COP 4531 Complexity & Analysis of Data Structures & Algorithms

Transcription

1 COP Complexity & Analysis of Data Structures & Algorithms Overview of Graphs Breadth irst Search, and Depth irst Search hanks to several people who contributed to these slides including Piyush Kumar and SH Poon (BS detailed example) and the text authors

2 Outline What are Graphs? erminology Representation (Adjacency matrices and Linked lists) Searching Breadth irst Search (BS) Depth irst Search (DS)

3 Graphs A graph G = (V,E) is composed of: V: set of vertices E V V: set of edges connecting the vertices An edge e = (u,v) is an ordered or unordered pair of vertices Directed graphs ordered: (u, v) Edge incident from or leaves u Edge incident to or enters v Undirected graphs unordered: {u, v}

4 Graphs with edge weights he initial graph definition defines whether an edge exists or not. Often we want to associate a weight w(e) to the edge: w: E R he weight can represent distance, time, etc.

5 Directed graphs

6 An undirected graph

7 A more complicated undirected graph

8 Some Graph Applications Graph Nodes Edges transportation street intersections highways communication computers fiber optic cables World Wide Web web pages hyperlinks social people relationships food web species predatorprey software systems functions function calls scheduling tasks precedence constraints circuits gates wires

9 a is adjacent to b iff (a,b) Ε. degree (a) = number of adjacent vertices (Self loop counted twice) Self Loop: (a,a)! erminology Could also define parallel edges: E = {...(a,b), (a,b)...} is a multiset a a b

10 erminology continued A Simple Graph is a graph with no self loops or parallel edges. Incidence: v is incident to e if v is an end vertex of e. v e

11 Degree of a vertex he number of edges incident on the vertex in an undirected graph or directed graphs we have outdegree and indegree (edges leaving or entering the vertex). he degree is indegree + outdegree Isolated vertex has degree Max degree vertex, min degree vertex

12 Example Max Degree =. Isolated vertices =. V =, E = Sum of degrees = = E = v V degree (v) Handshaking theorem

13 QUESION How many edges are there in a graph with vertices each of degree?

14 QUESION How many edges are there in a graph with vertices each of degree? otal degree sum = = E edges by the handshaking theorem.

15 Handshaking Corollary he number of vertices with odd degree is always even. Proof: Let V and V be the set of vertices of even and odd degrees, respectively (Hence V V =, and V V = V). Now we know that E = v V degree(v) = v V degree(v) + v V degree(v) Since degree(v) is odd for all v V, V must be even.

16 wo ways Representation Adjacency List ( as a linked list for each node in the graph to represent the edges) Adjacency Matrix (as a boolean matrix)

17 Representing Graphs Vertex Adjacent Vertices Initial Vertex erminal Vertices,,,,,,,,

18 adjacency list

19 adjacency matrix

20 Another example (directed graph)

21 Another example (undirected graph)

22 AL Vs AM AL: akes O( V + E ) space AM: akes O( V * V ) space Question: How much time does it take to find out if (v i,v j ) belongs to E? AM? AL?

23 AL Vs AM AL: akes O( V + E ) space AM: akes O( V * V ) space Question: How much time does it take to find out if (v i,v j ) belongs to E? AM : O() AL : O( V ) in the worst case.

24 Connectivity st connectivity problem. Given two node s and t, is there a path between s and t?! st shortest path problem. Given two node s and t, what is the length of the shortest path between s and t?!!!! Applications. Maze traversal. Kevin Bacon number / Erdos number ewest number of hops in a communication network. riendster.

25 BS/DS Breadthfirst search (BS) and depthfirst search (DS) are two distinct orders in which to visit the vertices and edges of a graph. BS: radiates out from a root to visit vertices in order of their distance from the root. hus closer nodes get visited first.

26 Breadth first search Question: Given G in AM form, how do we say if there is a path between nodes a and b? Note: Using AM or AL its easy to answer if there is an edge (a,b) in the graph, but not path questions. his is one of the reasons to learn BS/DS.

27 BS A Breadthirst Search (BS) traverses a connected component of a graph, and in doing so defines a spanning tree.

28 BS

29 Example Adjacency List Visited able (/) source Initialize visited table (all empty ) Q = { } Initialize Q to be empty

30 Example Adjacency List Visited able (/) source lag that has been visited. Q = { } Place source on the queue.

31 Example Adjacency List Visited able (/) source Neighbors Mark neighbors as visited. Q = {} {,, } Dequeue. Place all unvisited neighbors of on the queue

32 Example Adjacency List Visited able (/) source Neighbors Mark new visited Neighbors. Q = {,, } {,,, } Dequeue. Place all unvisited neighbors of on the queue. Notice that is not placed on the queue again, it has been visited!

33 Example Adjacency List Visited able (/) Neighbors source Mark new visited Neighbors. Q = {,,, } {,,,, } Dequeue. Place all unvisited neighbors of on the queue. Only nodes and haven t been visited yet.

34 Example Adjacency List Visited able (/) source Neighbors Q = {,,,, } {,,, } Dequeue. has no unvisited neighbors!

35 Example Adjacency List Visited able (/) source Neighbors Q = {,,, } {,, } Dequeue. has no unvisited neighbors!

36 Example Adjacency List Visited able (/) source Neighbors Q = {,, } {, } Dequeue. has no unvisited neighbors!

37 Example Adjacency List Visited able (/) source Neighbors Mark new visited Vertex. Q = {, } {, } Dequeue. place neighbor on the queue.

38 Example Adjacency List Visited able (/) source Neighbors Mark new visited Vertex. Q = {, } {, } Dequeue. place neighbor on the queue.

