24 Single-Source Shortest Paths

Transcription

1 Single-Source Shore Pah Profeor Parick wihe o find he hore poible roue from Phoeni o Indianapoli. Given a road map of he Unied Sae on which he diance beween each pair of adjacen inerecion i marked, how can he deermine hi hore roue? One poible wa would be o enumerae all he roue from Phoeni o Indianapoli, add up he diance on each roue, and elec he hore. I i ea o ee, however, ha even diallowing roue ha conain ccle, Profeor Parick would have o eamine an enormou number of poibiliie, mo of which are impl no worh conidering. For eample, a roue from Phoeni o Indianapoli ha pae hrough Seale i obvioul a poor choice, becaue Seale i everal hundred mile ou of he wa. In hi chaper and in Chaper, we how how o olve uch problem efficienl. In a hore-pah problem, wearegivenaweighed,direcedgraph G D.V; E/, wihweighfuncionw W E! R mapping edge o real-valued weigh. The weigh w.p/ of pah p Dh ; ;:::; k i i he um of he weigh of i coniuen edge: w.p/ D kx w. i ; i /: id We define he hore-pah weigh ı.u; / from u o b ( minfw.p/ W u p g if here i a pah from u o ; ı.u; / D oherwie : A hore pah from vere u o vere i hen defined a an pah p wih weigh w.p/ D ı.u; /. In he Phoeni-o-Indianapoli eample, we can model he road map a a graph: verice repreen inerecion, edge repreen road egmen beween inerecion, and edge weigh repreen road diance. Our goal i o find a hore pah from a given inerecion in Phoeni o a given inerecion in Indianapoli.

2 Chaper Single-Source Shore Pah Edge weigh can repreen meric oher han diance, uch a ime, co, penalie, lo, or an oher quani ha accumulae linearl along a pah and ha we would wan o minimie. The breadh-fir-earch algorihm from Secion. i a hore-pah algorihm ha work on unweighed graph, ha i, graph in which each edge ha uni weigh. Becaue man of he concep from breadh-fir earch arie in he ud of hore pah in weighed graph, ou migh wan o review Secion. before proceeding. Varian In hi chaper, we hall focu on he ingle-ource hore-pah problem: given agraphg D.V; E/, wewanofindahorepahfromagivenource vere V o each vere V. The algorihm for he ingle-ource problem can olve man oher problem, including he following varian. Single-deinaion hore-pah problem: Find a hore pah o a given deinaion vere from each vere. Breveringhedirecionofeachedgein he graph, we can reduce hi problem o a ingle-ource problem. Single-pair hore-pah problem: Find a hore pah from u o for given verice u and. Ifweolveheingle-ourceproblemwihourcevereu, we olve hi problem alo. Moreover, all known algorihm for hi problem have he ame wor-cae ampoic running ime a he be ingle-ource algorihm. All-pair hore-pah problem: Find a hore pah from u o for ever pair of verice u and. Alhoughwecanolvehiproblembrunningaingleource algorihm once from each vere, we uuall can olve i faer. Addiionall, i rucure i inereing in i own righ. Chaper addree he all-pair problem in deail. Opimal ubrucure of a hore pah Shore-pah algorihm picall rel on he proper ha a hore pah beween wo verice conain oher hore pah wihin i. (The Edmond-Karp maimum-flow algorihm in Chaper alo relie on hi proper.) Recall ha opimal ubrucure i one of he ke indicaor ha dnamic programming (Chaper ) and he greed mehod (Chaper ) migh appl. Dijkra algorihm, which we hall ee in Secion., i a greed algorihm, and he Flod- Warhall algorihm, which find hore pah beween all pair of verice (ee Secion.), i a dnamic-programming algorihm. The following lemma ae he opimal-ubrucure proper of hore pah more preciel.

