Detecting Change in Snapshot Sequences

Transcription

1 Dttn Cn n Snpsot Squns Mnzn S n Stpn Wntr Dprtnt o Gots, Unvrsty o Mourn, Vtor, 3010, ustr.s@pr.un.u.u, wntr@un.u.u strt. Wrss snsor ntwors r poy to ontor yn orp pnon, or ots, ovr sp n t. Ts ppr prsnts nw sptotpor t o or yn r ots n snsor ntwors. Our o supports or t rst t t nyss o n n squns o snpsots tt r ptur y rnt rnurty o osrvtons, n our o ows ot nrnt n nonnrnt ns. Ts ppr ouss on ttn quttv spt ns, su s r n spt o r ots. ntrz ort s vop, su tt spt ns n nty tt y n-ntwor rton o ntrz tsts. 1 Introuton Mny spt pnon r ontnuousy nn n t n sp [5]. Envronnt ontorn, or xp, ouss spy on rs o nvronnt sturn, su s rzn, rtzn, pouton, n on. Envronnt sturn s on o t y rvn ors or nvronnt ns, n rnt sturn vnts n rsut n vrs ns. For xp, on vnt n rsut n t rs o orst ovr n t n ro orst to t n. oon vnt n rsut n t nrs o so ostur n t trnsor ro t n to wtn [9]. Wrss snsor ntwors (WSNs) r nrsny n us n spt pptons to tt n n t nvronnts. WSN s ntwor o oputn vs tt n oort v ro ountons. WSN s so ntwor o osrvton vs, sn no n t WSN s qupp wt snsors tt n tty n-rn osrvton o t nvronnts. WSN s to osrv n n r-t, n n, wt ost ny tpor rnurty (ut typy orsr spt rnurty). In t ontxt o orp norton sn, snpsot-s n vnts ppros v n us to o wrss snsor ntwors. Snpsots ppro r oony us n pptons, wr snsor nos r ts to proy sns n trnst snpsots o n nvronnt y sttn t WSN to rtn tpor snsn rsouton (.., [16]). tou t snpsot ppro s or prt, tort stus (.., [18]) r or ntrst n vnt-s o o WSNs u to ts vnts o ttn snt ns or vnts.

2 () () () () F. 1. T vovn o spt ots ovr t. ()-() r t rnt t stps,.., t 1, t 2, t 3, n t 4. Howvr, prty so vnt-s os r sut to rnurty ts. y rtn spt n tpor rsouton o osrvton, vnts y not ror sowr, sot. Evn soon y ru tt ontnuous osrvton s poss wt utur tnoos, tr s st t pro o runny o osrvtons. Lr ount o nry n onsu to osrv n nvronnt wr no n s ourr. In ts rustn, ts ppr vops nw sptotpor t o tt norports ot snpsots n vnt-s ppros, su tt t o n pt to rnt rnurts o osrvtons, n n tt n nyz spt ns. In our sptotpor t o, WSN s o s st o pont ots wt pont osrvtons, n t prry strutur n our o r s. Es r portnt oponnts or spt ots. For xp, n r s n poyon tt s onsttut s squn o s. y snsor ntwor pont osrvtons, orp pnon w strt s r ots tt onsst o ponts, n ts ponts w tn ornz s squn o s. Sn tpor rnurty o osrvton s ow n our o, orp pnon r rprsnt s squns o snpsots. Our o supports t nyss o n n squns o snpsots tt s ptur y rnt rnurty o osrvtons. Fur 1 ustrts n xp out t rprsntton o yn r ots n snsor ntwor. Ts ppr s our or ontrutons. Frsty, w propos nw t strutur tt nsurs t ounrs o r ots r wys os trvrs trs. Sony, w propos t n nsrt s to rprsnt ns o r ots n snpsot squns. T squns o t n nsrt s wys or os trvrs trs, n tr r t rnt typs o ts os trs tt n us to stnus sx topoo ns n two non-topoo ns. Try, our ppro n support ot nrnt n non-nrnt ns. n ny, w vop ntrz ort to nty tt t t rnt typs o ns n snsor ntwor. 2 Rt Wor T n nvstt n ts ppr s t n o spt ntts. n ours wnvr spt ntts possss rnt spt ttruts t rnt

