Managing Moving Objects on Dynamic Transportation Networks

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1 Maagg Movg Objects o Dyamc Trasportato Networks Zhmg Dg Ralf Hartmut Gütg Praktsche Iformatk IV, Feruverstät Hage, D Hage, Germay {zhmg.dg, rhg}@feru-hage.de Abstract Oe of the key research ssues wth movg objects databases (MOD) s the modelg of movg objects. I ths paper, a ew movg objects database model, Movg Objects o Dyamc Trasportato Networks (MODTN), s proposed. I MODTN, movg objects are modeled as movg graph pots whch move oly wth predefed trasportato etworks. To express geeral evets of the system, such as traffc jams, temporary costructos, serto ad deleto of juctos or routes, the uderlyg trasportato etworks are modeled as dyamc graphs so that the state ad the topology of the graph system at ay tme stat ca be tracked ad uered. Besdes, to track the locato of etwork costraed movg objects, a locato update mechasm s provded, ad the correspodg ucertaty maagemet ssues are aalyzed. 1. Itroducto Wth the developmet of wreless commucatos ad postog techologes, the cocept of movg objects databases (MOD) has become creasgly mportat, ad has posed a great challege to the database commuty. Exstg database maagemet systems (DBMS s) are ot well eupped to hadle cotuously chagg data, such as the locatos of movg objects [19]. Therefore, ew modelg methods are eeded to solve ths problem. I recet years, a lot of research has bee focused o the MOD techology, ad may models ad algorthms have bee proposed. I [18, 19, 13], Wolfso et al. have proposed a Movg Objects Spato-Temporal (MOST) model whch s capable of trackg ot oly the curret, but also the ear future postos of movg objects. Su et al. [15] have preseted a data model for movg objects based o lear costrat databases. Cho et al. [1] have proposed a Space-Tme Grd Storage model for movg objects. I [7, 3, 8], Gütg et al. have preseted a data model ad data structures for movg objects based o abstract data types. Besdes, Pfoser ad Jese et al. [10, 12] have dscussed the dexg problem for movg object trajectores. However, early oe of these works have treated the teracto betwee movg objects ad the uderlyg trasportato etworks ay way. More recetly, creasg research terests are focused o modelg trasportato etworks ad etwork costraed movg objects. Vazrgas et al. [17] have dscussed movg objects o fxed road etworks, Papadas et al. [9] have preseted a framework to support spatal etwork databases. I [14], the authors have preseted a computatoal data model for etwork costraed movg objects. Besdes, the dex problems of etwork costraed movg objects have also bee studed [4, 11]. However, all these works have oly cosdered statc trasportato etworks, ad moreover, oe of these works have dealt wth the relatoshp betwee the dscrete presetato of movg objects ad the locato update polces. The ucertaty maagemet ssues for etwork costraed movg objects are ot dscussed ether. To explore these research ssues, we propose a ew movg objects database model, Movg Objects o Dyamc Trasportato Networks (MODTN), ths paper. MODTN s a exteso to our work preseted [6], whch deals wth the modelg of movg objects statc trasportato etworks at the abstract level. The remag part of ths paper s orgazed as follows. Secto 2 formally defes the MODTN model, Secto 3 dscusses the locato update mechasm of MODTN, Secto 4 aalyzes locato computato methods ad ucertaty maagemet ssues, ad Secto 5 fally cocludes the paper. 2. The MODTN Model The modelg of movg objects o dyamc trasportato etworks s composed of two relatvely depedet steps. The frst step s the modelg of the uderlyg trasportato etworks. Sce the trasportato etworks ca be subject to dscrete chages over tme, they should be modeled as dyamc graphs whch allow us to express state chages (such as traffc jams ad blockages caused by temporary costructos) ad topology chages (such as serto ad deleto of juctos or routes). For smplcty, dyamc

2 trasportato etworks ad dyamc graphs wll be used terchageably throughout ths paper. I modelg dyamc graphs, we utlze a state-based method. The basc dea s to assocate a temporal attrbute to every route or jucto of the graph system so that the state of the route or jucto at ay tme stat ca be retreved. Sce the chages to the graph system are dscrete, we ca use a seres of temporal uts to represet a temporal attrbute wth each temporal ut descrbg oe sgle state of the route or jucto durg a certa tme perod. As a result, the whole spato-temporal hstory of the graph system ca be stored ad uered. The secod step s the modelg of movg objects o the graph system whch has bee hadled the frst step. Sce most cases a movg object ca be vewed as a pot, movg objects are modeled as movg graph pots MODTN. A movg graph pot s a fucto from tme to graph pot, whch ca be represeted as a group of movg uts the dscrete model. The methodology proposed ths paper ca be easly exteded to deal wth more complcated stuatos where movg objects eed to be modeled as movg graph les or movg graph regos Overvew of the System Archtecture I the followg dscusso, we suppose that the whole MOD