3 Chaper Single-Source Shore Pah Lemma. (Subpah of hore pah are hore pah) Given a weighed, direced graph G D.V; E/ wih weigh funcion w W E! R, le p Dh ; ;:::; k i be a hore pah from vere o vere k and, for an i and j uch ha i j k, lep ij Dh i ; ic ;:::; j i be he ubpah of p from vere i o vere j.then,p ij i a hore pah from i o j. p Proof If we decompoe pah p ino i p ij p jk i j k,henwehaveha w.p/ D w.p i / C w.p ij / C w.p jk /.Now,aumehahereiapahpij from i o j wih weigh w.pij /<w.p p ij /. Then, i pij p jk i j k i a pah from o k whoe weigh i /Cw.pij /Cw.p jk/ i le han w.p/,whichconradic he aumpion ha p i a hore pah from o k. Negaive-weigh edge Some inance of he ingle-ource hore-pah problem ma include edge whoe weigh are negaive. If he graph G D.V; E/ conain no negaiveweigh ccle reachable from he ource, henforall V,hehore-pah weigh ı.; / remain well defined, even if i ha a negaive value. If he graph conain a negaive-weigh ccle reachable from, however,hore-pahweigh are no well defined. No pah from o a vere on he ccle can be a hore pah we can alwa find a pah wih lower weigh b following he propoed hore pah and hen ravering he negaive-weigh ccle. If here i a negaiveweigh ccle on ome pah from o,wedefineı.; / D. Figure. illurae he effec of negaive weigh and negaive-weigh ccle on hore-pah weigh. Becaue here i onl one pah from o a (he pah h; ai), we have ı.; a/ D w.;a/ D. Similarl, here i onl one pah from o b, andoı.; b/ D w.;a/ C w.a;b/ D C. / D. There are infiniel man pah from o c: h; ci, h; c; d; ci, h; c; d; c; d; ci, andoon. Becaue he ccle hc;d;ci ha weigh C. / D >,hehorepahfrom o c i h;ci,wihweighı.; c/ D w.;c/ D. Similarl,hehorepahfrom o d i h;c;di,wihweighı.; d/ D w.;c/cw.c;d/ D. Analogoul,here are infiniel man pah from o e: h; ei, h; e; f; ei, h; e; f; e; f; ei, ando on. Becaue he ccle he; f; ei ha weigh C. / D <,however,here i no hore pah from o e. B ravering he negaive-weigh ccle he; f; ei arbiraril man ime, we can find pah from o e wih arbiraril large negaive weigh, and o ı.; e/ D.Similarl,ı.; f / D.Becauegi reachable from f,wecanalofindpahwiharbirarillargenegaiveweighfrom o g, and o ı.; g/ D.Vericeh, i,andj alo form a negaive-weigh ccle. The are no reachable from, however,and oı.; h/ D ı.; i/ D ı.; j / D.

4 Chaper Single-Source Shore Pah a c e b d f 8 g h 8 j i Figure. Negaive edge weigh in a direced graph. The hore-pah weigh from ource appear wihin each vere. Becaue verice e and f form a negaive-weigh ccle reachable from, he have hore-pah weigh of. Becauevereg i reachable from a vere whoe horepah weigh i, i,oo,haahore-pahweighof. Vericeuchah, i, andj are no reachable from,andoheirhore-pahweighare,evenhoughhelieonanegaive-weigh ccle. Some hore-pah algorihm, uch a Dijkra algorihm, aume ha all edge weigh in he inpu graph are nonnegaive, a in he road-map eample. Oher, uch a he Bellman-Ford algorihm, allow negaive-weigh edge in he inpu graph and produce a correc anwer a long a no negaive-weigh ccle are reachable from he ource. Tpicall, if here i uch a negaive-weigh ccle, he algorihm can deec and repor i eience. Ccle Can a hore pah conain a ccle? A we have ju een, i canno conain a negaive-weigh ccle. Nor can i conain a poiive-weigh ccle, ince removing he ccle from he pah produce a pah wih he ame ource and deinaion verice and a lower pah weigh. Tha i, if p Dh ; ;:::; k i i a pah and c Dh i ; ic ;:::; j i i a poiive-weigh ccle on hi pah (o ha i D j and w.c/ > ), hen he pah p Dh ; ;:::; i ; j C ; j C ;:::; k i ha weigh w.p / D w.p/ w.c/ < w.p/,andop canno be a hore pah from o k. Tha leave onl -weigh ccle. We can remove a -weigh ccle from an pah o produce anoher pah whoe weigh i he ame. Thu, if here i a hore pah from a ource vere o a deinaion vere ha conain a -weigh ccle, hen here i anoher hore pah from o wihou hi ccle. A long a a hore pah ha -weigh ccle, we can repeaedl remove hee ccle from he pah unil we have a hore pah ha i ccle-free. Therefore, wihou lo of generali we can aume ha when we are finding hore pah, he have no ccle, i.e., he are imple pah. Since an acclic pah in a graph G D.V; E/