3 ts [2]. Grnon n St [4] ssy ntts n t sptotpor wor nto two tors: on s ontnunts, n t otr s ourrnts. T stuy o ts ppr s on t prsptv o ontnunts tt xst t vn t t vn v o rnurty n unro rnt typs o ns ovr t. Hornsy n Enor [5] sust quttv rprsntton o n. Tr noton o n s s on ot ntty, n st o oprtons tt tr prsrv or n ntty. Ts ppr pps tr onpt o ntty ssun tt snsor ntwor n st up wt rtn pproprt rnurty o osrvtons or rtn nvronnt pnonon, orn to t spn tor [14]. Grnon n St [4] propos SNP n SPN ontoos. T SNP ontooy s snpsot-s vw, wr ntts r ornz s tpor squns o snpsots. On t otr n, vnt-s sptotpor t os n SPN ontooy (.., [10, 11, 17]) n unrstoo s ron-s o tt s u to t snpsot-s o [3]. Snpsot-s n vnts ppros r oony us n wrss snsor ntwor pptons. Ln t. [6] propos rnt ounry tton ppro, wr snpsots o ontor ron r trnstt to snt ntr oputr v t sn o t ntwor t rtn t ntrvs. snpsot t vn t t s strt s ontour p, w onssts o rnt ounrs. Ony t nos on t rnt ounrs n to rport to t sn, w onstruts ontour p. unqu wor or otn snpsot t s propos y Sr t. [15]. Tr to s swp,.., wvront tt trvrss t wo ntwor n psss nos n t ntwor xty on. Ts ppros r t to stt snpsots. Ty nnot yny nyz snpsot squns to rv snt ns. Du t. [1] vops o to tr snt ns or vnts n t nvronnt. Ty o snsor ntwor s trnuton ntwor. T trnuton n n ovr t n rspons to t ovnt o spt pnon. Woroys n Du [18] so us trnutons, n spt ots r rprsnt s st o trns. Cns o spt ots r rprsnt s t nsrton or ton o trns ro t st. Howvr, tr ppro s t to nrnt ns,.., t n o sn trn t t stp. Sq n Du [13] nvstt rnt snsor ntwor struturs or ttn topoo ns. Svr oony-us noroo struturs, su s Duny trnuton, Gr rp, rtv noroo rp, n ry trnuton, v n tst. Tr ppro so ssus nrnt ns. Tr sston o spt ns s s on t tort stuy o Jn n Woroys [8]. r ount o xstn wor on quttv ppros to rtrz topooy n topoo ns n snsor ntwor xsts,.., [7]. Ts ppr s stnus ro otr rt wor y usn squns or quttv rsonn. W us t n nsrt s to rprsnt ns o r ots, n t squns o t n nsrt s r wys os trs rrss t nur o nos on t trs. Tus, our ppro ows or t rst t ot nrnt n non-nrnt ns.

