Distributed Online Matching Algorithm For Multi-Path Planning of Mobile Robots

Transcription

1 Proect Paper for embers: Seugkook Yu Sooho Park Dstrbuted Ole atchg Algorthm For ult-path Plag of oble Robots 1. Itroducto Curretly, we are workg o moble robots whch move oly o a specal structure: trusses (See Fg. ). Each robot has two grppers ad three ots for 3-d moto (See Fg.1). Ths robot s expected to move o the trusses ad help costructo of the structure future. We eed to develop a collaborato scheme for lots of robots o the same structure. Note that a robot ca oly move o the specfc pots o the trusses ad every robot has the detcal structure ad fuctos. Also, t must occupy at least oe pot to support tself. Fg.1 the moble robot ad ts structure: 3-ots ad grppers Fg. (a) truss structure ad a robot (b) example of path plag of a sgle robot: umbers deote odes o the trusses

2 1.1 Problem Statemet Our task s to deploy robots to request pots o the trusses by mmum umber of total moves. It s to mmze the eergy usage. For ths, we eed a collso-free m-cost path plag algorthm for mult-robots sce a robot ca't move through the other robot. The trusses ca be modeled by a graph, whch odes are the specfc pots where a grpper of the robot ca grasp ad edges deote whether a robot ca move betwee pars of odes. We wll assume that the graph structure, the target pots, ad the tal posto of robots are gve, ad also that the robots are collaboratg. The real challege s that we wat a dstrbuted algorthm. That s, we do ot wat a cetralzed cotroller whch kows everythg ad cotrols all the robots. Istead, a robot has lmted commucato ad sesg area whch s modeled by umber of edges from the ode occuped by the robot, ad t must decde whch way to go wth the lmted (sesed) formato. 1. Our Approach ad Results I ths paper, we show that the problem ca be solved by a perfect matchg betwee tal odes ad target odes. The matchg uses the fact that the robots are detcal. Ths s ot a offle matchg problem, but closer to the ole matchg. However, ths problem s dfferet from the ole matchg that a robot does ot kow how other robots are matched to the requests. From ow, we call the problem dstrbuted ole matchg. I the ole matchg, t s prove that the lower boud of the compettve rato of ay determstc algorthm s (k-1). The permutato algorthm [1] acheves ths boud. I our work, a dstrbuted matchg algorthm s devsed, usg a greedy ad the permutato algorthm. We prove that t has ( k k) 1 (3 ) k k -compettve rato for the case where each robot s deployed oe by oe, ad -compettve rato for the smultaeous deploymet. We also gve a dscusso about radomzato, ad coecture that the radomzato mght ot be helpful. 1.3 Clarfcato of the problem We clarfy our problem as follows: The truss structure s modeled by a udrected graph G, where odes are pots where a robot ca grasp ad there s a postve cost o each edge. Naturally, tragular equalty holds for G We model each robot as a pot o the graph G The umber of the robots s k, ad all robots are detcal The sesg rage of robots s assumed to be oe,.e. a robot ca commucate wth other robot f ad oly f they are adacet. Whe the commucato s possble, the robots ca share all formato they have. Itally, each robot s located r, ad R s a set of all the tal odes.

3 All the target odes are gve to each robot. T s a set of the target odes. t deotes each target ode. Note that s ot related wth dex of robots. T ca be overlapped wth R. The goal s for the robots to reach all target odes wth mmum cost. Ths paper s orgazed as follows: Secto descrbes the best offle algorthm ad how t ca be vewed as the matchg problem. I Secto 3, we propose the dstrbuted ole matchg algorthm ad show how t works. We gve dscussos about the boud secto 4. We wll gve some modfcato to the algorthm at Secto 5 to mprove the practcalty of the algorthm.. Optmal offle algorthm I ths secto, we dscuss about the optmum soluto of the cetralzed offle case where a cetral cotroller kows everythg o the graph ad cotrol each robot. We also show that we ca solve the problem wth a perfect matchg algorthm..1 Optmal soluto for the cetralzed offle case Ths problem ca be reduced to a m-cost dsot path problem wth vertex capactes sce at ay tme each vertex ca be occuped by oly oe robot. We solve ths problem as follows: - Expad the graph G to reflect the vertex capactes (as we solved the problem set). Note that vertex capacty s oe for all vertces. - Fd dsot paths of the expaded graph. I our case, we should use m-cost max flow rather tha ust max flow algorthm. Ths yelds the m cost paths wthout ay collso.. Ca ths be solved as a matchg problem? If a robot does ot have ts physcal sze, ay perfect matchg betwee R ad T wll gve a feasble soluto, ad each robot ca move o the shortest path to ts matched target ode. oreover, the m-cost perfect matchg s, of course, the optmum soluto. However, a real system, ths set of the shortest paths may rase two crtcal problems: the collso where robots' path cross ad the deadlock where paths form a cycle the graph. Now we propose a tutve selecto crtero to avod these two ssues...1 Selecto crtera: to exchage dettes Whe two paths cross, smply chagg dettes, whch meas to exchage all formato they have ad the acto they are executg ow, wll prevet the collso as show Fg. 3. O the other case whe adacet robots shares the same edge but ther paths do ot cross, let the hgher ID robots go frst ad the other robots wat utl robots wth the hgher ID pass the edge.

