Graphs. Shortest Path and Topological Sort

Transcription

1 Graphs Shortest Path ad Topological Sort

2 Example Relatioal Networks School Friedship Network (from Moody 2001) Yeast Metabolic Network (from Terrorist Network (by Valdis Krebs, Orget.com) Protei-Protei Iteractios (by Peter Uetz) 2

3 More Relatioal Networks Flickr Social Network (from gustavog/sets/164006/) Campaig Cotributios from Oil Compaies (from Geomic Associatios (from Sel et al., 2002) Seagrass Food Web (geerated at 3

4 Basic Graph Defiitios A graph G = (V,E) cosists of a fiite set of vertices, V, ad a fiite set of edges, E. Each edge is a pair (v,w) where v, w Î V. q V ad E are sets, so each vertex v Î V is uique, ad each edge e Î E is uique. q Edges are sometimes called arcs or lies. q Vertices are sometimes called odes or poits. 4

5 Graph Applicatios Graphs ca be used to model a wide rage of applicatios icludig Itersectios ad streets withi a city Roads/trais/airlie routes coectig cities/coutries Computer etworks Electroic circuits 5

6 Basic Graph Defiitios (2) A directed graph is a graph i which the edges are ordered pairs. That is, (u,v) ¹ (v,u), u, v Î V. Directed graphs are sometimes called digraphs. A udirected graph is a graph i which the edges are uordered pairs. That is, (u,v) = (v,u). A sparse graph is oe with few edges. That is E = O( V ) A dese graph is oe with may edges. That is E = O( V 2 ) 6

7 Udirected Graph All edges are two-way. Edges are uordered pairs. V = { 1, 2,3, 4, 5} 3 4 E = { (1,2), (2, 3), (3, 4), (2, 4), (4, 5), (5, 1) } 7

8 Directed Graph All edges are oe-way as idicated by the arrows. Edges are ordered pairs. V = { 1, 2, 3, 4, 5} E = { (1, 2), (2, 4), (3, 2), (4, 3), (4, 5), (5, 4), (5, 1) } 8

9 A Sigle Graph with Multiple Compoets

10 Basic Graph Defiitios (3) Vertex w is adjacet to vertex v if ad oly if (v, w) Î E. For udirected graphs, with edge (v, w), ad hece also (w, v), w is adjacet to v ad v is adjacet to w. A edge may also have: q weight or cost -- a associated value q label -- a uique ame The degree of a vertex, v, is the umber of vertices adjacet to v. Degree is also called valece. 10

11 Basic Graph Defiitios (4) For directed graphs vertex w is adjacet to vertex v if ad oly if (v, w) Î E. Idegree of a vertex w is the umber of edges (v,w). OutDegree of a vertex w is the umber of edges(w,v)

12 Paths i Graphs A path i a graph is a sequece of vertices w 1, w 2, w 3,, w such that (w i, w i+1 ) Î E for 1 i <. The legth of a path i a graph is the umber of edges o the path. The legth of the path from a vertex to itself is 0. A simple path is a path such that all vertices are distict, except that the first ad last may be the same. A cycle i a graph is a path w 1, w 2, w 3,, w, w Î V such that: q q q there are at least two vertices o the path w 1 = w (the path starts ad eds o the same vertex) if ay part of the path cotais the subpath w i, w j, w i, the each of the edges i the subpath is distict (i. e., o backtrackig alog the same edge) A simple cycle is oe i which the path is simple. A directed graph with o cycles is called a directed acyclic graph, ofte abbreviated as DAG 12

13 Paths i Graphs (2) How may simple paths from 1 to 4 ad what are their legths?

14 Coectedess i Graphs A udirected graph is coected if there is a path from every vertex to every other vertex. A directed graph is strogly coected if there is a path from every vertex to every other vertex. A directed graph is weakly coected if there would be a path from every vertex to every other vertex, disregardig the directio of the edges. A complete graph is oe i which there is a edge betwee every pair of vertices. A coected compoet of a graph is ay maximal coected subgraph. Coected compoets are sometimes simply called compoets. 14

15 Disjoit Sets ad Graphs Disjoit sets ca be used to determie coected compoets of a udirected graph. For each edge, place its two vertices (u ad v) i the same set -- i.e. uio( u, v ) Whe all edges have bee examied, the forest of sets will represet the coected compoets. Two vertices, x, y, are coected if ad oly if fid( x ) = fid( y ) 15

