Solving Evcuion Prolem Efficienly Erlie Arrivl Flow wih Muliple Source Ndine Bumnn Univeriä Dormund, FB Mhemik 441 Dormund, Germny ndine.umnn@mh.uni-dormund.de Mrin Skuell Univeriä Dormund, FB Mhemik 441 Dormund, Germny mrin.kuell@uni-dormund.de Arc Erlie rrivl flow cpure he eence of evcuion plnning. Given nework wih cpciie nd rni ime on he rc, ue of ource node wih upplie nd ink node, he k i o end he given upplie from he ource o he ink quickly poile. The ler requiremen i mde more precie y he erlie rrivl propery which require h he ol moun of flow h h rrived he ink i mximl for ll poin in ime imulneouly. I i clicl reul from he 1970 h, for he pecil ce of ingle ource node, erlie rrivl flow do exi nd cn e compued y eenilly pplying he Succeive Shore Ph Algorihm for minco flow compuion. While i h previouly een oerved h n erlie rrivl flow ill exi for muliple ource, he prolem of compuing one efficienly h een open for mny yer. We preen n exc lgorihm for hi prolem whoe running ime i rongly polynomil in he inpu plu oupu ize of he prolem. 1. Inroducion In ypicl evcuion iuion, he mo imporn k i o ge people ou of n endngered uilding or re f poile. Since i i uully no known how long uilding cn wihnd fire efore i collpe or how long dm cn rei flood efore i rek, i i dvile o orgnize n evcuion uch h much poile i ved no mer when he inferno will cully hppen. In he more rc eing of nework flow over ime, he ler requiremen i cpured y o-clled erlie rrivl flow. Before we dicu hi in more deil, we fir give hor nd decripive inroducion ino flow over ime. Thi work w uppored y DFG Focu Progrm 116, Algorihmic Apec of Lrge nd Complex Nework, grn no. SK 58/4-1 nd SK 58/5-3. Flow over ime. We conider nework N = (V,A) wih cpciie u e 0 nd rni ime τ e 0 on he rc e A. The cpciy of n rc ound he flow re (i.e., flow per ime) which flow cn ener he rc. The rni ime of n rc pecifie he moun of ime i ke for flow o rvel from he il o he hed of he rc. Moreover, here i e of ource node S + V nd e of ink node S V \ S +. Ech ource S + h upply v() > 0 nd ech ink S demnd v() > 0 uch h w S + S v(w) =0. A flow over ime 1 pecifie for ech rc e nd ech poin in ime he flow re which flow ener he rc (nd leve he rc gin τ e ime uni ler). Flow conervion conrin require h every poin in ime nd for every inermedie node w V \ (S + S ) he flow enering nd leving node w mu cncel ou ech oher. Flow over ime hve een inroduced y Ford nd Fulkeron [8] (ee lo [9]). Given nework wih ingle ource node, ingle ink node, nd ime horizon θ 0, hey conider he prolem of ending much flow poile from o wihin ime θ. I urn ou h mximl --flow over ime cn e deermined y ic min-co flow compuion where rni ime of rc re inerpreed co coefficien. Ford nd Fulkeron [8] lo inroduce he concep of ime-expnded nework h coni of one copy of he node e of he given nework for ech ime uni (we cll uch copy ime lyer). For ech rc e of he originl nework wih rni ime τ e he imeexpnded nework conin copie connecing ny wo ime lyer dince τ e. For more deil, we refer o [8, 5]. On he poiive ide, mo flow over ime prolem cn e olved y ic flow compuion in 1 There exi wo differen u cloely reled model for flow over ime dicree nd coninuou model. We conider he coninuou model u he preened reul lo hold in he dicree model. For more deil on hi iue we refer o [7]. In order o diinguih hem from flow over ime, we refer o clicl nework flow lo ic flow. 1
ime-expnded nework. On he negive ide, imeexpnded nework re huge in heory nd in prcice. In priculr, he ize of ime expnded nework i liner in he given ime horizon θ nd herefore exponenil (u ill peudopolynomil) in he inpu ize. Hoppe nd Trdo [15] conider he quicke rnhipmen prolem which i defined follow. Given nework wih everl ource nd ink node wih given upplie nd demnd, find flow over ime wih miniml ime horizon θ h ifie ll upplie nd demnd. Hoppe nd Trdo give rongly polynomil lgorihm for hi prolem. They preen heir reul for he dicree ime model. Fleicher nd Trdo [7] how h i lo hold in he coninuou ime