Minimum-Latency Beaconing Schedule in Multihop Wireless Networks

Transcription

1 This fll text paper was peer reiewed at the direction of IEEE Commnications Society sbject matter experts for pblication in the IEEE INFOCOM 009 proceedings Minimm-Latency Beaconing Schedle in Mltihop Wireless Networks Peng-Jn Wan Xiaoha X Lixin Wang Xiaoha Jia EK Park Abstract Minimm-latency beaconing schedle MLBS) in synchronos mltihop wireless networks seeks a schedle for beaconing with the shortest latency This problem is NP-hard een when the interference radis is eqal to the transmission radis All prior works assme that the interference radis is eqal to the transmission radis and the best-known approximation ratio for MLBS nder this special interference model is 7 In this paper we present a new approximation algorithm called strip coloring for MLBS nder the general protocol interference model Its approximation ratio is at most when the interference radis is eqal to transmission radis and is between and 6 in general the transmission range of bt is otside the interference range of any other transmitting node Sch interference model is referred to as the protocol interference model [7] and is widely sed becase of its generality and tractability I INTRODUCTION Beaconing in wireless networks is a primitie commnication task in which eery node locally broadcasts a packet to all its neighbors Assme that all commnications proceed in synchronos time-slots and each node can transmit at most one packet of a fixed size in each time-slot A beaconing schedle assigns a time-slot to eery node sbject to the constraint that the nodes assigned in each time-slot are interference free The latency of a beaconing schedle is the nmber of time-slots dring which at least one transmission occrs The problem of compting a beaconing schedle with minimm latency in a wireless network is referred to Minimm-Latency Beaconing Schedle MLBS) The problem MLBS is a classic and fndamental problem in wireless networks and ariants of this problem hae been extensiely stdied in [] [] [4] [0] [] [] [4] [] [6] In this paper we stdy the problem MLBS nder the following model for wireless networks All the networking nodes are located in a plane and are each eqipped with an omnidirectional antenna Each node has a fixed transmission radis which is normalized to one and an interference radis The commnication range and the interference range of a node are the two disks centered at of radis one and respectiely see Figre ) A node can receie the message sccessflly from a transmitting node if is within Department of Compter Science Illinois Institte of Technology s: wan@csiited xx@iited and wanglix@iited This work was spported in part by National Science Fondation of USA nder grant CNS- 088 Department of Compter Science City Uniersity of Hong Kong jia@cscityedhk This work was spported in part by Research Grants Concil of Hong Kong Grant No CityU 407 and NSF China Grant No School of Compting and Engineering Uniersity of Missori at Kansas City ekpark@mkced Fig The protocol interference mode: each node has a nit transmission radis and an interference radis Under the aboe networking model the commnication topology of a wireless network in a nit-disk graph [] in which there is an edge between two nodes if and only if their distances is at most one We assme that the commnication topology is always connected Then any pair of nodes with distance at most max { } interfere with each other and pair of nodes with distance greater than do not interfere with each other We also define the interference topology of a wireless network as follows: For any pair of nodes and there is an edge between and in the interference topology if one of the three conditions holds: ) and are within each other s commnication range see Figre a)) ) some node w other than and is within s commnication range and s interference range see Figre b)) and ) some node w other than and is within s commnication range and s interference range see Figre c)) Then a beaconing schedle for a wireless network is eqialent to a proper ertex coloring of its interference topology with the latency corresponding to the nmber of colors Hence the problem MLBS for a wireless network seeks the minimm ertex coloring of its interference topology All the prior works on MLBS [] [] [4] [0] [] [] [4] [] [6] implicitly assmed that When the interference topology is the sqare of a nit-disk graph representing the commnication topology Sen and Hson [] proed that minimm coloring of a sqare of a nit-disk graph /09/$ IEEE

