Minimum Spanning Trees Ch 23 Traversing graphs

Transcription

1 Net: Graph Algorithms Graphs Ch 22 Graph representations adjacenc list adjacenc matri Minimum Spanning Trees Ch 23 Traersing graphs Breadth-First Search Depth-First Search 7/16/2013 CSE

2 Graphs Definition A graph G = (V,E) is composed of: V: set of ertices E V V: set of edges connecting the ertices An edge e = (u,) is a pair of ertices (u,) is ordered, if G is a directed graph 7/16/2013 CSE

3 Graph Terminolog adjacent ertices: connected b an edge degree (of a erte): # of adjacent ertices V deg( ) = 2(# of edges) Since adjacent ertices each count the adjoining edge, it will be counted twice path: sequence of ertices 1, 2,... k such that consecutie ertices i and i+1 are adjacent 7/16/2013 CSE

4 Graph Terminolog (2) simple path: no repeated ertices 7/16/2013 CSE

5 Graph Terminolog (3) ccle: simple path, ecept that the last erte is the same as the first erte connected graph: an two ertices are connected b some path 7/16/2013 CSE

6 Graph Terminolog (4) subgraph: subset of ertices and edges forming a graph connected component: maimal connected subgraph. E.g., the graph below has 3 connected components 7/16/2013 CSE

7 Graph Terminolog (5) (free) tree - connected graph without ccles forest - collection of trees 7/16/2013 CSE

8 Data Structures for Graphs The Adjacenc list of a erte : a sequence of ertices adjacent to Represent the graph b the adjacenc lists of all its ertices Space =Θ ( n+ deg( )) =Θ ( n+ m) 7/16/2013 CSE

9 Data Structures for Graphs Adjacenc matri Matri M with entries for all pairs of ertices M[i,j] = true there is an edge (i,j) in the graph M[i,j] = false there is no edge (i,j) in the graph Space = O(n 2 ) 7/16/2013 CSE

10 Spanning Tree A spanning tree of G is a subgraph which is a tree contains all ertices of G 7/16/2013 CSE

11 Minimum Spanning Trees Undirected, connected graph G = (V,E) Weight function W: E R (assigning cost or length or other alues to edges) Spanning tree: tree that connects all ertices Minimum spanning tree: tree that connects all the ertices and minimizes wt ( ) = wu (, ) ( u, ) T 7/16/2013 CSE

12 Optimal Substructure MST T T 2 T 1 Remoing the edge (u,) partitions T into T 1 and T 2 wt ( ) = wu (, ) + wt ( ) + wt ( ) 1 2 We claim that T 1 is the MST of G 1 =(V 1,E 1 ), the subgraph of G induced b ertices in T 1 Also, T 2 is the MST of G 2 7/16/2013 CSE

13 Greed Choice Greed choice propert: locall optimal (greed) choice ields a globall optimal solution Theorem Let G=(V, E), and let S V and let(u,) be min-weight edge in G connecting S to V S Then (u,) T somemstof G 7/16/2013 CSE

14 Proof Greed Choice (2) suppose (u,) T look at path from u to in T swap(, ) the first edge on path from u to in T that crosses from S to V S this improes T contradiction (T supposed to be MST) S V-S u 7/16/2013 CSE

15 Generic MST Algorithm Generic-MST(G, w) w) 1 A // // Contains edges that belong to to a MST MST 2 while A does not not form a spanning tree do do 3 Find an an edge (u,) that is is safe for for A 4 A A {(u,)} 5 return A Safe edge edge that does not destro A s propert MoreSpecific-MST(G, w) w) 1 A // // Contains edges that belong to to a MST MST 2 while A does not not form a spanning tree do do Make a cut cut (S, (S, V-S) of of G that respects A Take the the min-weight edge (u,) connecting S to to V-S V-S 4 A A {(u,)} 5 return A 7/16/2013 CSE

16 Prim s Algorithm Verte based algorithm Grows one tree T, one erte at a time A cloud coering the portion of T alread computed Label the ertices outside the cloud with ke[] the minimum weight of an edge connecting to a erte in the cloud, ke[] =, if no such edge eists 7/16/2013 CSE

17 Prim s Algorithm (2) MST-Prim(G,w,r) 01 Q V[G] // Q ertices out of T 02 for each u Q 03 ke[u] 04 ke[r] 0 05 π[r] NIL 06 while Q do 07 u EtractMin(Q) // making u part of T 08 for each Adj[u] do 09 if Q and w(u,) < ke[] then 10 π[] u 11 ke[] w(u,) updating kes 7/16/2013 CSE

18 Prim Eample 7/16/2013 CSE

19 Prim Eample (2) 7/16/2013 CSE

20 Prim Eample (3) 7/16/2013 CSE

21 Priorit Queues A priorit queue is a data structure for maintaining a set S of elements, each with an associated alue called ke We need PQ to support the following operations BuildPQ(S) initializes PQ to contain elements of S EtractMin(S) returns and remoes the element of S with the smallest ke ModifKe(S,,newke) changes the ke of in S A binar heap can be used to implement a PQ BuildPQ O(n) EtractMin and ModifKe O(lg n) 7/16/2013 CSE

