G) Q new heap-based priority queue s a vertex of G for all v G. vertices() if v = s set. Distance(v, 0) else set. Distance(v, ) set. Parent(v, ) l Q. insert(get. Distance(v), v) set. Locator(v, l) while Q. is. Empty() l Q. remove. Min() u l. get. Value() for all e G. incident. Edges(u) z G. opposite(u, e) r weight(e) if r < get. Distance(z) set. Distance(z, r) set. Parent(z, e) Q. replace. Key(get. Entry(z), r) Minimum Spanning Trees 12
Example 2 7 B 0 2 B 5 C 0 © 2010 Goodrich, Tamassia 2 0 A 4 9 5 C 5 F 8 8 7 E 7 7 2 4 F 8 7 B 7 D 7 3 9 8 A D 7 5 F E 7 2 2 4 8 8 A 9 8 C 5 2 D E 2 3 7 Minimum Spanning Trees 7 B 0 3 7 7 4 9 5 C 5 F 8 8 A D 7 E 3 7 13 4
Example (contd. ) 2 2 B 0 4 9 5 C 5 F 8 8 A D 7 7 7 E 4 3 3 2 2 B 0 © 2010 Goodrich, Tamassia Minimum Spanning Trees 4 9 5 C 5 F 8 8 A D 7 7 7 E 3 3 14 4
Analysis q Graph operations n q Label operations n n q n 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 Prim-Jarnik’s algorithm runs in O((n + m) log n) time provided the graph is represented by the adjacency list structure n q We set/get the distance, parent and locator labels of vertex z O(deg(z)) times Setting/getting a label takes O(1) time Priority queue operations n q Method incident. Edges is called once for each vertex Recall that Sv deg(v) = 2 m The running time is O(m log n) since the graph is connected © 2010 Goodrich, Tamassia Minimum Spanning Trees 15
Baruvka’s Algorithm (Exercise) q q q Like Kruskal’s Algorithm, Baruvka’s algorithm grows many clusters at once and maintains a forest T Each iteration of the while loop halves the number of connected components in forest T The running time is O(m log n) Algorithm Baruvka. MST(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 © 2010 Goodrich, Tamassia Minimum Spanning Trees 16
Example of Baruvka’s Algorithm (animated) 2 Slide by Matt Stallmann included with permission. 4 3 8 4 9 1 5 8 5 3 6 7 9 6 4 2 4 3 4 9 1 5 © 2010 Goodrich, Tamassia 8 CSC 316 5 3 6 7 6 4 17
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