Preparation Data Structures 11 graphs

1. Data Structures Graphs Andres Mendez-Vazquez May 8, 2015 1 / 91

2. Images/cinvestav- Outline 1 Graphs Graphs Everywhere History Basic Theory 2 Graph Representation Matrix Representation Possible Code for This Representation Adjacency List Representation 3 Traversing the Graph Breadth-ﬁrst search Depth-First Search 4 Applications Finding a path between nodes Connected Components Spanning Trees Topological Sorting 2 / 91

4. Images/cinvestav- We are full of Graphs Maps 4 / 91

5. Images/cinvestav- We are full of Graphs State Machines for Branch Prediction in CPU’s 5 / 91

6. Images/cinvestav- We are full of Graphs Social Networks 6 / 91

8. Images/cinvestav- History Something Notable Graph theory started with Euler who was asked to ﬁnd a nice path across the seven Königsberg bridges The Actual City 8 / 91

9. Images/cinvestav- History Something Notable Graph theory started with Euler who was asked to ﬁnd a nice path across the seven Königsberg bridges The Actual City 8 / 91

10. Images/cinvestav- There is No-Solution Using Graphs The (Eulerian) path should cross over each of the seven bridges exactly once Can you ﬁnd Any possible solution? 9 / 91

11. Images/cinvestav- There is No-Solution Using Graphs The (Eulerian) path should cross over each of the seven bridges exactly once Can you ﬁnd Any possible solution? 9 / 91

12. Images/cinvestav- Necessary Condition Euler discovered that A necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree. Add an extra bridge 10 / 91

13. Images/cinvestav- Necessary Condition Euler discovered that A necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree. Add an extra bridge Add Extra Bridge 10 / 91

14. Images/cinvestav- Studying Graphs All the previous examples are telling us Data Structures are required to design structures to hold the information coming from graphs!!! Good Representations They will allow to handle the data structures with ease!!! 11 / 91

15. Images/cinvestav- Studying Graphs All the previous examples are telling us Data Structures are required to design structures to hold the information coming from graphs!!! Good Representations They will allow to handle the data structures with ease!!! 11 / 91

17. Images/cinvestav- Basic Theory Deﬁnition A Graph is composed of the following parts: Nodes and Edges Nodes They can represent multiple things: People Cities States of Being etc Edges They can represent multiple things too: Distance between cities Friendships Matching Strings Etc 13 / 91

20. Images/cinvestav- Basic Theory Deﬁnition A graph G = (V , E) consists of a set of vertices (or nodes) V and a set of edges E, where we assume that they are ﬁnite i.e. |V | = n and |E| = m. Example 14 / 91

21. Images/cinvestav- Basic Theory Deﬁnition A graph G = (V , E) consists of a set of vertices (or nodes) V and a set of edges E, where we assume that they are ﬁnite i.e. |V | = n and |E| = m. Example 14 / 91

22. Images/cinvestav- Properties Incident An edge ek = (vi , vj) is incident with the vertices vi and vj. A simple graph has no self-loops or multiple edges like below 15 / 91

23. Images/cinvestav- Properties Incident An edge ek = (vi , vj) is incident with the vertices vi and vj. A simple graph has no self-loops or multiple edges like below 15 / 91

24. Images/cinvestav- Some properties Degree The degree d(v) of a vertex V is its number of incident edges A self loop A self-loop counts for 2 in the degree function. Proposition The sum of the degrees of a graph G = (V , E) equals 2|E| = 2m (trivial). 16 / 91

27. Images/cinvestav- Empty Complete A complete graph Kn is a simple graph with all n(n−1)/2 possible edges, like the graph below for n = 2, 3, 4, 5. Example 17 / 91

28. Images/cinvestav- Empty Complete A complete graph Kn is a simple graph with all n(n−1)/2 possible edges, like the graph below for n = 2, 3, 4, 5. Example 17 / 91

29. Images/cinvestav- We need NICE representations First One Matrix Representation Second One Adjacency Representation 18 / 91

30. Images/cinvestav- We need NICE representations First One Matrix Representation Second One Adjacency Representation 18 / 91