system, there ca be multple graphs coexstg whle each graph s composed of a set of routes ad a set of juctos. For each route, ts geometry s descrbed by a polyle so that t ca actually assume a arbtrary shape stead of just a straght le. A jucto coects two or more routes of the graph system. The coected routes ca come from oe graph ( ths case, the jucto s called -graph jucto ), or belog to dfferet graphs ( ths case, the jucto s called tergraph jucto ). A jucto ca locate the mddle of a route, or at the begg/ed of the route. Fgure 1 llustrates a graph system whch s composed of two graphs. Graph1:Street-Net j4 r4 r4 j3 j5 Graph2:Tra-Net r3 j2 r3 j 2 j 1 Routes of Street-Net Routes of Tra-Net I-Graph Juctos Iter-Graph Juctos Fgure 1. Graph system cosstg of two graphs Movg objects ca move sde oe graph ad trasfer from oe route to aother va -graph juctos. r5 j1 r j j They ca also trasfer from oe graph to aother va tergraph juctos. I the system, both movg objects ad the uderlyg trasportato etworks are dyamc movg objects chage ther locatos cotuously, whle trasportato etworks chage ther states ad topologes dscretely. I order to evsage the above deas, we gve a example. Ths example shows how a moder logstc system works. We suppose that such a system, trasportato vehcles are uuely detfed ad each of them s eupped wth a portable computg platform ad some other tegrated locato trackg eupmets so that ts locato at ay tme stat ca be retreved. We assume that such a logstc system, whch s resposble for cargo delvery servces, exsts the Verkehrsverbud Rhe-Ruhr (VRR) area of Germay. The whole hghway etwork of the VRR area s expressed as a graph the database. Besdes, the street etwork of each cty ths area s also stored as a depedet graph. As a result, the whole system s composed of multple graphs whch ca overlap each other, as show Fgure 2. CtyStreetNet-2 Movg Object Movg Object HghwayNet loc. updates loc. updates Traffc Jam db. updates db. updates CtyStreetNet-1 Fgure 2. Archtecture of the MODTN system For a certa vehcle, t ca move ether by hghway betwee two ctes, or by street sde a cty, durg ts whole jourey. Therefore, t ca pass through several dfferet graphs durg oe trp. Sce both hstorcal ad curret locato formato s kept the database, the system ca support the followg ueres: tell me the locato of vehcle x310 at 2:00 PM of last Frday ad fd all vehcles that are curretly the Hageer street. Besdes, sce the geeral evets (such as traffc jams, car accdets, serto ad deleto of routes or juctos) of the graph system are stored dyamc graphs, the followg ueres: tell me the topology of the Hage street etwork at tme t ad fd the curret traffc jam the Hageer street ad the movg objects affected by t ca also be hadled. To speed up uery processg, both movg objects ad dyamc graphs should be dexed. The database records ad the dex structures cota locato formato coverg a tme perod from the past utl the future. Therefore, whe locato updates occur, both database records ad the dex eed to be modfed.

3 2.2. The Data Model Let s frst deal wth the trasportato etworks. I the MODTN model, the whole trasportato etworks are modeled as a dyamc graph system. Defto 1 (dyamc graph system) A dyamc graph system, GS, s defed as a set of dyamc graphs ad ter-graph juctos: GS = {G 1, G 2,, G, j 1, j 2,, j m } where m 1, m m 0, G (1 [ [ ) s a dyamc graph, ad j k (1 [ k [ m) s a ter-graph jucto (see Defto 5). Defto 2 (dyamc graph) A dyamc graph, G, s defed as a par: G = ( R, J ) where R s a set of dyamc routes ad J s a set of dyamc -graph juctos. Defto 3 (dyamc route) A dyamc route of graph G, deote by r, s defed as follows: r = (gd, rd, route, le, fdr, tp) where gd ad rd are detfers of G ad r respectvely, route s a polyle whch descrbes the geometry of r, le s the legth of the route, fdr c {0, 1, 2} s the traffc flow drectos allowed the route, ad tp s the temporal attrbute (see Defto 6) assocated wth r. The polyle route the above defto ca be defed as a seres of pots the Eucldea space. For smplcty, we suppose that the graph system s spatally embedded the X%Y plae so that the polyle ca be preseted as a seres of pots the X%Y plae. The polyle s cosdered drected, whose drecto s from the frst vertex to the last vertex, whch eables us to speak of the begg pot (or 0-ed) ad the ed pot (or 1-ed) of the route. The traffc flow drectos allowed a route ca have three possbltes, whch are specfed by fdr, whose value ca assume 0, 1, 2, whch correspods to from 0-ed to 1-ed, from 1-ed to 0-ed, ad both drectos allowed respectvely. Defto 4 (dyamc -graph jucto) A dyamc -graph jucto of graph G, deoted by j, s defed as follows: j = (gd, jd, loc, ((rd, pos )) = 1, m, tp) where gd ad jd are detfers of G ad j respectvely, loc s the locato of j whch ca be preseted as a pot value the X%Y plae, ((rd, pos )) = 1 descrbes the routes coected by j, m s the coectvty matrx of j, ad tp s the temporal attrbute assocated wth j. (rd, pos ) (1 [ [ ) the above defto dcates the th route coected by j, where rd s the detfer of the route ad pos c [0, 1] descrbes the posto of the jucto sde the route. We suppose that the