5 Chaper Single-Source Shore Pah conain a mo jv j diinc verice, i alo conain a mo jv j edge. Thu, we can reric our aenion o hore pah of a mo jv j edge. Repreening hore pah We ofen wih o compue no onl hore-pah weigh, bu he verice on hore pah a well. We repreen hore pah imilarl o how we repreened breadh-fir ree in Secion.. Given a graph G D.V; E/, wemainainfor each vere V a predeceor : ha i eiher anoher vere or NIL. The hore-pah algorihm in hi chaper e he aribue o ha he chain of predeceor originaing a a vere run backward along a hore pah from o. Thu, given a vere for which : NIL, heprocedureprint-path.g; ; / from Secion. will prin a hore pah from o. In he mid of eecuing a hore-pah algorihm, however, he value migh no indicae hore pah. A in breadh-fir earch, we hall be inereed in he predeceor ubgraph G D.V ;E / induced b he value. Here again, we define he vere e V o be he e of verice of G wih non-nil predeceor, plu he ource : V D f V W : NILg [ fg : The direced edge e E i he e of edge induced b he value for verice in V : E D f.:; / E W V fgg : We hall prove ha he value produced b he algorihm in hi chaper have he proper ha a erminaion G i a hore-pah ree informall, a rooed ree conaining a hore pah from he ource o ever vere ha i reachable from. Ahore-pahreeilikehebreadh-firreefromSecion.,bui conain hore pah from he ource defined in erm of edge weigh inead of number of edge. To be precie, le G D.V; E/ be a weighed, direced graph wih weigh funcion w W E! R, and aume ha G conain no negaive-weigh ccle reachable from he ource vere V,o ha hore pah are well defined. A hore-pah ree rooed a i a direced ubgraph G D.V ;E /, where V V and E E, uchha. V i he e of verice reachable from in G,. G form a rooed ree wih roo, and. for all V,heuniqueimplepahfrom o in G i a hore pah from o in G.

6 8 Chaper Single-Source Shore Pah (a) (b) (c) Figure. (a) Aweighed,direcedgraphwihhore-pahweighfromource. (b) The haded edge form a hore-pah ree rooed a he ource. (c) Anoher hore-pah ree wih he ame roo. Shore pah are no necearil unique, and neiher are hore-pah ree. For eample, Figure. how a weighed, direced graph and wo hore-pah ree wih he ame roo. Relaaion The algorihm in hi chaper ue he echnique of relaaion. For each vere V,wemainainanaribue:d, whichianupperboundonheweighof a hore pah from ource o. We call :d a hore-pah eimae. We iniialie he hore-pah eimae and predeceor b he following.v /-ime procedure: INITIALIZE-SINGLE-SOURCE.G; / for each vere G:V :d D : D NIL :d D Afer iniialiaion, we have : D NIL for all V, :d D, and:d Dfor V fg. The proce of relaing an edge.u; / coni of eing wheher we can improve he hore pah o found o far b going hrough u and, if o, updaing :d and :. A relaaion ep ma decreae he value of he hore-pah I ma eem range ha he erm relaaion i ued for an operaion ha ighen an upper bound. The ue of he erm i hiorical. The oucome of a relaaion ep can be viewed a a relaaion of he conrain :d u:d C w.u;/, which,bheriangleinequali(lemma.),mube aified if u:d D ı.; u/ and :d D ı.; /. Thai,if:d u:d C w.u;/, hereino preure o aif hi conrain, o he conrain i relaed.

7 Chaper Single-Source Shore Pah u v RELAX(u,v,w) u v (a) u v RELAX(u,v,w) u v (b) Figure. Relaing an edge.u; / wih weigh w.u;/ D. Thehore-paheimaeofeach vere appear wihin he vere. (a) Becaue :d >u:dcw.u;/ prior o relaaion, he value of :d decreae. (b) Here, :d u:d C w.u;/ before relaing he edge, and o he relaaion ep leave :d unchanged. eimae :d and updae predeceor aribue :. The following code perform a relaaion ep on edge.u; / in O./ ime: RELAX.u; ; w/ if :d >u:d C w.u;/ :d D u:d C w.u;/ : D u Figure. how wo eample of relaing an edge, one in which a hore-pah eimae decreae and one in which no eimae change. Each algorihm in hi chaper call INITIALIZE-SINGLE-SOURCE and hen repeaedl relae edge. Moreover, relaaion i he onl mean b which horepah eimae and predeceor change. The algorihm in hi chaper differ in how man ime he rela each edge and he order in which he rela edge. Dijkra algorihm and he hore-pah algorihm for direced acclic graph rela each edge eacl once. The Bellman-Ford algorihm relae each edge jv j ime. Properie of hore pah and relaaion To prove he algorihm in hi chaper correc, we hall appeal o everal properie of hore pah and relaaion. We ae hee properie here, and Secion. prove hem formall. For our reference, each proper aed here include he appropriae lemma or corollar number from Secion.. The laer five of hee properie, which refer o hore-pah eimae or he predeceor ubgraph, implicil aume ha he graph i iniialied wih a call o INITIALIZE- SINGLE-SOURCE.G; / and ha he onl wa ha hore-pah eimae and he predeceor ubgraph change are b ome equence of relaaion ep.