model. Erlie rrivl flow re moived y pplicion reled o evcuion. In he conex of emergency evcuion from uilding, Berlin [1] nd Chlme e l. [3] udy he quicke rnhipmen prolem in nework wih muliple ource nd ingle ink. Jrvi nd Rliff [19] 3 how h hree differen ojecive of hi opimizion prolem cn e chieved imulneouly: (1) Minimizing he ol ime needed o end he upplie of ll ource o he ink, () fulfilling he erlie rrivl propery, nd (3) minimizing he verge ime for ll flow needed o rech he ink. Hmcher nd Tufecki [13] udy n evcuion prolem nd propoe oluion which furher preven unnecery movemen wihin uilding. Erlie rrivl flow. Shorly fer Ford nd Fulkeron inroduced flow over ime, he more elore --erlie rrivl flow prolem w udied y Gle [10]. Here he gol i o find ingle --flow over ime h imulneouly mximize he moun of flow reching he ink up o ny ime θ 0. A flow over ime fulfilling hi requiremen i id o hve he erlie rrivl propery nd i clled erlie rrivl flow. Gle [10] howed h --erlie rrivl flow lwy exi. Miniek [6] nd Wilkinon [31] oh gve peudopolynomil-ime lgorihm for compuing erlie rrivl flow ed on he Succeive Shore Ph Algorihm [0, 16, ]. Hoppe nd Trdo [14] preen fully polynomil ime pproximion cheme for he erlie rrivl flow prolem h i ed on clever cling rick. In nework wih everl ource nd ink wih given upplie nd demnd, flow over ime hving he erlie rrivl propery do no necerily exi [4]. We give imple counerexmple wih one ource nd wo ink in Figure 1. For he ce of everl ource wih given upplie nd ingle ink, however, erlie rrivl flow do lwy exi [7]. Thi follow, for exmple, from he exience of lexicogrphiclly mximl flow in imeexpnded nework; ee, e.g., [6]. We refer o hi prolem he erlie rrivl rnhipmen prolem. Hjek nd Ogier [1] give he fir polynomil ime lgorihm for he erlie rrivl rnhipmen prolem wih zero rni ime. Fleicher [4] give n lgorihm wih improved running ime. Fleicher nd Skuell [6] ue condened ime-expnded nework o pproxime he erlie rrivl rnhipmen prolem for he ce of rirry rni ime. They give n FP- TAS h pproxime he ime dely follow: For every ime θ 0 he moun of flow h hould hve reched he ink in n erlie rrivl rnhipmen y ime θ, reche he ink le ime (1+ε)θ. Tjndr [30] how how o compue erlie rrivl rnhipmen in nework wih ime dependen upplie nd cpciie in ime polynomil in he ime horizon nd he ol upply ource. The reuling running ime i hu only peudopolynomil in he inpu ize. Our conriuion. While i h previouly een oerved h erlie rrivl rnhipmen exi in he generl muliple-ource ingle-ink eing, he prolem of compuing one efficienly h een open. All previou lgorihm rely on ime expnion of he nework ino exponenilly mny ime lyer. We olve hi open prolem nd preen n efficien lgorihm which, in priculr, doe no rely on ime expnion. Uing necery nd ufficien crierion for he feiiliy of rnhipmen over ime prolem y Klinz [1], we fir recurively conruc he erlie rrivl pern, h i, he piece-wie liner funcion h decrie he ime-dependen mximum flow vlue. Our lgorihm employ umodulr funcion minimizion wihin he prmeric erch frmework of Megiddo [4, 5]. A y-produc, we preen new proof for he exience of erlie rrivl flow h doe no rely on ime expnion. We finlly how how o urn he erlie rrivl pern ino n erlie rrivl flow ed on he quicke rnhipmen lgorihm of Hoppe nd Trdo [15]. The running ime of our lgorihm i polynomil in he inpu ize plu he numer of rekpoin of he erlie rrivl pern. Since he erlie rrivl pern i more or le explicily pr of he oupu of he erlie rrivl rnhipmen prolem, we cn y h he running ime of our lgorihm i polynomilly ounded in he inpu plu oupu ize. Ouline. In he nex ecion we e necery nd ufficien crierion for he feiiliy of rnhipmen over ime prolem nd pply i o our eing. In Secion 3 we give n in-deph nlyi of he rucure of he erlie rrivl pern nd preen recurive lgorihm o compue i. How o compue he cul erlie rrivl rnhipmen ou of he pern i finlly howninsecion4. 3 Sricly peking, Jrvi nd Rliff [19] only conider he ingle-ource ce u heir oervion lo pplie o he more generl ce wih muliple ource.