2 This fll text paper was peer reiewed at the direction of IEEE Commnications Society sbject matter experts for pblication in the IEEE INFOCOM 009 proceedings Fig and w b) a) Three possible scenarios of intereference between a pair of nodes is NP-hard The classical greedy coloring known as first-fit coloring in the smallest-degree-last ordering [8] was adopted by Sen and Malesinska [4] to prodce a coloring of the sqare of nit-disk graph While this greedy coloring has polynomial approximation ratio when applied to general graphs it is a 7- approximation when restricted to sqares of nit-disk graphs [6] The bond 7 is the best-known pper bond on the achieable approximation ratio for MLBS with It has been open for years whether there is an algorithm for MLBS with whose approximation ratio is smaller than 7 In this paper we propose an algorithm called strip coloring for MLBS with arbitrary which exploits a key strctral property of the interference topology For its approximation ratio is at most which is smaller than the best-known bond 7 on achieable approximation ratio For > its approximation ratio ranges from to 6 depending on the ale of The following notations will be sed throghot this paper The Eclidean distance between two nodes and is denoted by The disk of radis r centered at a node is denoted by D r) The topological bondary of a point set A is denoted by ALetG VE) be an arbitrary graph We se χ G) and ω G) to denote its chromatic nmber and cliqe nmber of G respectiely In general χ G) ω G) If χ G) ω G) then G is said to be perfect The remaining of this paper is organized as follows In Section II we introdce a fnction and present seeral properties of this fnction In Section III we describe the algorithm strip coloring and analyze its approximation ratio Finally we conclde this paper in Section III by discssing on some frther releant reslts II PRELIMINARIES Define a fnction h ) oer [ ) as follows: if < then c) h ) cos arccos arccos ; w if then h ) sin arccos arcsin ) ; if > then h ) ) sin arccos arcsin ) A straightforward calclation yields that if < then if h ) 8 4 then h ) if > then h ) ) ) ; ) 4 ) ; ) ) ) We first present the geometric interpretation of the ale h ) Lemma : Sppose that Letpq be a line segment with < pq < and are the two points on the opposite side of pq sch that p q q and p see Figre ) Then the distance between q and strictly increases with pq and eqals to h ) when pq Fig Figre for Lemma p q Proof: By law of cosine cos pq pq pq ) pq pq Since cos pq strictly increases with pq when 0 < pq < So pq strictly decreases with pq Clearly pq strictly decreases with pq So q strictly decreases with pq Ths the distance between q and strictly increases with pq

3 This fll text paper was peer reiewed at the direction of IEEE Commnications Society sbject matter experts for pblication in the IEEE INFOCOM 009 proceedings When pq wehae q pq pq arccos arccos Hence the distance between q and is cos q h ) Lemma : Sppose that Consider a right triangle p with p π p and p see Figre 4) Let q be a point on the circle D max { }) which lies on the different side of from p and q π Then the distance between q and strictly decreases with pq and eqals to h ) when pq p Fig 4 Figre for Lemma q max{ } h) Proof: By law of cosine pq strictly increases with pq Since q pq p and p is fixed q also strictly increases with pq and so does the distance between q and When pq q pq p max { } arccos arcsin and hence the distance between q and is max { } sin q h ) Next we gie some sefl properties of h ) Webegin with the following lemma Lemma : arccos arcsin strictly increases with when Proof: Let θ arccos arcsin Set t Then 0 <t and θ arccos t arcsin t and the deriatie of θ oer t is dθ dt ) t t < 0 t ) t)t) t t)t) ) t t t dθ Hence dt < 0 when 0 <t< This implies that θ strictly decreases with t when 0 < t and conseqently strictly increases with when increases It s easy to erify that h ) is continos The lemma below shows the monotonicity of h ) Lemma 4: h ) is strictly increasing In addition also strictly increasing when Proof: When h ) 8 4 Since increases when When ) ) h) is strictly increases when h ) strictly h ) sin arccos arcsin ) Hence h ) strictly increases when When > h ) strictly increases Since strictly increases and h) h ) sin arccos arcsin ) strictly increases with when > by Lemma Next we gie some ineqalities satisfied by h ) Lemma : The following statements are tre ) For any h ) < max { } ) For any < <h) < ) When is less than resp eqal to greater than) max { }) h ) is greater than resp eqal to less than)

4 This fll text paper was peer reiewed at the direction of IEEE Commnications Society sbject matter experts for pblication in the IEEE INFOCOM 009 proceedings Proof: ) By Lemma arccos arcsin strictly increases on [ ) Since lim arccos arcsin ) π we hae arccos arcsin < π for any Ths when h ) ) sin arccos arcsin ) < ) sin π ) Since h ) strictly increases on [ ] by Lemma 4 we hae h ) h ) < for any So the first part of the lemma holds ) We proe the second part in two cases Case : In this case h ) 8 ) ) 4 [ ] Let t Then t 0 We first show that h ) > Replacing with t wehae 4 ) ) 4 ) t ) ) ) ) t ) t 6t6 60t 78t 4 908t 64t 4t 76 6 t ) > 0 Hence 4 ) > ) ) which implies that 8 ) ) > 4 ) So Now we show that h ) > h ) < Replacing with t wehae ) ) ) ) ) t ) ) ) t t t t 6 t ) 0t 4 7t t 8t 48 ) We claim that t 0t 4 7t [ t 8t] 48 is negatie on [0 09] which contains 0 When t 0 t 0t 4 7t t is strictly conex and so is t 0t 4 7t t 8t 48 Ths the maximm of t 0t 4 7t t 8t 48 on [0 09] is achieed at the endpoints which are both negatie by straightforward calclation Ths or claim holds Hence ) ) > ) which implies that 8 ) ) < 4 ) 4 So Case : Ths h ) h ) h h ) < In this case 4 ) ) > 4 On the other hand ) ) 4 4 ) ) )) 4 6 > 0 which implies that ) 4 ) < Ths h ) < ) It s easy to erify that when h )