8 Chaper Single-Source Shore Pah Triangle inequali (Lemma.) For an edge.u; / E, wehaveı.; / ı.; u/ C w.u;/. Upper-bound proper (Lemma.) We alwa have :d ı.; / for all verice V,andonce:d achieve he value ı.; /, ineverchange. No-pah proper (Corollar.) If here i no pah from o,henwealwahave:d D ı.; / D. Convergence proper (Lemma.) If u! i a hore pah in G for ome u; V,andifu:d D ı.; u/ a an ime prior o relaing edge.u; /,hen:d D ı.; / a all ime aferward. Pah-relaaion proper (Lemma.) If p Dh ; ;:::; k i i a hore pah from D o k,andwerelahe edge of p in he order. ; /;. ; /; : : : ;. k ; k /,hen k :d D ı.; k /. Thi proper hold regardle of an oher relaaion ep ha occur, even if he are inermied wih relaaion of he edge of p. Predeceor-ubgraph proper (Lemma.) Once :d D ı.; / for all V,hepredeceorubgraphiahore-pah ree rooed a. Chaper ouline Secion. preen he Bellman-Ford algorihm, which olve he ingle-ource hore-pah problem in he general cae in which edge can have negaive weigh. The Bellman-Ford algorihm i remarkabl imple, and i ha he furher benefi of deecing wheher a negaive-weigh ccle i reachable from he ource. Secion. give a linear-ime algorihm for compuing hore pah from a ingle ource in a direced acclic graph. Secion. cover Dijkra algorihm, which ha a lower running ime han he Bellman-Ford algorihm bu require he edge weigh o be nonnegaive. Secion. how how we can ue he Bellman-Ford algorihm o olve a pecial cae of linear programming. Finall, Secion. prove he properie of hore pah and relaaion aed above. We require ome convenion for doing arihmeic wih infiniie. We hall aume ha for an real number a,wehavea CDCa D.Alo,o make our proof hold in he preence of negaive-weigh ccle, we hall aume ha for an real number a,wehavea C. / D. / C a D. All algorihm in hi chaper aume ha he direced graph G i ored in he adjacenc-li repreenaion. Addiionall, ored wih each edge i i weigh, o ha a we ravere each adjacenc li, we can deermine he edge weigh in O./ ime per edge.

9 . The Bellman-Ford algorihm. The Bellman-Ford algorihm The Bellman-Ford algorihm olve he ingle-ource hore-pah problem in he general cae in which edge weigh ma be negaive. Given a weighed, direced graph G D.V; E/ wih ource and weigh funcion w W E! R, he Bellman-Ford algorihm reurn a boolean value indicaing wheher or no here i anegaive-weighcclehaireachablefromheource. Ifhereiuchaccle, he algorihm indicae ha no oluion ei. If here i no uch ccle, he algorihm produce he hore pah and heir weigh. The algorihm relae edge, progreivel decreaing an eimae :d on he weigh of a hore pah from he ource o each vere V unil i achieve he acual hore-pah weigh ı.; /. The algorihm reurn TRUE if and onl if he graph conain no negaive-weigh ccle ha are reachable from he ource. BELLMAN-FORD.G; w; / INITIALIZE-SINGLE-SOURCE.G; / for i D o jg:vj for each edge.u; / G: E RELAX.u; ; w/ for each edge.u; / G: E if :d >u:d C w.u;/ reurn FALSE 8 reurn TRUE Figure. how he eecuion of he Bellman-Ford algorihm on a graph wih verice. Afer iniialiing he d and value of all verice in line, he algorihm make jv j pae over he edge of he graph. Each pa i one ieraion of he for loop of line and coni of relaing each edge of he graph once. Figure.(b) (e) how he ae of he algorihm afer each of he four pae over he edge. Afer making jv j pae, line 8 check for a negaive-weigh ccle and reurn he appropriae boolean value. (We ll ee a lile laer wh hi check work.) The Bellman-Ford algorihm run in ime O.VE/, inceheiniialiaionin line ake.v / ime, each of he jv j pae over he edge in line ake.e/ ime, and he for loop of line ake O.E/ ime. To prove he correcne of he Bellman-Ford algorihm, we ar b howing ha if here are no negaive-weigh ccle, he algorihm compue correc hore-pah weigh for all verice reachable from he ource.

10 Chaper Single-Source Shore Pah (a) (b) (c) 8 8 (d) (e) Figure. The eecuion of he Bellman-Ford algorihm. The ource i vere. The d value appear wihin he verice, and haded edge indicae predeceor value: if edge.u; / i haded, hen : D u. Inhipariculareample, eachparelaeheedgeinheorder.; /;.; /;.; /;.; /;.; /;.; /;. ; /;. ; /;.; /;.; /. (a) The iuaion ju before he fir pa over he edge. (b) (e) The iuaion afer each ucceive pa over he edge. The d and value in par (e) are he final value. The Bellman-Ford algorihm reurn TRUE in hi eample. Lemma. Le G D.V; E/ be a weighed, direced graph wih ource and weigh funcion w W E! R, andaumehag conain no negaive-weigh ccle ha are reachable from. Then, afer he jv j ieraion of he for loop of line of BELLMAN-FORD, wehave:d D ı.; / for all verice ha are reachable from. Proof We prove he lemma b appealing o he pah-relaaion proper. Conider an vere ha i reachable from, andlep Dh ; ;:::; k i,where D and k D, beanhorepahfrom o. Becauehorepahare imple, p ha a mo jv j edge, and o k jv j. EachofhejV j ieraion of he for loop of line relae all jej edge. Among he edge relaed in he ih ieraion, for i D ; ; : : : ; k, i. i ; i /.Bhepah-relaaionproper, herefore, :d D k :d D ı.; k / D ı.; /.