1 v() = τ =0 τ =0 v( 1)= 1 τ =1 v( )= 1 Figure 1. A nework wih one ource nd wo ink wih uni demnd for which n erlie rrivl flow doe no exi. All rc hve uni cpciy nd he rni ime re given in he drwing. Noice h one uni of flow cn rech ink 1 y ime 1; in hi ce, he econd uni of flow reche ink only y ime 3. Alernively we cn end one uni of flow ino ink 1 nd imulneouly y ime. Bu here doe no exi n erlie rrivl flow where oh flow uni mu hve reched heir ink y ime nd one of hem mu hve lredy rrived ime 1.. Preliminrie We conider nework wih cpciie nd rni ime on he rc, ource node S + nd ink node S wih upplie nd demnd v : S + S R. We mke ue of he following reul of Klinz [1] (ee lo [7]). Lemm.1 (Klinz [1]). For θ 0 nd X S + S le v(x) := w X v(w) nd le o θ (X) e he mximl moun of flow h cn e en from he ource S + X o he ink S \ X wihin ime θ (ignoring upplie nd demnd). There exi flow over ime wih ime horizon θ h ifie ll upplie nd demnd if nd only if o θ (X) v(x) for ll X S + S. For θ 0 nd X S + S,hevlueo θ (X) cn e oined y ic min-co flow compuion. Conider he exended nework N defined follow. Sring from N, inroduce uper ource h i conneced o ll ource S + X y n uncpcied rc wih rni ime zero nd uper ink h cn e reched from ll ink S \ X y uch n rc. By conrucion of N,hevlueo θ (X) i equl o he vlue of mximl --flow over ime in N wih ime horizon θ. Furher exend N y dding n uncpcied dummy rc from o. I follow from he work of Ford nd Fulkeron [8] h o θ (X) = min { co θ (x) x circulion in N }. (1) Here, co θ (x) denoe he co of circulion x where rni ime on rc re inerpreed co coefficien nd he co coefficien of dummy rc (, ) i θ. A conequence of (1), he funcion θ o θ (X) i he co funcion of prmeric min-co flow prolem. A uch, i i piecewie liner nd convex. Bed on he work of Megiddo [3], Hoppe nd Trdo [15] oerve h he funcion o θ : S + S R i umodulr, h i, o θ (X)+o θ (Y ) o θ (X Y )+o θ (X Y ) for ll X, Y S + S. In he following we reric o nework wih ingle ink. The erlie rrivl pern p : R + R + i defined y eing p(θ) o he mximl moun of flow h cn e en ino he ink y ime θ wihou violing upplie he ource. An erlie rrivl rnhipmen i flow over ime uch h p(θ) uni of flow hve rrived he ink y ime θ for ll θ 0 imulneouly. For he ce of ingle ource S + = {} wih unounded upply, he --erlie rrivl pern i p(θ) =o θ ({}) nd hu piecewie liner nd convex. For he ce of everl ource, he erlie rrivl pern p i ill piecewie liner (ee Corollry.3 elow) u no necerily convex. A imple exmple wih wo ource i given in Figure. Noice h in hi exmple he re of flow rriving he ink (i. e., he derivive of p) uddenly decree ince he enire upply of ource 1 h rrived nd hi ource h herefore run empy. In Secion 3 we will oerve hi effec in more generl conex. The following lemm i eenilly reformulion of Lemm.1 for he eing of erlie rrivl rnhipmen nd will ler urn ou o e ueful. The proof i echnicl nd will e conined in he full verion of he pper. Lemm.. Le θ, q 0. Thenp(θ) q if nd only if o θ (S ) q v(s + \ S ) for ll S S +. () A conequence of Lemm., we cn how h he erlie rrivl pern i piecewie liner funcion. Corollry.3. The erlie rrivl pern p i piecewie liner.
flow vlue v( 1)=1 1 3 1 v( )= 1 3 ime Figure. A imple exmple of grph wih wo ource, uni cpciie, nd uni rni ime where he opiml rrivl pern of feile erlie rrivl rnhipmen i piecewie liner nd non-convex. Proof. A reul of Lemm. we ge p(θ) = min{o θ (S )+v(s + \ S ) S S + }. Since θ o θ (S ) i piecewie liner (nd convex) funcion for ll S S +, he reul follow. In he nex ecion we how how we cn deermine he erlie rrivl pern of he erlie rrivl rnhipmen prolem. The erlie rrivl rnhipmen ielf cn hen e oined from he given erlie rrivl pern hown in Secion 4. 3. Conrucing he erlie rrivl pern Throughou hi ecion we ue he following exmple innce o illure he preened ide nd echnique. Exmple. Aume we re given nework on he lef hnd ide in Figure 3 wih uni rni ime nd uni cpciie. The upplie of he ource re given in he picure. 3.1. The rucure of he erlie rrivl pern We how h he erlie rrivl pern p i compoed of everl --erlie rrivl pern in exended nework wih n ddiionl uperource h i conneced o cerin ue of ource in S +. We r y conidering he exended nework N 0 h rie from connecing uperource o ll node in S + y n uncpcied, zero rni ime rc. The node in S + re no longer ource u ke he role of inermedie node in N 0 nd heir ol upply v(s + ) i hifed o he uperource. Thu, feile -flow over ime in he exended nework N 0 induce flow over ime in N where v(s + ) uni of flow re eing en from he ource in S + o ink. Noice, however, h he induced flow over ime in N migh viole individul upplie he ource node. The --erlie rrivl pern in N 0 i he funcion θ o θ (S + ).Foreveryθ 0 i hold h p(θ) o θ (S + ).Ifp(θ) =o θ (S + ) for ll θ 0, we re done ince we know how o oin he --erlie rrivl pern θ o θ (S + ). Oherwie, le θ 1 := up{θ p(θ) = o θ (S + )}. 4 We ue he following lemm o prove h p(θ) =o θ (S + ) for ll 0 θ θ 1. Lemm 3.1. Le S S S + nd 0 θ θ. Then, o θ (S ) o θ (S ) o θ (S ) o θ (S ). Proof. Conider n exended nework N wih n ddiionl ink h cn e reched from hrough n uncpcied rc (, ) wih rni ime θ θ.the underlying inuiion i h ll flow rriving efore ime θ cn e forwrded o he new ink where i rrive efore ime θ. For S S + {, } le ō θ ( S) denoe he mximum moun of flow h cn e en from he ource in S o he ink in (S + {, }) \ S y ime θ. By conrucion of N we ge for S S + he following equliie: ō θ ( S) = o θ ( S) nd ō θ ( S {}) = o θ ( S). (3) We cn now prove he emen of he lemm. By (3) nd umodulriy of ō θ ( ) we ge o θ (S ) o θ (S ) = ō θ (S {}) ō θ (S {}) ō θ (S ) ō θ (S ) = o θ (S ) o θ (S ). Thi conclude he proof. Corollry 3.. Le θ 1 =mx{θ p(θ) =o θ (S + )}. Then p(θ) =o θ (S + ) for ll 0 θ θ 1. Proof. Aume y conrdicion h p(θ) <o θ (S + ) for ome 0 θ < θ 1. By Lemm. here exi S S + wih o θ (S ) < o θ (S + ) v(s + \ S ). I follow from Lemm 3.1 h o θ1 (S ) < o θ1 (S + ) v(s + \ S ) uch h p(θ 1 ) <o θ1 (S + ) y Lemm.. Thi conrdic he choice of θ 1. 4 The upremum here i indeed mximum ince p(θ) nd o θ (S + ) re oh coninuou funcion of θ.
