13.1 Weighted Graphs and Their Representation

Transcription

1 Chpter Shortet Pth In thi hpter we will over prolem involving fining the hortet pth etween vertie in grph with weight (length) on the ege. One oviou pplition i in fining the hortet route from one re to nother, however hortet pth hve mny other pplition. We will look t verion of the prolem uming we re fining itne from ingle oure to ll other vertie in the grph, n two vrint epening on whether we llow negtive ege weight or not. We firt introue weighte grph.. Weighte Grph n Their Repreenttion Mny pplition of grph require oiting weight or other vlue with the ege of grph. Suh grph n e formlly efine follow. Definition. (Weighte n Ege-Lele Grph). A weighte grph or n egelele grph i triple G = (E, V, w) where w : E L i funtion mpping ege or irete ege to their vlue, n L i the et of poile vlue. In grph, if the t oite with the ege re rel numer (L R), we often ue the term weight to refer to the vlue, n ue the term weighte grph to refer to the grph. In the generl e, we ue the term ege lel or ege vlue to refer to the vlue. Weight or other vlue on ege oul repreent mny thing, uh itne, or pity, or the trength of reltionhip. 97

2 98 CHAPTER. SHORTEST PATHS Exmple.. An exmple irete weighte grph Chpter 9 erie three ifferent repreenttion of grph uitle for prllel lgorithm, ege et, jeny tle, n jeny equene. Quetion.. Cn you ee how we n exten thee repreenttion to upport ege vlue? We n exten ll of thee repreenttion to upport ege vlue y eprtely repreenting the funtion from ege to vlue uing tle (mpping) the tle mp eh ege (or r) to it vlue. Thi repreenttion llow looking up the ege vlue of n ege e = (u, v) y uing tle lookup. We ll thi n ege tle. Exmple.4. For the weighte grph in Exmple., the ege tle i: W = {(, ).7, (, 4)., (, 4)., (4, ).} A nie property of n ege tle i tht it work uniformly with ll repreenttion of the truture of grph, n i len ine it eprte the ege vlue from the truturl informtion. However keeping the ege tle eprtely rete reunny, wting pe n poily requiring extr work eing the ege vlue. The reunny n e voie y toring the ege vlue iretly with the ege informtion. For exmple, uing ege tle mke ege et reunnt o there i typilly no nee to keep eprte ege et. Ajeny tle n e extene to inlue the ege vlue y repling the et of neighor of vertex v with tle tht mp eh neighor u to the ege vlue w(v, u). Ajeny equene n e extene y reting equene of neighor-vlue pir for eh out ege of vertex.

3 .. SHORTEST WEIGHTED PATHS 99 Exmple.. For the weighte grph in Exmple., the jeny tle repreenttion i G = { {.7, 4.}, {4.}, 4 {.}}, n the jeny equene repreenttion i (uming equene re e): G = (,.7), (4,.),, (4,.), (,.).. Shortet Weighte Pth Conier weighte grph G = (V, E, w), w : E R. The grph n e either irete or unirete. For onveniene we efine w(u, v) = if (u, v) E. We efine the weight of pth the um of the weight of the ege long tht pth. Exmple.6. In the following grph the weight of the pth,,, e i 6. The weight of the pth,,, i. 7 e 4 6 For weighte grph G(V, E, w) hortet (weighte) pth from vertex u to vertex v i pth from u to v with minimum weight. There might e multiple pth with equl weight, n if o they re ll hortet weighte pth from u to v. We ue σ G (u, v) to inite the weight of hortet pth from u to v. In hortet pth prolem, we re require to fin the hortet (weighte) pth etween vertie, or perhp jut the weight of uh pth. Quetion.7. Wht i the (weighte) hortet pth from to e? In the grph ove, the hortet pth from to e i,,, e with weight 6. Quetion.8. Wht hppen if we hnge the weight of the ege (, ) from 7 to 7? If we hnge the weight of the ege (, ) to 7 the hortet pth etween n e h weight ine pth n keep going roun the yle,,, reuing it weight y 4 eh time

