Chapter 16. Shortest Paths Shortest Weighted Paths

Transcription

1 Chpter 6 Shortet Pth Given grph where ege re lle with weight (or itne) n oure vertex, wht i the hortet pth etween the oure n ome other vertex? Prolem requiring u to nwer uh querie re roly known hortet-pth prolem. Shortet-pth prolem ome in everl flvor. For exmple, the ingle-oure horetet pth prolem require fining the hortet pth etween given oure n ll other vertie; the ingle-pir hortet pth prolem require fining the hortet pth etween given oure n given etintion vertex; the ll-pir hortet pth prolem require fining the hortet pth etween ll pir of vertie. In thi hpter, we onier the ingle-oure hortet-pth prolem n iu two lgorithm for thi prolem: Dijktr n Bellmn-For lgorithm. Dijktr lgorithm i more effiient ut it i motly equentil n it work only for grph where ege weight re non-negtive. Bellmn-For lgorithm i goo prllel lgorithm n work for ll grph ut require ignifintly more work. 6. Shortet Weighte Pth Conier weighte grph G = (V, E, w), w : E R. The grph n either e 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. 7

2 76 CHAPTER 6. SHORTEST PATHS 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 δ G (u, v) to inite the weight of hortet pth from u to v. Quetion 6.. Wht hppen if we hnge the weight of the ege (, ) from 7 to 7? If we llow for negtive weight ege then it i poile to rete hortet pth of infinite length (in ege) n whoe weight i. In Exmple 6. 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 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. A riefly mentione efore, in thi hpter, we re interete in fining ingleoure hortet pth, efine follow. Prolem 6. (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 δ(u, v) of the hortet pth n not the pth itelf. In Chpter 4 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. April 9, (DRAFT, PPAP)

3 6.. DIJKSTRA S ALGORITHM 77 Quetion 6.4. Cn we ue BFS to olve the ingle-oure hortet pth prolem on weighte grph? Ignoring weight n uing BFS, unfortuntely, oe not work on weighte grph. Exmple 6.. To ee why BFS oe not work, onier the following irete grph with vertie: 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 6.6. 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 (t roun 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). We will ue imilr ie for weighte pth. 6. Dijktr Algorithm Conier vrint of the SSSP prolem, where ll the weight on the ege re non-negtive (i.e. w : E R ). We refer to thi the SSSP + prolem. Dijktr lgorithm olve effiiently the SSSP + prolem. Dijktr lgorithm i importnt eue it i effiient n lo eue it i very elegnt exmple of n effiient greey lgorithm tht gurntee optimlity (of olution) for nontrivil prolem. April 9, (DRAFT, PPAP)

4 78 CHAPTER 6. SHORTEST PATHS 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 leer weight thn the pth tht en t the firt viit to the vertex. When erhing for hortet pth, we thu hve to onier only the imple pth. Quetion 6.7. Give rute-fore lgorithm for fining the hortet pth? Let u 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 6.8. 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. 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. Exmple 6.9. If hortet pth from Pitturgh to Sn Frnio goe through Chigo, then tht hortet pth inlue the hortet pth from Pitturgh to Chigo. To ee how u-pth property n e helpful, onier the grph in Exmple 6. n uppoe tht n orle h tol u the hortet pth to ll vertie exept for the vertex v. We wnt to fin the hortet pth to v. Quetion 6.. Cn you fin the hortet pth to v? By inpeting the grph, we know tht the hortet pth to v goe through either one of,, or. Furthermore, y u-pth property, we know tht the hortet pth to v onit of the hortet pth to one of,, or, n the ege to v. Thu, ll we hve to o i to fin the u mong the in-neighor of v tht minimize the weight of the pth to v, i.e. δ G (, u) plu the itionl ege weight to get to v. Quetion 6.. Coul the hortet-pth e going through v? If o, then re we till gurntee to fin hortet pth? April 9, (DRAFT, PPAP)

5 6.. DIJKSTRA S ALGORITHM 79 Note tht we hve eie to fin only the hortet pth tht re imple, whih nnot go through v itelf. Exmple 6.. In the following grph G, uppoe tht we hve foun the hortet pth from the oure to ll the other vertie exept for v; eh vertex i lele with it itne to. 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, whih in thi e i vertex. X v 7 Let now onier the e where the orle gve u the hortet pth to ll ut two of the vertie u, v. Quetion 6.. Cn we fin the hortet pth to ny one of u or v? Let X enote the et of vertie for whih we know the hortet pth. We my think of thi et the viite et in grph erh. Let lulte for u (n v) the hortet pth tht goe through neighor in X n then goe to u (v). Conier now the hortet of thee two itne n ume without lo of generlity tht it i the pth to u. Quetion 6.4. Coul the hortet pth to u e thi pth? Sine oure i in X, we know tht the hortet pth to u ro out from X vi ome ege tht goe to either u or v. Sine we know tht the weight of uh pth to u i the mller, n ine ll ege weight re non-negtive, there nnot e horter pth tht goe to u n thn ome k to v gin. We thu onlue tht the hortet pth to v i the pth tht we hve foun y oniering ll inoming ege from X. Exmple 6. illutrte thi. April 9, (DRAFT, PPAP)

