3.1 Basic Definitions and Applications

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1 Graphs hapter hapter Graphs. Basic efinitions and Applications Graph. G = (V, ) n V = nodes. n = edges between pairs of nodes. n aptures pairwise relationship between objects: Undirected graph represents symmetric relation irected graph represents general binary relation n Graph size parameters: n = V, m =. n Simple: no loops and no multiple edges V = {,,,,,,, } = { -, -, -, -, -, -, -, -, -, - } n = m = World Wide Web cological Food Web Some Graph Applications Web graph. Food web graph. n Node: web page. n Node = species. Graph Nodes dges n dge: hyperlink from one page to another. n dge = from prey to predator. transportation street intersections highways communication computers fiber optic cables World Wide Web web pages hyperlinks cnn.com social people relationships food web species predator-prey software systems functions function calls netscape.com novell.com cnnsi.com timewarner.com scheduling tasks precedence constraints circuits gates wires hbo.com sorpranos.com Reference:

2 Graph Representation: Adjacency Matrix Graph Representation: Adjacency List Paths and onnectivity Adjacency matrix. n-by-n matrix with A uv = if (u, v) is an edge. n Two s of each edge for undirected graph. n Space proportional to n. n hecking if (u, v) is an edge takes () time. n Identifying all edges takes (n ) time. Adjacency list. Node indexed array of lists. n Two representations of each edge for undirected graphs. n Space proportional to m + n. degree = number of neighbors of u n hecking if (u, v) is an edge takes O(deg(u)) time. n Identifying all edges takes (m + n) time. ef. A path in a graph G = (V, ) is a sequence P of nodes v, v,, v k-, v k with the property that each consecutive pair (v i, v i+ ) is an edge in. ef. A path is simple if all nodes are distinct. ef. An undirected graph is connected if for every pair of nodes u and v, there is a path from u to v. ef. A directed graph is strongly connected if for every pair of nodes u and v, there is a path from u to v. ycles Trees Rooted Trees ef. A cycle is a path v, v,, v k-, v k in which v = v k, k >, and the first k- nodes are all distinct. ef. An undirected graph is a tree if it is connected and does not contain a cycle. Theorem. Let G be an undirected graph on n nodes. Any two of the following statements imply the third. n G is connected. n G does not contain a cycle. n G has n- edges. Rooted tree. A directed graph where its underlying undirected graph is a tree and there is a node r called root and each edge points away from r. Importance. Models hierarchical structure. root r parent of v v cycle = child of v a tree the same tree, rooted at

3 onnectivity Breadth First Search. Graph Traversal s-t connectivity problem. Given two node s and t, is there a path between s and t? BFS intuition. xplore outward from s in all possible directions, adding nodes one "layer" at a time. s-t shortest path problem. Given two node s and t, what is the length of the shortest path between s and t? Applications. n Maze traversal. n Fewest number of hops in a communication network. BFS algorithm. n L = { s }. s L L L n- n L = all neighbors of L. n L = all nodes that do not belong to L or L, and that have an edge to a node in L. n L i+ = all nodes that do not belong to an earlier layer, and that have an edge to a node in L i. Theorem. For each i, L i consists of all nodes at distance exactly i from s. There is a path from s to t iff t appears in some layer. Breadth First Search Breadth First Search: Analysis onnected omponent Property. Let T be a BFS tree of G = (V, ), and let (x, y) be an edge of G. Then the level of x and y differ by at most. Theorem. The above implementation of BFS runs in O(m + n) time if the graph is given by its adjacency representation. onnected component. Find all nodes reachable from s. Pf. n Runs in O(m + n) time: when we consider node u, there are deg(u) incident edges (u, v) total time processing edges is u V deg(u) = m each edge (u, v) is counted exactly twice in sum: once in deg(u) and once in deg(v) L L onnected component containing node = {,,,,,,, }. L L

4 onnected omponent Breadth-First Search xample Breadth-First Search xample onnected component. Find all nodes reachable from s. s R u it's safe to add v v Theorem. Upon termination, R is the connected component containing s. n BFS = explore in order of distance from s. Start search at vertex. Visit/mark/label start vertex and put in a FIFO queue. Breadth-First Search xample Breadth-First Search xample Breadth-First Search xample Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices;

5 Breadth-First Search xample Breadth-First Search xample Breadth-First Search xample Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices; Breadth-First Search xample Breadth-First Search xample Breadth-First Search xample Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices;

6 Breadth-First Search xample Breadth-First Search xample Breadth-First Search xample Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices; Breadth-First Search xample Breadth-First Search xample Breadth-First Search xample Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices;

7 Breadth-First Search xample Breadth-First Search xample Breadth-First Search xample Remove from Q; visit adjacent unvisited vertices; Remove from Q; visit adjacent unvisited vertices; Queue is empty. Search terminates. Time omplexity onnected omponents Spanning Tree ach visited vertex is put on (and so removed from) the queue exactly once. When a vertex is removed from the queue, we examine its adjacent vertices. O(n) if adjacency matrix used O(vertex degree) if adjacency lists used Total time O(kn), where k is number of vertices in the component that is searched (adjacency matrix) O(k+m), where m is the number of edges in the component (adjacency list). Start a breadth-first search at any as yet unvisited vertex of the graph. Newly visited vertices (plus edges between them) define a component. Repeat until all vertices are visited. O(n ) when adjacency matrix used O(n+m) when adjacency lists used (m is number of edges) A subgraph of a connected graph that contains every vertex and is a tree. Breadth-first search from vertex. Breadth-first spanning tree.