4 p r p 1 p1 r r r p 1 1 (a) (b) Fg. 3 exchagg dettes (a) two paths are crossg (b) after exchagg IDs.. Selecto crtera: to break a deadlock A deadlock s the status whch some robots ca ot move eve though ay paths of the robots are ot crossed as show Fg. 4. There are four robots {, r, r 3, r 4 } wth each path {p 1, p, p 3, p 4 }, whch form a rectagular cycle Fg. 4(a). To prevet ths deadlock, we troduce a commucato protocol push. A robot sed a push wth ts ID to aother robot whch s blockg ts way, ad t also cludes IDs of the robots whch are pushg the robot behd. Note that a cycle, every robot ca commucate each other sce they are adacet by defto of the cycle. If a robot fds that t s pushg tself as show Fg. 4(b), t s a cycle. I that case, a robot wth the lowest ID forces the blockg robot to exchage IDs utl a robot the push protocol does ot block t aymore. After ths forced exchage, the cycle wll be broke as show Fg. 4(c). Therefore, we do ot eed to worry about the deadlock wth ths selecto crtero. p 4 p 4 Push {1} p 4 p 1 p p 1 p r p p 1 p 4 p Push {1,,3,4} p 4 p Push {1,} p 4 p p 3 r 4 p 3 r 3 p 3 r 4 p 3 r 3 p 3 r 4 p 3 r 3 p Push {1,,3} p p (a) (b) (c) Fg. 4 How to break a deadlock (a) a deadlock (a cycle) happes (b) commucato protocol for prevetg the deadlock: push (c) the cycle s broke by exchagg dettes. The last questo s whether the process wll be termated. The aswer s deftely yes. Wheever at least oe robot s movg, the total dstace betwee the robots ad the target odes s strctly decreasg because there s o deadlock. Note that the selecto crtera ever crease the dstace. Therefore, f each robot has a dstct target ode, whch meas a perfect matchg, they wll evetually reach oe of them. I a dstrbuted system, as we wll see soo, more tha two robot ca tally have the same target ode. We wll make each robot have the dstct target the ext secto.

5 3. Dstrbuted matchg algorthm Here we propose our algorthm: a dstrbuted greedy permutato whch each robot goes to the earest target ode ad whe there s collso, the collded robots are recofgured by spreadg them to the local optmal vertces. We show that the algorthm has O(k ) compettve rato for both sequetal ad smultaeous deploymet of the robots. 3.1 A smple greedy algorthm s bad Let s cosder a tutve greedy algorthm where each robot successvely fds the earest target ode. A robot chooses the earest target from ts curret posto as the frst target ode. If the target ode has bee occuped, t wll choose the earest target ode from that target ode. Now, see Fg.5 where (k-1) robots have occuped (k-1) target odes ad oly oe robot s fdg ts target by the greedy algorthm. If the left edge has (1+ε) cost, where ε s a postve umber, the robot wll cotuously go to the rght ode ad fally retur to t k after vstg all the rght odes. The cost s ( k -1), whle the optmal cost s oly (1+ε). Therefore, ths case, the compettve rato s ( k -1) whe ε s small. I the appedx, we show that ( k -1) s the tght upper boud of the smple greedy algorthm. 1+ε 1 4 r k r r 3 k- r k-1 t k t 1 t t 3 t k-1 Fg.5 A bad example for a smple greedy algorthm 3. Basc cocepts We start wth a mportat lemma [1] about the mmal weght partal matchg whch s defed as the set of edges that form the mmal weght partal perfect matchg betwee the subset { r1, r,, r } ad subset of T wth a mmal umber of edges Defe T as the subset of T, vertces of T whch are cdet o. Lemma 3.1 The cost of the 's form a mootocally o-decreasg sequece. Lemma 3. For each, the set dfferece T -T -1 cotas exactly oe vertex. The proof of two lemmas are gve [1] We exted ths partal matchg to dstrbuted case. Let be the subset of R, cossted of robots whch have already collded at some target ode wth at least oe other robot the group. Let be the m-cost matchg of the sub-group. Note that s sequetally costructed as a ew robot colldes wth some robot of. Therefore, Lemma 3.1 ad 3. hold for. Now, we prove a mportat lemma about local mcost matchg of two dsots subsets of R.