flow vlue o θ ({ 1,, 3}) 8 v( 1 1)=1 v( )=1 v() =7 1 6 4 v( 3 3)=5 3 1 3 4 5 ime θ Figure 3. Exmple of nework N =(V,A), he nework expnded y uperource, nd he correponding --erlie rrivl pern. Exmple. In order o compue he --erlie rrivl pern for he nework given in he lef pr of Figure 3 we iner uperource depiced in he middle pr of Figure 3. Applying he Succeive Shore Ph Algorihm o hi nework yield, for exmple, he wo ph P 1 = (, 1,,) nd P = (, 3,,,), oh wih flow re 1. The reuling rrivl pern up o ime 6 i given in he righ pr of Figure 3. Noice h he flow rriving ink node fer ime 3 viole he upply of node 1 ince more hn one uni of flow h een en hrough ph P 1. On he oher hnd i cn eily e een h we cn reroue he flow gining ph decompoiion wih P 1 = (, 3,,), P = (, 1,,,), nd P 3 = (,,,,) where he flow re on ph P 1 i 1 nd he flow re on ph P nd P 3 re only 1/. Noice h he flow rriving over hee ph he ink doe no viole upplie up o ime 5 nd h ill he me rrivl pern. Furher, here i no oher wy of ending flow oeying he upplie of ource 1,, 3 for longer hn 5 ime uni. Afer ime 5 he lope of he erlie rrivl pern p decree ince no more flow ou of ource 1 nd cn rech he ink. In priculr, he vlue of θ 1 equl 5. In our exmple, ny flow over ime in N h end p(θ 1 ) uni ino he ink y ime θ 1 mu ue up he upplie of ource 1 nd.inoherword, he ounded upplie of hee ource re he reon why p(θ) < o θ (S + ) for θ > θ 1. The nex lemm illumine hi effec for generl innce. Lemm 3.3. There exi ue of ource S 1 S + uch h o θ1 (S 1 ) = o θ1 (S + ) v(s + \ S 1 ). Before we prove he lemm, we fir give n inuiive inerpreion of i emen. In n erlie rrivl rnhipmen, p(θ 1 ) = o θ1 (S + ) uni of flow rech he ink y ime θ 1. The lemm e h mo o θ1 (S + ) v(s + \ S 1 ) of hee uni cn origine from ource in S 1. The remining v(s + \ S 1 ) uni mu origine from ource in S + \ S 1.Thee ource herefore run empy nd cnno conriue o flow rriving fer ime θ 1 he ink. Proof. By conrdicion ume h o θ1 (S ) > o θ1 (S + ) v(s + \ S ) for ll S S +. Since o θ (S ) nd o θ (S + ) re coninuou funcion of θ, here exi ɛ>0 uch h o θ1+ɛ (S ) o θ1+ɛ (S + ) v(s + \ S ) for ll S S +. By Lemm. hi implie p(θ 1 + ɛ) o θ1+ɛ (S + ). Thi conrdic he choice of θ 1. We conider he reduced innce of he erlie rrivl rnhipmen prolem h i oined y eing he upplie of ll ource in S + \ S 1 o zero. The erlie rrivl pern of he modified innce i denoed y p. The following heorem i he min reul of hi ecion. Theorem 3.4. Le θ 1 =mx{θ p(θ) =o θ (S + )} nd S 1 S + uch h o θ1 (S 1 )=o θ1 (S + ) v(s + \ S 1 ) (ee Lemm 3.3). Le p denoe he erlie rrivl pern of he modified innce wih ource e S 1. Then, { o θ (S + ) if θ<θ 1, p(θ) = p (θ)+v(s + \ S 1 ) if θ θ 1. A reul of Theorem 3.4, we hve reduced he prolem of conrucing he erlie rrivl pern p o he prolem of compuing n --erlie rrivl pern nd compuing n erlie rrivl pern for mller numer of ource S 1. Proof. I follow from Corollry 3. h p(θ) = o θ (S + ) for θ θ 1. I remin o how h p(θ) = p (θ)+v(s + \ S 1 ) for ll θ θ 1. I i cler h hold ince y ime θ mo p (θ) nd v(s + \ S 1 ) uni of flow cn rech he ink origining from ource in S 1 nd S + \ S 1, repecively.