22 Prim s Running Time Time = V T(EtractMin) + O( E )T(ModifKe) Time = O( V lg V + E lg V ) = O( E lg V ) Q T(EtractMin) T(DecreaseKe) Total arra O( V ) O(1) O( V 2 ) binar heap O(lg V ) O(lg V ) O( E lg V ) Fibonacci heap O(lg V ) O(1) amortized O( V lg V + E ) 7/16/2013 CSE

23 Kruskal's Algorithm Edge based algorithm Add the edges one at a time, in increasing weight order The algorithm maintains A a forest of trees. An edge is accepted it if connects ertices of distinct trees We need an ADT that maintains a partition, i.e.,a collection of disjoint sets MakeSet(S,): S S {{}} Union(S i,s j ): S S {S i,s j } {S i S j } FindSet(S, ): returns unique S i S, where S i 7/16/2013 CSE

24 Kruskal's Algorithm The algorithm keeps adding the cheapest edge that connects two trees of the forest MST-Kruskal(G,w) 01 A 02 for each erte V[G] do 03 Make-Set() 04 sort the edges of E b non-decreasing weight w 05 for each edge (u,) E, in order b nondecreasing weight do 06 if Find-Set(u) Find-Set() then 07 A A {(u,)} 08 Union(u,) 09 return A 7/16/2013 CSE

25 Kruskal's Algorithm: eample 7/16/2013 CSE

26 Kruskal's Algorithm: eample (2) 7/16/2013 CSE

27 Kruskal's Algorithm: eample (3) 7/16/2013 CSE

28 Kruskal's Algorithm: eample (4) 7/16/2013 CSE

29 Kruskal running time Initialization O( V ) time Sorting the edges Θ( E lg E ) = Θ( E lg V ) (wh?) O( E ) calls to FindSet Union costs Let t() the number of times is moed to a new cluster Each time a erte is moed to a new cluster the size of the cluster containing the erte at least doubles: t() log V Total time spent doing Union t () Vlog V Total time: O( E lg V ) V 7/16/2013 CSE

30 Net: Graph Algorithms Graphs Graph representations adjacenc list adjacenc matri Traersing graphs Breadth-First Search Depth-First Search 7/16/2013 CSE

31 Graph Searching Algorithms Sstematic search of eer edge and erte of the graph Graph G = (V,E) is either directed or undirected Toda's algorithms assume an adjacenc list representation Applications Compilers Graphics Maze-soling Mapping Networks: routing, searching, clustering, etc. 7/16/2013 CSE

32 Breadth First Search A Breadth-First Search (BFS) traerses a connected component of a graph, and in doing so defines a spanning tree with seeral useful properties BFS in an undirected graph G is like wandering in a labrinth with a string. The starting erte s, it is assigned a distance 0. In the first round, the string is unrolled the length of one edge, and all of the edges that are onl one edge awa from the anchor are isited (discoered), and assigned distances of 1 7/16/2013 CSE

33 Breadth First Search (2) In the second round, all the new edges that can be reached b unrolling the string 2 edges are isited and assigned a distance of 2 This continues until eer erte has been assigned a leel The label of an erte corresponds to the length of the shortest path (in terms of edges) from s to 7/16/2013 CSE

34 7/16/2013 CSE Breadth First Search: eample 0 r s u t w 0 s Q r s u t w 1 w 1 r Q r s u t 2 w 2 t 1 r 2 Q r s u t 2 w 2 2 t 2 Q

35 7/16/2013 CSE Breadth First Search: eample r s u 3 t 2 w u Q r s u 3 t 2 w 3 u 2 3 Q r s u 3 t 2 w 3 3 u Q r s u 3 t 2 w 3 Q

36 Breadth First Search: eample r s t u w Q - 7/16/2013 CSE

37 BFS Algorithm BFS(G,s) 01 for each erte u V[G]-{s} 02 color[u] white 03 d[u] 04 π[u] NIL 05 color[s] gra 06 d[s] 0 07 π[u] NIL 08 Q {s} 09 while Q do 10 u head[q] 11 for each Adj[u] do 12 if color[] = white then 13 color[] gra 14 d[] d[u] π[] u 16 Enqueue(Q,) 17 Dequeue(Q) 18 color[u] black Init all ertices Init BFS with s Handle all u s children before handling an children of children 7/16/2013 CSE

38 BFS Algorithm: running time Gien a graph G = (V,E) Vertices are enqueued if there color is white Assuming that en- and dequeuing takes O(1) time the total cost of this operation is O( V ) Adjacenc list of a erte is scanned when the erte is dequeued (and onl then ) The sum of the lengths of all lists is O( E ). Consequentl, O( E ) time is spent on scanning them Initializing the algorithm takes O( V ) Total running time O( V + E ) (linear in the size of the adjacenc list representation of G) 7/16/2013 CSE

