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Introduction to Data Structure

Introduction to Data Structure. CHAPTER 6 GRAPHS. 6.1 The Graph Abstract Data Type 6.2 Elementary Graph Operations 6.3 Minimum Cost Spanning Trees 6.4 Shortest Paths and Transitive Closure 6.5 Activity Networks. Contents. Chapter 1 Basic Concepts Chapter 2 Arrays

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Introduction to Data Structure

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  1. Introduction to Data Structure CHAPTER 6 GRAPHS 6.1 The Graph Abstract Data Type 6.2 Elementary Graph Operations 6.3 Minimum Cost Spanning Trees 6.4 Shortest Paths and Transitive Closure 6.5 Activity Networks Chapter 6 Graphs

  2. Contents Chapter 1 Basic Concepts Chapter 2 Arrays Chapter 3 Stacks and Queues Chapter 4 Linked Lists Chapter 5 Trees Chapter 6 Graph Chapter 7 Sorting Chapter 8 Hashing Chapter 9 Heap Structures Chapter 10 Search Structures Chapter 6 Graphs

  3. 6.1 The Graph Abstract Data Type p. 263 structureGraph is objects: a nonempty set of vertices and a set of undirected edges, where each edge is a pair of vertices functions: for all graphGraph, v, v1, and v2Vertices Graph Create() ::= return an empty graph Graph InsertVertex(graph, v) ::= return a graph with v inserted. Graph InsertEdge(graph, v1, v2) ::= return a graph with a new edge (v1, v2). Graph DeleteVertex(graph, v) ::= return a graph with v and all its incident edges removed. Graph DeleteEdge(graph, v1, v2) ::= return a graph with the edge (v1, v2) removed. Boolean IsEmpty(graph) ::= if (graph==empty) return TRUE else return FALSE List Adjacent(graph, v) ::= return a list of all vertices adjacent to v. Chapter 6 Graphs

  4. C g c d e A D b a B f 6.1.1 Introduction • 1736, Euler Koenigberg Bridge Problem: • Euler showed: there is a walk starting at any vertex, going through each edge exactly once and terminating at the start vertex iff the degree of each vertex is even • called Eulerian walk. • the degree of a vertex: # of edges incident to a vertex • J. R. Newman: the world of Mathematics, 1956, p.573-580 attaching Chapter 6 Graphs

  5. V1 V2 1 2 3 6.1.2 Definitions • A Graph, G = (V, E), consists of two sets V and E • V: finite non-empty set of vertices • E: set of pairs of vertices, edges, e.g.(1,2) or 1,2 Note: V(G): set of vertices of graph G E(G): set of edges of graph G • Undirected Graph – the pair of vertices representing any edge is unordered. Thus, the pairs (V1,V2) and (V2,V1) represent the same edge. • Directed Graph –each edge is represented by a directed pairs V1, V2 Note: V1, V2 and V2, V1 represent two different edges. V(G) = {1, 2, 3} E(G) = {1,2 2,3 3,1} Chapter 6 Graphs

  6. 0 0 0 1 2 1 2 1 3 3 6 5 4 2 G3 Examples for Graph • Examples for Graph G2 G1 V(G1)={0,1,2,3} E(G1)={(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)} V(G2)={0,1,2,3,4,5,6} E(G2)={(0,1),(0,2),(1,3),(1,4),(2,5),(2,6)} V(G3)={0,1,2} E(G3)={0,1, 1,0, 1,2} • Note: Graph G2 is also a tree,  Tree is a special case of graph. Chapter 6 Graphs

  7. 0 0 2 1 3 1 2 Examples for Graphlike Structure (a) (b) Graph with a self edge Multigraph: multiple occurrences of the same edge Not consider as a graph in this book Chapter 6 Graphs

  8. 0 1 2 3 Complete Graph • A Complete Graph is a graph that has the maximum number of edges • For undirected graph with n vertices, the maximum number of edges is n(n-1)/2 • For directed graph with n vertices, the maximum number of edges is n(n-1) • Example: G1 is a complete graph n(n-1)/2 = 6 Chapter 6 Graphs

  9. 0 1 2 G3 0 1 2 3 Adjacent and Incident • If (v0, v1) is an edge in an undirected graph, • Adjacent: v0 and v1 are adjacent • Incident: The edge (v0, v1) is incident on vertices v0 and v1 • If <v0, v1> is an edge in a directed graph • Adjacent: v0 is adjacent to v1, and v1 is adjacent from v0 • Incident: The edge <v0, v1> is incident on v0 and v1 The vertices adjacent to vertex 2: 0, 1 and 2 The edge incident on vertex 2: (0,2), (1,2) and (2,3) The vertices adjacent to vertex 1: 0, The edge incident on vertex 1: <0,1>, <1,0> and <1,2> Chapter 6 Graphs