5 This fll text paper was peer reiewed at the direction of IEEE Commnications Society sbject matter experts for pblication in the IEEE INFOCOM 009 proceedings Since h ) strictly increases on [0 ] if < then if < then < h ) ; > h ) Next assme that > Consider a right triangle p with pw π p and p Letq be a point on the same side of p as sch that q and pq By Lemma the distance between q and eqals to h ) Since we hae q > pq p q q h ) > ) h ) So the third part of the lemma holds For each let k ) h ) q h ) In the remaining of this section we compte k ) It s easy to erify that k ) 4 By Lemma ) we hae k ) on ] By Lemma ) for any we hae By Lemma 4 h ) > > ) h) is strictly decreasing on [ ) Hence < h ) h ) < Let and be the ales greater than two at which h) are eqal to 4 and respectiely Then k ) if ); 4 if or [ ); if [ ); if [ ) Next we compte the nmerical ales of and A straightforward calclation yields that Using the qartic formla we obtain c 7c c 40 c 4c 976 7c where c c ) 8 The nmerical ales of and are and 446 respectiely III STRIP COLORING In this section we present an algorithm called strip coloring for compting a ertex coloring of the interference topology denoted by H Or algorithm design exploits a key strctral property of H Recall that a graph G VE) is said to be a cocomparability graph if there is a ertex ordering [ n ] of V sch that if i<j<kand i k E then either i j E or j k E Sch ordering is referred to as its cocomparability ordering Eery cocomparability graph is perfect [6] and its minimm ertex coloring can be compted in polynomial time by soling a maximm bipartite matching [] We discoer that H has the following restricted cocomparability: Lemma 6: Let U be any sbset of of nodes which lie in a horizontal strip of height h ) Then the sbgraph H [U] of H indced by U is a cocomparability graph with the lexicographic ordering being its cocomparability ordering Proof: Sppose that and are three nodes in U in the lexicographic order satisfying that and interfere with each other We proe by contradiction that either or interferes with Assme to the contrary that neither nor interferes with Then both and are greater than max { } Let w be the intersection point between the segment and the ertical line throgh Then w h ) max { } So w is shorter than both and which imply that both and are acte Frthermore the distance between and is no more than w and hence is at most h ) By Lemma ) > max { }) h ) > max { }

6 This fll text paper was peer reiewed at the direction of IEEE Commnications Society sbject matter experts for pblication in the IEEE INFOCOM 009 proceedings Ths either D ) D ) contains at least one node other than or D ) D ) contains at least one node other than By symmetry we assme the former holds We will proe that D ) D ) D ) which implies that and also interfere with each other Since > D ) and D ) intersect at two points Let p be the intersection point of D ) and D ) which lies on different side of from see Figre ) Then D ) D ) D ) max { } p q q max { } h) h) q q if and only if p Fori and letq i be the point on D i max { }) satisfying that ) the distance between q i and the line is h ))q i lies on the different side of from p and ) the perpendiclar foot q i of q i on the line is on the segment Since > max { }) h ) q q q is closer to than q Ths q > q max{ } This implies that q is otside D max { }) Similarly q is otside D max { }) Then lies inside the rectangle q q q q So p if the distances from p to the for ertices of the rectangle q q q q are all at most Clearly for i and pq i max { p p } Since both p and q lies on the same side of the perpendiclar bisector of which is also the perpendiclar bisector of q q wehae pq pq Therefore it is sfficient to show that pq Consider the triangle p q Since q and p hae fixed length pq strictly increases with p q by the law of cosine Since p q q p and q is also fixed pq also strictly increases with p We consider two cases Case : < By Lemma ) max { }) h ) > Hence p is always acte and p strictly decreases with So pq also strictly decreases with By Lemma when h ) pq Ths pq <in this case Case : By Lemma ) max { }) h ) Ths p achiees its maximm when p is perpendiclar to So pq achiees its maximm when p is Fig Figre for the proof of Lemma 6 perpendiclar to By Lemma the maximm of pq is Ths pq in this case Now we are ready to describe the algorithm strip coloring For simplicity each color is represented by a pair λ λ ) where 0 λ k ) and λ is a positie integer We first compte the minimal axis-parallel rectangle srronding all the networking nodes Then we partition sch rectangle into top-closed bottom-open horizontal strips in the manner that the pper bondary of the top-most strip aligns with the top of the rectangle the heights of all strips except the bottom-most one are all eqal to k) and the height of the bottom-most strip is at most k) see Figre 6) Nmber the sccessie strips from top to bottom sing integers 0 and let V i denote the set nodes in the strip i By the definition of k ) the height of each strip is at most h ) By Lemma 6 each H [V i ] is a cocomparable graph and hence a minimm coloring of H [V i ] with ω H [V i ]) colors can be compted in polynomial time So we compte a minimm coloring of each H [V i ] sing the colors {i mod k ))λ ): λ ω H [V i ])} The entire coloring of all nodes is alid since for any i and j with i<jand i j)modk ))0 any node in a strip i and any node in a strip j are separated by a distance greater than and ths are not adjacent in H Next we show that strip coloring ses at most k ))ω H) colors Since H [V i ] is a sbgraph of H ω H [V i ]) ω H) Therefore all colors sed by the strip coloring belong to the set {λ λ ):0 λ k ) λ ω H)} This implies that the strip coloring ses at most k ))ω H) colors In smmary we hae the following theorem Theorem 7: The strip coloring ses at most k ))ωh) colors and its approximation ratio is at most k )