6 Lemma 3.3 R, R (, R R = ), = + φ R+ R R R Proof. Assume that there exsts R + whch has less cost tha R +. The we ca fd the better partal matchg tha R ad by makg subsets of edges whch are oly cdet o R ad. Ths s a cotradcto. Therefore the cost of R + s the same as R +. Besdes, whe we add to R, augmeted umber of the edges are mmal by the defto of. 3.3 Dstrbuted matchg by mtatg partal optmal matchg ad usg commucato For smplcty, cosder a case where each robot s deployed sequetally, that s, a robot starts to move oly after ts predecessor reaches ts fal target ode. We wll deal wth a geeral case later. The algorthm works as follows: Algorthm Let a robot r move to ts earest target ode 1: If the target s empty ake a ew group R else (t must have collded wth a robot some group ad t may be already a member of aother group R l ) Get a ew target ode by comparso wth the partal optmal matchg + Rl Let every other robots R l + kow the ew robot ad ts target ode by vstg them ove r to the ew target ode Ed f : Repeat 1 utl r reaches empty target ode At the start, a robot behaves greedly. If the target ode s already occuped, we add the robot to the group to whch the collded robot belogs to. The, we compute a ew target ode whch s cdet o the partal optmal matchg of the group cludg the ew member ad whch s ot the target odes of the old group. There wll be exactly oe target ode to vst accordg to Lemma 3.. Oe thg we eed to do s to let every robot the group kow whch robots are the members of the group ad whch ther target odes are. Otherwse, oe robot ca be a member of multple groups because each member of a group may have the dfferet formato about the group to whch t belogs, ad that leads braches of the group. We call ths process commucato from ow. The other case s that a robot may collde wth aother group after og oe group. The algorthm works as the same: fd a ew target ode by comparso wth +, do commucato wth all members, ad go to the ew target ode. We wll provde some modfcatos at secto 5 to decrease ths commucato Rl

7 cost, but a smple worst case aalyss shows that the compettve rato s stll O(k ). Also, we preset a lower k boud for the umber of collsos, whch s O(k ) ad a emprcal proof showg that up to collsos happe the worst case wthout commucato. Now we show how much the proposed algorthm costs. For the cost of the tal greedy approach, we troduce a set of edges F whch s a collecto of the earest edges of. Lemma 3.4 R,cost( F ) cost( ) R R Proof. Assume that total m-robots are a group. WLOG, we ca reorder the robots order of the deploymet. Let f ( = 1,.., ) be the cost of each robot s frst approach edge. Sce the cost of the cdet edge o r ca ot be less tha f, cost( f) cost( R ). m From ow we use as the cost of the set t 1 t 1 t t -1 r a r t b t b r -1 t -1 r -1 t -1 r f e = - -1 t r f e t (a) (b) Fg. 6 matchg wth a ew robot a group: (a) a full traversg path of the robots the old group (b) a full traversg path of the ew robot Lemma 3.5 R, there exsts a path whch start at ay tal ode r, traverse every tal ode ad target ode at least oce, ad eds at a arbtrary target ode t. We call t a full-traversg path. The cost of the path s less tha 3 tmes of sum of all (3 ) = # robot where # robot from 1 robot to curret -robots, whch s deoted by = meas robots has bee deployed the group.

