Greedy algorithms. Shortest paths in weighted graphs. Tyler Moore. Shortest-paths problem. Shortest path applications.

Transcription

1 Shortet-path problem Greedy algorithm Shortet path in weighted graph Problem. Gien a digraph G = (V, E), edge length e 0, orce V, and detination t V, find the hortet directed path from to t. Tyler Moore CS, The Unierity of Tla orce 0 Some lide created by or adapted from Dr. Kein Wayne. For more information ee Some code reed from Python Algorithm by Magn Lie Hetland. 0 detination t length of path = = / Car naigation Shortet path application PERT/CPM. Map roting. Seam caring. Robot naigation. Textre mapping. Typeetting in LaTeX. Urban traffic planning. Telemarketer operator chedling. Roting of telecommnication meage. Network roting protocol (OSPF, BGP, RIP). Optimal trck roting throgh gien traffic congetion pattern. Reference: Network Flow: Theory, Algorithm, and Application, R. K. Ahja, T. L. Magnanti, and J. B. Orlin, Prentice Hall,. / /

2 The many cae of finding hortet path Weighted Graph Data Strctre Neted Adjacency Dictionarie w/ Edge Weight We e already een how to calclate the hortet path in an nweighted graph (BFS traeral) We ll now tdy how to compte the hortet path in different circmtance for weighted graph Single-orce hortet path on a weighted DAG Single-orce hortet path on a weighted graph with nonnegatie weight (Dijktra algorithm) a b d c e f g h N = { a : { b :, c :, d :, e :, f : }, b : { c :, e : }, c : { d : }, d : { e : }, e : { f : }, f : { c :, g :, h : }, g : { f :, h : }, h : { f :, g : } } >>> b i n N[ a ] # Neighborhood memberh Tre >>> l e n (N[ f ] ) # Degree >>> N[ a ] [ b ] # Edge w e i g h t f o r ( a, b ) / / Shortet path in DAG Recrie oltion to finding hortet path in DAG Recrie approach to finding the hortet path from a to z Ame we already know the ditance d() to z for each of a neighbor G[a] Select the neighbor that minimize d() + W (a, ) def r e c d a g p (W,, t ) : #S h o r t e t path from to #Memoize f def d ( ) : #D i t a n c e from to t i f == t : retrn 0#We r e t h e r e! # Retrn the b e t o f e e r y f i r t t e p retrn min (W[ ] [ ]+d ( ) f o r i n W[ ] ) retrn d ( ) #Apply f to a c t a l t a r t node / /

3 Shortet path in DAG: Iteratie approach Relaxing edge The iteratie oltion i a bit more complicated We mt tart with a topological ort Keep track of an pper bond on the ditance from a to each node, initialized to Go throgh each ertex and relax the ditance etimate by inpecting the path from the ertex to it neighbor In general, relaxing an edge (, ) conit of teting whether we can horten the path to fond o far by going throgh ; if we can, we pdate d[] with the new ale Rnning time: Θ(m + n) : d[] = : d[] = : W[][] = / 0 / Relaxing edge Iteratie oltion to finding hortet path in DAG : d[] = 0 : d[] = : W[][] = i n f = f l o a t ( i n f ) def r e l a x (W,,, D, P ) : d = D. get (, i n f ) + W[ ] [ ]# P o i b l e h o r t c t e t i m a t e i f d < D. get (, i n f ) : # I i t r e a l l y a h o r t c t? D[ ], P [ ] = d, # Update e t i m a t e and p a r e n t retrn Tre # There wa a change! def dag p (W,, t ) : #S h o r t e t path from to t d = { : f l o a t ( i n f ) f o r i n W} # D i t a n c e e t i m d [ ] = 0 #S t a r t node : Zero d i t a n c e f o r i n t o p o r t (W) : #I n top o r t e d o r d e r... i f == t : break #Hae we a r r i e d? f o r i n W[ ] : #For each ot edge... d [ ] = min ( d [ ], d [ ] + W[ ] [ ] ) # Relax the edg retrn d [ t ] #D i t a n c e to t ( from ) 0 / /

4 Shortet-path on weighted DAG example Shortet-path on weighted DAG: exercie b d a c e Topological ort: a, c, b, d, e d[node]: pper bd. dit. from a Node init. (=a) (=c) (=b) (=d) (=e) a b c d e 0 0 / / Bt what if there are cycle? With a DAG, we can elect the order in which to iit node baed on the topological ort With cycle we can t eaily determine the bet order If there are no negatie edge, we can traere from the tarting ertex, iiting node in order of their etimated ditance from the tarting ertex In Dijktra algorithm, we e a priority qee baed on minimm etimated ditance from the orce to elect which ertice to iit Rnning time: Θ((m + n) lg n) Dijktra algorithm combine approache een in other algorithm Node dicoery: bit like breadth-firt traeral Node iitation: elected ing priority qee Shortet path calclation: e relaxation a in algorithm for hortet path in DAG Dijktra' algorithm Greedy approach. Maintain a et of explored node S for which algorithm ha determined the hortet path ditance d() from to. Initialize S = { }, d() = 0. Repeatedly chooe nexplored node which minimize S d() e hortet path to ome node in explored part, followed by a ingle edge (, ) / /

5 Dijktra' algorithm Dijktra' algorithm: proof of correctne Greedy approach. Maintain a et of explored node S for which algorithm ha determined the hortet path ditance d() from to. Initialize S = { }, d() = 0. Repeatedly chooe nexplored node which minimize add to S, and et d() = π(). S d() e hortet path to ome node in explored part, followed by a ingle edge (, ) d() Inariant. For each node S, d() i the length of the hortet path. Pf. [ by indction on S ] Bae cae: S = i eay ince S = { } and d() = 0. Indctie hypothei: Ame tre for S = k. Let be next node added to S, and let (, ) be the final edge. The hortet path pl (, ) i an path of length π(). Conider any path P. We how that it i no horter than π(). Let (x, y) be the firt edge in P that leae S, and let P' be the bpath to x. P i already too long a oon a it reache y. (P) (P') + (x, y) nonnegatie length d(x) + (x, y) indctie hypothei π (y) definition of π(y) π () Dijktra choe intead of y S P' x y P / / Dijktra' algorithm: efficient implementation Dijktra algorithm Critical optimization. For each nexplored node, explicitly maintain π() intead of compting directly from formla: For each S, π () can only decreae (becae S only increae). More pecifically, ppoe i added to S and there i an edge (, ) leaing. Then, it ffice to pdate: Critical optimization. Ue a priority qee to chooe the nexplored node that minimize π (). π() = min e = (,) : S d() + e. π () = min { π (), d() + (, ) } from heapq import heapph, heappop def d i j k t r a (G, ) : D, P, Q, S = { : 0 }, {}, [ ( 0, ) ], et ( ) # Et., t r e e, q while Q: # S t i l l n p r o c e e d node?, = heappop (Q) # Node with l o w e t e t i m a t e i f i n S : contine # A l r e a d y i i t e d? Skip i t S. add ( ) # We e i i t e d i t now f o r i n G[ ] : # Go throgh a l l i t n e i g h b o r r e l a x (G,,, D, P) # Relax the ot edge heapph (Q, (D[ ], ) )# Add to qee, w/ e t. a p r i retrn D, P # F i n a l D and P r e t r n e d / /

6 Dijktra algorithm example Dijktra algorithm: exercie b d 0 a c e d[node]: pper bd. dit. from a Node init. (=a) (=c) (=e) (=b) (=d) a b 0 c d e 0 / /