4 CHAPTER. SHORTEST PATHS roun. Rell tht pth llow repete vertie - imple pth oe not. For thi reon, when omputing hortet pth we will nee to e reful out yle of negtive weight. A we will ee, even if there re no negtive weight yle, negtive ege weight mke fining hortet pth more iffiult. We will therefore firt onier the prolem of fining hortet pth when there re no negtive ege weight. Computing hortet pth i importnt in mny prtil pplition. In ft, there re everl vrint of thi prolem uh the ingle oure n the multiple oure verion, oth with or without negtive weight. Prolem.9 (Single-Soure Shortet Pth (SSSP)). Given weighte grph G = (V, E, w) n oure vertex, the ingle-oure hortet pth (SSSP) prolem i to fin hortet weighte pth from to every other vertex in V. Although there n e mny equl weight hortet pth etween two vertie, the prolem only require fining one. Alo ometime we only re out the weight of the hortet pth to eh vertex n not the pth itelf. We will refer to thi vrint of the SSSP prolem the SSSP δ prolem. Exerie.. For weighte grph G = (V, E, w), ume you re given the itne δ G, v for ll vertie V. For prtiulr vertex u, erie how you oul fin hortet pth from to u y only looking t the in neighor of eh vertex on the pth. Hint: trt t u. In Chpter we w how Breth-Firt Serh (BFS) n e ue to olve the ingle-oure hortet pth prolem on grph without ege weight, or, equivlently, where ll ege hve weight. Quetion.. Cn we ue BFS to olve the ingle-oure hortet pth prolem on weighte grph? BFS oe not work on weighte grph, unfortuntely, eue it oe not tke ege weight into ount.

5 .. DIJKSTRA S ALGORITHM Exmple.. To ee why BFS oe not work, onier the following irete grph with vertie: In thi exmple, BFS firt viit n then. When it viit, it ign it n inorret weight of. Sine BFS never viit gin, it will not fin the hortet pth going trough, whih hppen to e horter. Exmple.. Another grph where BFS fil to fin the hortet pth orretly The reon why BFS work on unweighte grph i quite intereting n helpful for unertning other hortet pth lgorithm. The key ie ehin BFS i to viit vertie in orer of their itne from the oure, viiting loet vertie firt, n the next loet, n o on. More peifilly, for eh frontier F i, BFS h the orret itne from the oure to eh vertex in the frontier. It then n etermine the orret itne for unenountere neighor tht re itne one further wy (on the next frontier).. Dijktr Algorithm Dijktr lgorithm olve the SSSP prolem when ll the weight on the ege re non-negtive (i.e. w : E R ). We will refer to thi vrint of SSSP the SSSP + prolem. Dijktr i n importnt lgorithm oth eue it i n effiient lgorithm for n importnt prolem, ut lo eue it i very elegnt exmple of n effiient greey lgorithm tht generte optiml olution on nontrivil tk. In thi etion, we re going to (re-)iover thi lgorithm y tking vntge of propertie of grph n hortet pth. Before going further we note tht ine no ege hve negtive weight, there nnot e negtive weight yle. Therefore one n never mke pth horter y viiting vertex twie i.e. pth tht yle k to vertex nnot hve mller weight thn the pth tht en t the firt viit to the vertex. Thi men we only nee to onier imple pth when erhing for hortet pth.