11 . The Bellman-Ford algorihm Corollar. Le G D.V; E/ be a weighed, direced graph wih ource vere and weigh funcion w W E! R, andaumehag conain no negaive-weigh ccle ha are reachable from. Then,foreachvere V,hereiapahfrom o if and onl if BELLMAN-FORD erminae wih :d < when i i run on G. Proof The proof i lef a Eercie.-. Theorem. (Correcne of he Bellman-Ford algorihm) Le BELLMAN-FORD be run on a weighed, direced graph G D.V; E/ wih ource and weigh funcion w W E! R. If G conain no negaive-weigh ccle ha are reachable from, henhealgorihmreurntrue, wehave:d D ı.; / for all verice V,andhepredeceorubgraphG i a hore-pah ree rooed a. If G doe conain a negaive-weigh ccle reachable from, hen he algorihm reurn FALSE. Proof Suppoe ha graph G conain no negaive-weigh ccle ha are reachable from he ource. Wefirproveheclaimhaaerminaion,:d D ı.; / for all verice V.Ifvere i reachable from,henlemma.provehi claim. If i no reachable from, henheclaimfollowfromheno-pahproper. Thu, he claim i proven. The predeceor-ubgraph proper, along wih he claim, implie ha G i a hore-pah ree. Now we ue he claim o how ha BELLMAN-FORD reurn TRUE. Aerminaion,wehaveforalledge.u; / E, :d D ı.; / ı.; u/ C w.u;/ (b he riangle inequali) D u:d C w.u;/ ; and o none of he e in line caue BELLMAN-FORD o reurn FALSE. Therefore, i reurn TRUE. Now, uppoe ha graph G conain a negaive-weigh ccle ha i reachable from he ource ; lehicclebec Dh ; ;:::; k i,where D k.then, kx w. i ; i /<: (.) id Aume for he purpoe of conradicion ha he Bellman-Ford algorihm reurn TRUE. Thu, i :d i :d C w. i ; i / for i D ; ; : : : ; k. Summing he inequaliie around ccle c give u

12 Chaper Single-Source Shore Pah kx i :d id D kx. i :d C w. i ; i // id kx i :d C id kx w. i ; i /: id Since D k,eachvereinc appear eacl once in each of he ummaion P k id i:d and P k id i :d, ando kx i :d D id kx i :d : id Moreover, b Corollar., i :d i finie for i D ; ; : : : ; k. Thu, kx w. i ; i /; id which conradic inequali (.). We conclude ha he Bellman-Ford algorihm reurn TRUE if graph G conain no negaive-weigh ccle reachable from he ource, and FALSE oherwie. Eercie.- Run he Bellman-Ford algorihm on he direced graph of Figure., uing vere a he ource. In each pa, rela edge in he ame order a in he figure, and how he d and value afer each pa. Now, change he weigh of edge. ; / o and run he algorihm again, uing a he ource..- Prove Corollar...- Given a weighed, direced graph G D.V; E/ wih no negaive-weigh ccle, le m be he maimum over all verice V of he minimum number of edge in a hore pah from he ource o. (Here,hehorepahibweigh,no he number of edge.) Sugge a imple change o he Bellman-Ford algorihm ha allow i o erminae in m C pae, even if m i no known in advance..- Modif he Bellman-Ford algorihm o ha i e :d o for all verice for which here i a negaive-weigh ccle on ome pah from he ource o.