total legth of ay route s 1, ad the every locato the route ca be preseted by a real umber p c [0, 1]. The matrx m descrbes the coectvty of the jucto. It cotas possble matches of traffc flows the routes coected by the jucto, ad the elemet value assocated wth each match ca assume ether 0 or 1, whch dcates whether movg objects ca trasfer from the traffc flow to the out traffc flow through ths jucto, as show Fgure 3. Fgure 3. A jucto ad ts coectvty matrx As llustrated Fgure 3, route allows movg objects rug both drectos so that t ca have two traffc flows, + ad -. Route s a oe-way street so that t has oly oe traffc flow, -. These three traffc flows ca have 9 combatos, ad the value correspodg to each combato descrbes the trasferablty of the two traffc flows. Defto 5 (dyamc ter-graph jucto) A dyamc ter-graph jucto, deoted by j, s defed as follows: j = (jd, loc, tp, ((gd, rd, pos )) = 1, m, tp) The defto of the ter-graph jucto s very smlar to that of the -graph jucto. The 3-tuple (gd, rd, pos ) (1 [ [ ) descrbes the routes coected by j, whch ca come from dfferet graphs. Defto 6 (temporal attrbute) The temporal attrbute assocated wth a jucto or a route, deoted by tp, descrbes the state hstory of the jucto or route, whch s defed as a seuece of the followg form: tp =((I, s )) = 1 where I s a tme terval, s s the state (see Defto 7) of the jucto or route durg I. (I, s ) (1 [ [ ) s called the th temporal ut of tp. For a vald temporal attrbute, the followg codtos should be satsfed: (1), j c {1, }, g j: I 3 I j = (2) c {1, -1}: I, I +1 (, meas before tme seres) (3) I = [ m( I 1), max( I)]. = 1 0-ed () ed 1-ed ed (out) a) a jucto ad the traffc flows t b) the coectvty matrx For a certa temporal ut (I, s ) (1 [ [ ), I s composed of two tme stat values, m(i ) ad max(i ),

4 whch dcate the startg pot ad the edpot of I respectvely. m(i ) must be a defed value whle max(i ) ca be ether defed or udefed. If max(i ) s a udefed value Ω, the I s called a ope temporal ut. Otherwse t s called a closed temporal ut. Sematcally, Ω meas utl ow. Therefore, f a jucto or route s stll actve the trasportato etwork, ts temporal attrbute wll cota exactly oe ope temporal ut, whch forms ts last temporal ut. Otherwse, f t has already bee deleted from the trasportato etwork, the ts temporal attrbute wll oly cota closed temporal uts. The serto ad deleto tme of a jucto or a route ca be decded by m(i 1 ) ad max(i ) respectvely. Fgure 4 llustrates a example temporal attrbute value. opeed blocked closed opeed t1 t2 t3 t4 tow Fgure 4. A example temporal attrbute value Defto 7 (state) A state of a jucto or a route, deoted by s, s defed as follows: t [t1, t2), (opeed, ) [t2, t3), (blocked, {(traffc-jam, [0.2,0.3])}) [t3, t4), (closed, ) [t4, Ω), (opeed, ) a) state chages of a route b) the correspodg temporal uts s = (σ, (br, BP ) = 1 ) where σ c {opeed, closed, blocked}. If σ = blocked, the s must be assocated wth a route, ad (br, BP ) = 1 s eeded ths stuato to descrbes the blockages of the route where br descrbes the reaso (traffc-jam, costructo, traffc-cotrol, etc.), ad BP ` [0, 1] descrbes the locato, of the th blockage of the route. I the above defto, we assume that the locato of the blockage s statc so that t ca be expressed as a closed terval over [0, 1], whose boudares dcate the locato of the borders of the blockage. I dyamc trasportato etworks, a jucto ca have two states: opeed ad closed, ad a route ca have three states: opeed, closed, ad blocked. If a jucto or a route s opeed, the t s etrely avalable to movg objects. If a jucto or a route s closed, the t s etrely uavalable to movg objects, whch meas that o movg objects are allowed to stay or move o ay part of t. A closed jucto or route s ot deleted from the system. Istead, t s oly temporarly uavalable to movg objects ad ca be reopeed afterwards. The blocked state s used to descrbe a specal kd of state of a route, whch meas partally avalable to movg objects. That s, the ublocked part of the route s stll avalable to movg objects, but o movg objects ca move through the blocked part. Fgure 5 gves a example blocked route. Blockage Fgure 5. A blocked route wth movg objects I the dyamc graph system, sce every jucto or route has a temporal attrbute assocated, we ca kow ts state at ay gve tme stat. Ths s very useful movg objects databases sce a lot of ueres ca oly be processed effcetly by accessg the states of the trasportato etworks. For stace, please tell me all the routes whch are curretly blocked by traffc jams ad the movg objects affected by them. Besdes, through the temporal attrbute, we ca also kow the lfe spa of ay jucto or route of the graph system so that the topology chages of the trasportato etworks ca also be expressed ad uered. For stace, fd the shortest path from a to b at tme stat t. Based o the above deftos for dyamc trasportato etworks, we ca the defe some useful data types, graph pot, graph route secto, graph le, ad graph rego, whch form the bass for the modelg ad ueryg of movg objects. Let graph(gd), juct(jd), juct(gd, jd), route(gd, rd) be fuctos whch retur the graph, the jucto, ad