Undirected Graphs. Hwansoo Han
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1 Undirected Graphs Hwansoo Han
2 Definitions Undirected graph (simply graph) G = (V, E) V : set of vertexes (vertices, nodes, points) E : set of edges (lines) An edge is an unordered pair Edge (v, w) = (w, v) v, w are adjacent each other Edge (v, w) is incident upon vertexes v and w v w 2
3 Definitions A path A sequence of vertexes: v 1, v 2,, v n s.t. (v i, v i+1 ) is an edge for 1 i < n Length of a path Number of arcs minus one on the path (e.g. n-1) The path v 1, v 2,, v n connects v 1 and v n A graph is connected, if every pair of its vertexes are 3 connected
4 Definitions Simple path All vertexes on the path are distinct Except v 1 and v n may be the same e.g. include cycle (Simple) cycle Simple path of length 3 or more that connects a vertex to itself Each of following case is not cycle v (length 0) : no edge v, v (length 1) : loop v, w, v (length 2) : move back and forth along one edge 4
5 Definitions Subgraph Let G = (V, E) be a graph A subgraph of G is a graph G = (V, E ) V is a subset of V E consists of edges (v, w) in E s.t. both v and w are in V If E consists of all edges (v, w) in E s.t. both v and w are in V, G is called an induced subgraph of G 5 Induced subgraph V = { a, b, c }
6 Definitions Connected Component of a graph G Maximal connected induced subgraph i.e. a connected induced subgraph that is not a proper subgraph of any other connected subgraph of G Example An unconnected graph with two connected components 6
7 Definitions Free tree A connected, acyclic graph A free tree can be made into an ordinary tree If we pick any vertex we wish as the root, and Orient each edge from the root Every free tree with n 1 vertexes contains exactly n-1 edges If we add any edge to a free tree, we get a cycle 7
8 Representations Adjacency matrix Matrix is symmetric (v, w) and (w, v) are both marked Adjacency list (v, w) and (w, v) are both listed Adjacency matrix Adjacency list 8
9 Minimum-Cost Spanning Trees Spanning tree of G = (V, E) A free tree that connects all the vertexes in V Minimum-cost spanning tree (MST) Weighted graph Each edge (v, w) has a cost c(v, w) attached to it Cost of a spanning tree is sum of costs of edges in tree MST property 9 Let G = (V, E) be a connected graph Let U be a proper subset of V If (u, v) is an edge of lowest cost s.t. u U and v V-U, there is a minimum-cost spanning tree that includes (u, v) as an edge
10 Minimum-Cost Spanning Trees - example 10
11 MST Algorithm Prim s algorithm Prim s algorithm Suppose V = { 1, 2,, n}, start with U = { 1 } Grows a spanning tree, one edge at a time Finds a shortest edge (u, v) that connects U and V-U, then adds v to U Repeat the above step until U = V History V. Jarnik (Czech mathematican) developed first in 1930 C. Prim independently developed later in 1957 E. Dijkstra rediscovered in
12 Prim s Algorithm Two arrays CLOSEST[i] Vertex in U that is closest to vertex i in V-U LOWCOST[i] Cost of edge (i, CLOSEST[i]) Time complexity : O(n 2 ) 12
13 Prim s Algorithm - example 13
14 MST Algorithm Kruskal s algorithm Kruskal s algorithm Suppose V = { 1, 2,, n}, starts with T = { V, } Build progressively larger components Examine edges from E in order of increasing cost If the edge connect two different connected components, add the edge to T Otherwise, discard the edge Repeat until all vertexes are in one components History J. Kruskal published the algorithm in 1956 Similar algorithm published by O. Boruvka in 1926 Rediscovered by Choquet in 1938, by Florek et al. in 1951, and again by Sollin in
15 Kruskal s Algorithm Priority queue for edges Select minimum cost edge from Partially ordered tree MFSet for components Find root and compare Iterate for each edge Time complexity: O(eloge) 15
16 Kruskal s Algorithm - example 16
17 Graph Traversal Graph problems frequently require to visit all vertexes Need to visit the vertexes of a graph systematically Depth-first search (DFS) Breadth-first search (BFS) DFS and BFS visit all connected vertexes to a given vertex 17
18 Depth-First Search DFS for undirected graphs The same algorithm dfs() for digraphs can be used Depth-first spanning forest dfs_visit() For undirected graphs, two kinds of edges tree edges and back edges Procedure dfs_visit (G: graph ); 18
19 Depth-First Search - example Undirected graph Depth-first search Tree edges Back edges 19
20 Breadth-First Search Search as broadly as possible by next visiting all the adjacent vertexes to the current vertex BFS also build a spanning tree Tree edge (x, y), if vertex y is first visited from x Other edges are all cross edges Breadth-first spanning forest Call bfs() for each unvisited vertex 20
21 Bread-First Search - example Undirected graph Breadth-first search Tree edges Cross edges 21
22 Breadth-First Search Use a QUEUE to visit in BFS 22
23 DFS and BFS Connected components The trees in DFSF and BFSF are connected components Test for cycles If G = (V, E) is V = n and E n, it must have a cycle Otherwise, it could have a cycle A back edge in DFS completes a cycle A cross edge in BFS completes a cycle with a common ancestor on BFS tree 23
24 Articulation Points An articulation point of a graph If remove a vertex v and all edges incident upon v, we break a connected component of the graph into two or more pieces. a and c are articulation points 24
25 Biconnected Components A graph is biconnected A connected graph with no articulation points Example of application in connectivity of graphs In a communication network, if a graph representing the network is biconnected, it can survive the failure of one node The higher the connectivity of a graph, the more likely the graph is to survive the failure of some of its vertexes 25
26 Finding Articulation Points Find all articulation points of a connected graph 1. Perform DFS, computing dfnumber[v] dfnumber[v] is the order of first visit (mark) the vertex v i.e. a preorder traversal of the depth-first spanning tree 2. For each vertex v, compute low[v] low[v] is the MIN(dfnumber[w]), for all w reachable from v by following down zero or more tree edges and one back edge low[v] = MIN (dfnumber[v], dfnumber[w]), where (u,w) is a back edge for some descendant u of v Algorithm to compute low[v] : postorder traversal, taking the minimum of - dfnumber[v] - dfnumber[z] for any vertex z, for which there is a back edge (v, z) - low[y] for any child y of v 26
27 Finding Articulation Points (cont d) 3. Now, find the articulation points as follows The root is an articulation points, if and only if it has two or more children Since there are no cross edges, deletion of the root disconnects the subtrees rooted at its children A vertex v other than the root is an articulation point, if and only if there is some child w of v such that low[w] dfnumber[v] Vertex v disconnects w and its descendants If low[w] < dfnumber[v], then there must be a way to get from w down the tree and back to a proper ancestor of v 27
28 Finding Articulation Points - example Articulation points a and c 28
29 Graph Matching bipartite graphs Matching problems on graphs E.g. Given a set of teachers and a set of courses, A teacher is qualified to teach certain courses Wish to assign a course to a qualified teacher Find course assign Graph representation Two sets of vertexes, V 1 and V 2 (two sets are disjoint) Each edge (v, w) having one in each set (v V 1, w V 2 ) Such graph is called bipartite 29
30 Bipartite Graphs - example Matching on bipartite graphs 30
31 Graph Matching Matching in general term For a graph G = (V, E), a subset of edges in E with no two edges incident upon the same vertex in V is called a matching Maximal matching problem Find a maximum subset of edges for matching Complete matching Every vertex is an endpoint of some edge in the matching Complete matching is a maximal matching 31
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