6 8 CHAPTER 6. SHORTEST PATHS Exmple 6.. In the following grph uppoe tht we hve foun the hortet pth from the oure to ll the vertie in X. The hortet pth re inite y lel next to the vertie. The hortet pth from the oure to ny vertex in Y i the pth to vertex with weight followe y the ege (, v) with weight 4 n hene totl 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 6 v u. Quetion 6.6. Cn you ee how we n generlize thi ie? Let try to generlize the rgument n onier the e where the orle tell u the hortet pth from to 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 one of the vertie in Y. Quetion 6.7. Wht woul e the point of oing tht? 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 6.8. Cn you ue thi property to ientify vertex v Y tht i no frther from the oure thn ny other vertex in Y? April 9, (DRAFT, PPAP)

7 6.. DIJKSTRA S ALGORITHM 8 Conier the vertex v Y tht h n ege to ome lrey viite vertex x X n tht minimize the um δx + w(x, v). Sine ll pth to Y mut go through the frontier when exiting X, n ine ege re non-negtive, u-pth nnot e longer thn the full pth. Thu, no other vertex in Y n e loer to the oure thn x. See Exmple 6.9. Furthermore, ine ll other vertie in Y re frther thn v n ine ll ege re non-negtive, the hortet pth for v i δx + w(x, v). Exmple 6.9. 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 the oure to ny vertex in Y i the pth to vertex with weight followe y the ege (, v) with weight 4 n hene totl 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 Y= V \ X v t 4 u. Thi intuition i tte n prove in Lemm 6.. 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. Dijktr lgorithm imply oe extly thi: it turn the rnk until ll vertie re viite. Quetion 6.. You might hve notie tht the terminology tht we ue in explining Dijktr lgorithm loely relte to tht of grph erh. Doe thi lgorithm remin you of prtiulr grph erh tht we iue? 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. In other wor, thi lgorithm i tully n intne of priority-firt erh. We re now rey to efine preiely Dijktr lgorithm. April 9, (DRAFT, PPAP)

8 8 CHAPTER 6. SHORTEST PATHS Lemm 6. (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 u-pth 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. Algorithm 6. (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 roun it viit unviite vertie tht hve the minimum hortet-pth weight from. April 9, (DRAFT, PPAP)

9 6.. DIJKSTRA S ALGORITHM 8 Remrk 6.. 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 roun. Thi i not true. To ee thi onier the exmple elow n onvine yourelf tht it oe not ontrit our reoning. Lemm 6.4. Dijktr lgorithm return, (v) = δ G (, v) for v rehle from. Proof. We how tht for eh tep in the lgorithm, 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 6., 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. 6.. 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 6.. 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 9 in the oe) n, if it hn t lrey een viite, viit it y ing (x ) to the tle of viite vertie (Line 6), 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 other vrint on Dijktr lgorithm uing Priority Queue. April 9, (DRAFT, PPAP)

10 84 CHAPTER 6. SHORTEST PATHS Algorithm 6. (Dijktr Algorithm uing Priority Queue). funtion ijktrpq(g, ) = let 4 % require: (x ) X, = δ G (, x) 6 {(, y) Q y V \ X} = {( + w(x, y), y) : (x ) X, y N + (x) \ X} 7 % return: {x δ G (, x) : x R G ()} 8 funtion ijktr (X, Q) = 9 e PQ.eleteMin (Q) of (, _) X ((, v), Q ) if (v, ) X then ijktr (X, Q ) 4 ele let 6 X = X {(v, )} 7 funtion relx (Q, (u, w)) = PQ.inert ( + w, u) Q 8 Q = iter relx Q N G (v) 9 in ijktr (X, Q ) en in ijktr ({}, PQ.inert (, ) {}) en Firtly we oul hek inie the relx funtion 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, whoe oe i hown in Algorithm 6.. Let fir onier the ADT tht we will ue long with their ot. For the priority queue, we ume PQ.inert n PQ.eleteMin hve O(log n) work n pn. To repreent the grph, we n either ue tle or n rry, mpping vertie to their neighor long with the weight of the ege in etween. For the purpoe of Dijktr lgorithm, it uffie to ue the tree e ot for tle. Note tht ine we on t upte the grph, we on t nee ingle three rry. To repreent the tle of itne to viite vertie, we n ue tle, n rry equene, or ingle three rry equene. April 9, (DRAFT, PPAP)