the route correspodg to the specfed detfers respectvely. Defto 8 (graph pot) A graph pot s a pot resdg the graph system. The set of graph pots of graph system GS, deoted by GP, s defed as follows: GP ={jd juct(jd) c terjucts(gs) } 4 {(gd, jd) juct(gd, jd) c jucts(gs) } 4 {(gd,rd,pos) route(gd,rd) c routes(gs), posc[0,1]} where terjucts(gs), jucts(gs) ad routes(gs) are the set of ter-graph juctos, the set of -graph juctos, ad the set of routes of GS respectvely. Defto 9 (graph route secto) A graph route secto s a part of a route. The set of graph route sectos of graph system GS, deoted by GRS, ca be defed as follows: GRS={(gd,rd,S) route(gd, rd) c routes(gs), S ` [0,1]} Defto 10 (graph le) A graph le s a polyle sde the graph system, whch ca be defed by just specfyg the vertces of the polyle. The set of graph les of graph system GS, deoted by GL, ca be defed as the followg form: GL = { (vertex ) = 1 m 2, vertex = (gd, rd, pos ) c GP, ad: (1) c {1, -1}: route(gd, rd ) = route(gd +1, rd +1 ) - Eucl(vertex ) = Eucl(vertex +1 ) = Eucl (getjuct (vertex, vertex +1 )); (2) {1, -1} : vable(vertex, vertex +1 ) }

5 I the above defto, the fucto Eucl(gp) returs the Eucldea value of a graph pot, getjuct(vertex, vertex +1 ) returs the jucto whch vertex, vertex +1 are located. vable(vertex, vertex +1 ) meas that through route(gd, rd ) or getjuct(vertex, vertex +1 ), movg objects ca trasfer from vertex to vertex +1. Graph les are cosdered as drected paths the trasportato etwork, whose drectos are determed by the vertex seres. Defto 11 (graph rego) A graph rego s defed as a set of juctos ad a set of route sectos. The set of graph regos of the graph system GS, deoted by GR, s defed as follows: GR = {(V, W) V ` GJ, W ` GRS } where GJ = {jd juct(jd) c terjucts(gs) } 4 {(gd, jd) juct(gd, jd) c jucts(gs) }. Dfferet from the graph le, a graph rego ca cota a arbtrary set of graph route sectos. Based o the above deftos of etwork costraed data types, we ca the model movg objects o the dyamc graph system. I MODTN, movg objects are modeled as movg graph pots. Defto 12 (movg graph pot) A movg graph pot, mgp, s defed as a fucto from tme to graph pot, that s: mgp = f: T GP where T s the doma of tme, ad GP s the doma of graph pot of the graph system. I the above deftos, most data types are defed dscretely so that they ca be mplemeted drectly. The oly excepto s the movg graph pot data type. I mplemetato, Defto 12 should be traslated to a dscrete represetato. That s, a movg graph pot s expressed as a set of movg uts, ad each movg ut descrbes oe sgle movg patter of the movg object for a certa perod of tme. Defto 13 (dscrete presetato of movg graph pot) a dscrete presetato of movg graph pot, dmgp, s defed as a seuece: dmgp = ((t, (gd, rd, pos ), vm )) = 1 where t s a tme stat, (gd, rd, pos ) = gp s a graph pot descrbg the locato of the movg object at tme t, ad vm s the speed measure of the movg object at tme t. (t, (gd, rd, pos ), vm ) = µ s called the th movg ut of dmgp. The speed measure vm s a real umber value. Its abstract value s eual to the speed of the movg object, whle ts sg (ether postve or egatve) depeds o the drecto of the movg object. If the movg object s movg from 0-ed towards 1-ed, the the sg s postve. Otherwse, f t s movg from 1-ed to 0-ed, the sg s egatve. For a vald dscrete presetato of movg graph pot, the followg codtos should be met: (1) c {1, -1} : route(gd, rd ) = route(gd +1, rd +1 ) - Eucl(gp ) = Eucl(gp +1 ) = Eucl(getjuct (gp,gp +1 )); (2) c {1, -1} : vable(gp, gp +1 ); (3) c {1, -1} : t < t +1, ad the movg object s assumed to move at eve speed betwee t ad t +1. Fgure 6 gves a example dscrete represetato of a movg graph pot. t1 r3 t5 t ow t2 r3 r6 r6 t4 r8 t3 r4 r5 t1, (g1,, 0 ), v1 t2, (g1,, 0.5), v2 t3, (g1,, 1), t4, (g1, r6, 0 ), v3 t5, (g1, r6, 0.7 ), v4 actve movg ut a) jourey of a movg object b) correspodg movg utes Fgure 6. A example movg graph pot value I Defto 13, we assume that betwee two cosecutve movg uts, movg objects move at eve speed. As a result, the locato of movg objects ca be computed by terpolato (see Subsecto 4.1). However, ths s oly a deal stuato. I real world MOD applcatos, the movg uts are geerated by locato updates (see Secto 3), ad the movg object s oly movg roughly at eve speed betwee two movg uts. As a result, the locato of movg objects s ucerta betwee two locato updates, ad we have to take the ucertaty problem to cosderato. Whe ucerta maagemet s cosdered, the above defto of movg pot s actually terpreted as a movg route secto. I ths paper, we stll call ths defto movg graph pot to keep cosstecy. For a rug movg object, ts last movg ut cotas predcted formato. We call the last movg ut actve movg ut, whch cotas key formato for locato update strateges (see Secto 3). 3. Locato Update Strateges of MODTN The above data model eables us to preset the formato of movg objects ad the uderlyg trasportato etworks databases. However, ths s stll ot eough because for a rug MOD system, t s mpossble to get all the real-tme formato from the system maually so that a mechasm should be provded to collect the data automatcally. For the trasportato etworks, sce the topology ad state chages are dscrete ad relatvely less freuet, the database records ca be mataed maually by ssug update commads. It s also possble that some of the state chages (such as traffc jams) ca be reported to the database automatcally. For stace, some techues