8 Proof. We prove the lemma by ducto. Whe a umber of robots s 1, t s obvously true. Assume the = lemma holds for a group of -1 robots, the optmum matchg of whch s deoted by R # robot 1. Whe the -th robot r comes wth ts fal target ode t (Note that frstly ths robot must collde wth a robot ), there are two cases the ew optmum R # robot= : r matches t (See Fg 6) or some t m the old wth -1 robots (Fg. 7). f s the frst approach edge to the earest target ode. t 1 t 1 r m r -1 t t m t m r a t b r m t b r -1 t -1 r -1 t -1 f r t r t (a) (b) Fg. 7 matchg wth a ew robot a group: (a) a full traversg path of the robots the old group (b) a full traversg path of the ew robot Frstly look at the former Fg. 6(a). Suppose there s a full traversg path from a arbtrary robot to ay target R = ode, deoted by r a ad t b # robot 1, whch s the red dotted le Fg 6(a), we ca make a ew full traversg path by oly addg the recprocatg partal path for the ew target ode, whch are f ad e draw as a red sold le. Note that R # robot= 1 + f R # robot=, because ay matchg cost wth a ew robot should be o less tha the earest edge cost. Also, whe r matches t, e = # robot= # robot= 1, because we do ot chage ay matchg the old but add the ew matchg from r to t. Therefore, f + e ( R # # 1 ) ( # # 1)) robot= robot= + robot= robot=. 4( ) R # robot= R # robot= 1 O the other had, for a full-traversg path from r, whch was ot R, we use r # robot= 1, the former match of

9 t whch s the target ode drected by f R # robot= 1. We ca make a full-traversg path from r to ay target ode { t1, t,, t 1} by the assumpto. We ca easly add t to the path by usg e. I ths case, we eed the followg path: f + e + { r t} + ( ) + # robot= # robot= 1 # robot= # robot= 1 # robot= 1 3 # robot= # robot= 1 Secodly, whe t matches aother robot r m R # robot=, r must match oe of the target odes, t m, the old group as show Fg 7. As we dd, we ca make a ew full traversal path from ay r a ad t b R by # robot= 1 addg { rm t} ad { r tm} two tmes the ew optmal matchg R. # robot = ({ r t } + { r t }) = m m # robot If we start at r, we use the full traversal path from r, whch was matched by the target t the old group. ad add the recprocal path of { r t }. That costs: m f + { r t} + { r t } + + m # robot= # robot= 1 # robot= 1 # robot= 3 # robot= Therefore, the worst addg cost amog all cases s bouded by 3 R # robot=, whch drects: P P # robot = P [3( 1) ] + 3 # robot= 1 # robot= [3( 1) ] + 3 = (3 ) # robot= # robot= Lemma 3.6 The cost of costructg a group R # robot= by sequetally addg oe robot s bouded by 1 (3 - ) # robot= Proof. Exame each step of addg a ew robot to the group. By the proposed algorthm, a robot r start movg wth fdg the earest target ode whch belogs to the target odes of R # robot= 1. The t has to traverse all the other members R # robot = 1 for commucato ad fally reaches the fal target pot. Note that ths s a path whch starts from r, vsts all the target odes of R, ad arrves t # robot = 1. By Lemma3.5, we ca fd the path wth the bouded cost. Therefore, the whole cost s:

10 COST (3 ) # robot # robot (3 ) # robot= # robot 3 1 ( - ) # robot= # robot= (3 ) Lemma 3.7 Whe mergg two groups of R ad # robot= R l # robot= m to a ew group # R robot= m+, where both groups are oly costructed by addg a sgle robot. The whole cost cludg the merge ad the costructo of two groups are bouded by 1 [3( ) m ( m )] + Rl + +. Proof. WLOG, assume that the last robot of R l hts a robot. The the cost of oly mergg s to traverse all robots R l + from the ht pot to the fal target pot +. Note that oly r m eeds to recofgure tself by Lemma3.3. Therefore the mergg cost s bouded by [3( + m) ] R + R, ad total cost cludg the costructos s: 1 1 COST (3 - ) R # (3 - ) # [3( ) ] robot= + m m Rl robot= m + + m + Rl 1 1 (3 - ) + (3m - m)( ) + [3( + m) ] 1 [ (3m - m) + 3( + m) ] R+ Rl # robot= + Rl # robot= + Rl We used 3.3 ad assumed m WLOG. Ad, m + m m m + + m [3( ) ( )] [ (3 - ) 3( ) ] 1 ( ) = + m m+ 1 [3( 1) 6 6 1] = + + m m + 1 [3( 1) (6 1)( 1)] 0 = + m Therefore, the Lemma s true. Rl l Lemma 3.8 Whe mergg two groups of R ad # = R l # robot= m to a ew group # robot R robot= m+ cost cludg the merge ad the costructo of two groups are bouded by [3( + m) ] R + R. l. The whole Proof. By Lemma3.7, we fd that the costructo cost cludg the merge s the same as makg a group by sequetally addg a sgle robot. Therefore, Proof for Lemma3.7 holds for the geeral merge.