16 Udirected Graph/Disjoit Set Example Sets represetig coected compoets { 1, 2, 3, 4, 5 } { 6 } { 7, 8, 9 } 9 16

17 DiGraph / Strogly Coected Compoets a b g d c f h e j i 17

18 A Graph ADT Has some data elemets q Vertices ad Edges Has some operatios q q q getdegree( u ) -- Returs the degree of vertex u (outdegree of vertex u i directed graph) getadjacet( u ) -- Returs a list of the vertices adjacet to vertex u (list of vertices that u poits to for a directed graph) isadjacetto( u, v ) -- Returs TRUE if vertex v is adjacet to vertex u, FALSE otherwise. Has some associated algorithms to be discussed. 18

19 Adjacecy Matrix Implemetatio Uses array of size V V where each etry (i,j) is boolea q q q TRUE if there is a edge from vertex i to vertex j FALSE otherwise store weights whe edges are weighted Very simple, but large space requiremet = O( V 2 ) Appropriate if the graph is dese. Otherwise, most of the etries i the table are FALSE. For example, if a graph is used to represet a street map like Mahatta i which most streets ru E/W or N/S, each itersectio is attached to oly 4 streets ad E < 4* V. If there are 3000 itersectios, the table has 9,000,000 etries of which oly 12,000 are TRUE. 19

20 Udirected Graph / Adjacecy Matrix

21 Directed Graph / Adjacecy Matrix

22 Weighted, Directed Graph / Adjacecy Matrix

23 Adjacecy Matrix Performace Storage requiremet: O( V 2 ) Performace: getdegree ( u ) isadjacetto( u, v ) getadjacet( u ) 23

24 Adjacecy List Implemetatio If the graph is sparse, the keepig a list of adjacet vertices for each vertex saves space. Adjacecy Lists are the commoly used represetatio. The lists may be stored i a data structure or i the Vertex object itself. q Vector of lists: A vector of lists of vertices. The i- th elemet of the vector is a list, L i, of the vertices adjacet to v i. If the graph is sparse, the the space requiremet is O( E + V ), liear i the size of the graph If the graph is dese, the the space requiremet is O( V 2 ) 24

25 Vector of Lists

26 Adjacecy List Performace Storage requiremet: Performace: getdegree( u ) isadjacetto( u, v ) getadjacet( u ) 26

27 Graph Traversals Like trees, graphs ca be traversed breadthfirst or depth-first. q Use stack (or recursio) for depth-first traversal q Use queue for breadth-first traversal Ulike trees, we eed to specifically guard agaist repeatig a path from a cycle. Mark each vertex as visited whe we ecouter it ad do ot cosider visited vertices more tha oce. 27

28 Graph Traversals Both take time: O(V+E) 28 Slide by Rose Hoberma (CMU)

29 Breadth-First Traversal void bfs() { Queue<Vertex> q; Vertex u, w; } for all v i V, d[v] = // mark each vertex uvisited q.equeue(startvertex); // start with ay vertex d[startvertex] = 0; // mark visited while (!q.isempty() ) { u = q.dequeue( ); for each Vertex w adjacet to u { if (d[w] == ) { // w ot marked as visited d[w] = d[u]+1; // mark visited path[w] = u; // where we came from q.equeue(w); } } } 29

30 Breadth-First Example v1 0 1v1 v3 q v1 v2 v3 v4 u v2 1v1 v5 v4 2v2 BFS Traversal v1 v2 v3 v4 30

31 Uweighted Shortest Path Problem Uweighted shortest-path problem: Give as iput a uweighted graph, G = ( V, E ), ad a distiguished startig vertex, s, fid the shortest uweighted path from s to every other vertex i G. After ruig BFS algorithm with s as startig vertex, the legth of the shortest path legth from s to i is give by d[i]. If d[i] =, the there is o path from s to i. The path from s to i is give by traversig path[] backwards from i back to s. 31

32 Recursive Depth First Traversal void dfs() { } for (each v Î V) dfs(v) void dfs(vertex v) { } if (!v.visited) { } v.visited = true; for each Vertex w adjacet to v) if (!w.visited ) dfs(w) 32

33 DFS with explicit stack void dfs() { } Stack<Vertex> s; Vertex u, w; s.push(startvertex); startvertex.visited = true; while (!s.isempty() ) { } u = s.pop(); for each Vertex w adjacet to u { } if (!w.visited) { w.visited = true; s.push(w); 33