6 trasportato egeerg eable the system to detect car accdets ad ther locatos automatcally by examg the chage of traffc flows so that the database ca be updated real-tme to reflect up-to-date stuatos. However, t s early mpossble to track the real-tme locato of movg objects a rug MOD system by maually ssug database commads. Therefore, we eed a locato update mechasm to track movg objects automatcally. I [18, 19] Wolfso et al. have dscussed the locato update polces for the MOST model. The basc dea ca be summarzed as predct ad compare. That s, the system makes a predcto accordg to the curret movg patter of the movg object. Durg ts move, the movg object compares ts actual posto measured by GPS wth the computed posto. Wheever the dfferece betwee them reaches a certa threshold, a locato update s trggered to modfy the database formato. Ths dea provdes a geeral prcple for locato update polces MOD system. I MODTN model, however, ths basc prcple ca be optmzed sce the movg objects are ot modeled the Eucldea space drectly, but modeled predefed trasportato etworks Basc Ideas of Locato Update MODTN I MODTN, we suppose that every movg object s eupped wth a portable computg platform whch s coected wth some other tegrated devces such as the sesor commucator, the wreless terface, ad the mlemeter whch measures the dstace covered by the movg object a certa route. I every jucto of the trasportato etworks, there are a group of sesors stalled so that wheever a movg object trasfers from oe route to aother va the jucto, t gets a otfcato whch wll trgger a locato update. The sesors are assocated wth some formato exchagg eupmets whch wll sed the route detfer, the locato of the jucto sde the route, ad the sg for speed measures to the movg object whe t eters to a ew route. (As a alteratve to the sesors ad mlemeters, GPS ca also be used to fulfll the locato update purpose. Wth a local algorthm, the movg object ca trasform the locato formato from the GPS (wth the (x, y) format) to the (rd, pos) form, where rd s the detfer of the route where the movg object s located, ad pos c [0, 1] s the locato of the movg object sde the route. I ths paper, we focus o the sesor alteratve. The same locato update strateges are suted for the GPS case). The basc dea behd the locato update polcy of MODTN s the Ierta Prcple. That s, the system assumes that the movg object wll cotue to move alog the curret route at roughly steady speed for some more tme, ad wheever ths assumpto becomes vald, the movg object wll tate a locato update so that the up-to-date formato of the movg object ca be reported to the database server. Whe a movg object mo tates ts jourey the MOD system from a jucto, t eeds to sed a locato update message msgu to the server, whch cotas the followg formato: msgu = (md, t u, (gd u, rd u, pos u ), vm u ) where md s the detfer of mo, t u s the tme whe the locato update s trggered, (gd u, rd u, pos u ) = gp u s the locato (expressed as a graph pot) of mo at tme t u, ad vm u s the speed measure of mo at tme t u. The sg of vm u ca be determed from the formato receved from the sesor whe the movg object eters to a ew route va a jucto. Whe recevg ths frst locato update message, the server wll extract the formato cotaed t ad geerate a correspodg movg ut. Ths movg ut wll be saved to the movg graph pot value of the movg object (at ths momet, the movg ut s also the actve movg ut). The movg object also eeds to keep the actve movg ut for locato update purposes. Durg ts move, the movg object wll compare ts actual movg parameters (curret route detfer, locato, ad speed) wth the movg patter cotaed the actve movg ut. Wheever certa codtos are met, a locato update wll be trggered so that the locato of the movg object ca be tracked. I MODTN, there are 3 kds of locato updates, the ID-Trggered Locatos Update (ITLU), the Dstace-Threshold- Trggered Locato Update (DTTLU), ad the Speed- Threshold-Trggered Locato Update (STTLU). Amog them, oly ITLU ad DTTLU are basc oes whle STTLU s optoal ad s eeded oly whe ucertaty maagemet s volved (see Subsecto 4.2). Whe recevg a locato update message, the server wll extract the formato cotaed the message ad geerate a correspodg movg ut. Ths ew ut s the appeded to the correspodg movg graph pot value of the movg object ID-Trggered Locato Update (ITLU) As stated earler, MODTN, every jucto s eupped wth a group of sesors so that wheever a movg object trasfers from oe route to aother, a locato update wll be trggered to reflect the chage of route detfers, as show Fgure 7 (we draw the sesors accordg to ther fuctoaltes. I real-world applcatos some of them ca be combed). I Fgure 7(a), movg objects m1 ad m2 wll trgger a ITLU respectvely, whle m3 wll ot, sce t does t chage to aother route eve though t passes through a jucto.