  10. 0 1 2 0 3 1 2 1 2 3 0 (i) (ii) (iii) (iv) G1 0 1 2 3 Subgraph • Subgraph • A subgraph of G is a graph G’ → • V(G’)  V(G) • E(G’)  E(G) • e.g. some of the subgraphs of G1: Chapter 6 Graphs

  11. 0 0 0 0 1 1 1 1 0 2 2 2 (i) (ii) (iii) (iv) G3 Subgraph (cont.) • e.g. some of the subgraphs of G3: • If G is connected with n vertices then its connected subgraph at least has n-1 edges. • Why? Chapter 6 Graphs

  12. 1 • 1,2,4 is a path • 1,2,3 is not a path • 1,2,4,2 is a path 2 3 4 Path • Path: from vp to vq in G is a sequence of vertices vp, vi1, vi2,...,vin, vp  (vp, vi1), (vi1, vi2),..., (vin, vq)  E(G) if G is undirectedor vp, vi1, vi1, vi2,..., vin, vq  E(G) if G is directed • eg. • Length: The length of a path is the number of edges on it eg. Length of path 1,2,4 is 2 Chapter 6 Graphs

  13. 1 2 3 Simple Path • Simple Path: is a path in which all vertices except possibly the first and last are distinct. • e.g.1. 1,2,4 is a simple path, path: 1,2,4,2 is not a simple path. • e.g.2. 1 • 1,2,4 is a path • 1,2,3 is not a path • 1,2,4,2 is a path 2 3 4 1,2,3,1 also a simple path with length 3. 1,2,3,1,2? path, length 4, not simple. 1,2,3,1,3 not a path. Chapter 6 Graphs

  14. 0 1 2 3 6 5 4 1 2 3 Cycle • Cycle: is a simple path, first and last vertices are same. • eg. 1,2,3,1. • Note: Directed cycle, Directed path. • Acyclic graph G2 tree (acyclic graph) Chapter 6 Graphs

  15. 1 2 3 不是 strongly connected 4 strongly connected connected connected Connected • Connected • Two vertices V1 and V2 are Connected: if in an undirected graph G,  a path in G from V1 to V2(or from V2 to V1 ∵undirected) • Strongly Connected • V1, V2 are Strongly Connected: if in a directed graph (digraph) G’,  a path in G from V1 to V2 and also from V2 to V1 • Graph (Strongly) Connected • graph connected (graph strongly connected) if  Vi, Vj  V(G),  a path from Vi to Vj in G Chapter 6 Graphs

  16. Connected Component • Connected Component(Strongly Connected Component): is a maximal connected subgraph. eg. for graph for digraph 1 1 3 2 3 G1 2 4 4 Two connected components of G1 1 1 G2 2 3 2 3 Two strongly connected components of G2 Chapter 6 Graphs

  17. in-degree out-degree in-degree=1 out-degree=2 degree=3 Degree • Degree of Vertex • is the # of edges incident to that vertex eg. • In Direct Graph • If G has n vertices and e edge, di = degree of Vi, then degree of V3 = 3 3 Chapter 6 Graphs

  18. Network • Network • A graph with weighted edges is called. • eg. 台北 90 120 新竹 花蓮 Diagraph G = (V,E) Network N = (G,W) is a diagraph G together with a real valued for w: E  R. R is a real number w(u,v) is the weight of edge (arc) (u,v)  E. 150 100 台中 210 高雄 Chapter 6 Graphs

  19. 6.1.3 Graph Representations • Method 1: Adjacency Matrix • Method 2: Adjacency Lists • Method 3: Adjacency Multilists Chapter 6 Graphs

  20. eg. 1 2 3 4 degree = di A 1 • symmetric • space need n2 bits, but half can be saved 1 0 1 1 0 2 2 3 2 1 0 0 0 1 4 3 1 0 0 1 2 6 / 2 = 3 ... edges 4 0 0 1 0 1 6.1.3.1 Adjacency Matrix • Adjacency Matrix • Let G = (V, E) with n vertices, n 1.The adjacency matrix of G is a 2-dimensional n  n matrix, A A(i, j) = 1 iff (vi, vj) ( vi, vj for a diagraph)  E(G), A(i, j) = 0 otherwise. Chapter 6 Graphs

  21. 0 1 2 0 • not be symmetric 0 0 1 0 1 1 1 1 0 1 2 2 2 0 0 0 0 out-degree = = dj 1 1 1 in-degree Adjacency Matrix: Examples • Eg. 2. Chapter 6 Graphs

  22. 4 5 0 1 2 6 3 7 Adjacency Matrix: Examples • Examples for Adjacency Matrix G4 Chapter 6 Graphs