6 CHAPTER. SHORTEST PATHS Quetion.4. Cn you think of rute-fore lgorithm for fining the hortet? pth. Let trt with rute fore lgorithm for the SSSP + prolem, tht, for eh vertex, onier ll imple pth etween the oure n the vertex n elet the hortet uh pth. Quetion.. How mny imple pth n there e etween two vertie in grph? Unfortuntely there n e n exponentil numer of pth etween ny pir of vertie, o ny lgorithm tht trie to look t ll pth i not likely to le eyon very mll intne. We therefore hve to try to reue the work. Towr thi en we note tht the u-pth of hortet pth mut lo e hortet pth etween their en vertie, n look t how thi n help. Thi u-pth property i key property of hortet pth. Exmple.6. If hortet pth from Pitturgh to Sn Frnio goe through Chigo, then tht hortet pth inlue the hortet pth from Pitturgh to Chigo. Exerie.7. Prove tht the u-pth property hol for ny grph, lo in the preene of negtive weight. We re going to ue thi property oth now to erive Dijktr lgorithm, n lo gin in the next etion to erive Bellmn-For lgorithm for the SSSP prolem on grph tht o llow negtive ege weight. To ee how thi property n e helpful, uppoe n orle h tol you the hortet pth to ll vertie exept for one vertex, v. Quetion.8. Uing the u-pth property, n you fin the hortet pth to v? We n fin the hortet pth to v, eue we know the hortet upth form the oure to the vertex u immeitely efore v. All we nee to o i fin whih u mong the in-neighor of v minimize the weight of the pth to u, i.e. δ G (, u) plu the itionl ege weight to get to v. We note tht thi rgument relie on the ft tht hortet pth mut e imple o nnot go through v itelf.

7 .. DIJKSTRA S ALGORITHM Exmple.9. In the following grph G, uppoe tht we hve foun the hortet pth from the oure to ll the other vertie exept for v. The weight of the hortet pth to v i min δ G (, ) +, δ G (, ) + 6, δ G (, ) +. The hortet pth goe through the vertex (,, or ) tht minimize the weight. Viite Set 6 v Let try to generlize the rgument to the e where inte of the orle telling u the hortet pth from to ll ut one of the vertie, it tell u the hortet pth from to ome uet of the vertie X V with X. Alo let efine Y to e the vertie not in X, i.e. V \ X. The quetion i whether we n effiiently etermine the hortet pth to ny vertie in Y. If we oul o thi, then we woul hve rnk to repetely new vertie to X until we re one. A in grph erh, we ll the et of vertie tht re neighor of X ut not in X, i.e. N + (X) \ X, the frontier. It houl e ler tht ny pth tht leve X h to go through frontier vertex on the wy out. Therefore for every v Y the hortet pth from mut trt in X, ine X, n then leve X vi vertex in the frontier. Quetion.. Do you ee how we n ue thi property to ientify vertex v Y tht i t let loe to the oure ny other vertex in Y? Billy the intuition now i tht t let one pth tht goe from X iretly to vertex on the frontier, i n overll hortet pth to ny vertex in Y. Thi i eue ll pth to Y mut go through the frontier when exiting X, n ine ege re non-negtive, upth nnot e longer thn the full pth.

8 4 CHAPTER. SHORTEST PATHS Exmple.. In the following grph uppoe tht we hve foun the hortet pth from the oure to ll the vertie in X (mrke y numer next to the vertie). The hortet pth from ny vertex in X iretly to vertex in Y i the pth to followe y the ege (, v) with weight 9. If ege weight re non-negtive there nnot e ny horter wy to get to v, whtever w(u, v) i, therefore we know tht δ(, v) = 9. X 8 Y= V \ X u? 4 v Thi intuition i formlize in Lemm. long with proof. The Lemm tell u tht one we know the hortet pth to et X we n more vertie to X with their hortet pth. Thi give u rnk tht n hurn out t let one new vertex on eh roun. Note tht thi pproh i tully n intne of priority-firt erh. Quetion.. Do you rell how priority-firt erh work? Rell tht priority-firt erh i grph erh in whih on eh tep we viit the frontier vertie with the highet priority. In our e the viite et i X, n the priority i efine in term of p(v), the hortet-pth weight oniting of pth to x X with n itionl ege from x to v. Thi give u the following rther onie efinition of Dijktr lgorithm. Algorithm.4 (Dijktr Algorithm). For weighte grph G = (V, E, w) n oure, Dijktr lgorithm i priority-firt erh on G trting t with () =, uing priority p(v) = min((x) + w(x, v)) (to e minimize), n etting (v) = p(v) x X when v i viite. Note tht Dijktr lgorithm will viit vertie in non-ereing hortet-pth weight ine on eh tep it viit unviite vertie tht hve the minimum hortet-pth weight from.

