Introduction to Algorithms A graph G =(V, E) V = set of vertices E = set of edges

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1 Introuction to Algorithms A graph G =(V, E) V = set of vertices E = set of eges In an unirecte graph: ege(u, v) = ege(v, u) Chapter 22: Elementary Graph Algorithms In a irecte graph: ege(u,v) goes from u to v, notate u v ege(u, v) is not the same as ege(v, u) 2 B A D C B A D C Ajacent vertices: connecte by an ege Vertex v is ajacent to u if an only if (u, v) E. In an unirecte graph with ege (u, v), an hence (v, u), v is ajacent to u an u is ajacent to v. v a b a b Directe graph: Unirecte graph: c c V = {A, B, B C, C D} E = {(A,B), (A,C), (A,D), (C,B)} V = {A, B, B C, C D} E = {(A,B), (A,C), (A,D), (C,B), (B,A), (C,A), (D,A), (B,C)} Vertex a is ajacent to c an c is ajacent to a e Vertex c is ajacent to a, but a is NOT ajacent to c e 3 4

A Path in a graph from u to v is a sequence of eges between vertices w 0, w 1,, w k, such that (w, i w i+1 ) E, E u = w 0 an v = w k, for 0 i < k The length of the path is k, the number of eges on

5 Loops If the graph contains an ege (v, v) from a to itself, then the path v, v is sometimes referre to as a loop. a b c e The graphs we will consier will generally be loopless.

2 A Path in a graph from u to v is a sequence of eges between vertices w 0, w 1,, w k, such that (w, i w i+1 ) E, E u = w 0 an v = w k, for 0 i < k The length of the path is k, the number of eges on the path a b a b c c e e abece is a path. ace is a path. ceb is a path. abec is NOT a path. bca is NOT a path. 5 Loops If the graph contains an ege (v, v) from a to itself, then the path v, v is sometimes referre to as a loop. a b c e The graphs we will consier will generally be loopless. A simple path is a path such that all vertices are istinct, except that the first an last coul be the same. a c b e abec is a simple path. cec is a simple path. abece is NOT a simple path. simple path: no repeate vertices Cycles A cycle in a irecte graph is a path of length at least 2 such that the first on the path is the same as the last one; if the path is simple, then the cycle is a simple cycle. a c b e abea is a simple cycle. abecea is a cycle, but is NOT a simple cycle. abec is NOT a cycle. A cycle in a unirecte graph A path of length at least 3 such hthat tthe first on the path is the same as the last one. The eges on the path are istinct. a c b e aba is NOT a cycle. abecea is NOT a cycle. abecea is a cycle, but NOT simple. abea is a simple cycle. 7 8

A complete graph is a graph with an ege between every pair of vertices. A graph is calle complete graph if every is ajacent to every other.

3 If each ege in the graph carries a value, then the graph is calle weighte graph. A weighte graph is a graph G = (V, E, W), where each ege, e E is assigne a real value weight, W(e). A complete graph is a graph with an ege between every pair of vertices. A graph is calle complete graph if every is ajacent to every other. Complete Unirecte Graph has all possible eges n = 1 n = 2 n = 3 n = 4 10 connecte graph: any two vertices are connecte by some path An unirecte graph is connecte if, for every pair of vertices u an v there is a path from u to v. tree - connecte graph without cycles forest - collection of trees 12

4 En vertices (or enpoints) of an ege a U an V are the enpoints of a Eges incient on a V a,, an b are incient on V V Ajacent vertices U an V are ajacent U X Degree of a X X has egree 5 a b c e W g h i Z j Parallel eges h an i are parallel l eges f Y Self-loop j is a self-loop Path sequence of alternating vertices an eges begins with a ens with a Simple path P 1 path such that all its vertices U X an eges are istinct. P 2 Examples c e P 1 = (V, X, Z) is a simple path. P 2 = (U, W, X, Y, W, V) is a W g path that is not simple. a V f b Y h Z Cycle circular sequence of alternating vertices an eges Simple cycle V cycle such that all its vertices a b an eges are istinct U X Examples C 2 h e C 1 = (V,X,Y,W,U,V)isa V) c C 1 simple cycle W g C 2 = (U, W, X, Y, W, V, U) is a cycle that is not simple f Y Z 15 1

5 In-Degree of a Vertex in-egree is number of incoming eges inegree(2) = 1, inegree(8) = 0 Out-Degree of a Vertex out-egree is number of outboun eges outegree(2) = 1, outegree(8) = Applications: Communication Network = city, ege = communication link Driving Distance/Time Map e = city, cy, ege weight = istance/time