I remin o how h hold, h i, p (θ) + v(s + \ S 1 ) uni of flow cn e en ino he ink y ime θ θ 1 wihou exceeding upplie he ource. We check he condiion given in Lemm.. For S S + nd θ θ 1 we ge y umodulriy of o θ ( ): o θ (S ) o θ (S S 1 )+o θ (S S 1 ) o θ (S 1 ) y Lemm 3.1: o θ (S S 1 )+o θ1 (S S 1 ) o θ1 (S 1 ) y Lemm. nd Lemm 3.3: ( ) p (θ) v(s 1 \ S ) + (o θ1 (S + ) v ( S + \ (S S 1 ) )) ( ) o θ1 (S + ) v(s + \ S 1 ) = p (θ) v(s 1 \ S ) v ( S + \ (S S 1 ) ) + v(s + \ S 1 ) = p (θ) v(s + \ S )+v(s + \ S 1 ). The reul now follow from Lemm.. Exmple. For our exmple given in Figure 3 we hve lredy een h up o ime θ 1 = 5 flow of vlue 5 including he ol upply of he ource 1 nd cn e en ino he ink. In priculr, i hold h o θ (S ) o θ (S + ) v(s + \ S ) for ll S S + nd θ θ 1. For he e S 1 := { 3 } S + nd θ = θ 1 hi inequliy i igh. The funcion θ o θ (S + ) i lredy known (ee he righ pr of Figure 3). For he rericed erlie rrivl prolem wih ource S 1 = { 3 }, he erlie rrivl pern p i given in he lef pr of Figure 4. By Theorem 3.4, he reuling erlie rrivl pern p of he originl innce i he lower envelop of he wo funcion depiced in he righ pr of Figure 4. 3.. Compuing he erlie rrivl pern Wih Theorem 3.4 we hve reduced he prolem of compuing he erlie rrivl pern o n -erlie rrivl flow prolem nd n erlie rrivl rnhipmen prolem on reduced innce wih ricly mller e of ource. Applying hi reul recurively o he reduced innce finlly yield Algorihm 1 which compue he erlie rrivl pern p. For he undernding of he lgorihm i i helpful o oerve h θ i <θ i+1 for ll i 0. The emen i cler for i =0ince he ource in S + \ S 1 Algorihm 1: Compuing he erlie rrivl pern. Inpu: (G, S +,) Oupu: Erlie rrivl pern p. 1 e i := 0, S i := S +,ndθ i := 0; while S i do 3 compue he mximl vlue θ i+1 0 uch h o θ i+1 (S ) o θ i+1 (S i) v(s i \ S ) for ll S S i; 4 compue n incluion-wie miniml S i+1 S i wih o θ i+1 (S i+1) = o θ i+1 (S i) v(s i \ S i+1) ; (4) 5 compue he funcion θ o θ (S i) on he inervl [θ i,θ i+1) nd e p(θ) :=o θ (S i)+v(s + \ S i) for θ [θ i,θ i+1); 6 i := i +1; 7 e p(θ) :=v(s + ) for ll θ θ i; hve poiive upply nd herefore cnno run empy ime θ 0 =0. For i 1 ume y conrdicion h θ i+1 θ i. Thi yield y Lemm 3.1: o θi (S i+1 ) o θi (S i )+o θi+1 (S i+1 ) o θi+1 (S i ) y (4): y (4) wih i := i 1: = o θi (S i ) v(s i \ S i+1 ) = o θi (S i 1 ) v(s i 1 \ S i ) v(s i \ S i+1 ) = o θi (S i 1 ) v(s i 1 \ S i+1 ) which conrdic he miniml choice of S i S i+1 in ep 4 of he lgorihm. Theorem 3.5. Algorihm 1 compue he erlie rrivl pern nd cn e implemened o run in rongly polynomil ime in he inpu plu oupu ize. In order o prove hi heorem, we need he following echnicl lemm which give ound on he compuionl complexiy of ep 5. Lemm 3.6. For 0 θ i θ i+1 nd S S +, he piecewie liner funcion g : [θ i,θ i+1 ) R wih g(θ) :=o θ (S ) cn e compued in ime polynomil in he inpu ize plu he numer of rekpoin. Proof. In order o compue g(θ) =o θ (S ), we conider he exended nework N h i oined follow. Add uperource h i conneced o ll ource in S y n uncpcied rc wih rni ime zero nd h cn e reched from y n uncpcied dummy rc (, ). A lredy ed in (1), g(θ) i equl o he co of min-co circulion in N
flow vlue p(θ) flow vlue p (θ) 8 6 4 1 10 8 6 4 1 3 4 5 6 7 8 9 ime θ 1 3 4 5 6 7 8 9 ime θ Figure 4. Opiml pern p for he prolem wih he reduced e of ource S 1 (lef) nd he comined pern p he lower ound of he line egmen (righ). where he co coefficien of he dummy rc (, ) i e o τ (,) = θ. We denoe he co of n rirry circulion x in hi nework y co θ (x). We r y compuing min-co circulion x in N for θ = θ i. Le N x denoe he reidul nework of x nd le θ e he lengh of hore -ph in N x. Since here i he uncpcied dummy rc (, ) of co θ i in N x, opimliy of x implie θ θ i. Moreover, for ll θ [θ i,θ ],hecirculion x i ill min-co circulion nd g(θ) = co θ (x). Since he co of x depend linerly on θ, he funcion g i hu liner on he inervl [θ i,θ ]. If θ θ i+1, hen we re done. Oherwie we hve dicovered rekpoin of g θ. Noice h x i no longer opiml for θ>θ ince he co cn e reduced y ugmening flow on negive cycle formed y hore --ph of lengh θ in N x nd he dummy rc (, ) of