9 .. DIJKSTRA S ALGORITHM Lemm. (Dijktr Property). Conier (irete) weighte grph G = (V, E, w), w : E R, n oure vertex V. Conier ny prtitioning of the vertie V into X n Y = V \ X with X, n let then min p(y) = min δ G(, y). y Y y Y p(v) min x X (δ G(, x) + w(x, v)) X Y=V \ X v x v t v m In Englih: The overll hortet-pth weight from vi vertex in X iretly to neighor in Y (in the frontier) i hort ny pth from to ny vertex in Y. Proof. Conier vertex v m Y uh tht δ G (, v m ) = min v Y δ G (, v), n hortet pth from to v m in G. The pth mut go through n ege from vertex v x X to vertex v t in Y (ee the figure). Sine there re no negtive weight ege, n the pth to v t i upth of the pth to v m, δ G (, v t ) nnot e ny greter thn δ G (, v m ) o it mut e equl. We therefore hve min y Y p(y) δ G (, v t ) = δ G (, v m ) = min y Y δ G (, y), ut the leftmot term nnot e le thn the rightmot, o they mut e equl. Implition: Thi give u wy to eily fin δ G (, v) for t let one vertex v Y. In prtiulr for ll v Y where p(v) = min y Y p(y), it mut e the e tht p(v) = δ G (, v) ine there nnot e horter pth. Alo we nee only onier the frontier vertie in lulting p(v), ine for other in Y, p(y) i infinite. Remrk.. It my e tempting to think tht Dijktr lgorithm viit vertie tritly in inreing orer of hortet-pth weight from the oure, viiting vertie with equl hortet-pth weight on the me tep. Thi i not true. To ee thi onier the exmple elow n onvine yourelf tht it oe not ontrit our reoning.

10 6 CHAPTER. SHORTEST PATHS Lemm.6. Dijktr lgorithm return, (v) = δ G (, v) for v rehle from. Proof. We how the invrint for eh tep tht for ll x X (the viite et), (x) = δ G (, x). Thi i true t the trt ine X = {} n () =. On eh tep the erh vertie v tht minimize P (v) = min x X ((x) + w(x, v)). By our umption we hve tht (x) = δ G (, x) for x X. By Lemm., p(v) = δ G (, v), giving (v) = δ G (, v) for the newly e vertie, mintining the invrint. A with ll priority-firt erhe, it will eventully viit ll rehle v. Exerie.7. Argue tht if ll ege in grph hve weight then Dijktr lgorithm erie viit extly the me et of vertie in eh roun BFS oe. Implementing Dijktr Algorithm with Priority Queue We now iu how to implement thi trt lgorithm effiiently uing priority queue to mintin P (v). We ue priority queue tht upport eletemin n inert. The priorityqueue e lgorithm i given in Algorithm.8. Thi vrint of the lgorithm only one vertex t the time even if there re multiple vertie with equl itne tht oul e e in prllel. It n e me prllel y generlizing priority queue, ut we leve thi n exerie. The lgorithm mintin the viite et X tle mpping eh viite vertex u to (u) = δ G (, u). It lo mintin priority queue Q tht keep the frontier prioritize e on the hortet itne from iretly from vertie in X. On eh roun, the lgorithm elet the vertex x with let itne in the priority queue (line 8 in the oe) n, if it hn t lrey een viite, viit it y ing (x ) to the tle of viite vertie (line ), n then ll it neighor v to Q long with the priority (x) + w(x, v) (i.e. the itne to v through x) (line 7 n 8). Note tht neighor might lrey e in Q ine it oul hve een e y nother of it in-neighor. Q n therefore ontin uplite entrie for vertex, ut wht i importnt i tht the minimum itne will lwy e pulle out firt. line hek to ee whether vertex pulle from the priority queue h lrey een viite n ir it if it h. Thi lgorithm i jut onrete implementtion of the previouly erie Dijktr lgorithm. We note tht there re ouple vrint on Dijktr lgorithm uing Priority Queue. Firtly we oul hek inier the relx funtion whether whether u i lrey in X n if o not inert it into the Priority Queue. Thi oe not ffet the ymptoti work oun ut proly woul give ome improvement in prtie. Another vrint i to eree the priority of the neighor inte of ing uplite to the priority queue. Thi require more powerful priority queue tht upport ereekey funtion. Cot of Dijktr Algorithm. We now onier the work of the Priority Queue verion of Dijktr lgorithm. The lgorithm i equentil o the pn i the me. We nlyze the work y ounting up the numer of opertion. In the lgorithm (Figure.8) inlue ox roun