32. Images/cinvestav- Adjacency Matrix Representation This is the simplest one If we number the nodes of an undirected graph: 20 / 91

33. Images/cinvestav- Adjacency Matrix Representation In a natural way the edges can be identiﬁed by the nodes For example, the edge between 1 and 4 nodes gets named as (1,4) Then How, we use this to represent the graph through a Matrix or and Array of Arrays??!!! 21 / 91

34. Images/cinvestav- Adjacency Matrix Representation In a natural way the edges can be identiﬁed by the nodes For example, the edge between 1 and 4 nodes gets named as (1,4) Then How, we use this to represent the graph through a Matrix or and Array of Arrays??!!! 21 / 91

35. Images/cinvestav- What about the following? How do we indicate that an edge exist given the following matrix 1 2 3 4 5 6 1 − − − − − − 2 − − − − − − 3 − − − − − − 4 − − − − − − 5 − − − − − − 6 − − − − − − You say it!! Use a 0 for no-edge Use a 1 for edge 22 / 91

36. Images/cinvestav- What about the following? How do we indicate that an edge exist given the following matrix 1 2 3 4 5 6 1 − − − − − − 2 − − − − − − 3 − − − − − − 4 − − − − − − 5 − − − − − − 6 − − − − − − You say it!! Use a 0 for no-edge Use a 1 for edge 22 / 91

37. Images/cinvestav- We have then... Deﬁnition 0/1 N × N matrix with N =Number of nodes or vertices A(i, j) = 1 iﬀ (i, j) is an edge 23 / 91

38. Images/cinvestav- We have then... For the previous example 1 2 3 4 5 6 1 0 0 0 1 0 0 2 0 0 0 1 0 0 3 0 0 0 1 0 0 4 1 1 1 0 1 0 5 0 0 0 1 0 1 6 0 0 0 0 1 0 24 / 91

39. Images/cinvestav- Properties of the Matrix for Undirected Graphs Property One Diagonal entries are zero. Property Two Adjacency matrix of an undirected graph is symmetric: A (i, j) = A (j, i) for all i and j 25 / 91

40. Images/cinvestav- Properties of the Matrix for Undirected Graphs Property One Diagonal entries are zero. Property Two Adjacency matrix of an undirected graph is symmetric: A (i, j) = A (j, i) for all i and j 25 / 91

41. Images/cinvestav- What about direct Graphs!!! Similar idea Use a 0 for no-edge Use a 1 for directed edge 26 / 91

42. Images/cinvestav- Example We have that 3 1 2 4 5 6 7 1 2 3 4 5 6 7 1 0 1 1 1 0 0 0 2 0 0 0 1 1 0 0 3 0 0 0 0 0 1 0 4 0 0 1 0 0 1 1 5 0 0 0 1 0 0 1 6 0 0 0 0 0 0 0 7 0 0 0 0 0 1 0 27 / 91

44. Images/cinvestav- What about the code? Partial Code p u b l i c c l a s s Graph{ // P r i v a t e s t o r a g e p r i v a t e boolean adjacencyMatrix [ ] [ ] ; p r i v a t e i n t certexCount ; // Co nstruct or p u b l i c Graph ( i n t vertexCount ) { t h i s . vertexCount = vertexCount ; adjacencyMatrix = new boolean [ vertexCount ] [ vertexCount ] ; } //Some Methods p u b l i c void addEdge ( i n t i , i n t j ) { i f ( i >= 0 && i < vertexCount && j > 0 && j < vertexCount ) { adjacencyMatrix [ i ] [ j ] = t r u e ; adjacencyMatrix [ j ] [ i ] = t r u e ; } } p u b l i c void removeEdge ( i n t i , i n t j ) { i f ( i >= 0 && i < vertexCount && j > 0 && j < vertexCount ) { adjacencyMatrix [ i ] [ j ] = f a l s e ; adjacencyMatrix [ j ] [ i ] = f a l s e ; } } } 29 / 91