4 3 Cn Rprsntton n Snsor Ntwor snsor ntwor n o s rt pnr rp G = (V,E), w s ut y Duny trnuton [12]. In ts ppr w ssu tt t snsor ntwor s ns nou to nsur t onstruton o trnuton rwor. In t rp G, V s st o nos n E s st o s,.., (v,v ), w rprsnt t rt ounton ns twn t nos v,v V. W ssu tt E s sytr,.., (v,v ) E, tn (v,v) E. Not tt t rton o rt w ustrt n ur wn t s rvnt, otrws t rprsntton o G n sp s n Fur 2(). T st o nors o v V s not s nr(v) = {v : (v,v ) E}. For xp n Fur 2(), nr(v) = {,,,,}, n t st nr(v) s sort nto ows orr. E no ony stors t out ts n ts t nors. v () v () v () v () F. 2. Snsor ntwor strutur. () T st nr(v) o no v s sort nto ows orr. () T rt s n ys o no v. () T trvrs o ounry s s on t ows ru. () ron wt o n tt y t trvrs ornttons. T st o rt s nt to no v s not s (v). Sn E s sytr, t st (v) n rv ro nr(v),.., (v) = {(v,v ) : v nr(v)} {(v,v) : v nr(v)}. In Fur 2(), or xp, t no v s v nors, so tt v s tn rt s. T st (v) s so sort n ows orr, n or s norn no v, t (v,v) s wys t nxt o t (v,v ), s n Fur 2(). no v n ts nors r ornz nto rt ys, or spy ys. y s not s (v,,,v), wr t tr nos v, n r stnt, n tr s n or ny two onsutv nos n t y,.., (v,), (,), (,v) E. Sn nr(v) s n ows orr, t ys r ountrows, n n xp,.., (v,,,v), s sown n Fur 2(). s on t ov nton, w n n t rprsntton o r ots n snsor ntwor: Dntons: rt (v,v ) E s n ot, ot v n v r ot n n r ot. y (v,,,v) s n ot y, t vrtxs o t y,.., v,,, r n n r ot. n ot (v,v ) E

5 v () t 0 v () t 1 v () t 2 F. 3. Insrt n t ounry s r r y t so-n rrows n t s-n rrows rsptvy. Ony ounry s r ustrt n t ur. s non-ounry, t ons to n ot y. n ot (v,v ) E s ounry, t os not on to ny ot y. r ots n rprsnt s squn o ounry s. n xp s sown n Fur 2(), n w t nos,, n v r ot n n r ot. T s (,v), (v,), (,), (,v), (v,) r ounry s, sn ty o not on to ny ot ys. Usuy ony ounry s w ustrt n t urs, s n Fur 2 () n (). Trvrss on ounry s o r ots wys or os trs. For xp, n Fur 2(), suppos t trvrs s strt t no, tn t trvrs w os tr oow t ows ru: v v. Fory, t pt o st o onnt ounry s s Eurn tr, sn t nur o ounry s n no s wys vn. n portnt proprty o ts trs s t trvrs orntton. Gvn ron wt o, or xp n Fur 2(), xtrn trvrss o t ron r ows n ntrn trvrss o t o r ountrows. Sn t r o poyon s postv t vrtxs o t poyon r rrn n ountrows orr, n ntv ty r n ows orr, t rsut o r uton n us to trn t orrn o t vrts o poyon, n tus t orntton o trvrs. In our o, t n o r ots w ptur s squns o snpsots n t snsor ntwor. In snpsot, squns o ounry s r us to rprsnt r ots. Sn r ots r vovn ovr t, t squns o ounry s wou yny n t rnt snpsots, s n Fur 3()-(). T n o ounry s twn two onsutv snpsots s rprsnt y t nsrton n ton o ounry s: Dntons: ounry (v,v ) E s n nsrt ounry, or spy n nsrt, t t t t s not ounry t prvous t t 1, ut o ounry t t t. ounry (v,v ) E s t t t t t s ounry t prvous t t 1, ut s not ounry t t t. W ssu t snsor rn s nry,.., {0,1}, wr t rn o 1 rprsnts tt t snsor no s ot n n r ot. W n tt no v ns ts snsor rns,.., ro 1 to 0, t t stp t, tn v