lengh θ. We oin he nex liner piece of g ring θ follow. Compue he unework N x of he reidul nework N x h i formed y ll rc h lie on ome hore --ph. Compue mximum --flow in N x nd urn i ino circulion y in N x y ending ll flow from ck o on he dummy rc (, ). Augmening x ccording o y yield new circulion x. The new circulion i opiml for ll θ [θ,θ ] where θ >θ i he lengh of hore --ph in he new reidul nework N x nd deermine he nex rekpoin of g. The decried proce i iered unil he lengh of hore --ph in he reidul nework i le θ i+1. Noice h he overll running ime i domined y he iniil min-co flow compuion plu numer of rekpoin mny mx-flow compuion. Exmple. In our exmple depiced in Figure 3 we cn find he funcion g : [θ i,θ i+1 ) R decried ove. For he inervl [θ 0,θ 1 ) we ge he nework N,N x, nd N x follow. Nework N i conruced y dding uperource conneced o ll ource y uncpcied, zero rni ime rc nd n uncpcied rc (, ) wih rni ime τ (,) =. Thi i depiced in Figure 5.1. In hi nework we compue min-co (mximum flow) circulion y ending one uni of flow for exmple over cycle,,,,. Thi yield he reidul nework N x which i depiced in Figure 5.. There he hore -ph, for exmple ph, 3,,,, h lengh θ =3. In he unework N x coniing of ll rc eing pr of ome hore --ph we now compue mximum --flow. Such ph flow, 1,,, i depiced in Figure 5.3. Reconidering nework N x ogeher wih new circulion of one uni of flow long cycle, 1,,,, reul in he new reidul nework N x(new) which i depiced in Figure 5.4. There no (hore) --ph remin nd herefore he rni ime of rc (, ) i e o infiniy which i ricly greer hn θ 1. Thu we hve found funcion g on he inervl [θ 0,θ 1 ) which i of he form hown in Figure 5.5. Proof of Theorem 3.5. The correcne of he lgorihm follow from Secion 3.1 nd in priculr from Theorem 3.4. I hu remin o prove he ed ound on he running ime. Fir noice h he numer of ierion of he while-loop in ep i ounded y he numer of ource ince le one ource i elimined from S i in every ierion. Since ep 5 cn e done in rongly polynomil ime, i remin o how h ep 3 nd 4 cn lo e done in rongly polynomil ime. We r wih he compuion of θ i+1 in ep 3. For θ 0 we define he funcion f θ : Si R y f θ (S ):=o θ (S ) o θ (S i )+v(s i \ S ) for S S i. Compuing θ i+1 hu moun o finding he mximl vlue θ 0 uch h f θ (S ) 0 for ll S S i. (5) Since o θ i umodulr nd he funcion S v(s i \ S ) o θ (S i )
1.) N τ (,) = θ 0 =.) N x τ (,) = θ 0 = 1 1 3 3 3.) N x 4.) N x (new) τ (,) = θ 0 = 1 1 3 5.) co θ (x) 3 8 6 4 θ 0 =θ =3 θ 1 =5 ime Figure 5. Nework ued o compue he funcion g :[θ 0,θ 1 ) R decried in he proof of Lemm 3.6 for he exmple given in Figure 3. i modulr, f θ i umodulr. According o (1), compuing f θ (S ) for ome S S i require wo minco flow compuion where he co coefficien depend linerly on he prmeer θ. I w hown y Gröchel, Lováz, nd Schrijver [11] h here i rongly polynomil lgorihm for minimizing umodulr funcion 5. I cn herefore e eed in rongly polynomil ime wheher (5) i fulfilled for fixed vlue θ. Emedding hi lgorihm ino Megiddo prmeric erch frmework (ee [4, 5]) give procedure for ep 3 whoe running ime i rongly polynomil in he inpu ize of our prolem (more deil cn e found in [15]). We finlly dicu how o compue S i+1 in ep 4 in rongly polynomil ime. Noice h (4) rnle o f θi+1 (S i+1 ) = 0,hi,S i+1 minimize he umodulr funcion f θi+1. By umodulriy of f θi+1, here exi unique incluion-wie miniml ue S i+1 which cn e oined follow 6 (ee, e.g., [9, Chper 45]). Iniilize S i+1 := S i. For ech S i, check wheher he minimum vlue 5 Cominoril lgorihm chieving rongly polynomil running ime re given y Iw, Fleicher, nd Fujihige [18] nd y Schrijver [8]. A fully cominoril lgorihm i given y Iw [17]. 6 For he purpoe of our lgorihm i i of coure dvngeou o chooe he miniml ue S i+1 in order o reduce he numer of ource fr poile. of f θi+1 over ll ue of S i+1 \{} i zero. If o, ree S i+1 := S i+1 \{}. Doing hi for ll elemen of S i finlly yield he unique incluion-wie miniml ue S i+1 wih f θi+1 (S i+1 )=0.Afer lgorihm for compuing he incluion-wie miniml ue S i+1 w recenly given y McCormick nd Queyrnne []. 4. Turning he erlie rrivl pern ino n erlie rrivl rnhipmen In hi ecion we ume h we re given he piecewie liner erlie rrivl pern p of he erlie rrivl rnhipmen prolem y i rekpoin (x 0,f 0 ), (x 1,f 1 ),...,(x k,f k ),hi, 0 if θ x 0, if x i θ, p(θ) = f i + θ xi x i+1 x i (f i+1 f i ) θ x i+1, 0 i<k, f k if θ x k. An illurion i given in Figure 6. Noice h he vlue x i deermine poin in ime nd he vlue f i deermine n moun of flow for ll i. Furher noice h x 0 <x 1 < <x k