8 Spanning Tree epth-first Search for Undirected Graph epth-first Search xample Start a breadth-first search at any vertex of the graph. If graph is connected, the n- edges used to get to unvisited vertices define a spanning tree (breadth-first spanning tree). All visited vertices have the shortest steps to the start vertex in this spanning tree. Time depthfirstsearch(v) { Mark vertex v as reached. for (each vertex u adjacenct from v) if (unreached(u)) depthfirstsearch(u); // (v, u) is a tree edge. // else either (u, v) is a true edge or (v, u) is a back edge. } O(n ) when adjacency matrix used O(n+m) when adjacency lists used (m is number of edges) Start search at vertex. Label vertex and do a depth first search from either or. Suppose that vertex is selected. epth-first Search xample epth-first Search xample epth-first Search xample Label vertex and do a depth first search from either,, or. Suppose that vertex is selected. Label vertex and do a depth first search from either,, or. Suppose that vertex is selected. Label vertex and do a depth first search from either or. Suppose that vertex is selected.

9 epth-first Search xample epth-first Search xample epth-first Search xample Label vertex and return to vertex. From vertex do a dfs(). Label vertex and do a depth first search from either or. Suppose that vertex is selected. Label vertex and return to. From vertex do a dfs(). epth-first Search xample epth-first Search xample epth-first Search xample Label vertex and return to. Return to. Return to. o a dfs().

10 epth-first Search xample epth-first Search xample epth-first Search xample Label and return to. Return to. Return to. Return to the invoking method. epth-first Search Properties Bipartite Graphs Same complexity as BFS. Same properties with respect to path finding, connected components, and spanning trees. dges used to reach unlabeled vertices define a depth-first spanning tree when the graph is connected. There are problems for which BFS is better than FS and vice versa.. Testing Bipartiteness ef. An undirected graph G = (V, ) is bipartite if the nodes can be colored red or blue such that every edge has one red and one blue end. Applications. n Stable marriage: men = red, women = blue. n Scheduling: machines = red, jobs = blue. All edges are classified as either tree edges or back edges. Theorem: For any back edge (v, u), (u, v) lies in a cycle. a bipartite graph

11 v v v v v Testing Bipartiteness An Obstruction to Bipartiteness Obstruction to Bipartiteness Testing bipartiteness. Given a graph G, is it bipartite? n Many graph problems become: easier if the underlying graph is bipartite (matching) tractable if the underlying graph is bipartite (independent set) n Before attempting to design an algorithm, we need to understand structure of bipartite graphs. Lemma. If a graph G is bipartite, it cannot contain an odd length cycle. Pf. Not possible to -color the odd cycle, let alone G. orollary. A graph G is bipartite iff it contain no odd length cycle. v -cycle v v v v v v v v bipartite (-colorable) not bipartite (not -colorable) bipartite (-colorable) not bipartite (not -colorable) a bipartite graph G another drawing of G Testing Bipartiteness using epth-first Search Graph Search depthfirstsearch(v) { Mark vertex v as reached. for (each vertex u adjacent from v) if (unreached(u)) depthfirstsearch(u); // (v, u) is a tree edge. // else either (u, v) is a tree edge or (v, u) is a back edge. } boolean bipartitefs(v, color) { Mark vertex v as color color = oppositeolor(color) // white = oppositeolor(black), for (each vertex u adjacent from v) if (unmarked(u)) // (v, u) is a tree edge. if (! bipartitefs(u, color)) return false; else if (color(v) == color(u)) // (v, u) is a back edge. return false return true }. onnectivity in irected Graphs irected reachability. Given a node s, find all nodes reachable from s. irected s-t shortest path problem. Given two node s and t, what is the length of the shortest path between s and t? Graph search. BFS extends naturally to directed graphs. Web crawler. Start from web page s. Find all web pages linked from s, either directly or indirectly.