6 r xpnson ppr prt-spt spt ontrton sppr s-r F. 4. Sx topoo ns n two non-topoo ns. s n tv no t t. For xp, n Fur 3, nos,, n v r tv nos t t 1, n no s t ony tv no t t 2. W n tt twn two onsutv t stps t n t 1, n o r ot s nrnt tr s on n ony on tv no t t stp t. In Fur 3, t n twn t 0 n t 1 s non-nrnt n t n twn t 1 n t 2 s nrnt. Our o supports ot nrnt n non-nrnt ns. 4 Dttn Cn T trvrss on t n/or nsrt s n us to stnus t rnt typs o ns o r ots. Ts t typs o ns nu sx topoo ns n two non-topoo ns. T sx topoo ns r pprn, spprn, r, spt, s-r n prt-spt [8, 13]. T two non-topoo ns r xpnson n ontrton. Exps o t t typs o ns r sown n Fur 4. s w suss, rnt typs o ns w v rnt typs o trvrss on t/nsrt s, n o t trvrss w or os trvrs trs. W suss t t typs o ns n our roups. 4.1 pprn n Dspprn Tr s os trvrs tr or t pprn o r ots. In Fur 5(), n r ot pprs, n st o ounry s s n nsrt. T st o nsrt ounry s ors os tr. Suppos t trvrs strts t no, tn t tr s:. n strt rprsntton o t trvrs n Fur 5() s sown n (), wr t so-n rrow rprsnts trvrs on nsrt s, n t r rprsnts no tt s no ntty o. Not tt Fur 5 ustrts t st wn ns v our ut r ot ntts v not yt n upt. In Fur 5(), n r ot spprs, n t ounry s wt r ot ntty o v n t. n n Fur 5(), t s-n rrow nots trvrs on t s. Not tt r ot ntty s ntry stor n ounry o t r ot. 4.2 Expnson n Contrton For t xpnson n ontrton o r ots, t os tr w nu two snts. On snt o t trvrs onssts o nsrt s,

7 n notr snt onssts o t s. In Fur 5(), or xp, n r ot xpns. T nsrt s or on snt o t trvrs:. T t s or notr snt:. Not tt t son snt o t trvrs s n rvrs rton. Fur 5() ustrts tt t two snts o t trvrs r onnt y two nos n. T two nos n r t trnston nos o t trvrs. Snt 2 s wr t r ot ntty, w Snt 1 o not v r ot ntty yt. Sry, n Fur 5() n (), n r ot ontrts, n tr r two snts tt s onnt y two trnston nos n. Ston 5 w onstrt ow xpnson n ontrton n stnus: t trvrs o xpnson onssts o (1) nsrt s n (2) t s, n t trvrs o ontrton onssts o (1) t s n (2) nsrt s. 4.3 Mr n Spt T os tr or t r o two r ots nus our snts. Fur 5() sows r o two r ots n. T rst snt o t trvrs s on nsrt s ro no to no :. Two trnston nos n r ot n rnt r ots,.., n rsptvy. T son snt o t trvrs s on t t s o r ot ro no to no :, n t t s v t r ot ntty, s sown n Fur 5() n (). T tr snt w strt ro r ot n rturn to r ot v nsrt s:. Fny, t ourt snt o t trvrs w rturn to t orn no v t t s o r ot :. Not tt Snts 2 n 4 r n rvrs rton. Ts our snts o trvrs r onnt y our trnston nos,,, n, n w two nos r ot n r ot, n t otr two r n r ot. T trvrs o spt so nus our snts, n t our snts r n t squns o (1) t s n (2) nsrt s n (3) t s n (4) nsrt s, s n Fur 5() n (). T our snts o trvrs r so onnt y our trnston nos,,, n. Snts 1 n 3 on t s r wr t r ot ntty. In oprson, t our trvrs snts o r r n rnt orr: (1) nsrt s n (2) t s n (3) nsrt s n (4) t s, s n Fur 5(). ot o t trvrss n Fur 5() n () strt t no. T ston o t trvrs strtn no w suss n Ston S-r n Prt-spt In Fur 5(), n r ot s-rs nto ron wt o. T trvrs o s-r s sr strutur s r: (1) nsrt s n (2) t s n (3) nsrt s n (4) t s, s sown n Fur 5(n). Snts 2 n 4 o s-r v t s r ot ntty, s ustrt n Fur 5() n (n). In oprson, Snt 2 n

8 ppr () 1 () sppr () () 1 xpn () () 1 2 ontrt () () 1 2 r () () spt () () (n) n n () s r (p) n n (o) spt prt F. 5. Et trvrss or t rnt os trvrs trs