7 s7 s8 s1 s2 m3 s6 m1 s5 m2 s4 s3 t1, (g1, ra, 1), v1 : : t4, (g2,, 0), v4 t5, (g2,, 0.35), t6, (g2,, 0.7), v5 a) ID-trggered locato update b) correspodg movg utes of m1 Fgure 7. ID-trggered locato update For m1, suppose that t passes by sesor s1 at tme t5, ad passes by sesor s6 at tme t6. The locato update wll be trggered at tme t6, ad two locato update messages wll be set to the server va oe commucato package through the wreless terface. The server ca the extract two movg uts ad save them to the correspodg movg graph pot value of m1, as show Fgure 7(b). At tme t6, except the locato update, some other computatos ad adjustmets are also eed for movg object m1. For stace, the formato exchagg eupmet assocated wth sesor s6 wll sed the detfer of the ew route, the locato of the jucto sde the ew route, ad the sg of speed measures to m1. Besdes, the mlemeter of m1 s refreshed accordg to the ewly receved formato Dstace-Threshold-Trggered Locato Update (DTTLU) Durg ts move alog a certa route, mo wll compare ts actual posto measured by the mlemeter wth the computed posto derved from the actve movg ut. The computed posto at the curret tme stat t ow, deoted by pos ow, ca be computed wth the followg formula: pos ow = pos + vm (t ow t ) where pos ad t are the posto ad the tme correspodg to the last locato update respectvely. If the dfferece betwee the actual posto ad pos ow, deoted by ε, exceeds a certa predefed threshold ξ (for stace, 0.5 klometer) the a ew locato update s trggered to report the actual locato of the movg object, as show Fgure 8. last loc. update computed loc. ε actual loc. t1, (g1, ra, 1), v1 : t5, (g2,, 0.35), t6, (g2,, 0.7), v5 t7, (g2,, 0.3), v6 2 ew movg uts from ITLU actve movg ut a) dstace-threshold-trggered loc. update b) correspodg movg utes Fgure 8. Dst.-threshold-trggered loc. update Durg ths process, some trasformato s eeded sce the speed ad the mlemeter are usg real : 1 ew movg ut from DTTLU actve movg ut measurg utes (such as klometer) whle the graph postos are expressed wth a real umber value p c [0, 1]. However, ths trasformato s trval sce the legth of every route s avalable from the database. Whe evaluatg the computed posto, a specal case should be cosdered whe the movg object s ear the ed of the route ad the actual speed s lower tha the predcted oe. I ths case, the computed posto ca exceed the scope of [0, 1] ad we ca terpret these extra values as the dstace covered by the movg object other routes after the curret route s fshed, so that the locato update polcy does ot eed to be chaged. Whe recevg a dstace threshold trggered locato update message, the server wll extract oe movg ut from t ad wll apped the ut to the correspodg movg graph pot value of the movg object, as llustrated Fgure 8(b) Speed-Threshold-Trggered Locato Update (STTLU) If we compute the locatos of movg objects through terpolato (see Subsecto 4.1), ITLU ad DTTLU are eough to track the locatos of movg objects. However, for the sake of ucertaty maagemet (see Subsecto 4.2), especally to reduce ucertaty as more as possble, we eed aother kd of locato update, Speed- Threshold-Trggered Locato Update (STTLU). As stated earler, the actve movg ut of the movg object, µ, cotas the curret movg patter of the movg object, ad the movg object s expected to move roughly at the speed dcated µ. Durg ts move, the movg object wll compare ts actual speed wth the speed cotaed µ. Wheever the dfferece betwee them exceeds a certa predefed threshold ψ (for stace, 10 klometer/hour), the a locato update s trggered. I ths way, we ca be assured that betwee ay two cosecutve locato updates (suppose the correspodg movg uts are µ = (t, (gd, rd, pos ), vm ) ad µ +1 = (t +1, (gd +1, rd +1, pos +1 ), vm +1 ), the speed of the movg object s betwee ( vm - ψ) ad ( vm + ψ) ( vm s the abstract value of vm ). Whe recevg a speed-threshold-trggered locato update message, the server wll extract oe movg ut from t ad apped the ew movg ut to the correspodg movg graph pot value of the movg object. 4. Queryg the Locato of Movg Objects I ths secto, we dscuss how the locato of a movg object ca be computed from ts movg uts. We suppose that the correspodg movg graph pot value of a actve movg object, mo, s as follows:

8 mgpot = ((t, (gd, rd, pos ), vm )) = 1 Let µ = (t, (gd, rd, pos ), vm ) (1 [ [ ) be the th movg ut of the movg object, ad gp =(gd, rd, pos ) be the locato of the movg object at tme t. For the sake of smplcty, we assume that the speed measure vm s postve, whch meas that the movg object s movg from 0-ed towards 1-ed alog route(gd, rd ). The methodology ca be easly adapted to the stuato whe the speed measure s egatve. Let v max = vm + ψ ad v m = vm ψ where ψ s the speed threshold Computg the Locatos of Movg Objects through Iterpolato The locato of a movg object ca be computed through terpolato. By terpolato, the move of the movg object betwee ay two locato updates s approxmated to a eve speed move. For the uery: where s movg object mo at tme t?, the aswer, whch s a graph pot gp = (gd, rd, pos ), ca be computed the followg way. Case 1. c {1,, } : t = t I ths case, t happes to be a locato update tme, ad the locato formato cotaed µ ca be retured drectly as the result. That s: gp = (gd, rd, pos ) Case 2. c {1,, 1} : t < t < t +1 I ths case, t s betwee two cosecutve locato updates, ad the correspodg movg uts µ, µ +1 eed to be further checked ths stuato. If route(gd, rd ) = route(gd +1, rd +1 ), the accordg to the locato update