11 6.. DIJKSTRA S ALGORITHM 8 Exmple 6.6. 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 April 9, (DRAFT, PPAP)

12 86 CHAPTER 6. SHORTEST PATHS Opertion Line # of ll PQ Tree Tle Arry ST Arry eletemin Line 9 O(m) O(log m) inert Line 7 O(m) O(log m) Priority Q totl O(m log m) fin Line O(m) - O(log n) O() O() inert Line 6 O(n) - O(log n) O(n) O() Ditne totl - O(m log n) O(n ) O(m) N G (v) Line 8 O(n) - O(log n) O() - iter Line 8 O(m) - O() O() - Grph e totl - O(m + n log n) O(m) - Figure 6.: The ot for the importnt tep in the lgorithm ijktrpq. To nlyze the work, we lulte the work for eh ifferent kin of opertion n um them up to fin the totl work. Figure 6. ummrize the ot of the opertion, long with the numer of ll me to eh opertion. The lgorithm in inlue ox roun 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 fely ignore it. Of the remining opertion, The iter n N G (v) on Line 8 re on the grph, Line n 6 re on the tle of viite vertie X, n Line 9 n 7 re on the priority queue Q. We n lulte the totl numer of ll to eh opertion y noting tht the oy of the let trting on Line i only run one for eh vertex. Thu, Line 6 n N G (v) on Line 8 re only lle O(n) time. All other opertion re performe one for eh ege. The totl work for Dijktr lgorithm uing tree tle i therefore O(m log m + m log n + m + n log n). Sine m n, the totl work i O(m log n). Sine the lgorithm i equentil, the pn i the me the work. Be on the tle one houl note tht when uing either tree tle or ingle three equene, 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; there might, however, e ifferene in ontnt ftor. One houl lo note tht uing regulr purely funtionl rry i not goo ie, eue the ot i then ominte y the inertion n the lgorithm run in Θ(n ) work. 6. 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. When oniering itne on mp for exmple, negtive weight o not mke ene (t let without time trvel), ut mny other prolem reue to hortet pth, n in thee reution April 9, (DRAFT, PPAP)

13 6.. THE BELLMAN FORD ALGORITHM 87 negtive weight o 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 to the ingleoure hortet pth prolem, iue erlier. In uh e, we woul like the lgorithm to inite tht uh yle exit n terminte. Exerie 6.7. 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. Exerie 6.8. In your olution to the previou exerie n you get negtive weight yle? If o, wht oe thi men? Quetion 6.9. 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. More peifilly, Dijktr lgorithm i e on the umption tht the hortet pth to the vertex v in the frontier tht i loet to the et of viite vertie, whoe itne hve een etermine, n e etermine y oniering jut the inoming ege of v. With negtive ege weight, thi i not true nymore, eue there n e horter pth tht venture out of the frontier n then ome k to v. Exmple 6.. To ee where Dijktr property fil with negtive ege weight onier the following exmple Dijktr lgorithm woul viit then n leve with itne of inte of the orret itne. The prolem i tht when Dijktr viit, it fil to onier the poiility of there eing horter pth from to (whih i impoile with nonnegtive ege weight). April 9, (DRAFT, PPAP)

14 88 CHAPTER 6. SHORTEST PATHS Quetion 6.. How n we fin hortet pth on grph with negtive weight? Quetion 6.. 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, however, i tht the u-pth of hortet pth themelve re hortet. Dijktr lgorithm took vntge of thi property y uiling longer pth from horter one, i.e., y uiling hortet pth in non-ereing orer. With negtive ege weight thi oe not work nymore, eue pth n get horter we ege. But there i nother wy to ue the me property: uiling pth tht ontin more n more ege. To ee how, uppoe tht you hve foun the hortet pth from oure to ll vertie with k or fewer ege. Quetion 6.. How n you upte the hortet pth for k + ege? Tht i you wnt to fin the hortet pth tht ontin k + or fewer ege. We n ompute the hortet pth with k + or fewer ege y extening ll pth y one ege if thi i enefiil. More peifilly, ll we nee to o i onier eh vertex v n ll it inoming ege n pik the hortet pth to tht vertex tht rrive t neighor u uing k or fewer ege n then tke the ege from u to v. To mke thi more preie, let efine δg k (, t) the hortet weighte pth from to t uing t mot k ege. April 9, (DRAFT, PPAP)