9 o n () xpn o n () F. 6. Otr typs o trvrss n onsr s t ontons o t t s trvrs typs. T trvrs n () s onton o two trvrss n Fur 5() n (). 4 o r n Fur 5() n () v rnt r ot ntts,.., n rsptvy. Tus, rs n s-rs n stnus y t r ot ntts o t trvrs snts, s n Fur 5() n (n). In Fur 5(o), n r ot wt o prt-spts. T r ot wt o onssts o two os trs: on s ntrn or t o n t otr s xtrn or t r ot, s ry ntrou n Ston 3. T r ot ntty or o o n r ot s sp s. Fur 5(o) sows tt t ntrn s wt or os tr: n. Sr to spts, t trvrs or prt-spt s our snts, n t orr s (1) t s n (2) nsrt s n (3) t s n (4) nsrt s. For spt, Snts 1 n 3, n Fur 5(), v t s r ot ntty, w n t s o prt-spt n Fur 5(p), Snt 1 s n r ot ntty o n Snt 3 s rnt ntty. 4.5 Sury s sury, tr r t rnt typs o s trvrss on t n nsrt s. Ts t trvrss n stnus s on tr snts. E typ o trvrs n unquy nty on typ o n. Tr r poss otr typs o trvrss, n ty n onsr s t ontons o t t s trvrs typs. For xp, n Fur 6, n r ot xpns, n t t n nsrt s v n sprt nto two os trs:.., o n n. T trvrss n Fur 6 n onsr s onton o t two trvrss n Fur 5() n (). Our orts r to n t onton o t t s trvrss su tt rnt n typs n nt. 5 ort s suss n Ston 4, tr r t rnt typs o trvrss tt n us to stnus t typs o ns. Ts ston w prov ntrz

10 orts or trvrs ornzton n n tton. T s o our ort s to ntz trvrs t vn no v wt ss, n t ss w pss ro on no to notr urn t trvrs. so, ntrz t w rt nto t ss urn t trvrs. Sn t trvrss or os trs, t rt ss w rturn to ts orn no v, so tt no v w to tt rnt typs o ns s on t rt ss. 5.1 Msss Sn r ot ntts o trvrs snts n us to ntty rnt typs o ns s n Fur 5, rvnt r ot ntts w rt urn t trvrs. ss w ntz t t nnn o trvrs or t rton n t ntwor. ss ntz t no v s not s: s(v) = (p 1,p 2,p 3,p 4 ). T our nts,., p 1, p 2, p 3, n p 4, not t r ot ntts o t our snts n trvrs. W us s(v).p 1, or xp, to rprsnt t nt p 1 n s(v). In t nnn o trvrs, t nts n ss r pty, n t ss n rprsnt s: s(v) = (,,, ). T trnston nos r rspons to upt t nts o ss urn trvrs. For xp, t os tr or spt n Fur 5() onssts o our trnston nos,,, n, n our snts. I no ntzs trvrs wt ss s(), tn no w upt t nt s().p 1, n,, n w upt s().p 2, s().p 3, n s().p 4 rsptvy. Fny, t trvrs w rturn to t nt no wt ss tt ontns t norton out t os tr. t s n t s snt v t s r ot ntty. For xp, n Fur 5(), t s (,), (,) n (,) n t rst snt v t s ntty o, n tus s().p 1 =. Sry, s().p 2 = I, s().p 3 =, n s().p 4 = I. Not tt s().p 2 = I n s().p 4 = I r us to spy tt t son n ourt snts o t trvrs r on nsrt s n r ot ntts v not n prov. In Fur 5(), opt ss or t trvrs strt t no wou : s() = (,I,,I). 5.2 snt Trvrs Snts In so trvrss, t s poss tt tr s no n snt, n t snt s rr s n snt snt. For xp, n Fur 7(), no strts trvrs, tn t trvrs wou ony ontns tr snts: on nsrt s, on nsrt s, n on t. Sn trvrs o r s (1) nsrt s n (2) t s n (3) nsrt s n (4) t s, s n Ston 4.3, w rr t son snt on t s s n snt snt t no. T r ot ntty or t snt snt sou prov y no. s sown n Fur 7(), no n qur t r ot ntty,..,, ro ts ounry s