flow vlue p(θ) (f 1 f 0) f 6 f 5 x 6 x 1 x 6 x x 6 x 3 (f f 1) (f 3 f ) f 4 f 3 x 6 x 4 x 6 x 5 (f 4 f 3) f f 1 x 6 x 6 (f 5 f 4) x 0 x 1x x 3 x 4 x 5 x 6 ime θ (f 6 f 5) Figure 6. On he lef hnd ide we drw he erlie rrivl pern p wih rekpoin (x i,f i ), i = 1,,...,k =6. On he righ hnd ide he modified nework i depiced. The cpciy of rc e i =(, i ) i e o (f i f i 1 )(x i x i 1 ). nd x 0 i he fir poin in ime when flow cn rech he ink (i. e., x 0 i he rni ime of hore ph leding from ny ource o he ink). Moreover, 0=f 0 f 1 f k = v(s + ). We how h he prolem of finding n erlie rrivl rnhipmen cn e reduced o finding rnhipmen over ime in lighly modified nework N wih k ddiionl rc leding from o k new ink node 1,..., k. An illurion of he modificion i given in Figure 6. Node i no longer ink u ju n inermedie node of he modified nework N. For i =1,...,k, he demnd of ink i i e o (f i f i 1 ) uch h he ol demnd f k of he ink nd he ol upply v(s + ) he ource cncel ou ech oher. The rc leding from o ink i i clled e i. The rni ime of rc e i i defined o e τ ei := x k x i, i cpciy i (f i f i 1 )/(x i x i 1 ) nd hu equl o he derivive of p wihin he inervl [x i 1,x i ]. Noice h he cpciy of e i i choen uch h he demnd of ink i i fulfilled if flow i eing en mximl re ino rc e i wihin ime inervl [x i 1,x i ). A conequence of hi oervion, we cn e he following lemm. Lemm 4.1. An erlie rrivl flow in N wih erlie rrivl pern p nurlly induce feile rnhipmen over ime wih ime horizon x k ifying ll upplie nd demnd in N Proof. Tke n erlie rrivl flow in N nd urn i ino rnhipmen over ime in N y ending ll flow rriving in ime inervl [x i 1,x i ) o i long rc e i. The revere direcion of Lemm 4.1 lo hold. Due o pce limiion, we omi hi proof in hi exended rc. I will e conined in he full verion of he pper. Lemm 4.. A rnhipmen over ime wih ime horizon x k h ifie ll upplie nd demnd in he modified nework N nurlly induce n erlie rrivl rnhipmen in N. We finlly prove h rnhipmen over ime wih ime horizon x k h ifie ll upplie nd demnd in he modified nework N cully exi. A conequence of Lemm 4., hi yield new proof for he exience of n erlie rrivl rnhipmen in N. Lemm 4.3. There exi rnhipmen over ime wih ime horizon x k ifying ll upplie nd demnd in N. Proof. We denoe he e of ource in N y S + nd he e of ink y S = { 1,..., k }. For n rirry S S + S le ō θ (S ) denoe he mximum moun of flow h cn e en wihin ime θ from ource S + S o ink S \ S. By Lemm.1 we hve o how h ō θ (S ) v(s ) for θ = x k. Le o θ (S + S ) denoe he mximum moun of flow h cn e en wihin ime θ from ource S + S o. By Lemm 3.1 we ge o θ (S + S )+v(s + \ S ) p(θ) for ll θ 0. Thi inequliy cn e inerpreed follow: If we ume h he ol upply v(s + \ S ) of he ource S + \ S i lredy in from ime zero on, hen we cn end v(s + S ) ddiionl flow uni from he ource in S + S (ignoring heir individul upplie) ino uch h he moun of flow i le p(θ) ny ime θ 0. By forwrding flow from o he ink in S (imilr o he proof of Lemm 4.1), we ge flow over ime wih ime horizon x k h ifie he demnd of ll ink in S. From hi flow over ime we now remove he v(s + \ S ) flow uni h we umedoein ime zero. Thi yield flow over ime wih ime horizon x k from he ource in S + S