11 Lemma 3.9 The proposed dstrbuted algorthm for the sequetal deploymet has 1 (3 ) k k compettve rato. Proof. After all robots are deployed to the target odes by the proposed algorthm, there may exst a several depedet (o-collded) groups. The worst case s the oly oe group remas because the total costructo cost of mergg s always bgger tha the separated costructo cost of mult groups by Lemma3.7. At the last 1 merge, total umber of robots must be k, therefore, total cost s o greater tha (3 k k ) R. Note that cost( R ) s the optmum. Lemma 3.10 The proposed dstrbuted algorthm for the smultaeous deploymet gves rato. ( k k) compettve Proof. Ulke the ole matchg, the deploymet of robots takes a physcal tme. Therefore, the smultaeous deploymet may cause a case that a robot colldes wth a group whch s recofgurg ow. To elmate ths cofuso, we set a prmary robot each group whch frstly costructed the group. Whe a ew robot colldes wth ay robot of the group, we force t to vst the prmary robot frst, the let t do the same process to vst all target odes the group ad to settle the ew target ode - as a robot does the sequetal deploymet. Now we ca hadle the recofguratos order of vstg the prmary robots. Note that we ca always recofgure oe by oe order of vstg the prmary robot, by Lemma3.. Ths vstg s show to take aother f + R # robot= 1 R # robot= by ducto lke Fg. 6. Therefore the boud Lemma 3.5 s modfed to (4 3). Ad ths cosequetally yelds total ( ) k k. 4. Dscussos 4.1 Is the boud tght? I our problem, the commucato cost s domat. If t s ot ecessary, that s, a group member ca share the formato after the frst collso by ay method, we acheve O(k) compettve rato as the ole matchg does. O the other had, f there exsts a way to reduce the cost of the commucato or a good algorthm wthout the commucato although t makes the problem more hard to aalyze, we may reduce the order of the compettve rato. Ayhow, t seems hard to prove the tght boud whle the ole matchg s easly prove to have (k-1) boud for ay determstc algorthm [1]. We provde a proof that ths boud may be tght the ext secto. 4. O(k ) collsos wthout commucato We suspected that the umber of collso betwee robots should be decreased to reduce the cost. Wthout commucato, robot oly shares formato whe there s collso. We wll prove a lower boud o the worst case umber of collsos frst. It was hard to aalyze the umber of collsos betwee robots the worst case,

12 so we mplemeted the brute-force search algorthm to see the emprcal result. Frst, we prove that at least O(k ) collsos happes the worst case. Let s cosder the sequetal deploymet case. Whe a ew robot colldes wth a settled oe, they share the formato so that the ew robot wo t vst a target ode where t kows some robot s occupyg. See Fg. 8. We ca represet a robot as a ode ad formato as a lk ad a umber. The umber o the robot ode represets the umber of other robots locato that the robot kows. The lk represet that the two robots coected by a lk collded drectly oce. The robots at hgher posto the graph kows less tha the oes below. I a bad case, a ew robot colldes wth the robot wth the least formato ad cosecutvely colldes wth the robots that t has o formato about. The robot does t kow about the robot whch s odes away. So, whe there are k robots settled dow k k + 1 ad a ew robot s deployed, there ca be collsos for eve k, ad collsos for odd k. Therefore, f we deploy k robots sequetally, we get O(k ) collsos. Also, a brute-force search for the worst case gave us k ( ) collsos for k robots Fg. 8 collsos betwee robots for the sequetal deploymet of robots: It dvdes to two cases where the umber of robots s odd, ad eve. 4.3 Could radomzed algorthm help? As far as we have searched, there are oly a few results o the radomzed ole matchg. The best compettve rato of the radomzed oe[] s O(log 3 k) whle a determstc s O(k). They use a specal graph, HST tree, rather tha a geerc graph. Ad t s show that a geerc graph ca be modfed to a HST tree. We dd ot try ths approach to our problem. However, we coecture that a radomzed algorthm would ot work well our case because t s much harder to arrow the probablty the choces. For stace, for a uform metrc graph where odes are fully coected ad every edge cost s 1, the proposed radomzed ole

