An efficient iterative source routing algorithm

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1 An effcent teratve source routng algorthm Gang Cheng Ye Tan Nrwan Ansar Advanced Networng Lab Department of Electrcal Computer Engneerng New Jersey Insttute of Technology Newar NJ 7 {gc yt Ansar}@ntedu Abstract: Fndng a feasble path subect to multple constrants n a networ s an NP-complete problem cannot be solved n polynomal tme Thus many proposed source routng algorthms tacle ths problem by transformng t nto the shortest path problem whch s P-complete wth an ntegrated cost functon that maps mult-constrants nto a sngle cost However ths approach may fal to fnd a feasble path even when t exsts Some algorthms mprove ther success rato of fndng a feasble path by performng multple shortest path searches; each search s assocated wth a dfferent cost functon Thus how to vary the cost functon s a crtcal ssue for ths nd of approach In ths paper we propose an effcent algorthm to expedte the multple shortest path searches based on the analyss of the cost functon Smulatons show that our algorthm outperforms ts contender n terms of computatonal complexty success rato of fndng a feasble path Introducton One of the challengng ssues for hgh-speed pacet swtchng networs to facltate varous applcatons s to select feasble paths that satsfy dfferent qualty-of-servce requrements Ths problem s nown as QoS routng In general two ssues are related to QoS routng: state dstrbuton routng strategy [] State dstrbuton addresses the ssue of exchangng the state nformaton throughout the networ [] Routng strategy s used to fnd a feasble path that meets the QoS requrements In ths paper we focus on the latter tas assume that accurate networ state nformaton s avalable to each node A number of research wors have also addressed naccurate nformaton [3 7] whch s however beyond the scope of ths paper QoS constrants can be categorzed nto three types: concave addtve multplcatve Snce concave parameters set the upper lmts of all the lns along a path such as bwdth we can smply prune all the lns nodes that do not satsfy the QoS constrants We can also convert multplcatve parameters nto addtve parameters by usng the logarthm functon For nstance we can tae -log(- p ) as the replacement for loss rate p Thus we focus only on addtve constrants n ths paper It has been proved that multple addtvely constraned QoS routng s NP-complete [8] Hence taclng ths problem requres heurstcs In [9] a heurstc algorthm was proposed based on a lnear cost functon for two addtve constrants; ths s a CP (ultple Constraned Path Selecton) [] problem wth two addtve constrants A bnary search strategy for fndng the approprate value of n the lnear cost functon w( p) + w( p) or w( p) + w( p) where w ( p ) ( = ) are two respectve weghts of the path p was proposed a herarchcal Dstra algorthm was ntroduced to fnd the path It was shown that the worst-case complexty of the algorthm s Ο (log Bm ( + nlog n)) where B s the upper bound of the parameter m s the number of lns n s the number of nodes The authors n [] smplfed the multple constraned QoS routng problem nto the shortest path selecton problem n whch the Weghted Far Queung (WFQ) servce dscplne s assumed Hence ths routng algorthm cannot be appled to networs where other servce dscplnes are employed Smlar to [9] Lagrange Relaxaton Based Aggregated Cost (LARAC) was proposed n [] for the Delay Constraned Least Cost path problem (DCLC) Ths algorthm s based on a lnear cost functon cλ = c+ λd where c denotes the cost d the delay λ an adustable parameter It dffers from [9] on how λ s defned: λ s computed by Lagrange Relaxaton nstead of the bnary search It was shown that the computatonal 4 complexty of ths algorthm was Ο ( m log m) However n [] for the same problem (DCLC) a non-lnear cost functon was proposed after consderng the shortcomng of the lnear cost functon any researchers have posed the QoS routng problem as the -shortest path problem but the computatonal complexty s generally very hgh [34] To solve the delay-cost-constraned routng problem Chen Nahrstedt proposed an algorthm [5] whch maps each constrant from a postve real number to a postve nteger By dong so the mappng offers a coarser resoluton of the orgnal problem the postve nteger s used as an ndex n the algorthm The computatonal complexty s