  23. Adjacency Matrix: Determine Edges • Questions: • How many edges e are there in G? • Determine the # of edges in G? • Is G connected? • Answers: • Need check entries of the matrix. check n2 – n time. spend O(n2) time to determine e • Note: << n2/2 (n: Diagonal entries are zero, need not check it) e = 7 << n2/2 = 32 Chapter 6 Graphs

  24. 6.1.3.2 Adjacency Lists List Orthogonal list • Adjacency Lists • Adjacency matrices for G3 • Adjacency list: edge e  2e nodes. • Easy random access for any particular vertex 0 G3 1 2 Head Nodes(sequential)   Chapter 6 Graphs

  25. [0] [1] [2] [3] 1 2 3 0 2 3 0 0 1 3 1 2 0 1 2 3 Adjacency Lists: Example G1 • An undirected graph with n vertices and e edges n head nodes and 2e list nodes Chapter 6 Graphs

  26. 4 [0] [1] [2] [3] [4] [5] [6] [7] 1 2 0 3 0 3 5 0 1 2 5 1 2 6 6 4 3 5 7 6 7 Adjacency Lists: Example G4 G4 Chapter 6 Graphs

  27. Adjacency Lists: Interesting Operations • Number of nodes in a graph: = the number of nodes in adjacency list • Number of edges in a graph = determined in O(n+e) • Out-degree of a vertex in a directed graph = the number of nodes in its adjacency list • In-degree of a vertex in a directed graph: ??? traverse the whole data structure 0 G3 1 2 n vertices # of edges vs. O(n2) in adjacency matrix Chapter 6 Graphs

  28. 1 NULL    [0] [1] [2] 0 NULL 0 1 NULL 1 2 Adjacency Lists: Inverse Adjacency Lists • Inverse Adjacency List  determine in-degree of a vertex in a fast way. Chapter 6 Graphs

  29. 0 0 tail head column link for head row link for tail Adjacency Lists: Orthogonal Representation • Orthogonal representation Sparse Matrix ??? head node 2 1 0 1 1 2 1 1 0 2 Chapter 6 Graphs

  30. N2 N1 N3 N4 markvertex1 vertex2 path1 path2 N5 N6 0 2 N3 N4 2 3 0 3 N5 1 2 N5 N6 0 1 N2 N4 1 3 N6 [0] N1 edge(0,1) [1] edge(0,2) N2 [2] N3 edge(0,3) [3] edge(1,2) N4 0 edge(1,3) N5 1 2 edge(2,3) N6 3 6.1.3.3 Adjacency Multilists • Adjacency Multilists (以 edge 為主) vertex 0: N1N2N3 vertex 1: N1N4N5 vertex 2: N2N4N6 vertex 3: N3N5N6 Chapter 6 Graphs

  31. markvertex1 vertex2 Link 1 for V1 Link 2 for V2 0 1 N2 N4 2 3 1 3 N6 1 2 N5 N6 0 2 N3 N4 0 3 N5 N2 N1 N3 [0] N1 edge(0,1) N4 [1] N5 N6 edge(0,2) N2 [2] N3 edge(0,3) [3] edge(1,2) N4 0 edge(1,3) N5 1 2 edge(2,3) N6 3 Adjacency Multilists (cont.) • Node Structures vertex 0: N1N2N3 vertex 1: N1N4N5 vertex 2: N2N4N6 vertex 3: N3N5N6 Chapter 6 Graphs

  32. N2 N1 N3 N4 N5 N6 e=6  12 nodes 0 1 2 3 Adjacency MultiLists vs. Adjacency List • Adjacency Multilists (以 edge 為主) e=6  6 nodes • Adjacency List 0 1 2 3 Chapter 6 Graphs

  33. 4 5 0 1 2 6 G4 3 7 6.2 Elementary Graph Operations • TraversalGiven G = (V, E) and vertex v, find or visit all w V, such that w connects v. • Method 1: Depth First Search (DFS) preorder tree traversal • Method 2: Breadth First Search (BFS) level order tree traversal • Connected Components (Application 1 of Graph traversal) • Spanning Trees (Application 2 of Graph traversal) • Minimum Cost Spanning Tree (Application 3 of Graph traversal) • … Chapter 6 Graphs

  34. Graph Operations: Graph Traversal • Graph TraversalGiven: an undirected graph G = (V, E), a vertex v V(G)Interested in: visiting all vertices connected to v. (reachable)Approach: 1. Depth First Search (DFS). 2. Breadth First Search (BFS). stack DFS: 1,2,4,5,3,6 BFS: 1,2,3,4,5,6 1 2 3 Queue 6 4 5 Chapter 6 Graphs

  35. [0] [1] [2] [3] [4] [5] [6] [7] 1 2 0 3 0 4 6 0 5 2 1 1 7 6 5 4 3 1 7 2 7 7 2 7 3 4 5 6 6.2.1 Depth First Search • Example Adjacency Lists Chapter 6 Graphs