34 DFS Example v1 v3 s v1 v2 v4 v3 u v2 v5 v4 DFS Traversal v1 v3 v2 v4 34

35 Traversal Performace What is the performace of DF ad BF traversal? Each vertex appears i the stack or queue exactly oce i the worst case. Therefore, the traversals are at least O( V ). However, at each vertex, we must fid the adjacet vertices. Therefore, df- ad bftraversal performace depeds o the performace of the getadjacet operatio. 35

36 GetAdjacet Method 1: Look at every vertex (except u), askig are you adjacet to u? List<Vertex> L; for each Vertex v except u if (v.isadjacetto(u)) L.push_back(v); Assumig O(1) performace for push_back ad isadjacetto, the getadjacet has O( V ) performace ad traversal performace is O( V 2 ); 36

37 GetAdjacet (2) Method 2: Look oly at the edges which impige o u. Therefore, at each vertex, the umber of vertices to be looked at is D(u), the degree of the vertex This approach is O( D( u ) ). The traversal performace is O( V å i= 1 D( vi )) = O ( E ) sice getadjacet is doe O( V ) times. However, i a discoected graph, we must still look at every vertex, so the performace is O( V + E ). 37

38 Number of Edges Theorem: The umber of edges i a udirected graph G = (V,E ) is O( V 2 ) Proof: Suppose G is fully coected. Let p = V. The we have the followig situatio: vertex coected to 1 2,3,4,5,, p 2 1,3,4,5,, p p 1,2,3,4,,p-1 q There are p(p-1)/2 = O( V 2 ) edges. So O( E ) = O( V 2 ). 38

39 Weighted Shortest Path Problem Sigle-source shortest-path problem: Give as iput a weighted graph, G = ( V, E ), ad a distiguished startig vertex, s, fid the shortest weighted path from s to every other vertex i G. Use Dijkstra s algorithm Keep tetative distace for each vertex givig shortest path legth usig vertices visited so far. Record vertex visited before this vertex (to allow pritig of path). At each step choose the vertex with smallest distace amog the uvisited vertices (greedy algorithm). 39

40 Dijkstra s Algorithm The pseudo code for Dijkstra s algorithm assumes the followig structure for a Vertex object class Vertex { public List adj; //Adjacecy list public boolea kow; public DisType dist; //DistType is probably it public Vertex path; //Other fields ad methods as eeded } 40

41 Dijkstra s Algorithm void dijksra(vertex start) { for each Vertex v i V { v.dist = Iteger.MAX_VALUE; v.kow = false; v.path = ull; } start.distace = 0; } while there are ukow vertices { v = ukow vertex with smallest distace v.kow = true; for each Vertex w adjacet to v if (!w.kow) if (v.dist + weight(v, w)< w.distace){ decrease(w.dist to v.dist+weight(v, w)) w.path = v; } } 41

42 Dijkstra Example v1 3 v2 v v4 v3 1 v6 3 v v5 v10 1 v9 42

43 Correctess of Dijkstra s Algorithm The algorithm is correct because of a property of shortest paths: If P k = v 1, v 2,..., v j, v k, is a shortest path from v 1 to v k, the P j = v 1, v 2,..., v j, must be a shortest path from v 1 to v j. Otherwise P k would ot be as short as possible sice P k exteds P j by just oe edge (from v j to v k ) Also, P j must be shorter tha P k (assumig that all edges have positive weights). So the algorithm must have foud P j o a earlier iteratio tha whe it foud P k. i.e. Shortest paths ca be foud by extedig earlier kow shortest paths by sigle edges, which is what the algorithm does. 43

44 Ruig Time of Dijkstra s Algorithm The ruig time depeds o how the vertices are maipulated. The mai while loop rus O( V ) time (oce per vertex) Fidig the ukow vertex with smallest distace (iside the while loop) ca be a simple liear sca of the vertices ad so is also O( V ). With this method the total ruig time is O ( V 2 ). This is acceptable (ad perhaps optimal) if the graph is dese ( E = O ( V 2 ) ) sice it rus i liear time o the umber of edges. If the graph is sparse, ( E = O ( V ) ), we ca use a priority queue to select the ukow vertex with smallest distace, usig the deletemi operatio (O( lg V )). We must also decrease the path legths of some ukow vertices, which is also O( lg V ). The deletemi operatio is performed for every vertex, ad the decrease path legth is performed for every edge, so the ruig time is O( E lg V + V lg V ) = O( ( V + E ) lg V ) = O( E lg V ) if all vertices are reachable from the startig vertex 44