2 reduced to pseudo-polynomal tme the performance of the algorthm can be mproved by adustng a parameter but wth a larger overhead As revewed above some proposed algorthms perform multple shortest path searches by varyng cost functon Note that the executon of any shortest path searchng algorthm such as Dstra Bellman-Ford algorthm gets a same path f the same cost functon s used there s only one shortest path for a specfc networ Thus the way of changng the cost functon drectly affects the performance of the correspondng mult-constraned routng algorthm In ths paper we propose a source routng algorthm that adusts the cost functon teratvely effcently Through smulatons the performance of our algorthm outperforms ts contenders n terms of success rato computatonal complexty Problem formulaton notatons Defnton : ultple Addtve Constrants Path Selecton (ACP): Assume a networ s modeled as a drected graph GNE ( ) where N s the set of all nodes E s the set of all lns Each ln connected from node u to v denoted by = ( u v) E s assocated wth addtve parameters: w ( u v) = Gven a set of constrants ( c c c ) a par of nodes s t fnd a path p from s to t subect to W ( p ) = w ( u v ) < c = e uv p Defnton : Any path selected by ACP s a feasble path that s any path p from s to t that meets the requrement W ( p ) = w ( u v ) c e uv p = s a feasble path Notatons: f( x ): Cost functon where x = ( x x x ) C : The vector representaton of the QoS constrants ( c c c ) W( p ): The weght vector of path p e ( W ( p) W ( p) W ( p )) where W( p) = w( u v) as p C( p: ) The cost of path p whch can be wrtten f( w ( ) w ( ) w( )) where f () s the p cost functon Note that C( p) f( W( p)) because f( W( p )) = f( w( uv ) w( uv ) w ( uv )) p p p However f f( x ) s lnear C( p) = f( W( p)) It should also be noted that to be a cost functon f( x ) f( x) should have the property that f x x = ; e the cost functon s ncreasng wth respect to each addtve parameter 3 The proposed algorthm Snce multple QoS constrants routng problem s NP-complete no algorthms can ensure that the termnaton condton can be met n polynomal tme Here the termnaton condton s referred to as any of the followng condtons: A feasble path s found It s certan that no feasble path exsts Thus le [9] multple searches whch ncorporate an algorthm for adustng the cost functon parameters are necessary n order to ncrease the success rato of fndng a feasble path We use the followng cost functon as the ntal cost functon: x f( x) = = c th Let assume the functon for the search s f ( x x x ) = β x β = Thus β = = c = We shall next present some theorems showng the motvaton behnd our algorthm Theorem : No feasble path exsts f the least cost path of the th search has the cost no less than f ( C) Proof: By contradcton Assume path p satsfes the constrant C the least cost among all paths s no less than f ( C) ; that s C( p) f ( C) p C( p) f ( C) Also f ( x x x ) = f ( W( p)) = C( p) x Thus f ( W( p)) f ( C) f However snce ( x) path p satsfes x the constrant C W( p) < c { } f ( W( p)) < f ( C) whch contradcts f ( W( p)) f ( C) thus Theorem s proved Lemma : Path p s a feasble path only f f ( W( p)) < f ( C) = Proof: Ths can be readly derved from Theorem Based on Theorem Lemma a welldesgned algorthm for adustng the cost functon

3 referred to as QA (Quc-Adustng Algorthm) s proposed to expedte reachng the termnaton condton We frst execute the shortest path searchng algorthm wth the cost functon = β x = ( β = c = ) as the ntal cost functon Thus there are 3 possble outcomes: A feasble path s found A feasble path s not found the least satsfes C( p) f ( C) 3 A feasble path s not found the least satsfes C( p) < f ( C) For the frst case the termnaton condton s met the search termnates For the second one the least satsfes C( p) f ( C) thus for any path p C( p) f ( C) By Theorem no feasble path exsts (assumng that the probablty of havng another least cost path that s feasble wth cost exactly equal to f ( C ) s neglgbly zero f not zero) the search termnates here For the thrd case the shortest path searchng