Simple strategy Computes shortest paths from the start note

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Marion Watkins
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1 Leture Notes for I500 Fall 2009 Fundamental Computer Conepts of Informatis by Xiaoqian Zhang (Instrutor: Predrag Radivoja) Last time: dynami programming Today: Elementary graph algorithm (textbook Chapter 22) We express a graph as G = (V, E) that means a set of verties and edges vertexes edges There are always two standard ways to represent a graph: 1 2 (1,3)(2,3) 3 1, an adjaeny matrix , a olletion of adjaeny lists They an store either direted or undireted graphs. Next we will talk about two types of graph searhing algorithms: 1. Breath-first algorithms Simple strategy Computes shortest paths from the start note 1

2 Expands frontier, all nodes at distane k are expanded before any node at distane k+1 is expanded Take the word puzzle example mentioned before: TRAIN C R CRANE -- T -- C R C -- T C -- T C T -- Example: We finish all the possibilities of this layer before go to next one a b Steps of searhing algorithm: We need a queue (Q), 2 arrays (olor, parent) soure = s to help not go bak to node whih has disovered to reord predeessor of eah node (at most one) 0) Plae a in Q, a is disovered Q:a 1) Take top node(here is a) in Q, go through its adjaent list Q:a b Put in Q if doesn t disovered Color these as d Color top node as and remove top node from Q Q:a b 2) b is top node, remove Q:b 2

3 3) is top node find d dequeued Q: d 4) Q:e (d is expanded) 5) Q:--(e is expanded) Illustration: olor[a]= d Q: a d d Q: b d d Q: d Q: d d d Q: e d d 3

4 Q: - d Searh tree: if node b go first: if node go first: b 1 1 a a b Complexity of the algorithm: enqueue/dequeue s onstant time every node expanded one O( V ) to enqueue/dequeue and olor expanded node O( E ) to handle all neighbors during all expansions Total running time of BFS is O( V + E ) Example of applying BFS: Beam searh(m-algorithm) It goes in BFS fashion, only keeps the first best n th results, it is muh faster than dynami programming. 2. Depth-first searh gives priority to longer nodes only when all of v s desendents are explored, do we explore some shorter node u we will have two kinds of nodes disovered(the nodes in searh) 4

5 Pseudo ode(ontains two funtions: MAIN and DFS_VIST ): DFS_VISIT(u) olor[u] disovered time time + 1 dt[u] time for v in u s adjaent list if olor[v] == undisovered parent[v] = u DFS_VISIT(v) end end olor[u] finished time time +1 dt[u] time MAIN for u V olor u undisovered parent u NILL time 0 for u V if olor u undisovered DFS_VISIT u finished(adjaent list has been examined ompletely) Illustration: we assume white olor of notes as undisovered;gray as disovered but not finished; blak as finished 1/ 5

6 3/ 3/ 4/ 3/ 4/ 5/ 3/ 4/ 5/6 3/ 4/ 7 5/6 6

7 3/ 4/ 7 5/6 3/ 8 4/ 7 5/6 9 3/ 8 4/ 7 5/6 1/10 2/9 3/ 8 4/ 7 5/6 The searh tree is: If we get more omplexity graph, add a node f, the proedure of the algorithm should be: 7

8 10/11 1/12 2/9 f 3/8 4/7 5/6 The omplexity of DFS is obviously O( V + E ) There are several properties of DFS algorithm: 1, Sine disovery and finishing time of eah nodes have parenthesis struture, we an express searh proedure as: [ a [ b [ [ d [ e ] ] ] ] [ f ] ] it will help us to store the searh result in tree 2, DFS ould help us find loops in graph 1 a b 2/7 3 e 6 d 4/5 When the searh goes to the 7 th step, we will find b was unfinished but has disovered, then we know that there is a loop in this graph. 3, DFS ould help us find forward edges a 1 b 2/7 3/6/8 d 4/5 When we goes to 8 th step, we find a finished node, so we an tell that there is a forward edge in this graph (a ). On other priority we searh first, we also an ome out that there exists a forward edge in this graph. 8

9 4,DFS an be used perform Topologial Sort: 11/12 1/10 soks undershorts 17/18 wath 2/9 pants 7/8 shoes 3/6 belt 13/16 shirt 4/5 14/15 jaket tie We an first use Depth-first algorithm to searh for the whole graph and ome out one possible solution: wath, shirt, tie, soks, undershorts, pants, shoes, belt, jaket 9

Graph Representation

Graph Representation Adjacency list representation of G = (V, E) An array of V lists, one for each vertex in V Each list Adj[u] contains all the vertices v such that there is an edge between u and v Adj[u]