Introduction to Graphs. CS2110, Spring 2011 Cornell University

Transcription

1 Introduction to Graphs CS2110, Spring 2011 Cornell University

2 A graph is a data structure for representing relationships.

3 Each graph is a set of nodes connected by edges.

4 Synonym Graph Hostile Slick Icy Chilly Direct Nifty Cool Abrupt Sharp Composed

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10 Goals for Today Learn the formalisms behind graphs. Learn different representations for graphs. Learn about paths and cycles in graphs. See three ways of exploring a graph. Explore applications of graphs to real-world problems. Explore algorithms for drawing graphs.

11 Formalisms A (directed) graph is a pair G = (V, E) where V are the vertices (nodes) of the graph. E are the edges (arcs) of the graph. Each edge is a pair (u, v) of the start and end (or source and sink) of the edge.

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13 CAN MAN RAN MAT CAT SAT RAT

14 Directed and Undirected Graphs A graph is directed if its edges specify which is the start and end node. Encodes asymmetric relationship. A graph is undirected if the edges don't distinguish between the start and end nodes. Encodes symmetric relationship. An undirected graph is a special case of a directed graph (just add edges both ways).

15 How Big is a Graph G = (V, E)? Two measures: Number of vertices: V (often denoted n) Number of edges: E (often denoted m) E can be at most O( V 2 ) A graph is called sparse if it has few edges. A graph with many edges is called dense.

16 Navigating a Graph B D A F C E

17 Navigating a Graph B D A F C E

18 Navigating a Graph B D A F A B D F C E

19 Navigating a Graph B D A F A C F C E

20 A path from v 0 to v n is a list of edges (v 0, v 1 ), (v 1, v 2 ),, (v n-1, v n ).

21 The length of a path is the number of edges it contains.

22 Navigating a Graph B D A F C E

23 Navigating a Graph B D A F C E

24 A node v is reachable from node u if there is a path from u to v.

25 Navigating a Graph B D A F C E

26 Navigating a Graph B D A F C E

27 Navigating a Graph B D A F B D B C E

28 Navigating a Graph B D A F B D B D B C E

29 Navigating a Graph B D A F C E

30 Navigating a Graph B D A F A B D B D F C E

31 A cycle in a graph is a set of edges (v 0, v 1 ), (v 1, v 2 ),, (v n, v 0 ) that starts and ends at the same node.

32 A simple path is a path that does not contain a cycle. A simple cycle is a cycle that does not contain a smaller cycle

33 Properties of Nodes B D A F C E

34 The indegree of a node is the number of edges entering that node. The outdegree of a node is the number of edges leaving that node. In an undirected graph, these are the same and are called the degree of the node.

35 Summary of Terminology A path is a series of edges connecting two nodes. The length of a path is the number of edges in the path. A node v is reachable from u if there is a path from u to v. A cycle is a path from a node to itself. A simple path is a path without a cycle. A simple cycle is a cycle that does not contain a nested cycle. The indegree and outdegree of a node are the number of edges entering/leaving it.

36 Representing Graphs

37 Adjacency Matrices n x n grid of boolean values. B D Element A ij is 1 if edge from i to j, 0 else. Memory usage is O(n2 ) A C E F Can check if an edge exists in O(1). Can find all edges entering or leaving a node in O(n). A B C D E F A B C D E F

38 Adjacency Lists List of edges leaving each node. A B D F Memory usage is O(m+n) C E Find edges leaving a node in O(d + (u)) Check if edge exists in O(d + (u)) A B C D B D E B C F F E F C E