polces descrbed Secto 3, we ca be assured that the movg object s o route(gd, rd ) at tme t, ad ts locato at t s a graph pot gp = (gd, rd, pos ) where pos ca be computed wth the followg formula: pos pos pos = pos + (t - t ) t t If route(gd, rd ) g route(gd +1, rd +1 ), the from the locato update strateges descrbed Secto 3 we kow that µ ad µ +1 are geerated by a ITLU, Eucl(gp ) = Eucl(gp +1 ), ad at tme t the movg object s jucto getjuct(gp, gp +1 ). The graph pot correspodg to ths jucto wll be retured as the fal result. Case 3. t < t t ow I ths case, we kow that the movg object s stll o route(gd, rd ). Otherwse, there would be a ITLU trggered after t. Therefore, the locato of the movg object at tme t s a graph pot gp = (gd, rd, pos ) where pos ca be computed as follows: pos = pos + v (t t ) By computg the locato of a movg object through terpolato, the locato of the movg object at ay tme stat ca be smply preseted as a graph pot. As a result, the uery processg mechasm ad the uery laguage of the MOD system ca be smplfed. Besdes, the locato update mechasm ca also be smplfed sce the thrd kd of locato update, STTLU, s ot ecessary ths case. The result from the above computg method s oly a approxmate descrpto of the actual locato, ad the error troduced s closely related to the dstace threshold ξ Queryg Movg Objects wth Ucertaty Cosdered Eve though a lot of MOD applcatos, the terpolato techue descrbed Subsecto 4.1 s suffcet, a better soluto s to take the ucertaty brought about by the locato update polcy to cosderato. As stated [19], the locato of a movg object other tha locato update tme s actually ucerta. Therefore, we should troduce the cocept of possble locato presetg the locato of the movg object stead of just expressg t as a precse pot. I MODTN, the ucertaty maagemet problem ca be better solved because the possble locato of a movg object at ay hstorcal or preset tme stat s reduced to a route secto (See Fgure 9). I the followg dscusso, we suppose that ξ, v max, ad v m have already bee trasformed to the [0, 1] scope accordg to the legth of the correspodg route. Besdes, we wll focus o the ucertaty caused by samplg method aloe so that we assume the ucertaty caused by other factors to be eglgble. possble loc. at t (t <t <t +1) 0-ed th loc. update (+1)th loc. update the last loc. update 1-ed possble loc. at t (t<t tow) Fgure 9. Possble locatos of a movg object Case 1. c {1,, } : t = t I ths case, the possble locato of the movg object s a graph pot gp = (gd, rd, pos ). Case 2. c {1,, 1} : t < t < t +1 If route(gd, rd ) = route(gd +1, rd +1 ), the we ca be assured that the movg object s o route(gd, rd ), ad

9 ts possble posto s a graph route secto grs = (gd, rd, seg ) where seg ` [0, 1] ad satsfes the followg codtos: 1) seg ` [ pos ξ, pos + ξ], where pos = pos + vm (t t ). Otherwse there would be a DTTLU betwee t ad t +1 ; 2) seg ` [pos + v m ( t t ), pos + v max ( t t )], where v m ( t t ) ad v max ( t t ) are the shortest ad the logest dstaces the movg object ca cover durg <t (<t = t t ) tme wthout trggerg a STTLU; 3) seg ` [pos +1 v max (t +1 t ), pos +1 v m (t +1 t )]. Otherwse, the movg object would ot be able to arrve at gp +1 tme wthout trggerg a speed trggered locato update. To sum up, the possble locato of the movg object s: seg = [0, 1] 4 [ [pos + v m [pos +1 v max pos ξ, pos + ξ] 4 ( t t ), pos + v max ( t t )] 4 (t +1 t ), pos +1 v m (t +1 t )] where pos = pos + vm (t t ). If route(gd, rd ) g route(gd +1, rd +1 ), the we kow that µ ad µ +1 are geerated by a ITLU. I ths case, the possble locato of the movg object s a graph pot whch correspods to getjuct(gp, gp +1 ). Case 3. t < t t ow I ths case, we kow that the movg object s stll o route(gd, rd ). Otherwse, there would be a ITLU trggered after the last locato update. Therefore, the locato of the movg object at tme t s a graph route secto grs = (gd, rd, seg ) where seg ` [0, 1] ad satsfes the followg codtos: 1) seg ` [ pos ξ, pos + ξ], where pos = pos + vm (t t ). Otherwse there wll be a DTTLU trggered after t ; 2) seg ` [pos + v m ( t t ), pos + v max ( t t )]. Otherwse there wll be a STTLU trggered after t ; Therefore, seg ca be computed as follows: seg = [0, 1] 4 [ pos ξ, pos + ξ] 4 [pos + v m ( t t ), pos + v max ( t t )] where pos = pos + vm (t t ). By computg the locato of the movg object wth ucertaty volved, the movg graph pot defed Defto 13 s actually terpreted as a movg graph route secto. We stll call the defto movg graph pot ths paper just for the sake of cosstecy. Besdes, whe ucertaty s volved, we eed to adapt related operatos to the ucertaty cotext. For stace, the sde operato ca be exteded to sde_possbly ad sde_deftely, as stated [16] Predcto of the Future Locatos of Movg Objects Sce we assume that movg objects ca oly move sde the predefed trasportato etworks, we ca predct ther future locatos more accurately. Suppose the uery s: tell me the locato of movg object mo at t (t > t ow ). The predcted locato of the movg object ca be computed the followg way (we assume that movg objects do ot submt movg plas proactvely). Frst, the algorthm eeds to search the graph system ad to decde a foreseeable future path, ffp, accordg to the possble posto of the movg object at tme t ow (see Subsecto 4.2). ffp s a graph le whch starts from the computed posto of the movg object at tme t ow (see Subsecto 3.3) ad fshes at the frst jucto after whch multple cosecutve traffc flows exst, as show Fgure 10. r4 the last loc. update t ow