13 . Single-ource hore pah in direced acclic graph.-? Le G D.V; E/ be a weighed, direced graph wih weigh funcion w W E! R. Give an O.VE/-ime algorihm o find, for each vere V,hevalueı./ D min uv fı.u; /g..-? Suppoe ha a weighed, direced graph G D.V; E/ ha a negaive-weigh ccle. Give an efficien algorihm o li he verice of one uch ccle. Prove ha our algorihm i correc.. Single-ource hore pah in direced acclic graph B relaing he edge of a weighed dag (direced acclic graph) G D.V; E/ according o a opological or of i verice, we can compue hore pah from aingleourcein.v C E/ ime. Shore pah are alwa well defined in a dag, ince even if here are negaive-weigh edge, no negaive-weigh ccle can ei. The algorihm ar b opologicall oring he dag (ee Secion.) o impoe a linear ordering on he verice. If he dag conain a pah from vere u o vere,henu precede in he opological or. We make ju one pa over he verice in he opologicall ored order. A we proce each vere, we rela each edge ha leave he vere. DAG-SHORTEST-PATHS.G; w; / opologicall or he verice of G INITIALIZE-SINGLE-SOURCE.G; / for each vere u, akeninopologicalloredorder for each vere G:AdjŒu RELAX.u; ; w/ Figure. how he eecuion of hi algorihm. The running ime of hi algorihm i ea o anale. A hown in Secion., he opological or of line ake.v C E/ ime. The call of INITIALIZE- SINGLE-SOURCE in line ake.v / ime. The for loop of line make one ieraion per vere. Alogeher, he for loop of line relae each edge eacl once. (We have ued an aggregae anali here.) Becaue each ieraion of he inner for loop ake./ ime, he oal running ime i.v C E/,whichilinear in he ie of an adjacenc-li repreenaion of he graph. The following heorem how ha he DAG-SHORTEST-PATHS procedure correcl compue he hore pah.

14 Chaper Single-Source Shore Pah r (a) r (c) r (e) r (b) r (d) r (f) r (g) Figure. The eecuion of he algorihm for hore pah in a direced acclic graph. The verice are opologicall ored from lef o righ. The ource vere i. Thed value appear wihin he verice, and haded edge indicae he value. (a) The iuaionbefore he firieraion of he for loop of line. (b) (g) The iuaion afer each ieraion of he for loop of line. The newl blackened vere in each ieraion wa ued a u in ha ieraion. The value hown in par (g) are he final value. Theorem. If a weighed, direced graph G D.V; E/ ha ource vere and no ccle, hen a he erminaion of he DAG-SHORTEST-PATHS procedure, :d D ı.; / for all verice V,andhepredeceorubgraphG i a hore-pah ree. Proof We fir how ha :d D ı.; / for all verice V a erminaion. If i no reachable from, hen:d D ı.; / D b he no-pah proper. Now, uppoe ha i reachable from, ohahereiahor- e pah p D h ; ;:::; k i,where D and k D. Becaue we pro-

15 . Single-ource hore pah in direced acclic graph ce he verice in opologicall ored order, we rela he edge on p in he order. ; /;. ; /; : : : ;. k ; k /. The pah-relaaion proper implie ha i :d D ı.; i / a erminaion for i D ; ; : : : ; k. Finall, b he predeceorubgraph proper, G i a hore-pah ree. An inereing applicaion of hi algorihm arie in deermining criical pah in PERT char anali. Edge repreen job o be performed, and edge weigh repreen he ime required o perform paricular job. If edge.u; / ener vere and edge.; / leave,henjob.u; / mu be performed before job.; /. Apahhroughhidagrepreenaequenceofjobhamubeperformedina paricular order. A criical pah i a longe pah hrough he dag, correponding o he longe ime o perform an equence of job. Thu, he weigh of a criical pah provide a lower bound on he oal ime o perform all he job. We can find acriicalpahbeiher negaing he edge weigh and running DAG-SHORTEST-PATHS, or running DAG-SHORTEST-PATHS, wih he modificaion ha we replace b inlineofinitialize-single-source and > b < inhe RELAX procedure. Eercie.- Run DAG-SHORTEST-PATHS on he direced graph of Figure., uing vere r a he ource..- Suppoe we change line of DAG-SHORTEST-PATHS o read for he fir jv j verice, aken in opologicall ored order Show ha he procedure would remain correc..- The PERT char formulaion given above i omewha unnaural. In a more naural rucure, verice would repreen job and edge would repreen equencing conrain; ha i, edge.u; / would indicae ha job u mu be performed before job. We would hen aign weigh o verice, no edge. Modif he DAG- SHORTEST-PATHS procedure o ha i find a longe pah in a direced acclic graph wih weighed verice in linear ime. PERT i an acronm for program evaluaion and review echnique.