39 Graph Algorithms

40 Representing Prerequisites Graph Path Cycle Degree Path Length Simple Path Reachability Simple Cycle

41 A directed acyclic graph (DAG) is a directed graph with no cycles.

42 Examples of DAGs

43 Examples of DAGs

44 Examples of DAGs

45 Traversing a DAG Graph Path Cycle Degree Path Length Simple Path Reachability Simple Cycle

46 Traversing a DAG Graph Path Cycle Degree Path Length Simple Path Reachability Simple Cycle

47 Traversing a DAG Graph Path Cycle Degree Path Length Simple Path Reachability Simple Cycle

48 Traversing a DAG Graph Path Cycle Degree Path Length Simple Path Reachability Simple Cycle

49 Traversing a DAG Graph Cycle Path Degree Path Length Simple Path Reachability Simple Cycle

50 Traversing a DAG Graph Cycle Path Degree Path Length Simple Path Reachability Simple Cycle

51 Traversing a DAG Graph Cycle Simple Cycle Path Degree Path Length Simple Path Reachability

52 Traversing a DAG Graph Cycle Simple Cycle Degree Path Path Length Simple Path Reachability

53 Traversing a DAG Graph Cycle Simple Cycle Degree Path Path Length Simple Path Reachability

54 Traversing a DAG Graph Cycle Simple Cycle Degree Path Path Length Simple Path Reachability

55 Traversing a DAG Graph Cycle Simple Cycle Degree Path Path Length Simple Path Reachability

56 Traversing a DAG Graph Cycle Simple Cycle Degree Path Path Length Reachability Simple Path

57 Traversing a DAG Graph Cycle Simple Cycle Degree Path Path Length Simple Path Reachability

58 Traversing a DAG Graph Cycle Graph Simple Cycle Degree Path Cycle Degree Path Path Length Path Length Simple Path Reachability Simple Cycle Simple Path Reachability

59 Topological Sort Order the nodes of a DAG so no node is picked before its parents. Algorithm: Find a node with no incoming edges (indegree 0) Remove it from the graph. Add it to the resulting ordering. Not necessarily unique. Question: When is it unique?

60 Analyzing Topological Sort Assumes at each step that the DAG has a node with indegree zero. Is this always true? Claim one: Every DAG has such a node. Proof sketch: If this isn't true, then each node has at least one incoming edge. Start at any node and keep following backwards across that edge. Eventually you will find the same node twice and have found a cycle. Claim two: Removing such a node leaves the DAG a DAG. Proof sketch: If the resulting graph has a cycle, the old graph had a cycle as well.

61 Traversing an Arbitrary Graph

62 Traversing an Arbitrary Graph

63 Traversing an Arbitrary Graph

64 Traversing an Arbitrary Graph

65 Traversing an Arbitrary Graph

66 Traversing an Arbitrary Graph

67 Traversing an Arbitrary Graph

68 Traversing an Arbitrary Graph

69 Traversing an Arbitrary Graph

70 Traversing an Arbitrary Graph

71 Traversing an Arbitrary Graph

72 Traversing an Arbitrary Graph

73 Traversing an Arbitrary Graph

74 Traversing an Arbitrary Graph

75 Traversing an Arbitrary Graph

76 Traversing an Arbitrary Graph

77 Traversing an Arbitrary Graph

78 Traversing an Arbitrary Graph

79 Traversing an Arbitrary Graph

80 Traversing an Arbitrary Graph

81 Traversing an Arbitrary Graph

82 Traversing an Arbitrary Graph

83 Traversing an Arbitrary Graph

84 Traversing an Arbitrary Graph

85 Traversing an Arbitrary Graph

86 Traversing an Arbitrary Graph

87 Traversing an Arbitrary Graph

88 Traversing an Arbitrary Graph

89 Traversing an Arbitrary Graph

90 Traversing an Arbitrary Graph

91 Traversing an Arbitrary Graph

92 Traversing an Arbitrary Graph

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94 Traversing an Arbitrary Graph

95 General Graph Search Algorithm Maintain a collection C of nodes to visit. Initialize C with some set of nodes. While C is not empty: Pick a node v out of C. Follow all outgoing edges from v, adding each unvisited node found this way to C. Eventually explores all nodes reachable from the starting set of nodes. (Why?)

96 Depth-First Search Specialization of the general search algorithm where nodes to visit are put on a stack. Explores down a path as far as possible, then backs up. Simple graph search algorithm useful for exploring a complete graph. Useful as a subroutine in many important graph algorithms. Runs in O(m + n) with adjacency lists, O(n 2 ) with adjacency matrix.

97 Depth-first search A B C D E F

98 Depth-first search A B C D E F Stack

99 Depth-first search A B C D E F A Stack

100 Depth-first search A B C D E F Stack

101 Depth-first search A B C D E F E B Stack

102 Depth-first search A B C D E F E B Stack

103 Depth-first search A B C D E F B Stack

104 Depth-first search A B C D E F C F D B Stack

105 Depth-first search A B C D E F C F D B Stack

106 Depth-first search A B C D E F F D B Stack

107 Depth-first search A B C D E F F D B Stack

108 Depth-first search A B C D E F F D B Stack

109 Depth-first search A B C D E F D B Stack

110 Depth-first search A B C D E F D B Stack

111 Depth-first search A B C D E F B Stack

112 Depth-first search A B C D E F B Stack

113 Depth-first search A B C D E F Stack

114 Depth-first search A B C D E F Stack

115 Implementing DFS DFS(Node v, Set<Node> visited) { if (v is in visited) return; Add v to visited; } for (Node u connected to v) DFS(u, visited);