r3 r5 foreseeable future path possble pos. at t ow Fgure 10. Predctg future locatos The, the system eeds to predct a future speed for the movg object. Ths ca be fulflled ether by usg the speed cotaed µ (f t s ot too far away from t ow ), or by computg a average speed from the speed formato cotaed the movg uts of the movg object. I the latter case, the speed ca be computed lke ths: 1 1 v = ( vw ( t + t)) + vw ( tow t) = 1 tow t1 where vm s the abstract value of vm. Wth v kow, we ca predct the dstace the movg object ca cover <t tme (<t = t - t ow ) as follows: d = v <t From ffp ad d we ca get a graph pot value gp whch s the predcted posto of the movg object at tme t. Accordg to dfferet applcatos, gp ca be ether preseted drectly as the fal result, or exteded to a graph le value. I the latter case, we eed to mpose a gp r5 possble loc. after ffp computed pos. at t ow predcted pos. at t

10 error-factor, whch ca be a fucto of <t, to gp so that the fal result ca be a graph le value. I MODTN we cofe the predcto to the scope of the foreseeable future path ffp, sce after ffp the movg object ca have multple possble drectos, so that ts possble posto ca explode, as show Fgure Coclusos Oe of the key research ssues wth movg objects databases (MOD) s the modelg of movg objects. I ths paper, a ew movg objects database model, Movg Objects o Dyamc Trasportato Networks (MODTN), s proposed. I MODTN, trasportato etworks are modeled as dyamc graphs ad movg objects are modeled as movg graph pots. Besdes, a locato update mechasm s provded ad the related ucertaty maagemet ssues are aalyzed. We have desged a rch set of data types ad operatos for movg objects ad the uderlyg dyamc graphs (whch wll be preseted aother paper). These data types ad operatos have bee partly mplemeted C++ as three algebra modules, spatal algebra, dyamc graph algebra, ad movg object algebra, the Secodo system [2]. Secodo s a ew geerc evromet supportg the mplemetato of database systems for a wde rage of data models ad uery laguages. Besdes, a graphcal user terface, whch ca dsplay spatal objects, trasportato etworks, ad movg objects, has bee mplemeted Java. Compared wth other movg object models, MODTN has the followg features: 1) the system s eabled to support logc road ames, whle ueres based o Eucldea space ca also be supported; 2) both hstory ad curret locato formato ca be uered, ad the system ca also support future locato ueres based o the predcted formato; 3) locato update polces ca be optmzed sce the chage of drecto aloe wll ot trgger a locato update; 4) ucertaty problem ca be better maaged because the possble posto of a movg object at ay hstorcal or preset tme stat s reduced to a movg route secto; ad 5) geeral evets of the system, such as blockages ad topology chages ca also be expressed so that the system s eabled to deal wth the teracto betwee the movg objects ad the uderlyg trasportato etworks. Ackowledgemets Ths research was supported by the Deutsche Forschugsgemeschaft (DFG) research project Databases for Movg Objects uder the grat umber Gu-293/8-1. Refereces [1] Cho H D, Agrawal D, Abbad A E. Usg Space-Tme Grd for Effcet Maagemet of Movg Objects, Proc. of MobDE 2001, CA, USA, [2] Deker S, Gütg R H, Plug ad Play wth Query Algebras: SECONDO. A Geerc DBMS Developmet Evromet. Proc. of IDEAS 2000, Yokohoma, Japa, [3] Forlzz L, Gütg R H, Nardell E, Scheder M. A Data Model ad Data Structures for Movg Objects Databases. Proc. ACM SIGMOD Coferece, TX, USA, [4] Fretzos E. Idexg objects movg o fxed etworks, Proc. of SSTD 03, Sator slad, Greece, July, [5] Gütg R H, Secod-Order Sgature: A Tool for Specfyg Data Models, Query Processg, ad Optmzato. Proc. ACM SIGMOD Coferece. Washgto, USA, [6] Gütg R H, Almeda V T, Dg Z, Modelg ad Queryg Movg Objects Networks, Feruverstät Hage, Iformatk-Report 308, [7] Gütg R H, Böhle M H, Erwg M, Jese C S, Loretzos N A, Scheder M, Vazrgas M. A Foudato for Represetg ad Queryg Movg Objects. ACM Trasactos o Database Systems, 25(1), [8] Lema J A C, Forlzz L, Gütg R H, Nardell E, Scheder M, Algorthms for Movg Objects Databases. The Computer Joural, 46(6), 2003 [9] Papadas D, Zhag J, Mamouls N, Tao Y. Query processg spatal etwork databases, Proc. of VLDB 03, Berl, Germay, [10] Pfoser D, Jese C S, Theodords Y. Novel Approach to the Idexg of Movg Object Trajectores. Proc. of VLDB 00, Caro, Egypt, [11] Pfoser D, Jese C S. Idexg of Network-Costraed Movg Objects, Proc. of GIS 03, Lousaa, USA, 2003 [12] Saltes S, Jese C S, Leuteegger S T, Lopez M A. Idexg the Posto of Cotuously Movg Objects. Proc. of ACM SIGMOD 2000, TX, USA, [13] Sstla A P, Wolfso O, Chamberla S, Dao S. Modelg ad ueryg Movg Objects. Proc. of ICDE 1997, Brmgham, UK, [14] Specys L, Jese C S, Klgys A. Computatoal data modelg for etwork-costraed movg objects, Proc. of GIS 03, Lousaa, USA, 2003 [15] Su J, Xu H, Ibarra O. Movg Objects: Logcal Relatoshps ad Queres, Proc. of SSTD 01, CA, USA, [16] Trajcevsk G, Wolfso O, Chamberla S, Zhag F, The Geometry of Ucertaty Movg Objects Databases, Proc. of EDBT 02, Prague, Czech Republc, March [17] Vazrgas M, Wolfso O. A Spatotemporal Query Laguage for Movg Objects o Road Networks, Proc. of SSTD 01, CA, USA, [18] Wolfso O, Chamberla S, Dao S, Jag L. Locato Maagemet Movg Objects Databases. Proc.of WOSBIS 97, Budapest, Hugary, [19] Wolfso O, Xu B, Chamberla S, Jag L. Movg Object Databases: Issues ad Solutos. Proc. of SSDBM 98, Capr, Italy, July 1998.

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