45 Dijkstra ad Negative Edges Note i the previous discussio, we made the assumptio that all edges have positive weight. If ay edge has a egative weight, the Dijkstra s algorithm fails. Why is this so? Suppose a vertex, u, is marked as kow. This meas that the shortest path from the startig vertex, s, to u has bee foud. However, it s possible that there is egatively weighted edge from a ukow vertex, v, back to u. I that case, takig the path from s to v to u is actually shorter tha the path from s to u without goig through v. Other algorithms exist that hadle edges with egative weights for weighted shortest-path problem. 45

46 Directed Acyclic Graphs A directed acyclic graph is a directed graph with o cycles. A strict partial order R o a set S is a biary relatio such that q for all aîs, ara is false (irreflexive property) q for all a,b,c ÎS, if arb ad brc the arc is true (trasitive property) To represet a partial order with a DAG: q represet each member of S as a vertex q for each pair of vertices (a,b), isert a edge from a to b if ad oly if arb 46

47 Example DAG Uderwear Pats Belt Shirt Tie Socks Shoes Watch a DAG implies a orderig o evets Jacket 47 Slide by Rose Hoberma (CMU)

48 Example DAG Uderwear Pats Belt Shirt Tie Jacket Socks Shoes Watch I a complex DAG, it ca be hard to fid a schedule that obeys all the costraits. 48 Slide by Rose Hoberma (CMU)

49 More Defiitios Vertex i is a predecessor of vertex j if ad oly if there is a path from i to j. Vertex i is a immediate predecessor of vertex j if ad oly if ( i, j ) is a edge i the graph. Vertex j is a successor of vertex i if ad oly if there is a path from i to j. Vertex j is a immediate successor of vertex i if ad oly if ( i, j ) is a edge i the graph. The idegree of a vertex, v, is the umber of edges (u, v), i.e. the umber of edges that come ito v. 49

50 Topological Orderig A topological orderig of the vertices of a DAG G = (V,E) is a liear orderig such that, for vertices i, j ÎV, if i is a predecessor of j, the i precedes j i the liear order, i.e. if there is a path from v i to v j, the v i comes before v j i the liear order 50

51 Topological Sort For a directed acyclic graph G = (V,E) A topological sort is a orderig of all of G s vertices v 1, v 2,, v such that... Formally: for every edge (v i,v k ) i E, i<k. Visually: all arrows are poitig to the right 51 Slide by Rose Hoberma (CMU)

52 Topological Sort There are ofte may possible topological sorts of a give DAG Topological orders for this DAG : 1,2,5,4,3,6,7 2,1,5,4,7,3,6 2,5,1,4,7,3,6 Etc Each topological order is a feasible schedule. 52 Slide by Rose Hoberma (CMU)

53 Topological Sorts for Cyclic Graphs? Impossible! If v ad w are two vertices o a cycle, there exist paths from v to w ad from w to v. Ay orderig will cotradict oe of these paths 53 Slide by Rose Hoberma (CMU)

54 Topological sort algorithm Algorithm q Assume idegree is stored with each ode. q Repeat util o odes remai: Choose a root ad output it. Remove the root ad all its edges. Performace q O(V 2 + E), if liear search is used to fid a root. 54 Slide by Rose Hoberma (CMU)

55 Better topological sort Algorithm: q Sca all odes, pushig roots oto a stack. q Repeat util stack is empty: Pop a root r from the stack ad output it. For all odes such that (r,) is a edge, decremet s idegree. If 0 the push oto the stack. O( V + E ), so still O(V 2 ) i worst case, but better for sparse graphs. Q: Why is this algorithm correct? 55 Slide by Rose Hoberma (CMU)

56 Correctess Clearly ay orderig produced by this algorithm is a topological order But... Does every DAG have a topological order, ad if so, is this algorithm guarateed to fid oe? 56 Slide by Rose Hoberma (CMU)

57 Exercise Prove: q This algorithm ever gets stuck, i.e. if there are uvisited odes the at least oe of them has a idegree of zero. Approach: q Prove that if at ay poit there are usee vertices but oe of them have a idegree of 0, a cycle must exist, cotradictig our assumptio of a DAG. 57 Based o slide by Rose Hoberma (CMU)

58 Topological Sort 58

59 Ruig Time of TopSort 1. At most, each vertex is equeued just oce, so there are O( V ) costat time queue operatios. 2. The body of the for loop is executed at most oce per edges = O( E ) 3. The iitializatio is proportioal to the size of the graph if adjacecy lists are used = O( E + V ) 4. The total ruig time is therefore O ( E + V ) 60