116 Graph Search Trees

117 Graph Search Trees

118 Graph Search Trees

119 Graph Search Trees

120 Graph Search Trees

121 Graph Search Trees

122 Graph Search Trees

123 Graph Search Trees

124 Graph Search Trees

125 Graph Search Trees

126 Graph Search Trees

127 Graph Search Trees

128 Graph Search Trees

129 Graph Search Trees

130 Graph Search Trees

131 Graph Search Trees

132 Graph Search Trees

133 Graph Search Trees

134 Graph Search Trees

135 Graph Search Trees

136 Graph Search Trees

137 Mazes as Graphs

138 Mazes as Graphs

139 Mazes as Graphs

140 Mazes as Graphs

141 Creating a Maze with DFS Create a grid graph of the appropriate size. Starting at any node, run a depth-first search, adding the arcs to the stack in random order. The resulting DFS tree is a maze with one solution.

142 Problems with DFS Useful when trying to explore everything. Not good at finding specific nodes.

143 Problems with DFS Useful when trying to explore everything. Not good at finding specific nodes. Stack

144 Problems with DFS Useful when trying to explore everything. Not good at finding specific nodes. Stack

145 Problems with DFS Useful when trying to explore everything. Not good at finding specific nodes. A C B Stack

146 Problems with DFS Useful when trying to explore everything. Not good at finding specific nodes. A C B A B C Stack

147 Problems with DFS Useful when trying to explore everything. Not good at finding specific nodes. A C B A B C Stack

148 Breadth-First Search Specialization of the general search algorithm where nodes to visit are put into a queue. Explores nodes one hop away, then two hops away, etc. Finds path with fewest edges from start node to all other nodes. Runs in O(m + n) with adjacency lists, O(n 2 ) with adjacency matrix.

149 Breadth-first search A B C D E F

150 Breadth-first search A B C D E F Queue

151 Breadth-first search A B C D E F Queue A

152 Breadth-first search A B C D E F Queue

153 Breadth-first search A B C D E F Queue B E

154 Breadth-first search A B C D E F Queue B E

155 Breadth-first search A B C D E F Queue E

156 Breadth-first search A B C D E F Queue E C

157 Breadth-first search A B C D E F Queue E C

158 Breadth-first search A B C D E F Queue C

159 Breadth-first search A B C D E F Queue C D F

160 Breadth-first search A B C D E F Queue C D F

161 Breadth-first search A B C D E F Queue D F

162 Breadth-first search A B C D E F Queue D F

163 Breadth-first search A B C D E F Queue F

164 Breadth-first search A B C D E F Queue F

165 Breadth-first search A B C D E F Queue

166 Breadth-first search A B C D E F Queue

167 Implementing BFS BFS(Node v, Set<Node> visited) { Create a Queue<Node> of nodes to visit; Add v to the queue; while (The queue is not empty) { Dequeue a node from the queue, let it be u; if (u has been visited) continue; Add u to the visited set; } } for (Node w connected to u) Enqueue w in the queue;

168 Classic Graph Algorithms

169 Graph Coloring 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.

170 Graph Coloring is Hard. Determining whether a graph can be colored with k colors (for k > 2) is NP-complete. It is not known whether this problem can be solved in polynomial time. Want $1,000,000? Find a polynomial-time algorithm or prove that none exists.

171 Matching A matching in a graph is a subset of the edges that don't share any endpoints. Intuitively, pairing up nodes in the graph.

172 Matching A matching in a graph is a subset of the edges that don't share any endpoints. Intuitively, pairing up nodes in the graph.

173 Applications of Matching Unlike graph coloring, matching can be done quickly. Sample application: divvying up desserts.

174 Divvying Up Desserts

175 Divvying Up Desserts

176 Divvying Up Desserts

177 Divvying Up Desserts

178 Divvying Up Desserts

179 Divvying Up Desserts 3 7

180 Divvying Up Desserts

181 Divvying Up Desserts

182 Divvying Up Desserts

183 Divvying Up Desserts

184 Drawing Graphs

185 Hostile Slick Icy Chilly Direct Nifty Cool Abrupt Sharp Composed

186 Chilly Slick Composed Icy Direct Sharp Hostile Nifty Cool Abrupt

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191 Idea: Treat the graph as a physical system that exerts forces on itself.

192 This is called a force-directed layout algorithm.

193 Summary Graphs are a powerful abstraction for modeling relationships and connectivity. Adjacency lists and adjacency matrices are two common representations of graphs. Directed acyclic graphs can be visited via a topological sort. Depth-first search is a simple graph exploration algorithm. Breadth-first search searches a graph one layer at a time. There are many classic algorithms on graphs: Graph coloring tries to color nodes so no two nodes of the same color are connected. Matchings represent pairing up of graph elements. Graph drawing seeks to render aesthetically-pleasing graphs.