11 xpn () r () F. 7. It s poss tt tr r snt snts n trvrs. Trnston nos r r y rs. () snt on t s s snt t no. () snt on t s s snt t no. {(,),(,)}. T opt ss ntz y no wou : s() = (I,,I,). Gnry, n trvrss o r, spt, s-r n prt-spt, t rst n tr snts wys xst, w t son n ourt snts y snt. In t trvrss or xpnson n ontrton, t rst snt wys xsts, w t son snt y snt, s n xp n Fur 7(). 5.3 Trnston Es Sn our ort sou to nty ornz trvrss or ntrz t rton, ony s sust o snsor nos sou nont or ntzn trvrss. W n tt n nsrt or t s trnston t onnts n tv no n non-tv no. For xp, n Fur 5(), t t (,) s trnston tt s nt non-tv no n trn tv no. so (,) s trnston, ut t ns t n tv no n ns t non-tv no. Trnston s r wys n prs. I tr s trnston wt n nt non-tv no n trn tv no, tn tr wys xsts notr trnston wt n nt tv no n trn non-tv no. pr o trnston s s wys onnt y tr. For xp, n Fur 5(), (,) n (,) r onnt y t tr. I no v s non-tv no n s t nt no o trnston, tn no v s t rsponsty to ntz trvrs (typ 1) wt ss s(v). For xp, n Fur 5(), no s non-tv no, n t s t nt no o trnston (,), so sou strt trvrs wt nw ss s(). T trvrs w vst nos,,,,,,, n, n rturn to t strtn no wt opt ss s(). Sry, n Fur 5(), t non-tv no s t nt no o (,), n tus no sou so ntz trvrs. In t s o pprn or spprn o n r ot, t nos n t r ot r tv nos, n tus tr s no trnston. notr typ o trvrs (typ 2) s rqur or ts ss. tou our ort nus t ornzton n onton o trvrss typ 1 n typ 2, t ts r not urtr suss.

12 ort 1: Dttn Cn 1 Vrs: no v; 2 v rvs ss s(v) ntz y ts tn 3 s(v).p 1 n s(v).p 2 = tn 4 s(v).p 1 = I tn v tts n pprn; 5 s(v).p 1 I tn v tts spprn; 6 s s(v).p 2 n s(v).p 3 = tn 7 s(v).p 1 = I tn v tts n xpnson; 8 s(v).p 1 I tn v tts ontrton; 9 s s(v).p 4 tn 10 s(v).p 1 = I tn 11 s(v).p 2 s(v).p 4 tn v tts r; 12 s(v).p 2 = s(v).p 4 tn v tts s-r; 13 s s(v).p 1 I tn 14 s(v).p 1 s(v).p 3 tn v tts prt-spt; 15 s(v).p 1 = s(v).p 3 tn v tts spt; 5.4 Dttn Cn I no v rvs ss s(v) ntz y ts, tn no v s to tt rnt typs o ns s on t ss s(v) (s ort 1, n 2). I t ss ontns ony on snt,.., s(v).p 1 n s(v).p 2 =, tn no v tts n pprn or spprn (ns 3-5). I t snt s on nsrt s,.., s(v).p 1 = I, tn no v tts n pprn (n 4). I t snt s on t s,.., s(v).p 1 I, tn no v tts spprn (n 5). I t ss ontns two snts,.., s(v).p 2 n s(v).p 3 =, tn no v wou tt n xpnson or ontrton (ns 6-8). s ustrt n Fur 5() n (), t rst snt o xpnson s on nsrt s (n 7), w t rst snt o ontrton s on t s (n 8). Tr r our typs o ns,.., r, spt, s-r, n prt-spt tt v our trvrs snts (ns 9-15). In t ss o r n sr, t rst snt s on nsrt s (ns 10-12). Mr n s-r n tn stnus y t son n ourt snts o trvrss. T son n ourt snts v rnt r ot ntts or r, n v t s r ot ntty or s-r. W or spt n prt-spt, t rst snt s on t s (ns 13-15). T rst n tr snts v t s r ot ntty or spt, n v rnt r ot ntts or prt-spt. 6 Evuton T ntrz ort n Ston 5 ws vut n suton nvronnt. Rpst (ttp://rpst.souror.nt/) ws us or suton. T