12 Strong onnectivity ef. Nodes u and v are mutually reachable if there is a path from u to v and also a path from v to u. ef. A graph is strongly connected if every pair of nodes is mutually reachable. Lemma. Let s be any node. G is strongly connected iff every node is reachable from s, and s is reachable from every node. Strong onnectivity: Algorithm Theorem. an determine if G is strongly connected in O(m + n) time. Pf. n Pick any node s. n Run FS from s in G. reverse orientation of every edge in G n Run FS from s in G rev. n Return true iff all nodes reached in both FS executions. n orrectness follows immediately from previous lemma. Strongly onnected omponents Any directed graph can be partitioned into a unique set of strong components. Pf. Follows from definition. Pf. Path from u to v: concatenate u-s path with s-v path. Path from v to u: concatenate v-s path with s-u path. ok if paths overlap s u strongly connected not strongly connected v Strongly onnected omponents (continued) Strongly onnected omponents (example) Strongly onnected omponents (continued) The algorithm for finding the strong components of a directed graph G uses the transpose of the graph. n The transpose G T has the same set of vertices V as graph G but a new edge set consisting of the edges of G but with the opposite direction.. xecute the depth-first search dfs() for the graph G which creates the list dfslist consisting of the vertices in G in the reverse order of their finishing times.. Generate the transpose graph G T.. Using the order of vertices in dfslist, make repeated calls to dfs() for vertices in G T. The list returned by each call is a strongly connected component of G. dfslist: [A, B,,,, G, F] Using the order of vertices in dfslist, make successive calls to dfs() for graph G T Vertex A: dfs(a) returns the list [A,, B] of vertices reachable from A in G T. Vertex : The next unvisited vertex in dfslist is. alling dfs() returns the list []. Vertex : The next unvisited vertex in dfslist is ; dfs() returns the list [, F, G] whose elements form the last strongly connected component.

13 Running Time of strongomponents() Recall that the depth-first search has running time O(V+), and the computation for G T is also O(V+). It follows that the running time for the algorithm to compute the strong components is O(V+).. AGs and Topological Ordering irected Acyclic Graphs ef. An AG is a directed graph that contains no directed cycles. x. Precedence constraints: edge (v i, v j ) means v i must precede v j. ef. A topological order of a directed graph G = (V, ) is an ordering of its nodes as v, v,, v n so that for every edge (v i, v j ) we have i < j. v v v v v v v v v v v v v v a AG a topological ordering Precedence onstraints irected Acyclic Graphs irected Acyclic Graphs Precedence constraints. dge (v i, v j ) means task v i must occur before v j. Lemma. If G has a topological order, then G is a AG. Lemma. If G has a topological order, then G is a AG. Applications. n ourse prerequisite graph: course v i must be taken before v j. n ompilation: module v i must be compiled before v j. n Pipeline of computing jobs: output of job v i needed to determine input of job v j. Topological Order: A listing of all nodes v, v,, v n, such that all edge (v i, v j ) has the property that i < j. Applications. n A study plan with course prerequisite graph. n A schedule of modules for compilation. Pf. (by contradiction) n Suppose that G has a topological order v,, v n and that G also has a directed cycle. Let's see what happens. n Let v i be the lowest-indexed node in, and let v j be the node just before v i ; thus (v j, v i ) is an edge. n By our choice of i, we have i < j. n On the other hand, since (v j, v i ) is an edge and v,, v n is a topological order, we must have j < i, a contradiction. the directed cycle v v i v j v n Q. oes every AG have a topological ordering? Q. If so, how do we compute one? the supposed topological order: v,, v n

14 irected Acyclic Graphs irected Acyclic Graphs Graph Algorithms: Topological Sort Lemma. If G is a AG, then G has a node with no incoming edges. Pf. (by contradiction) n Suppose that G is a AG and every node has at least one incoming edge. Let's see what happens. n Pick any node v, and begin following edges backward from v. Since v has at least one incoming edge (u, v) we can walk backward to u. n Then, since u has at least one incoming edge (x, u), we can walk backward to x. n Repeat until we visit a node, say w, twice. n Let denote the sequence of nodes encountered between successive visits to w. is a cycle. w x u v Lemma. If G is a AG, then G has a topological ordering. Pf. (by induction on n) n Base case: true if n =. n Given AG on n > nodes, find a node v with no incoming edges. n G - { v } is a AG, since deleting v cannot create cycles. n By inductive hypothesis, G - { v } has a topological ordering. n Place v first in topological ordering; then append nodes of G - { v } n in topological order. This is valid since v has no incoming edges. AG v A The topological sorting algorithm: Key idea: initialize and maintain a queue (or stack) holding pointers to the vertices with predecessors B F A B F B Graph Algorithms: Topological Sort Graph Algorithms: Topological Sort Graph Algorithms: Topological Sort The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. A B F A B F B B F A B F No scan is required, so O(). Output: A B A B F Output: A F

15 Graph Algorithms: Topological Sort Graph Algorithms: Topological Sort Graph Algorithms: Topological Sort The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. The topological sorting algorithm: As each vertex is removed, update the predecessor counts, and for any vertex whose count has become zero, put it in the queue. A B F Output: A F B A B F Output: A F B A B F Output: A F B Topological Sorting Algorithm: Running Time Theorem. Algorithm finds a topological order in O(m + n) time. Pf. n Maintain the following information: count[w] = remaining number of incoming edges S = set of remaining nodes with no incoming edges n Initialization: O(m + n) via single scan through graph. n Update: to delete v remove v from S decrement count[w] for all edges from v to w, and add w to S if c count[w] hits this is O() per edge