45. Images/cinvestav- What about Time Complexity? Operations on a Graph Most of the basic operations in a graph are: Adding an edge – O(1) Deleting an edge – O(1) Answering the question “is there an edge between i and j” – O(1) Finding the successors of a given vertex – O(N) Finding (if exists) a path between two vertices – O(N2) 30 / 91

52. Images/cinvestav- Space Drawbacks of This Representation We need the following amount of space If you have N integers of 4 bytes each, we requiere 4 × N × N = 4N2 space Which is a killer!!! If your graph does not have an edge between any two pair of nodes So What to do? 31 / 91

55. Images/cinvestav- Possible Solutions If you have an undirected graph For an undirected graph, may store only lower or upper triangle (exclude diagonal). Space used (Assume Integers): 4 × N(N−1) 2 = 2N (N − 1) Better Use Sparse Matrix Representations 32 / 91

58. Images/cinvestav- We use the sparse Representation of Matrices!!! Example: Array of Row Chains 33 / 91

59. Images/cinvestav- We use the sparse Representation of Matrices!!! Example: Orthogonal Lists 34 / 91

60. Images/cinvestav- Code For Sparse Representation Node and Link Representations p u b l i c c l a s s Node { p r i v a t e Node next ; p r i v a t e i n t c o l ; p r i v a t e i n t Value // Node c o n s t r u c t o r p u b l i c Node (){ next . . . c o l . . . Value . . . } // Here the e x t r a // P u b l i c methods . . . } p u b l i c c l a s s L i n k e d L i s t { p r i v a t e Node head ; p r i v a t e i n t l i s t C o u n t ; // L i n k e d L i s t c o n s t r u c t o r p u b l i c L i n k e d L i s t () { head = new Node ( n u l l ) ; l i s t C o u n t = 0;} 35 / 91

61. Images/cinvestav- Code For Sparse Representation Graph Representation p u b l i c c l a s s GraphAdjListSparse { p u b l i c i n t NumNodes ; p u b l i c L i n k L i s t [ ] graph ; // c o n s t r u c t p u b l i c GraphAdjListSparse ( i n t n ) { NumNodes = n ; graph = new L i n k L i s t [ NumNodes ] ; f o r ( i n t i = 0; i < nodes ; i ++) graph [ i ] = new L i n k L i s t ( ) ; } // More Methods Here . . . 36 / 91

63. Images/cinvestav- Adjacency List Representation Deﬁnition Adjacency list for vertex i is a linear list of vertices adjacent from vertex i. Basically An array of N adjacency lists. Thus Each adjacency list is a chain. 38 / 91

66. Images/cinvestav- Adjacency List Representation For the previous example 1 2 3 4 5 6 4 4 6 4 5 1 2 2 3 5 39 / 91

67. Images/cinvestav- Properties Space for storage For undirected or directed graphs O (V + E) Search: Successful or Unsuccessful O(1 + degree(v)) In addition Adjacency lists can readily be adapted to represent weighted graphs Weight function w : E → R The weight w(u, v) of the edge (u, v) ∈ E is simply stored with vertex v in u’s adjacency list 40 / 91

71. Images/cinvestav- Possible Disadvantage When looking to see if an edge exist There is no quicker way to determine if a given edge (u,v) 41 / 91

72. Images/cinvestav- Traversing the Graph Why? Do you have any examples? Yes Search for paths satisfying various constraints Shortest Path Visit some sets of vertices Tours Search for subgraphs Isomorphisms 42 / 91

80. Images/cinvestav- Breadth-first search Definition Given a graph G = (V , E) and a source vertex s, breadth-first search systematically explores the edges of Gto “discover” every vertex that is reachable from the vertex s Something Notable A vertex is discovered the first time it is encountered during the search 44 / 91

81. Images/cinvestav- Breadth-first search Definition Given a graph G = (V , E) and a source vertex s, breadth-first search systematically explores the edges of Gto “discover” every vertex that is reachable from the vertex s Something Notable A vertex is discovered the first time it is encountered during the search 44 / 91

82. Images/cinvestav- Deﬁning Some Basic Terms Color Given a node u, we have that color is a ﬁeld indicating WHITE Never visited node GRAY Node pointer is at the Queue Q BLACK Node has been visited and processed 45 / 91