11 .. DIJKSTRA S ALGORITHM 7 Algorithm.8 (Dijktr Algorithm uing Priority Queue). funtion ijktrpq (G, ) = let % require: 4 (x ) X = δ G (, x) {(, y) Q y V \ X} = {( + w(x, y), y) : (x ) X, y N + (x) \ X} 6 % return: {x δ G (, x) : x R G ()} 7 funtion ijktr (X, Q) = 8 e PQ.eleteMin (Q) of 9 (NONE,_) X (SOME(,v), Q ) if (v, _) X then ijktr (X, Q ) ele 4 let X = X {(v, )} 6 funtion relx (Q, (u, w)) = 7 PQ.inert ( + w, u) Q 8 Q = iter relx Q N G (v) 9 in ijktr (X, Q ) en in ijktr ({}, PQ.inert (, ) {}) en eh opertion on the grph G, the et of viite vertie X, or the priority queue PQ. The PQ.inert in line i lle only one, o we n ignore it. Of the remining opertion, The iter n N G (v) on line 8 re on the grph, line n re on the tle of viite vertie, n line 8 n 7 re on the priority queue. For the priority queue opertion, we hve only iue one ot moel whih, for queue of ize n, require O(log n) work n pn for eh of PQ.inert n PQ.eleteMin. We hve no nee for mel opertion here. For the grph, we n either ue tree-e tle or n rry to e the neighor There i no nee for ingle three rry ine we re not upting the grph. For the tle of itne to viite vertie we n ue tree tle, n rry equene, or ingle three rry equene. The following tle ummrize the ot of the opertion, long with the numer of ll me to eh opertion. There i no prllelim in the lgorithm o we only nee to onier the equentil exeution of the ll. We oul lo ue hh tle, ut we hve not yet iue them.

12 8 CHAPTER. SHORTEST PATHS Exmple.9. An exmple run of Dijktr lgorithm. Note tht fter viiting,, n, the queue Q ontin two itne for orreponing to the two pth from to iovere thu fr. The lgorithm tke the hortet itne n it to X. A imilr itution rie when i viite, ut thi time for. Note tht when i viite, n itionl itne for i e to the priority queue even though it i lrey viite. Reunnt entrie for oth re remove next efore viiting. The vertex e i never viite it i unrehle from. Finlly, notie tht the itne in X never eree. X @ e e e e e e Q e e