15 6.. THE BELLMAN FORD ALGORITHM 89 Exmple 6.4. In the following grph G, uppoe tht we hve foun the hortet pth from the oure to vertie uing k or fewer ege; eh vertex u i lele with it k-itne to, written δg k (, u). The weight of the hortet pth to v uing k + or fewer ege i i min δ G (, v), min δ G (, ) +, δ G (, ) + 6, δ G (, ) + The hortet pth with t mot k + ege h weight n goe through vertex v 7-7 Be on thi ie, we n ontrut pth with more n more ege. We trt y etermining δg (, v) for ll v V. Sine no vertex other thn the oure i rehle with pth of length, δg (, v) = for ll vertie other thn the oure. Then we etermine δ G (, v) for ll v V, n itertively, δ k+ G (, v) e on ll δk G (, v) for k =,.... To lulte the upte itne with t mot k + ege the lgorithm, we n ue the hortet pth with k ege to eh of it in-neighor n then in the weight of the one itionl ege. More preiely, for eh vertex v, 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)) ). Quetion 6.. Cn the itne for vertex ty the me? Sine the new itne for vertex i lulte in term of the itne from mot reent itertion, if the itne for ll vertie remin unhnge, then they will ontinue remining unhnge fter one more itertion. It i thu unneery to ontinue iterting fter the itne onverge. Quetion 6.6. When houl we top? We n thu top when the itne onverge. April 9, (DRAFT, PPAP)

16 9 CHAPTER 6. SHORTEST PATHS Algorithm 6.9 (Bellmn For). funtion BellmnFor(G = (V, E), ) = let % require: v V, D v = δg k (, v) 4 funtion BF(D, k) = let 6 7 in D = {v min(d[v], min u N G (v)(d[u] + w(u, v))) : v V } 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 Quetion 6.7. I onvergene gurntee? Convergene, however i not gurntee. For exmple, if we hve negtive yle rehle from the oure then, the itne will keep getting mller n mller t eh itertion, whih i expete. Quetion 6.8. Cn we etet uh itution? We n etet negtive yle y notiing tht the itne o not onverge even fter V itertion. Thi i eue, imple (yli) pth in grph n inlue t mot V ege n, in the ene of negtive-weight yle, there lwy exit n yli hortet pth. Algorithm 6.9 efine the Bellmn For lgorithm e on thee ie. 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 6.. Theorem 6.4. Given irete weighte grph G = (V, E, w), w : E R, n oure, the BellmnFor lgorithm return either. δ G (, v) for ll vertie rehle from, or. if there i negtive weight-yle in the grph rehle from. April 9, (DRAFT, PPAP)

17 6.. THE BELLMAN FORD ALGORITHM pth length = pth length pth length - - pth length pth length 4 Figure 6.: Step of the Bellmn For lgorithm. The numer with qure inite wht hnge on eh tep. 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. To nlyze the ot of Bellmn-For we onier two ifferent repreenttion for grph, one uing tle n the other uing equene. For tle-e repreenttion of the grph, we ue tle mpping eh vertex to tle of neighor long with their rel-vlue weight. We repreent the itne D tle mpping vertie to their itne. 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 ) 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 April 9, (DRAFT, PPAP)

18 9 CHAPTER 6. SHORTEST PATHS n = V n m = E, the work i W = O log n + N G (v) + v V = O ((n + m) log n). ( + log n) u N G (v) Similrly, pn i ( ( )) S = O mx log n + log N G (v) + mx ( + log n) v V u N(v) = O(log n). The work n pn of 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. Sine 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) Let now onier the ot with equene repreenttion of grph. If we ume the vertie re the integer {,,..., V } then we n ue equene to repreent the grph. Inte of uing fin for tle, whih require O(log n) work, we n ue nth (inexing) 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. April 9, (DRAFT, PPAP)

19 6.4. PROBLEMS Prolem Exerie 6.4. We mentione tht we re ometime only interete in the weight of hortet pth, rther thn the pth itelf. For weighte grph G = (V, E, w), ume you re given the itne δ G (, v) for ll vertie v V. For prtiulr vertex u, erie how you oul etermine the vertie P in hortet pth from to u in O(p) work, where p = v P (v). Exerie 6.4. Prove tht the u-pth property hol for ny grph, lo in the preene of negtive weight. Exerie 6.4. 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. April 9, (DRAFT, PPAP)

20 94 CHAPTER 6. SHORTEST PATHS. April 9, (DRAFT, PPAP)