16 8 Chaper Single-Source Shore Pah.- Give an efficien algorihm o coun he oal number of pah in a direced acclic graph. Anale our algorihm.. Dijkra algorihm Dijkra algorihm olve he ingle-ource hore-pah problem on a weighed, direced graph G D.V; E/ for he cae in which all edge weigh are nonnegaive. In hi ecion, herefore, we aume ha w.u;/ for each edge.u; / E. A we hall ee, wih a good implemenaion, he running ime of Dijkra algorihm i lower han ha of he Bellman-Ford algorihm. Dijkra algorihm mainain a e S of verice whoe final hore-pah weigh from he ource have alread been deermined. The algorihm repeaedl elec he vere u V S wih he minimum hore-pah eimae, add u o S, andrelaealledgeleavingu. Inhefollowingimplemenaion,weuea min-priori queue Q of verice, keed b heir d value. DIJKSTRA.G; w; / INITIALIZE-SINGLE-SOURCE.G; / S D; Q D G:V while Q ; u D EXTRACT-MIN.Q/ S D S [ fug for each vere G:AdjŒu 8 RELAX.u; ; w/ Dijkra algorihm relae edge a hown in Figure.. Line iniialie he d and value in he uual wa, and line iniialie he e S o he emp e. The algorihm mainain he invarian ha Q D V S a he ar of each ieraion of he while loop of line 8. Line iniialie he min-priori queue Q o conain all he verice in V ;inces D;a ha ime, he invarian i rue afer line. Each ime hrough he while loop of line 8, line erac a vere u from Q D V S and line add i o e S,herebmainainingheinvarian.(Thefir ime hrough hi loop, u D.) Vere u, herefore,hahemallehore-pah eimae of an vere in V S. Then,line 8relaeachedge.u; / leaving u, hu updaing he eimae :d and he predeceor : if we can improve he hore pah o found o far b going hrough u. Oberve ha he algorihm never iner verice ino Q afer line and ha each vere i eraced from Q

17 . Dijkra algorihm 8 (a) (b) (c) (d) (e) (f) Figure. The eecuion of Dijkra algorihm. The ource i he lefmo vere. The hore-pah eimae appear wihin he verice, and haded edge indicae predeceor value. Black verice are in he e S,andwhievericeareinhemin-prioriqueueQ D V S. (a) The iuaion ju before he fir ieraion of he while loop of line 8. The haded vere ha he minimum d value and i choen a vere u in line. (b) (f) The iuaion afer each ucceive ieraion of he while loop. The haded vere in each par i choen a vere u in line of he ne ieraion. The d value and predeceor hown in par (f) are he final value. and added o S eacl once, o ha he while loop of line 8 ierae eacl jv j ime. Becaue Dijkra algorihm alwa chooe he lighe or cloe vere in V S o add o e S,weahaiueagreedraeg.Chapereplain greed raegie in deail, bu ou need no have read ha chaper o underand Dijkra algorihm. Greed raegie do no alwa ield opimal reul in general, bu a he following heorem and i corollar how, Dijkra algorihm doe indeed compue hore pah. The ke i o how ha each ime i add a vere u o e S, wehaveu:d D ı.; u/. Theorem. (Correcne of Dijkra algorihm) Dijkra algorihm, run on a weighed, direced graph G D.V; E/ wih nonnegaive weigh funcion w and ource, erminaewihu:d D ı.; u/ for all verice u V.

18 Chaper Single-Source Shore Pah p S p u Figure. The proof of Theorem.. Se S i nonemp ju before vere u i added o i. We decompoe a hore pah p from ource o vere u ino p! p u, where i he fir vere on he pah ha i no in S and S immediael precede. Verice and are diinc, bu we ma have D or D u. Pahp ma or ma no reener e S. Proof We ue he following loop invarian: A he ar of each ieraion of he while loop of line 8, :d D ı.; / for each vere S. I uffice o how for each vere u V,wehaveu:d D ı.; u/ a he ime when u i added o e S. Oncewehowhau:d D ı.; u/, werelonheupper-bound proper o how ha he equali hold a all ime hereafer. Iniialiaion: Iniiall, S D;, andoheinvarianiriviallrue. Mainenance: We wih o how ha in each ieraion, u:d D ı.; u/ for he vere added o e S. For he purpoe of conradicion, le u be he fir vere for which u:d ı.; u/ when i i added o e S. We hall focu our aenion on he iuaion a he beginning of he ieraion of he while loop in which u i added o S and derive he conradicion ha u:d D ı.; u/ a ha ime b eamining a hore pah from o u. Wemuhaveu becaue i he fir vere added o e S and :d D ı.; / D a ha ime. Becaue u, we alo have ha S ;ju before u i added o S. There mu be ome pah from o u, foroherwieu:d D ı.; u/ Db he no-pah proper, which would violae our aumpion ha u:d ı.; u/. Becaue here i a lea one pah, here i a hore pah p from o u. Prioroaddingu o S, pah p connec a vere in S,namel,oavereinV S,namelu. Leu conider he fir vere along p uch ha V S, andle S be predeceor along p. Thu,aFigure.illurae,wecandecompoepahp ino p! p u. (Eiherofpahp or p ma have no edge.) We claim ha :d D ı.; / when u i added o S. To prove hi claim, oberve ha S. Then, becaue we choe u a he fir vere for which u:d ı.; u/ when i i added o S, wehad:d D ı.; / when wa added