13 .4. THE BELLMAN FORD ALGORITHM 9 Opertion Line # of ll PQ Tree Tle Arry ST Arry eletemin 8 O(m) O(log m) inert 7 O(m) O(log m) Priority Q totl O(m log m) fin O(m) - O(log n) O() O() inert O(n) - O(log n) O(n) O() Ditne totl - O(m log n) O(n ) O(m) N G (v) 8 O(n) - O(log n) O() - iter 8 O(m) - O() O() - Grph e totl - O(m + n log n) O(m) - We n lulte the totl numer of ll to eh opertion y noting tht the oy of the let trting on line 4 i only run one for eh vertex. Thi men tht line n N G (v) on line 8 re only lle O(n) time. Everything ele i one one for every ege. Be on the tle one houl note tht when uing either tree tle or ingle three rry the ot i no more thn the ot of the priority queue opertion. Therefore there i no ymptoti vntge of uing one over the other, lthough there might e ontnt ftor peeup tht i not inignifint. One houl lo note tht uing regulr purely funtionl rry i not goo ie ine then the ot i ominte y the inertion n the lgorithm run in Θ(n ) work. The totl work for Dijktr lgorithm uing tree tle O(m log m + m log n + m + n log n) = O(m log n) ine m n..4 The Bellmn For Algorithm We now turn to olving the ingle oure hortet pth prolem in the generl e where we llow negtive weight in the grph. One might k how negtive weight mke ene. If tlking out itne on mp, they proly o not, ut vriou other prolem reue to hortet pth, n in thee reution negtive weight how up. Before proeeing we note tht if there i negtive weight yle (the um of weight on the yle i negtive) rehle from the oure, then there nnot e olution. Thi i eue every time we go roun the yle we get horter pth, o to fin hortet pth we woul jut go roun forever. In the e tht negtive weight yle n e rehe from the oure vertex, we woul like olution to the SSSP prolem to return ome initor tht uh yle exit n terminte. Exerie.. Conier the following urreny exhnge prolem: given the et urrenie, et of exhnge rte etween them, n oure urreny, fin for eh other urreny v the et equene of exhnge to get from to v. Hint: how n you onvert multiplition to ition.

14 CHAPTER. SHORTEST PATHS Exerie.. In your olution to the previou exerie n you get negtive weight yle? If o, wht oe thi men? Quetion.. Why n t we ue Dijktr lgorithm to ompute hortet pth when there re negtive ege? Rell tht in our evelopment of Dijktr lgorithm we ume non-negtive ege weight. Thi oth llowe u to only onier imple pth (with no yle) ut more importntly plye ritil role in rguing the orretne of Dijktr property. To ee where Dijktr property fil with negtive ege weight onier the following very imple exmple: Dijktr lgorithm woul viit then n leve with itne of inte of the orret itne. The prolem i tht the overll hortet pth iretly from the viite et to the frontier i not neerily the hortet pth to tht vertex. There n e horter pth tht firt tep further wy (to in the exmple), n then reue the length with the negtive ege weight. Quetion.. How n we fin hortet pth on grph with negtive weight? Quetion.4. Rell tht for Dijktr lgorithm, we trte with the rute-fore lgorithm n relize key property of hortet pth. Do you rell the property? A property we n till tke vntge of i tht the upth of hortet pth themelve re hortet. Uing thi property, we n uil hortet pth in lightly ifferent wy: y ouning the numer of ege on pth. Quetion.. Suppoe tht you hve ompute the hortet pth with l or le ege from to ll vertie. Cn you ome up with n lgorithm to upte the hortet pth for l + ege?

15 .4. THE BELLMAN FORD ALGORITHM Algorithm.6 (Bellmn For). funtion BellmnFor(G = (V, E), ) = let % require: v V D v = δg k (, v) 4 funtion BF(D, k) = let 6 D = {v min(d v, min u N u + w(u, v))) : v V } G 7 in 8 if (k = V ) then 9 ele if (ll{d v = D v : v V }) then D ele BF (D, k + ) en D = {v if v = then ele : v V } in BF (D, ) en If we hve ompute the hortet pth with l or le ege from to ll vertie, we n omputer the hortet pth for l + ege y oniering extening ll pth y one ege. To o thi ll we nee to o i onier ll inoming ege of eh vertex. Thi i the ie ehin the Bellmn-For lgorithm. We efine the following δg l (, t) = the hortet weighte pth from to t uing t mot l ege. We n trt y etermining δg (, v) for ll v V, whih i infinite for ll vertie exept itelf, for whih it i. Then perhp we n ue thi to etermine δg (, v) for ll v V. In generl we wnt to etermine δ k+ G (, v) e on ll δk G (, v). The quetion i how o we lulte thi. It turn out to e ey ine to etermine the hortet pth with t mot k + ege to vertex v ll tht i neee i the hortet pth with k ege to eh of it in-neighor n then to in the weight of the one itionl ege. Thi give u δ k+ (v) = min(δ k (v), min x N (v) Rememer tht N (v) inite the in-neighor of vertex v. (δ k (x) + w(x, v)) ). Algorithm.6 efine the Bellmn For lgorithm e on thi ie. It ume we hve e zero weight elf loop on the oure vertex. In Line 8 the lgorithm return (unefine) if there i negtive weight yle rehle from. In prtiulr ine no imple pth n e longer thn V, if the itne i till hnging fter V roun, then there mut e negtive weight yle tht w entere y the erh. An illutrtion of the lgorithm over everl tep i hown in Figure.. Theorem.7. Given irete weighte grph G = (V, E, w), w : E R, n oure, the BellmnFor lgorithm return either