87. Images/cinvestav- Deﬁning Some Basic Terms Distance d Given a node u, we have that d is a ﬁeld indicating the distance from the node source s so far 46 / 91

88. Images/cinvestav- Deﬁning Some Basic Terms Predecesor π Given a node u, we have that π is a ﬁeld indicating who is the immediate predecessor of u in the path from s to u 47 / 91

89. Images/cinvestav- Pseudo Code BFS(G, s) 1 for each vertex u ∈ G.V − {s} 2 u.color = WHITE 3 u.d = ∞ 4 u.π = NIL 5 s.color =GRAY 6 s.d = 0 7 s.π = NIL 8 Q = ∅ 9 Enqueue(Q, s) 10 while Q = ∅ 11 u =Dequeue(Q) 12 for each v ∈ G.Adj [u] 13 if v.color ==WHITE 14 v.color =GRAY 15 v.d = u.d + 1 16 Enqueue(Q, v) 17 u.color = BLACK 48 / 91

106. Images/cinvestav- Change the Order of Recursion It is like a wave going from a node 1 2 3 4 5 6 49 / 91

107. Images/cinvestav- Example What do you see? 50 / 91

115. Images/cinvestav- Complexity What about the outer loop? O(V ) Enqueue / Dequeue operations – Each adjacency list is processed only once. What about the inner loop? The sum of the lengths of f all the adjacency lists is Θ(E) so the scanning takes O(E) 58 / 91

116. Images/cinvestav- Complexity What about the outer loop? O(V ) Enqueue / Dequeue operations – Each adjacency list is processed only once. What about the inner loop? The sum of the lengths of f all the adjacency lists is Θ(E) so the scanning takes O(E) 58 / 91

117. Images/cinvestav- Complexity Overhead of Creation O (V ) Then Total complexity O (V + E) 59 / 91

118. Images/cinvestav- Complexity Overhead of Creation O (V ) Then Total complexity O (V + E) 59 / 91

119. Images/cinvestav- Properties Something Notable Breadth-first search constructs a breadth-first tree, initially containing only its root, which is the source vertex s Thus We say that u is the predecessor or parent of v in the breadth-first tree. 60 / 91

120. Images/cinvestav- Properties Something Notable Breadth-first search constructs a breadth-first tree, initially containing only its root, which is the source vertex s Thus We say that u is the predecessor or parent of v in the breadth-first tree. 60 / 91

121. Images/cinvestav- Breadth-First Search Property Something Notable All vertices reachable from the start vertex (including the start vertex) are visited. 61 / 91

123. Images/cinvestav- Depth-ﬁrst search Given G Pick an unvisited vertex v, remember the rest. Recurse on vertices adjacent to v 63 / 91

124. Images/cinvestav- The Pseudo-code Code for DFS DFS(G) 1 for each vertex u ∈ G.V 2 u.color = WHITE 3 u.π = NIL 4 time = 0 5 for each vertex u ∈ G.V 6 if u.color = WHITE 7 DFS-VISIT(G, u) 64 / 91

131. Images/cinvestav- Now, we have the following Code for DFS-VISIT DFS-VISIT(G, u) 1 time = time + 1 2 u.d = time 3 u.color = GRAY 4 for each vertex v ∈ G.Adj [u] 5 if v.color == WHITE 6 v.π = u 7 DFS-VISIT(G, v) 8 u.color = BLACK 9 time = time + 1 10 u.f = time 65 / 91

141. Images/cinvestav- Example What do we do? 66 / 91

149. Images/cinvestav- Complexity Analysis 1 The loops on lines 1–3 and lines 5–7 of DFS take Θ (V ). 2 The procedure DFS-VISIT is called exactly once for each vertex v ∈ V . 3 During an execution of DFS-VISIT(G, v) the loop on lines 4–7 executes |Adj (v)| times. 4 But v∈V |Adj (v)| = O (E) we have that the cost of executing g lines 4–7 of DFS-VISIT is Θ (E) . Then DFS complexity is Θ (V + E) 74 / 91