  36. [0] [1] [2] [3] [4] [5] [6] [7] 1 2 0 3 0 4 6 0 5 2 1 1 7 6 5 4 3 1 7 2 7 7 2 7 3 4 5 6 Depth First Search: Example • Example 6.1: Depth first search v0, v1, v3, v7, v4, v5, v2, v6 • Program 6.1, p.273 2 5 4 7 7 stack 3 3 1 1 0 0 Chapter 6 Graphs

  37. VISITED 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Depth First Search: Algorithm • Depth First SearchProcedure DFS(v)(* Array VISITED(n)  0 initially and VISITED(i)  1, if has visited*) VISITED(v)  1 print(v) for each vertex w adjacent to v do if VISITED(w) = 0 then call DFS(w) end end DFS Output DFS: 1,2,4,8,5,6,3,7 V1 V2 V3 V2 V4 V5 V4 V8 V8 V5 V6 V7 V5 V6 V3 V3 V7 V7 Chapter 6 Graphs

  38. Depth First Search: Time Complexity • Time complexity: Adjacency list • Time complexity: Adjacency matrix Adjacency list: O(e) e: # of edges e edges 2e nodes Adjacency matrix: O(n2) n: # of vertex Chapter 6 Graphs

  39. [0] [1] [2] [3] [4] [5] [6] [7] 1 2 0 0 3 4 6 0 5 2 1 1 7 6 5 4 3 1 7 7 2 7 2 7 3 4 5 6 6.2.2 Breadth First Search • Example 6.2: Breadth first search v0, v1,v2, v3, v4, v5, v6, v7 • Program 6:2 , p.275 Chapter 6 Graphs

  40. Breadth First Search: Algorithm • Breadth First Search procedure BFS(v) VISITED (v)  1 print (v) initialize Q with v while Q not =  do call DELETE Q(v,Q) for all vertices w adjacent to v do if VISITED (w)=0 then [ call ADDQ(w,Q); VISITED(w)  1, print(w)] end end end BFS Q visited Chapter 6 Graphs

  41. Breadth First Search: Example • eg. V1 V3 V2 V6 V7 V4 V5 Print: V1 , V2 , V3 , V4 , V5 , V6 , V7 , V8 . V8 Chapter 6 Graphs

  42. Breadth First Search: Time Complexity • Time complexity: by using Adjacency Matrix while for n n O(n2) Chapter 6 Graphs

  43. Breadth First Search: Time Complexity • Time complexity: by using Adjacency List: Total: d1 + d2 +...+ dn = O(e) where di = degree (vi)  n while for di: degree of (vi) di O(e) Chapter 6 Graphs

  44. Applications of Graph Traversal • Application 1. Finding components of a graph • Application 2. Finding a spanning tree of a connected graph • Application 3. Minimum cost spanning tree Chapter 6 Graphs

  45. VISITED 1 5 ... 1 1 1 1 2 3 6 7 1 2 3 4 ... 8 4 8 DFS: 1 2 4 3 || 5 6 7 8 || BFS: 1 2 3 4 || 5 6 7 8 || 6.2.3 Connected Components Application 1. Finding Components of a Graph Procedure COMP(G) (* determine the connected components of G *) for i  1 to n do VISITED(i)  0 for i  1 to n do if VISITED(i) = 0 then call DFS(i); print(“||”) end end COMP • Determine connected graph • Invoke DFS or BFS Chapter 6 Graphs

  46. 6.2.4 Spanning Trees • Application 2. Finding a Spanning Tree of a Connected Graph • Def. • Application 2.1– obtaining circuit equations for an electrical network • Application 2.2 – minimum cost spanning tree • Def. • Spanning TreeAny tree consisting only edges in G and including all vertices in G is called. • i.e. minimal subgraph G’ of G, such that V(G’) =V(G) and G’ is connected. Chapter 6 Graphs

  47. Spanning Trees (cont.) eg. • How to find it? • Modify BFS • Procedure BFS(v) If VISITTED(w)=0 then If VISITTED(w)=0 then 1.call ADDQ(w,Q) 1. 2.VISITTED(w)←1 2. 3.print(w) 3. 4.T = T∪{(v, w)} ...(How many?) Ans：16 (EX.) Chapter 6 Graphs

  48. Spanning Trees (cont.) • Modify Procedure DFS：同modify Procedure BFS • e.g. • Application 1– obtaining circuit equations for an electrical network BFS: 1,2,3,4 DFS: 1,2,4,3 1 1 1 2 3 2 3 2 3 4 4 4 BF spanning tree DF spanning tree Chapter 6 Graphs

  49. 0 0 2 2 1 1 DFS Spanning 6 6 5 5 4 4 3 3 0 7 7 2 1 6 5 4 3 7 DFS Spanning Tree vs. BFS Spanning Tree BFS Spanning Chapter 6 Graphs

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