7 This fll text paper was peer reiewed at the direction of IEEE Commnications Society sbject matter experts for pblication in the IEEE INFOCOM 009 proceedings Fig 6 Illstration of strip coloring )/ k) Note that 6 if ); if or [ k ) ); 4 if [ ); if [ ) For the strip coloring is a -approximation an improement pon the best-known 7-approximation IV DISCUSSIONS We can still apply the greedy first-fit coloring on the interference topology H For > we can proe that its approximation ratio is bonded by a packing parameter η ) which is the maximal nmber of points whose pairwise distances are greater than max { } in a half-disk of radis While still being bonded by a constant η ) is at least 7 which is worse than the bond k ) on the approximation ratio of the strip coloring REFERENCES [] I Chlamtac and A Farago Making Transmission Schedles Immne to Topology Changes in Mlti-Hop Packet Radio Networks IEEE/ACM Transactions on Networking Vol No pp -9 Feb 994 [] I Chlamtac and S Ktten A spatial rese tdma/fdma for mobile mltihop radio nertworks in IEEE INFOCOM pp March 98 [] BN Clark CJ Colborn and DS Johnson Unit disk graphs Discrete Mathematics 86: [4] A Ephremides and TV Trong Schedling Broadcasts in Mltihop Radio Networks IEEE Transactions on Commnications ol 8 no 4 pp April 990 [] L Ford and D Flkerson Flows in Networks Princeton Uniersity Press Princeton NJ 96 [6] MC Golmbic Algorithmic graph theory and perfect graphs Academic Press New York NY 980 [7] P Gpta and PKmar The capacity of wireless networks IEEE Trans Inform Theory ol 46 pp Mar 000 [8] D W Matla and L L Beck Smallest-last ordering and clstering and graph coloring algorithms Jornal of the Association of Compting Machinery 0): [9] RH Möhring Algorithmic aspects of comparability graphs and interal graphs In I Rial editor Graphs and Orders pp 4-0 Reidel Dordrecht 98 [0] R Nelson and L Kleinrock Spatial-TDMA: A collision-free mltihop channel access protocol IEEE Transactions on Commnications ol no 9 pp Sep 98 [] S Ramanathan and EL Lloyd Schedling algorithms for mlti-hop radio networks IEEE/ACM Transactions on Networking ol pp 66-7 April 99 [] R Ramaswami and K K Parhi Distribted schedling of broadcasts in a radio network in IEEE INFOCOM pp [] A Sen and M L Hson A New Model for Schedling Packet Radio Networks ACM/Baltzer Jornal Wireless Networks 997) pp 7-8 [4] A Sen and E Malesinska Approximation Algorithms for Radio Network Schedling Proceedings of th Allerton Coneference on Commnication Control and Compting Champaign Illinois pp 7-8 October 997 [] DS Steens and MH Ammar Ealation of slot allocation strategies for TDMA protocols in packet radio networks IEEE Military Commnications Conference pp: [6] P-J Wan C-W Yi X Jia and D Kim: Approximation Algorithms for Conflict-Free Channel Assignment in Wireless Ad Hoc Networks Wiley Jornal on Wireless Commnications and Mobile Compting 6):0- March 006 Using the similar argment sed in the proof of Lemma 6 we can proe that when all the networking nodes lie in a strip of height at most ) max { } max { } the interference topology is cocomparable and ths its minimm coloring can be soled in polynomial time