154. Images/cinvestav- Applications We have several Finding a path between nodes Strongly Connected Components Spanning Trees Topological Sort - The Program (or Project) Evaluation and Review (PERT) Computer Vision Algorithms Artiﬁcial Intelligence Algorithms Importance in Social Network Rank Algorithms for Google Etc. 75 / 91

164. Images/cinvestav- Finding a path between nodes We do the following Start a breadth-ﬁrst search at vertex v. Terminate when vertex u is visited or when Q becomes empty (whichever occurs ﬁrst). Time Complexity O V 2 when adjacency matrix used. O(V + E) when adjacency lists used. 77 / 91

168. Images/cinvestav- This allow to use the Algorithm for ﬁnding The Shortest Path Clearly This is the unweighted version or all weights are equal!!! We have the following function δ (s, v)= shortest path from s to v We claim that Upon termination of BFS, every vertex v ∈ V reachable from s has distance(v) = δ(s, v) 78 / 91

172. Images/cinvestav- Connected Components Deﬁnition A connected component (or just component) of an undirected graph is a subgraph in which any two vertices are connected to each other by paths. Example 80 / 91

173. Images/cinvestav- Connected Components Deﬁnition A connected component (or just component) of an undirected graph is a subgraph in which any two vertices are connected to each other by paths. Example 80 / 91

174. Images/cinvestav- Procedure First Start a breadth-ﬁrst search at any as yet unvisited vertex of the graph. Thus Newly visited vertices (plus edges between them) deﬁne a component. Repeat Repeat until all vertices are visited. 81 / 91

177. Images/cinvestav- Time O (V 2 ) When adjacency matrix used O(V + E) When adjacency lists used (E is number of edges) 82 / 91

180. Images/cinvestav- Spanning Tree Deﬁnition A spanning tree of a graph G = (V , E) is a acyclic graph where for u, v ∈ V , there is a path between them Example 84 / 91

181. Images/cinvestav- Spanning Tree Deﬁnition A spanning tree of a graph G = (V , E) is a acyclic graph where for u, v ∈ V , there is a path between them Example 84 / 91

182. Images/cinvestav- Procedure First Start a breadth-first search at any vertex of the graph. Thus If graph is connected, the n − 1 edges used to get to unvisited vertices define a spanning tree (breadth-first spanning tree). 85 / 91

183. Images/cinvestav- Procedure First Start a breadth-first search at any vertex of the graph. Thus If graph is connected, the n − 1 edges used to get to unvisited vertices define a spanning tree (breadth-first spanning tree). 85 / 91

187. Images/cinvestav- Topological Sorting Deﬁnitions A topological sort (sometimes abbreviated topsort or toposort) or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge (u, v) from vertex u to vertex y, u comes before v in the ordering. From Industrial Engineering The canonical application of topological sorting (topological order) is in scheduling a sequence of jobs or tasks based on their dependencies. Topological sorting algorithms were ﬁrst studied in the early 1960s in the context of the PERT technique for scheduling in project management (Jarnagin 1960). 88 / 91

188. Images/cinvestav- Topological Sorting Deﬁnitions A topological sort (sometimes abbreviated topsort or toposort) or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge (u, v) from vertex u to vertex y, u comes before v in the ordering. From Industrial Engineering The canonical application of topological sorting (topological order) is in scheduling a sequence of jobs or tasks based on their dependencies. Topological sorting algorithms were ﬁrst studied in the early 1960s in the context of the PERT technique for scheduling in project management (Jarnagin 1960). 88 / 91

189. Images/cinvestav- Then We have that The jobs are represented by vertices, and there is an edge from x to y if job x must be completed before job y can be started. Example When washing clothes, the washing machine must ﬁnish before we put the clothes to dry. Then A topological sort gives an order in which to perform the jobs. 89 / 91

192. Images/cinvestav- Algorithm Pseudo Code TOPOLOGICAL-SORT 1 Call DFS(G) to compute ﬁnishing times v.f for each vertex v. 2 As each vertex is ﬁnished, insert it onto the front of a linked list 3 Return the linked list of vertices 90 / 91

196. Images/cinvestav- Example Dressing 91 / 91

Preparation Data Structures 11 graphs

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