Minimum Spanning Trees

Transcription

1 Minimum Spanning Trees

2 We now know how to compute the minimum cost path from a start node to every other node in a graph. What other interesting problems can we solve using graphs?

3 onstructing a Network

4 onstructing a Network

5 onstructing a Network

6 ycle

7 A cycle in an undirected graph is a path that starts and ends at the same node.

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12 Spanning Trees

13 A spanning tree in an undirected graph is a set of edges, with no cycles, that connects all nodes.

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18 A minimum spanning tree (or MST) is a spanning tree with the least total cost.

19 Applications Electric Grids Given a collection of houses, where do you lay wires to connect all houses with the least total cost? This was the initial motivation for studying minimum spanning trees in the early 920's. (work done by zech mathematician Otakar Borůvka) Data lustering More on that later... Maze Generation More on that later...

20 Kruskal's Algorithm Kruskal's algorithm is an efficient algorithm for finding minimum spanning trees. Idea is as follows: Remove all edges from the graph. Place all edges into a priority queue based on their length. While the priority queue is not empty: Dequeue an edge from the priority queue. If the endpoints of the edge aren't already connected to one another, add in that edge. Otherwise, skip the edge.

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38 A graph graph can can have have many many minimum minimum spanning spanning trees. trees. Here, Here, the the choice choice of of which which length- length- edge edge we we visit visit first first leads leads to to different different results. results. 2

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63 Very simple algorithm, tricky proof that it produces a Minimal Spanning Tree

64 Maintaining onnectivity The key step in Kruskal's algorithm is determining whether the two endpoints of an edge are already connected to one another. Typical approach: break the nodes apart into clusters. Initially, each node is in its own cluster. Whenever an edge is added, the clusters for the endpoints are merged together into a new cluster.

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101 Kruskal's with lusters Place every node into its own cluster. Place all edges into a priority queue. While there are two or more clusters remaining: Dequeue an edge from the priority queue. If its endpoints are not in the same cluster: Merge the clusters containing the endpoints. Add the edge to the resulting spanning tree. Return the resulting spanning tree.

102 Kruskal's with lusters Specialized data structures exist for maintaining the clusters in Kruskal's algorithm. One such structure: disjoint-set forest. Not particularly complicated. heck Wikipedia for details. Easy extra credit on the last assignment (details in a bit.)

103 Applications of Kruskal's Algorithm

104 Data lustering

105 Data lustering

106 Data lustering Given a set of points, break those points apart into clusters. Immensely useful across all disciplines: luster individuals by phenotype to try to determine what genes influence which traits. luster images by pixel color to identify objects in pictures. luster essays by various features to see how students learn to write.

107 Data lustering

108 Data lustering

109 Data lustering

110 What makes a clustering good?

111 Maximum-Separation lustering Maximum-separation clustering tries to find a clustering that maximizes the separation between different clusters. Specifically: Maximize the minimum distance between any two points of different clusters. Very good on many data sets, though not always ideal.

112 Maximum-Separation lustering

113 Maximum-Separation lustering

114 Maximum-Separation lustering It is extremely easy to adopt Kruskal's algorithm to produce a maximum-separation set of clusters. Suppose you want k clusters. Given the data set, add an edge from each node to each other node whose length depends on their similarity. Run Kruskal's algorithm until only k clusters remain. The pieces of the graph that have been linked together are k maximally-separated clusters.

115 Maximum-Separation lustering

116 Maximum-Separation lustering

117 Maximum-Separation lustering

118 Mazes with Kruskal's Algorithm

119 Mazes with Kruskal's Algorithm

120 Mazes with Kruskal's Algorithm

121 Mazes with Kruskal's Algorithm

122 Mazes with Kruskal's Algorithm

123 Mazes with Kruskal's Algorithm

124 Mazes with Kruskal's Algorithm

125 Mazes with Kruskal's Algorithm

126 Mazes with Kruskal's Algorithm

127 Mazes with Kruskal's Algorithm

128 Mazes with Kruskal's Algorithm

129 Mazes with Kruskal's Algorithm

130 Mazes with Kruskal's Algorithm

131 Mazes with Kruskal's Algorithm

132 Mazes with Kruskal's Algorithm

133 Mazes with Kruskal's Algorithm

134 Mazes with Kruskal's Algorithm

135 Mazes with Kruskal's Algorithm

136 Mazes with Kruskal's Algorithm

137 Mazes with Kruskal's Algorithm

138 Mazes with Kruskal's Algorithm

139 Mazes with Kruskal's Algorithm

140 Mazes with Kruskal's Algorithm

141 Mazes with Kruskal's Algorithm

142 Mazes with Kruskal's Algorithm

143 Mazes with Kruskal's Algorithm

144 Mazes with Kruskal's Algorithm

145 Mazes with Kruskal's Algorithm

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148 Mazes with Kruskal's Algorithm

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154 Mazes with Kruskal's Algorithm

155 Other ool Graph Problems

156 Graph oloring Given a graph G, assign colors to the nodes so that no edge has endpoints of the same color. The chromatic number of a graph is the fewest number of colors needed to color it.

157 Graph oloring

158 Graph oloring is Useful

159 Graph oloring is Useful

160 Graph oloring is Useful

161 Graph oloring is Useful

162 Graph oloring is Useful

163 Graph oloring is Useful

164 Graph oloring is Useful

165 Graph oloring is Hard. No efficient algorithms are known for determining whether a graph can be colored with k colors for any k > 2. Want $,000,000? Find a polynomialtime algorithm or prove that none exists.

166 Minimum ut 0_beta/libs/graph/doc/stoer_wagner_imgs/stoer_wagner-example-min-cut.gif

167 Probabilistic Graphical Models

168 Summary of Graphs Graphs are an enormously flexible framework for encoding relationships between structures. MANY uses for graphs Want to learn more? Take S!