o he ink S uch h he ol moun of flow en i v(s + S ) ndnoinkins ge more hn i demnd. Therefore he flow rriving ink in S \ S i le v(s + S )+v(s S )=v(s ). We hve hu hown h ō θ (S ) v(s ) for θ = x k.thi conclude he proof. A conequence we cn e he following heorem. Theorem 4.4. Given he erlie rrivl pern p wih k rekpoin for nework N, n erlie rrivl rnhipmen in N cn e oined y compuing rnhipmen over ime in modified nework N wih k ddiionl node nd rc. In order o compue rnhipmen over ime in he modified nework N we cn ue he lgorihm of Hoppe nd Trdo [15]. Since he running ime of hi lgorihm i ounded y polynomil in he encoding ize of he inpu N nd ince he encoding ize of N i of he me order he encoding ize of N plu he encoding ize of p, he required running ime i polynomil in he inpu plu oupu ize of he erlie rrivl flow prolem on N. Acknowledgemen. The uhor wn o hnk Li Fleicher, Bein Klinz, nd Ekkehrd Köhler for inereing nd helpful dicuion on he opic of hi pper. Reference [1] G. N. Berlin. The ue of direced roue for eing ecpe poenil. Nionl Fire Proecion Aociion, Boon, MA, 1979. [] R. G. Buker nd P. J. Gowen. A procedure for deermining miniml-co nework flow pern. Technicl Repor 15, Operionl Reerch Office, John Hopkin Univeriy, Blimore, MD, 1961. [3] L. G. Chlme, R. L. Frnci, nd P. B. Sunder. Nework model for uilding evcuion. Mngemen Science, 8:86 105, 198. [4] L. K. Fleicher. Fer lgorihm for he quicke rnhipmen prolem. SIAM Journl on Opimizion, 1:18 35, 001. [5] L. K. Fleicher nd M. Skuell. Quicke flow over ime. SIAM Journl on Compuing. To pper. [6] L. K. Fleicher nd M. Skuell. The quicke mulicommodiy flow prolem. In W. J. Cook nd A. S. Schulz, edior, Ineger Progrmming nd Cominoril Opimizion, volume 337 of Lecure Noe in Compuer Science, pge 36 53. Springer, Berlin, 00. [7] L. K. Fleicher nd É. Trdo. Efficien coninuouime dynmic nework flow lgorihm. Operion Reerch Leer, 3:71 80, 1998. [8] L. R. Ford nd D. R. Fulkeron. Conrucing mximl dynmic flow from ic flow. Operion Reerch, 6:419 433, 1958. [9] L. R. Ford nd D. R. Fulkeron. Flow in Nework. Princeon Univeriy Pre, Princeon, NJ, 196. [10] D. Gle. Trnien flow in nework. Michign Mhemicl Journl, 6:59 63, 1959. [11] M. Gröchel, L. Lováz, nd A. Schrijver. Geomeric Algorihm nd Cominoril Opimizion, volume of Algorihm nd Cominoric. Springer, Berlin, 1988. [1] B. Hjek nd R. G. Ogier. Opiml dynmic rouing in communicion nework wih coninuou rffic. Nework, 14:457 487, 1984. [13] H. W. Hmcher nd S. Tifecki. On he ue of lexicogrphic min co flow in evcuion modeling. Nvl Reerch Logiic, 34:487 503, 1987. [14] B. Hoppe nd É. Trdo. Polynomil ime lgorihm for ome evcuion prolem. In Proceeding of he 5h Annul ACM SIAM Sympoium on Dicree Algorihm, pge 433 441, Arlingon, VA, 1994. [15] B. Hoppe nd É. Trdo. The quicke rnhipmen prolem. Mhemic of Operion Reerch, 5:36 6, 000. [16] M. Iri. A new mehod of olving rnporionnework prolem. Journl of he Operion Reerch Sociey of Jpn, 6:7 87, 1960. [17] S. Iw. A fully cominoril lgorihm for umodulr funcion minimizion. Journl of Cominoril Theory, Ser. B, 84:03 1, 00. [18] S. Iw, L. Fleicher, nd S. Fujihige. A cominoril rongly polynomil lgorihm for minimizing umodulr funcion. Journl of he ACM, 48:761 777, 001. [19] J. Jrvi nd H. Rliff. Some equivlen ojecive for dynmic nework flow prolem. Mngemen Science, 8:106 108, 198. [0] P. A. Jewel. Opiml flow hrough nework. Technicl Repor 8, Operion Reerch Cener, MIT, Cmridge, MA, 1958. [1] B. Klinz. Cied peronl communicion (1994) in [15]. [] S. T. McCormick nd M. Queyrnne. Finding ll opiml oluion for umodulr funcion minimizion. Tlk given Oerwolfch workhop ID 0545 on Cominoril Opimizion, Novemer 005. [3] N. Megiddo. Opiml flow in nework wih muliple ource nd ink. Mhemicl Progrmming, 7:97 107, 1974. [4] N. Megiddo. Cominoril opimizion wih rionl ojecive funcion. Mhemic of Operion Reerch, 4:414 44, 1979. [5] N. Megiddo. Applying prllel compuion lgorihm in he deign of eril lgorihm. Journl of he ACM, 30:85 865, 1983. [6] E. Miniek. Mximl, lexicogrphic, nd dynmic nework flow. Operion Reerch, 1:517 57, 1973. [7] D. Richrdon nd É. Trdo. Cied peronl communicion (00) in [5]. [8] A. Schrijver. A cominoril lgorihm minimizing umodulr funcion in rongly polynomil ime. Journl of Cominoril Theory, Serie B, 80:346 355, 000. [9] A. Schrijver. Cominoril Opimizion: Polyhedr nd Efficiency. Springer, Berlin, 003. [30] S. Tjndr. Dynmic Nework Opimizion wih Applicion o he Evcuion Prolem. PhD hei, Univeriä Kierluern, Shker Verlg, Achen, 003. [31] W. L. Wilkinon. An lgorihm for univerl mximl dynmic flow in nework. Operion Reerch, 19:160 161, 1971.