6 Street Map Some streets are one way A biirectional link represente by 2 irecte ege (5, 9) (9, 5) Computer Networks Electronic c circuits cu Printe circuit boar cslab1a cslab1b math.brown.eu Computer networks Local area network Internet Web cs.brown.eu brown.eu att.net cox.net John qwest.net 7 Paul Davi Graphs We will typically y express running times in terms of V = number of vertices, an E = number of eges If E V 2 the graph is ense If E V the graph is sparse If you know you are ealing with ense or sparse graphs, ifferent ata structures may make sense Graph Search Methos Many graph problems solve using a search metho Path from one to another Is the graph connecte? etc. Commonly use search methos: Breath-first search Depth-first search 23 24

7 Graph Search Methos A u is reachable from v iff there is a path from v to u. u A search metho starts at a given v an visits every that is reachable from v Breath-First Search Visit start (s) an put into a FIFO queue. Repeately remove a from the queue, visit its unvisite ajacent vertices, put newly visite vertices into the queue. All vertices reachable from the start (s) (incluing the start ) are visite Breath-First Search Again will associate colors to guie the algorithm White vertices have not been iscovere All vertices start out white Green vertices are iscovere but not fully explore They may be ajacent to white vertices Black vertices are iscovere anully explore They are ajacent only to black an green vertices Explore vertices by scanning ajacency list of green vertices Breath-First Search BFS(G, s) { // initialize vertices; 1 for each u V(G) {s}{ 2 o color[u] = WHITE 3 [u] = // istance from s to u 4 p[u] = NIL // preecessor or parent of u } 5 color[s] = GREEN [s] = 0 7 p[s] = NIL 8 Q = Empty; 9 Enqueue (Q,s); // Q is a queue; initialize to s 10 while (Q not empty) { 11 u = Dequeue(Q); 12 for each v aj[u] { 13 if (color[v] == WHITE) 14 color[v] = GREEN; 15 [v] = [u] + 1; What oes [v] represent? 1 p[v] = u; What oes p[v] represent? 17 Enqueue(Q, v); } 18 color[u] = BLACK; } } 27 28

8 Breath-First Search Lines 1-4 paint every white, set [u] to be infinity for each (u), an set p[u] the parent of every to be NIL. Line 5 paints the (s) green. Line initializes [s] to 0. Line 7 sets the parent of the to be NIL. Lines 8-9 initialize Q to the queue containing just the (s). The while loop of lines iterates t as long as there remain green vertices, which are iscovere vertices that have not yet ha their ajacency lists fully examine. This while loop maintains the test in line 10, the queue Q consists of the set of the green vertices. Breath-First Search Prior to the first iteration in line 10, the only green, an the only in Q, is the (s). Line 11 etermines the green (u) at the hea of the queue Q an removes it from Q. The for loop of lines consiers each (v) in the ajacency list of (u). If (v) is white, then it has not yet been iscovere, an the algorithm iscovers it by executing lines It is first greene, an its istance [v] is set to [u]+1. Then, u is recore as its parent. Finally, it is place at the tail of the queue Q. When all the vertices on (u s) ajacency list have been examine, u is blackene in line Breath-First Search: Example Breath-First Search: Example 0 Q: s 31 32

9 Breath-First Search: Example 1 0 Breath-First Search: Example Q: w r Q: r t x Breath-First Search: Example Breath-First Search: Example Q: t x v Q: x v u 35 3

10 Breath-First Search: Example Breath-First Search: Example Q: v u y Q: u y Breath-First Search: Example Breath-First Search: Example Q: y Q: Ø 39 40

11 Depth-First Search Depth-first search is another strategy for exploring a graph Explore eeper in the graph whenever possible Eges are explore out of the most recently iscovere v that still has unexplore eges When all of v s eges have been explore, backtrack to the from which v was iscovere Depth-First Search Initialize color all vertices white Visit each an every white using DFS- Visiti Each call to DFS-Visit(u) roots a new tree of the epth-first t forest at u A is white if it is uniscovere A is green if it has been iscovere but not all of its eges have been iscovere A is black after all of its ajacent vertices have been iscovere (the aj. list was examine completely) l Discovery time Finishing i time

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Outlines: Graphs Part-2

Elementary Graph Algorithms PART-2 1 Outlines: Graphs Part-2 Graph Search Methods Breadth-First Search (BFS): BFS Algorithm BFS Example BFS Time Complexity Output of BFS: Shortest Path Breath-First Tree