algorthm needs to be executed agan wth a dfferent cost functon Wth the new cost functon = β x = there are 4 possble outcomes for ths new search: A feasble path s found A feasble path s not found the least satsfes C ( p) f ( C) 3 A feasble path s not found the least satsfes C ( p ) < f ( C) p p 4 A feasble path s not found the least satsfes C ( p ) < f ( C) p = p Here C ( p ) represents the cost of path p usng cost functon f () Smlar to the frst search the search termnates for the frst two cases However snce p s the least cost path n the frst search p may be the least cost path agan (case 4) Nevertheless f there exsts a feasble path case 4 wll not occur Ths s because β s are set such that C ( p ; f p = p then C ( p mplyng that no feasble paths exst accordng to Theorem Smlarly f after < searches a feasble path s not found β s are set such that C ( p ) = C ( p ) = = C ( p Here p s a least cost path of the th search C ( p ) s the cost of path p wth f () as the cost functon f () s the cost functon for the ( + ) th search An example of how to compute β s gven later Usng ths procedure there are only 3 possble outcomes for the ( + ) th search: A feasble path s found A feasble path s not found the least + satsfes C ( p + ) f ( C) 3 A feasble path s not found the least + satsfes C ( p) < f ( C) p+ p = Obvously assumng there exsts a feasble path the larger the s the less possble case 3 wll occur because each teraton elmnates one path resultng n a contnuous decrease n the search space After a few teratons only a lmted number of paths are left n the search space from the source to the destnaton Thus wth ths method we can gradually ncrease the possblty of case or e ths method can speed up the occurrence of the termnaton condton (7) (8) (54) 3 (9) (9) 4 (43) Fg Networ topology When the soluton ( β s) that satsfes all the lnear equatons C ( p = may not exst e t s over-determned Thus we only need to conduct at most searches usng the above procedure However t s possble that a feasble path exsts but cannot be found by the above method For example fndng a feasble path for the networ shown n Fg wth constrant () f( x y) = x+ y as the ntal cost functon wll fal usng the above method Therefore we can loosen our restrcton to crcumvent ths case for the last teraton e = That s when = f a feasble path s stll not found β s are set such that C ( p ) = C ( p ) = = C ( p ) nstead of C ( p ) = C ( p ) = = C ( p By dong so the feasble path n the last example can be found The pseudo-code for the QA algorthm s shown n Fg Assume the computatonal complexty of the shortest path search algorthm s O( α ) where α s usually a functon of the number of nodes lns of the networ specfc to ths search algorthm Thus the overall computatonal complexty of our 5

4 algorthm s O(( + ) α) Note that the computatonal complexty ntroduced by computng the parameters ( β ) of the cost functon s not ncluded Snce β s wll only be computed at most tmes for the overall routng procedure n fndng a feasble path from a source to a destnaton ths computatonal overhead whch s related to the number of the constrants s thus neglgble Algorthm QA(GstC) Intal = β = c = β x = whle( ) 3 = + 4 Execute Shortest Path Search wth cost functon f ( x) get the shortest path p 5 f p s a feasble path 6 return SUCCESS 7 else f C ( p ) f ( C) 8 return FAIL (by Theorem ) 9 else f < compute β to mae C ( p ( = ) else f = compute β to mae end f end whle C ( p ) = C ( p ) ( = ) Smlar to the above t can be proved that β > So for the case of two addtve constrants the computatonal complexty ntroduced by computng β s s trval neglgble as compared to the overall computatonal complexty For the general case e multple addtve constrants any set of β s can be chosen as long as C ( p 4 Smulatons We evaluate our algorthm by ncorporatng QA wth the Dstra algorthm comparng t wth the algorthm n [9] whch s n the same category of our algorthm the best so far reported n the lterature The networ topology (Fg 3) presented n [9] [5] s adopted for comparson purposes In the smulatons the ln weghts are ndependent unformly dstrbuted from to all data are obtaned by runnng requests The performance of all algorthms are evaluated by the success rato (SR) defned below: Total number of success request of the algorthm SR = Total number of success request of the optmal algorthm Fg The QA algorthm We shall llustrate how to compute β s n QA for the case of two addtve constrants Assume the least cost path of the frst search s not a feasble path ( w w ) are ts weghts Thus n order to acheve C ( p for the case that w c let β = β β = β ( w c)/( c w) For the case of w = c we can set β = β = Thus t can be observed that f ( w w) = f ( c c) Note that only when w c w c > or w c > w c the second search s needed (otherwse a feasble path s found or the termnaton condton s reached) So β s nonnegatve Also assume the least cost path of the second ' ' search s not a feasble path ts weght s ( w w ) ' For the case that w w let β = β ' ' β = β( w w)/( w w) ; for the case that ' w = w let β = β = Thus ' ' f ( w w ) = f ( w w ) Fg 3 Networ Topology The algorthm that can always locate a feasble path as long as t exsts s refereed to as the optmal algorthm Here t s acheved smply by floodng In the smulatons two QoS constrants are set to be equal ncrease from 5 to 39 wth an ncrement of As shown n Fg 4 the lower bound of the success rato of our algorthm s 9935% whle that of Kormaz et al s algorthm s 995% Note that the worst case computatonal complexty of our algorthm s only three tmes of that of Dstra Algorthm whle that of Kormaz et al s algorthm s Ο (log Bm ( + nlog n)) here B = So our algorthm outperform Kormaz et al s algorthm n terms of computatonal complexty success rato of fndng a feasble path

5 SR SR Networ Sze:3 QA Kormazs Constrant Fg 4 SR n the 3-Node Networ 5 Conclusons In ths paper we have presented an teratve source routng algorthm for solvng mult-constraned QoS routng problem QA QA expedtes the search of a feasble path n the way that t teratvely reduces the search space by varyng the cost functon The ey ssue s that QA ensures that the prevous search result whch s not a feasble path wll not be the output of the future searches f a feasble path exsts Through smulatons we demonstrate that our algorthm outperforms ts contenders n terms of computatonal complexty success rato of fndng a feasble path 6 References [] S Chen K Nahsted An overvew of qualty of servce routng for next-generaton hgh-speed networ: problems solutons IEEE Networ vol no 6 pp December 998 [] A Shah J Rexford KG Shn Evaluatng the mpact of stale ln state on qualty-of-servce routng IEEE/AC Transactons on Networng vol 9 no pp 6-76 Aprl [3] R Guern A Orda QoS based routng n networs wth naccurate nformaton: theory algorthms Proceedngs of the INFOCO 97 pp [4] J Wang W Wang J Chen S Chen A romzed QoS routng algorthm on networs wth naccurate lnstate nformaton Proceedng of WCC-ICCT vol pp 67-6 [5] DH Lorenz A Orda QoS routng n networs wth uncertan parameters Proceedngs of INFOCO 98 vol pp [6] DH Lorenz A Orda QoS routng n networs wth uncertan parameters IEEE/AC Transactons on Networng vol 6 no 6 pp December 998 [7] S Chen K Nahrstedt Dstrbuted QoS routng wth mprecse state nformaton Proceedngs of 7 th Internatonal Conference on Computer Communcatons Networs pp [8] Z Wang J Crowcroft Qualty of Servce routng for supportng multmeda applcatons IEEE Journal on Selected Areas on Communcatons vol 4 no 7 pp 8-34 September 996 [9] T Kormaz Krunz S Tragoudas An effcent algorthms for fndng a path subect to two addtve constrants Proceedngs of the AC SIGETRICS pp June [] Pomavalz G Charaborty N Shrator QoS based routng algorthm n ntegrated servces pacet networs Proceedngs of IEEE 997 Conference on Networ Protocols pp [] Juttner B Szyatovsz I ecs Rao Lagrange releaxaton based method for the QoS routng problem Proceedngs of IEEE INFOCO vol pp [] L Guo I atta Search space reducton n QoS routng Proceedngs of 9 th IEEE Internatonal Conference on Dstrbuted Computng Systems pp [3] L Gang KG Ramarshnan A prune: an algorthm for fndng shortest paths subect to multple constrants Proceedngs of IEEE INFOCO vol pp [4] Eppsten Fndng the shortest path Proceedngs of 35 th Annual Symposum on Foundatons of Computer Scence pp [5] S Chen K Nahrsted On fndng mult-constraned path Proceedngs of IEEE ICC 98 vol pp

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