19 . Dijkra algorihm o S. Edge.; / wa relaed a ha ime, and he claim follow from he convergence proper. We can now obain a conradicion o prove ha u:d D ı.; u/. Becaue appear before u on a hore pah from o u and all edge weigh are nonnegaive (noabl hoe on pah p ), we have ı.; / ı.; u/, andhu :d D ı.; / ı.; u/ (.) u:d (b he upper-bound proper). Bu becaue boh verice u and were in V S when u wa choen in line, we have u:d :d. Thu,hewoinequaliiein(.)areinfacequaliie, giving :d D ı.; / D ı.; u/ D u:d : Conequenl, u:d D ı.; u/, whichconradicourchoiceofu. Weconclude ha u:d D ı.; u/ when u i added o S, andhahiequaliimainaineda all ime hereafer. Terminaion: A erminaion, Q D;which, along wih our earlier invarian ha Q D V S,impliehaS D V.Thu,u:d D ı.; u/ for all verice u V. Corollar. If we run Dijkra algorihm on a weighed, direced graph G D.V; E/ wih nonnegaive weigh funcion w and ource, hen a erminaion,he predeceor ubgraph G i a hore-pah ree rooed a. Proof Immediae from Theorem. and he predeceor-ubgraph proper. Anali How fa i Dijkra algorihm? I mainain he min-priori queue Q b calling hree priori-queue operaion: INSERT (implici in line ), EXTRACT-MIN (line ), and DECREASE-KEY (implici in RELAX, whichicalledinline8).the algorihm call boh INSERT and EXTRACT-MIN once per vere. Becaue each vere u V i added o e S eacl once, each edge in he adjacenc li AdjŒu i eamined in he for loop of line 8 eacl once during he coure of he algorihm. Since he oal number of edge in all he adjacenc li i jej, hifor loop ierae a oal of jej ime, and hu he algorihm call DECREASE-KEY a mo jej ime overall. (Oberve once again ha we are uing aggregae anali.) The running ime of Dijkra algorihm depend on how we implemen he min-priori queue. Conider fir he cae in which we mainain he min-priori

20 Chaper Single-Source Shore Pah queue b aking advanage of he verice being numbered o jv j. We impl ore :d in he h enr of an arra. Each INSERT and DECREASE-KEY operaion ake O./ ime, and each EXTRACT-MIN operaion ake O.V / ime (ince we have o earch hrough he enire arra), for a oal ime of O.V C E/ D O.V /. If he graph i ufficienl pare in paricular, E D o.v = lg V/ we can improve he algorihm b implemening he min-priori queue wih a binar minheap. (A dicued in Secion., he implemenaion hould make ure ha verice and correponding heap elemen mainain handle o each oher.) Each EXTRACT-MIN operaion hen ake ime O.lg V/.Abefore,herearejV j uch operaion. The ime o build he binar min-heap i O.V /. EachDECREASE-KEY operaion ake ime O.lg V/, andhereareillamojej uch operaion. The oal running ime i herefore O..V C E/lg V/,whichiO.E lg V/if all verice are reachable from he ource. Thi running ime improve upon he raighforward O.V /-ime implemenaion if E D o.v = lg V/. We can in fac achieve a running ime of O.V lg V C E/ b implemening he min-priori queue wih a Fibonacci heap (ee Chaper ). The amoried co of each of he jv j EXTRACT-MIN operaion i O.lg V/,andeachDECREASE- KEY call, of which here are a mo jej, akeonlo./ amoried ime. Hioricall, he developmen of Fibonacci heap wa moivaed b he obervaion ha Dijkra algorihm picall make man more DECREASE-KEY call han EXTRACT-MIN call, o ha an mehod of reducing he amoried ime of each DECREASE-KEY operaion o o.lg V/ wihou increaing he amoried ime of EXTRACT-MIN would ield an ampoicall faer implemenaion han wih binar heap. Dijkra algorihm reemble boh breadh-fir earch (ee Secion.) and Prim algorihm for compuing minimum panning ree (ee Secion.). I i like breadh-fir earch in ha e S correpond o he e of black verice in a breadh-fir earch; ju a verice in S have heir final hore-pah weigh, o do black verice in a breadh-fir earch have heir correc breadh-fir diance. Dijkra algorihm i like Prim algorihm in ha boh algorihm ue a minpriori queue o find he lighe vere ouide a given e (he e S in Dijkra algorihm and he ree being grown in Prim algorihm), add hi vere ino he e, and adju he weigh of he remaining verice ouide he e accordingl. Eercie.- Run Dijkra algorihm on he direced graph of Figure., fir uing vere a he ource and hen uing vere a he ource. In he le of Figure., how he d and value and he verice in e S afer each ieraion of he while loop.