Lecture 13: Graphs I: Breadth First Search

Transcription

1 Leture 13 Grphs I: BFS Fll 2011 Leture 13: Grphs I: Bredth First Serh Leture Overview Applitions of Grph Serh Grph Representtions Bredth-First Serh Rell: Grph G = (V, E) V = set of verties (ritrry lels) E = set of edges i.e. vertex pirs (v, w) ordered pir = direted edge of grph unordered pir = undireted e.g. V = {,,,d} E = {{,},{,}, {,},{,d}, {,d}} V = {,,} E = {(,),(,), (,),(,)} d UNDIRECTED DIRECTED Figure 1: Exmple to illustrte grph terminology Grph Serh Explore grph, e.g.: find pth from strt vertex s to desired vertex visit ll verties or edges of grph, or only those rehle from s 1

2 Leture 13 Grphs I: BFS Fll 2011 Applitions: There re mny. we rwling (how Google finds pges) soil networking (Feook friend finder) network rodst routing grge olletion model heking (finite stte mhine) heking mthemtil onjetures solving puzzles nd gmes Poket Cue: Consider Ruik s ue Configurtion Grph: vertex for eh possile stte edge for eh si move (e.g., 90 degree turn) from one stte to nother undireted: moves re reversile Dimeter ( God s Numer ) 11 for 2 2 2, 20 for 3 3 3, Θ(n 2 / lg n) for n n n [Demine, Demine, Eisenstt Luiw Winslow 2011] solved possile first moves rehle in two steps ut not one hrdest onfigs... redthfirst tree 2

3 Leture 13 Grphs I: BFS Fll 2011 # verties = 8! 3 8 = 264, 539, 520 where 8! omes from hving 8 uelets in ritrry positions nd 3 8 omes s eh uelet hs 3 possile twists. This n e divided y 24 if we remove ue symmetries nd further divided y 3 to ount for tully rehle onfigurtions (there re 3 onneted omponents). Grph Representtions: (dt strutures) Adjeny lists: Arry Adj of V linked lists for eh vertex u V, Adj[u] stores u s neighors, i.e., {v V (u, v) E}. (u, v) re just outgoing edges if direted. (See Fig. 2 for n exmple.) Adj Figure 2: Adjeny List Representtion: Spe Θ(V + E) in Python: Adj = ditionry of list/set vlues; vertex = ny hshle ojet (e.g., int, tuple) dvntge: multiple grphs on sme verties Impliit Grphs: Adj(u) is funtion ompute lol struture on the fly (e.g., Ruik s Cue). This requires Zero Spe. 3

4 Leture 13 Grphs I: BFS Fll 2011 Ojet-oriented Vritions: ojet for eh vertex u u.neighors = list of neighors i.e. Adj[u] In other words, this is method for impliit grphs Inidene Lists: n lso mke edges ojets e. e e. u.edges = list of (outgoing) edges from u. dvntge: store edge dt without hshing Bredth-First Serh Explore grph level y level from s level 0 = {s} level i = verties rehle y pth of i edges ut not fewer s... level0 level1 level2 lst level Figure 3: Illustrting Bredth-First Serh 4

5 Leture 13 Grphs I: BFS Fll 2011 uild level i > 0 from level i 1 y trying ll outgoing edges, ut ignoring verties from previous levels Bredth-First-Serh Algorithm Exmple BFS (V,Adj,s): See CLRS for queue-sed implementtion level = { s: 0 } prent = {s : None } i = 1 frontier = [s] # previous level, i 1 while frontier: next = [ ] # next level, i for u in frontier: for v in Adj [u]: if v not in level: # not yet seen level[v] = i = level[u] + 1 prent[v] = u next.ppend(v) frontier = next i + =1 level 1 level s 2 d 3 f 2 z 1 x 2 3 v frontier 0 = {s} frontier 1 = {, x} frontier 2 = {z, d, } frontier 3 = {f, v} (not x,, d) level 2 level 3 Figure 4: Bredth-First Serh Frontier Anlysis: vertex V enters next (& then frontier) only one (euse level[v] then set) se se: v = s 5