13 matchg algorthm - whch fds the earest oe ad f there s te (mult odes wth the same cost), t radomly select oe gves O(logk) expected compettve rato whle a greedy oe does O(k). However, our case, the same algorthm also yelds O(k), whch s the same order as that of a greedy oe. Although oe example ca ot tell everythg, we guess ths s a ht that a radomzed algorthm mght ot work better maly because of the requred commucato-equvalet-cost. 5. Improvg the practcalty of the algorthm We came up wth two modfcatos to decrease the umber of collsos, whch ca be used as a combato as well. However, a smple worst case aalyss shows that the methods do t mprove the compettve rato more tha a costat factor mprovemet, though t wll deftely work better practce. 5.1 Commucato oly after a ew robot fds the empty ode The proposed algorthm gve at secto 3.3 uses commucato everytme whe there s a collso betwee a ew robot ad a group. It leads to multple commucatos f the ew robot colldes wth aother group aga. We ca easly modfy ths so that there wll be commucato oly after the ew robot fds ts place. I ths case, the robot wll merge several groups at oe tme. It surely decreases the commucato cost, the umber of collsos betwee robots. However, the compettve rato s stll O(k ). Group Group 1 New robot Fg. 9 a traversal of a ew robot 5. Usg a prmary robot for each group ad uo-set data structure The other method s to use a prmary robot for each group so that a ew robot colldes wth the group oly eeds to commucate wth the prmary robot. We wll also use the uo-set data structure to maage the merges betwee groups. Now, each group s a set wth a prmary robot as the set-represetatve. If a ew robot colldes wth a member of a group, t wll go to the prmary robot of the group ad get a target ode. The prmary robot kows the group members ad ts target odes so that t ca calculate the partal optmal matchg. Whe groups are merged, the prmary robot of oe group wll sed ew robots to the other group s prmary robot the future. It s smlar to uo-set data structure.

14 Fg. 9 shows a possble traversal of a ew robot whe there are two groups of robots (each composed of 4 members ad 1 prmary robot). It colldes wth a robot the frst group ad vsts the prmary robot. The prmary robot seds the ew robot to a target ode, whch s actually occuped by a robot the secod group. It goes to the prmary robot of the secod group. Now, the two groups eed to be merged. It ca be doe by settg the prmary robot oe group to sed ew robots to other group s prmary robot the future or actually updatg the lks of each robot. We dd t aalyze the costs fully. However, a smple aalyss showed that t does t mprove the compettve rato sce the path to the prmary robot could clude all the members the group so that the umber of collso wll be smlar. 6. Cocluso I our paper, we propose a ew dstrbuted matchg algorthm for path-plag algorthm of moble robots. We show that ths problem ca be see as a perfect matchg by some selecto crtera. It s prove that the algorthm has O(k ) compettve rato whle a smple greedy gves a expoetal oe. Refereces [1] B. Kalayaasudaram ad K. Pruhs. Ole Weghted atchg, Joural of Algorthms 14(3), 1993 [] A. eyerso, A. Naavat, ad L. Poplawsk, Radomzed Ole Algorthms for mum etrc Bpartte atchg, AC-SIA Symposum o Dscrete Algorthms (SODA), 006

15 Appedx. A smple greedy algorthm for our problem. Ths algorthm works as a robot cotuously fd the earest eghbor target ode. t 1 r t r -1 t -1 r f c t Fg.10 a smple greedy algorthm Lemma. A smple greedy has k -1 compettve rato. Proof. By ducto wth a umber of the robots. Whe #robot=1, t s obvous. Say that ths holds for -1 robots. That s, SG ( 1) There are two cases: If r does ot collde aythg after ts deploymet, the SG SG 1 + c 1 ( 1) < ( -1) I the secod case, the maxmum umber of collso s -1. The total addg cost to SG 1 s bouded as follows: 3 cost c + ( c + f1) + ( c + f1) + + ( c + f1) ( c + f ) 1 Ths cost s maxmzed whe f1 = c, ad that case f1 = c cost( ). Therefore, the ducto holds for as follows: SG SG ( 1) 1 + ( -1)