39 Example Adjacency List Visited able (/) source Neighbors Q = {, } { } Dequeue. no unvisited neighbors of.

40 Example Adjacency List Visited able (/) source Neighbors Q = { } { } Dequeue. no unvisited neighbors of.

41 Example Adjacency List Visited able (/) source Q = { } SOP!!! Q is empty!!! Neighbors What did we discover?! Look at visited tables.! here exist a path from source vertex to all vertices in the graph!

42 ime Complexity of BS (Using adjacency list) Assume adjacency list n = number of vertices m = number of edges O(n + m) No more than n vertices are ever put on the queue. How many adjacent nodes will we ever visit. his is related to the number of edges. How many edges are there? Σ vertex v deg(v) = m* *Note: this is not per iteration of the while loop. his is the sum over all the while loops!

43 ime Complexity of BS (Using adjacency matrix) Assume adjacency matrix n = number of vertices m = number of edges O(n ) So, adjacency matrix is not good for BS!!! No more than n vertices are ever put on the queue. O(n) Using an adjacency matrix. o find the neighbors we have to visit all elements In the row of v. hat takes constant time O(n)!

44 Path Recording BS only tells us if a path exists from source s, to other vertices v.! It doesn t tell us the path! We need to modify the algorithm to record the path. Not difficult Use an additional predecessor array pred[..n] Pred[w] = v Means that vertex w was visited by v

45 BS + Path inding Set pred[v] to (let means no path to any vertex) Record who visited w

46 Example Adjacency List Visited able (/) source Pred Q = { } Initialize visited table (all empty )! Initialize Pred to Initialize Q to be empty

47 Example Adjacency List source Visited able (/) Q = { } lag that has been visited. Place source on the queue. Pred

48 Example source Q = {} {,, } Neighbors Adjacency List Dequeue. Place all unvisited neighbors of on the queue Visited able (/) Mark neighbors as visited.! Record in Pred who was visited by. Pred

49 Example Adjacency List Visited able (/) source Neighbors Pred Mark new visited Neighbors.! Q = {,, } {,,, } Record in Pred Dequeue. who was visited Place all unvisited neighbors of on the queue. by. Notice that is not placed on the queue again, it has been visited!

50 Example source Q = Neighbors {,,, } {,,,, } Adjacency List Dequeue. Place all unvisited neighbors of on the queue. Only nodes and haven t been visited yet. Visited able (/) Pred Mark new visited Neighbors.! Record in Pred who was visited by.

51 Example Adjacency List Visited able (/) source Neighbors Pred Q = {,,,, } {,,, } Dequeue. has no unvisited neighbors!

52 Example Adjacency List Visited able (/) source Neighbors Pred Q = {,,, } {,, } Dequeue. has no unvisited neighbors!

53 Example Adjacency List Visited able (/) source Neighbors Pred Q = {,, } {, } Dequeue. has no unvisited neighbors!

54 Example Adjacency List Visited able (/) source Q = {, } {, } Dequeue. place neighbor on the queue. Neighbors Pred Mark new visited Vertex.! Record in Pred who was visited by.

55 Example Adjacency List Visited able (/) source Q = {, } {, } Dequeue. place neighbor on the queue. Neighbors Pred Mark new visited Vertex.! Record in Pred who was visited by.

56 Example Adjacency List Visited able (/) source Neighbors Pred Q = {, } { } Dequeue. no unvisited neighbors of.

57 Example Adjacency List source Visited able (/) Q = { } { } Dequeue. no unvisited neighbors of. Neighbors Pred

58 Example Adjacency List source Visited able (/) Q = { } SOP!!! Q is empty!!! Neighbors Pred now stores the path! Pred

59 Pred array represents paths nodes visited by ry some examples. Path() > Path() > Path() >

60 BS tree We often draw the BS paths as a mary tree, where s is the root. Question: What would a level order traversal tell you?

61 Can also keep track of the distance from the source Assumption: if an edge (u, v) exists, then distance is from u to v In the next example code, each vertex initially has a color (white). When it is enqueued, color is set to gray. When it is dequeued color is set to black.

62 BS(G, s) for each vertex u G, V {s} u.color = white; u.d = ; u.π = nil s.color = gray; s.d = ; s.π = nil, Q = ; Enqueue(Q, s) while Q u = Dequeue(Q) for each v G.Adj[u] if v.color == white v.color = gray v.d = u.d + v. π = u Enqueue(Q, v) u.color = black

63 BS Breadth first search is a basis for many algorithms including spanning trees and single source shortest path Next, we will explore depth first search and applications

64 Another example BS

65 Depth irst Search Recall that in breadth first search, the discovery of nodes proceeds as a wavefront hitting all nodes distance from the source, then, etc. In depth first search we instead search deeper into the graph whenever possible dfs explores the edges out of the most recently discovered vertex first dfs creates a forest of trees that are reachable from the tree roots

66 he DS algorithm Input: G = (V, E) directed or undirected. No source vertex is assigned Output: timestamps for each vertex v.d = discovery time v.f = finishing time Explore every edge start from different vertices as needed as soon as a vertex is discovered, explore from it keep track of a predecessor tree by keeping track of a predecessor function v.π for each node As algorithm progresses vertex colors represent white: undiscovered node gray: discovered node but not finished exploring black: finished exploring from this node

67 DS pseudocode DS(G) for each vertex u G.V u.color = white; u.π = nil; time = for each vertex u G.V if u.color == white DSvisit(G, u)! DSvist(G, u) time = time + ; u.d = time; u.color = gray for each v G.Adj[u] if v.color == white v.π = u; DSvisit(G, v) u.color = black; time = time + ; u.f = time

68 Example DS