6 Leture 13 Grphs I: BFS Fll 2011 = Adj[v] looped through only one time = Adj[V ] = = O(E) time v V { E for direted grphs 2 E for undireted grphs O(V + E) ( LINEAR TIME ) to lso list verties unrehle from v (those still not ssigned level) Shortest Pths: f. L15-18 for every vertex v, fewest edges to get from s to v is { level[v] if v ssigned level else (no pth) prent pointers form shortest-pth tree = union of suh shortest pth for eh v = to find shortest pth, tke v, prent[v], prent[prent[v]], et., until s (or None) 6

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8 Leture 14 Grphs II: DFS Fll 2011 Leture 14: Grphs II: Depth-First Serh Leture Overview Depth-First Serh Edge Clssifition Cyle Testing Topologil Sort Rell: grph serh: explore grph e.g., find pth from strt vertex s to desired vertex djeny lists: rry Adj of V linked lists For exmple: for eh vertex u V, Adj[u] stores u s neighors, i.e., {v V (u, v) E} (just outgoing edges if direted) Adj Figure 1: Adjeny Lists Bredth-first Serh (BFS): Explore level-y-level from s find shortest pths 1

9 Leture 14 Grphs II: DFS Fll 2011 Depth-First Serh (DFS) This is like exploring mze. s Figure 2: Depth-First Serh Frontier Depth First Serh Algorithm follow pth until you get stuk ktrk long redrums until reh unexplored neighor reursively explore reful not to repet vertex strt v finish v prent = {s: None} DFS-visit (V, Adj, s): for v in Adj [s]: if v not in prent: prent [v] = s DFS-visit (V, Adj, v) DFS (V, Adj) prent = { } for s in V: if s not in prent: prent [s] = None DFS-visit (V, Adj, s) } } serh from strt vertex s (only see stuff rehle from s) explore entire grph (ould do sme to extend BFS) Figure 3: Depth-First Serh Algorithm 2

10 Leture 14 Grphs II: DFS Fll 2011 Exmple forwrd edge S 1 1 S ross edge k edge d e f 3 7 k edge Figure 4: Depth-First Trversl Edge Clssifition tree edges (formed y prent) nontree edges k edge: to nestor forwrd edge: to desendnt ross edge (to nother sutree) Figure 5: Edge Clssifition to ompute this lssifition (k or not), mrk nodes for durtion they re on the stk only tree nd k edges in undireted grph Anlysis DFS-visit gets lled with vertex s only one (euse then prent[s] set) = time in DFS-visit = Adj[s] = O(E) s V DFS outer loop dds just O(V ) = O(V + E) time (liner time) 3

11 Leture 14 Grphs II: DFS Fll 2011 Cyle Detetion Grph G hs yle DFS hs k edge Proof (<=) tree edges is yle k edge: to tree nestor (=>) onsider first visit to yle: v 2 v 3 v k v 1 v 0 FIRST! efore visit to v i finishes, will visit v i+1 (& finish): will onsider edge (v i, v i+1 ) = visit v i+1 now or lredy did = efore visit to v 0 finishes, will visit v k (& didn t efore) = efore visit to v k (or v 0 ) finishes, will see (v k, v 0 ) s k edge Jo sheduling Given Direted Ayli Grph (DAG), where verties represent tsks & edges represent dependenies, order tsks without violting dependenies 4

12 Leture 14 Grphs II: DFS Fll 2011 G H I A B C F 1 D 6 E 5 Figure 6: Dependene Grph: DFS Finishing Times Soure: Soure = vertex with no inoming edges = shedulle t eginning (A,G,I) Attempt: BFS from eh soure: from A finds A, BH, C, F from D finds D, BE, CF slow... nd wrong! from G finds G, H from I finds I Topologil Sort Reverse of DFS finishing times (time t whih DFS-Visit(v) finishes) DFS-Visit(v)... order.ppend(v) order.reverse() 5

13 Leture 14 Grphs II: DFS Fll 2011 Corretness For ny edge (u, v) u ordered efore v, i.e., v finished efore u u v if u visited efore v: efore visit to u finishes, will visit v (vi (u, v) or otherwise) = v finishes efore u if v visited efore u: grph is yli = u nnot e rehed from v = visit to v finishes efore visiting u 6

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