16 CHAPTER. SHORTEST PATHS pth length = pth length pth length - - pth length pth length 4 Figure.: Step of the Bellmn For lgorithm. The numer with re qure inite wht hnge on eh tep.. δ G (, v) for ll vertie rehle from, or. if there i negtive weight-yle in the grph rehle from. Proof. By inution on the numer of ege k in pth. The e e i orret ine D =. For ll v V \, on eh tep hortet (, v) pth with up to k + ege mut onit of hortet (, u) pth of up to k ege followe y ingle ege (u, v). Therefore if we tke the minimum of thee we get the overll hortet pth with up to k + ege. For the oure the elf ege will mintin D =. The lgorithm n only proee to n roun if there i rehle negtive-weight yle. Otherwie hortet pth to every v i imple n n onit of t mot n vertie n hene n ege. Cot of Bellmn For. We now nlyze the ot of the lgorithm. Firt we ume the grph i repreente tle of tle, we uggete for the implementtion of Dijktr. We then onier repreenting it equene of equene. For tle of tle we ume the grph G i repreente (R vtxtle) vtxtle, where vtxtle mp vertie to vlue. The R re the rel vlue weight on the ege. We ume the itne D re repreente R vtxtle. Let onier the ot of one ll to BF, not inluing the reurive ll. The only nontrivil omputtion re on line 6 n 9. Line 6 onit of tulte over the vertie. A the ot peifition for tle inite, to lulte the work we tke the um of the work for eh vertex, n for the pn we tke the mximum of the pn, n O(log n). Now onier wht the lgorithm oe for eh vertex. Firt it h to fin the neighor in the grph (uing fin G v). Thi require O(log V )

17 .4. THE BELLMAN FORD ALGORITHM work n pn. Then it involve mp over the neighor. Eh intne of thi mp require fin in the itne tle to get D u n n ition of the weight. The fin tke O(log V ) work n pn. Finlly there i reue tht tke O( + N G (v) ) work n O(log N G (v) ) pn. Uing n = V n m = E, the overll work n pn re therefore W = O log n + N G (v) + ( + log n) v V u N G (v) = O ((n + m) log n) ( ( )) S = O mx log n + log N G (v) + mx ( + log n) v V u N(v) = O(log n) Line 9 i impler to nlyze ine it only involve tulte n reution. It require O(n log n) work n O(log n) pn. Now the numer of ll to BF i oune y n, iue erlier. Thee ll re one equentilly o we n multiply the work n pn for eh ll y the numer of ll giving: W (n, m) = O(nm log n) S(n, m) = O(n log n) Cot of Bellmn For uing Sequene If we ume the vertie re the integer {,,..., V } then we n ue rry equene to implement vtxtle. Inte of uing fin, whih require O(log n) work, we n ue nth requiring only O() work. Thi improvement in ot n e pplie for looking up in the grph to fin the neighor of vertex, n looking up in the itne tle to fin the urrent itne. By uing the improve ot we get: W = O + N G (v) + v V u N G (v) = O(m) ( ( )) S = O mx + log N G (v) + mx v V u N(v) = O(log n) n hene the overll omplexity for BellmnFor with rry equene i: W (n, m) = O(nm) S(n, m) = O(n log n) By uing rry equene we hve reue the work y O(log n) ftor.

18 4 CHAPTER. SHORTEST PATHS.