39 BFS Algorithm: properties Gien a graph G = (V,E), BFS discoers all ertices reachable from a source erte s It computes the shortest distance to all reachable ertices It computes a breadth-first tree that contains all such reachable ertices For an erte reachable from s, the path in the breadth first tree from s to, corresponds to a shortest path in G 7/16/2013 CSE

40 BFS Tree Predecessor subgraph of G G = ( V, E ) π π π { : [ ] } { } {( [ ], ) : {}} V = V π NIL s π E = π E V s π G p is a breadth-first tree V p consists of the ertices reachable from s, and for all V p, there is a unique simple path from s to in G p that is also a shortest path from s to in G The edges in G p are called tree edges π 7/16/2013 CSE

41 Depth-first search (DFS) A depth-first search (DFS) in an undirected graph G is like wandering in a labrinth with a string and a can of paint We start at erte s, ting the end of our string to the point and painting s isited (discoered). Net we label s as our current erte called u Now, we trael along an arbitrar edge (u,). If edge (u,) leads us to an alread isited erte we return to u If erte is unisited, we unroll our string, moe to, paint isited, set as our current erte, and repeat the preious steps 7/16/2013 CSE

42 Depth-first search (2) Eentuall, we will get to a point where all incident edges on u lead to isited ertices We then backtrack b unrolling our string to a preiousl isited erte. Then becomes our current erte and we repeat the preious steps Then, if all incident edges on lead to isited ertices, we backtrack as we did before. We continue to backtrack along the path we hae traeled, finding and eploring uneplored edges, and repeating the procedure 7/16/2013 CSE

43 Depth-first search algorithm Initialize color all ertices white Visit each and eer white erte using DFS- Visit Each call to DFS-Visit(u) roots a new tree of the depth-first forest at erte u A erte is white if it is undiscoered A erte is gra if it has been discoered but not all of its edges hae been discoered A erte is black after all of its adjacent ertices hae been discoered (the adj. list was eamined completel) 7/16/2013 CSE

44 Depth-first search algorithm (2) Init all ertices Visit all children recursiel 7/16/2013 CSE

45 Depth-first search eample u 1/ w u 1/ 2/ w u 1/ 2/ w z z 3/ z u w u w u w 1/ 2/ 1/ 2/ B 1/ 2/ B 4/ 3/ z 4/ 3/ z 4/5 3/ z 7/16/2013 CSE

46 Depth-first search eample (2) u w u w u w 1/ 2/ B 4/5 3/6 z 1/ 2/7 B 4/5 3/6 z 1/ 2/7 F B 4/5 3/6 z u w u w u w 1/8 2/7 F B 4/5 3/6 z 1/8 2/7 F B 4/5 3/6 9/ z 1/8 2/7 F B 4/5 3/6 C 9/ z 7/16/2013 CSE

47 Depth-first search eample (3) u w u w u w 1/8 2/7 F B 4/5 3/6 C 9/ 10/ z 1/8 2/7 F B 4/5 3/6 C 9/ 10/ B z 1/8 2/7 F B 4/5 3/6 C 9/ 10/11 B z u w 1/8 2/7 F B 4/5 3/6 C 9/12 10/11 B z 7/16/2013 CSE

48 Depth-first search eample (4) When DFS returns, eer erte u is assigned a discoer time d[u], and a finishing time f[u] Running time the loops in DFS take time Θ(V) each, ecluding the time to eecute DFS-Visit DFS-Visit is called once for eer erte its onl inoked on white ertices, and paints the erte gra immediatel for each DFS-isit a loop interates oer all Adj[] the total cost for DFS-Visit is Θ(E) the running time of DFS is Θ(V+E) Adj[] =Θ( E) 7/16/2013 CSE V

49 Predecessor Subgraph Defined slightl different from BFS G π {( [ ], ) : and [ ] NIL} E = π E V π π = ( V, E ) π The PD subgraph of a depth-first search forms a depth-first forest composed of seeral depth-first trees The edges in G p are called tree edges 7/16/2013 CSE

50 DFS Timestamping The DFS algorithm maintains a monotonicall increasing global clock discoer time d[u] and finishing time f[u] For eer erte u, the inequalit d[u] < f[u] must hold 7/16/2013 CSE

51 Verte u is DFS Timestamping white before time d[u] gra between time d[u] and time f[u], and black thereafter Notice the structure througout the algorithm. gra ertices form a linear chain correponds to a stack of ertices that hae not been ehaustiel eplored (DFS-Visit started but not et finished) 7/16/2013 CSE

52 DFS Parenthesis Theorem Discoer and finish times hae parenthesis structure represent discoer of u with left parenthesis "(u" represent finishin of u with right parenthesis "u)" histor of discoeries and finishings makes a wellformed epression (parenthesis are properl nested) Intuition for proof: an two interals are either disjoint or enclosed Oerlaping interals would mean finishing ancestor, before finishing descendant or starting descendant without starting ancestor 7/16/2013 CSE