13 rt pnr rp strutur ws ut y Duny trnuton n Rpst. T pnr rp onssts o 500 snsor nos tt r poy n ron o squr unts, wt ounton rn o 80 unts. In t xprnt, w out 2000 suton runs, n typ o n out 250 runs. n r ot s sut s on or svr r r squr oxs, s xps n Fur 8. Fro t 1st to t 2000t suton runs, t prtr r ruy nrs, n tus t nur o tv nos sou so ruy nrs. t suton run, sp typ o spt n n our t rnoz otons wtn t snsor ntwor. Fur 8 () n () sow xps o r n prt-spt tt our t rnt otons o t snsor ntwor. Our orts wr nst n snsor no n t ntwor. In t suton runs, t t rnt typs o ns wr orrty tt y t rt pnr rp strutur, s on s t rnurty o osrvton s st up pproprty s on t rnurty o r ots. T xprnt nvstt t sty o t ort. T sty ws sur y nur o sss snt wt nrsn nur o tv nos. T rsuts o powr rrsson nyss or pprn, xpnson, n r, or xp, s sown n Fur 9(), (), n () rsptvy. Fur 9() sts t rsuts o rrsson nyss or t t typs o ns. powr rrsson (y = x ) s us to t t 250 pott rsuts or typ o n. t rrsson rsuts v oonss o t, wt t rn o R 2 ro to s n Fur 9(), tr s < 1 or t t typs o ns, n tus our ort sou v n orr o O(n) or ss. T xprnt sows tt our ntrz ort s y s. To opr wt our ntrz ort, w so pnt ntrz sns-n-trnst ort, n w tv no w spy orwr ts snsor rn to sn y ut-op routns. T rsuts o t ntrz ort s sown n t st row o Fur 9(). y oprn n n y = x, t s r tt our ntrz ort s or ss nt n s tn ntrz ort. r prt spt () () F. 8. Spt ns our t rnoz otons wtn snsor ntwor.

14 Msss snt ppr y = x R 2 = Msss snt Expn y = x R 2 = Msss snt Nur o tv nos () pprn () r Mr y = x R 2 = Nur o tv nos Nur o tv nos () xpnson Typs o n R 2 pprn Dspprn Expnson Contrton Mr Spt S-r Prt-spt Cntrz () rrsson nyss y = x F. 9. Sty o t ort. 7 Conuson In ts ppr, w v vop nw sptotpor t o tt ns t tton o quttv spt ns n snpsot squns. In our o, ounrs o r ots r os trvrs trs. Et typs o ns o r ots n so rprsnt n stnus y t rnt os trs tt onsst o nsrt n/or t s. W v so vop ntrz ort or WSNs to tt ns y n-ntwor rtons. T xprnt provs tt our ort s s n nt, n s to tt ns wt rnt nurs o tv nos. In t utur wor, our t o n urtr vop to support ntrz spt qurs, or xp, t qury o t topoo rtons twn two spt rons. Rrns 1. M. Du, S. Ntt, n M.F. Woroys. Montorn yn spt s usn rsponsv osnsor ntwors. In Prons o 13t nnu CM Intrnton Worsop on Gorp Inorton Systs (GIS05), ps 51-60, Gton. Quttv Spt Cn. Oxor Unvrsty Prss, Gton. Fs n ots n sp, t, n sp-t. Spt Conton n Coputton, 4(1):39-67, 2004.

15 4. P. Grnon n. St. SNP n SPN: Towrs yn spt ontooy. Spt Conton n Coputton, 4(1):69-104, K. Hornsy n M. Enor. Quttv Rprsntton o Cn. In S. Hrt n.u. Frn, s.: COSIT 1997, Vou 1329 o Ltur Nots n Coputr Sn, ps Sprnr, Hr, J. Ln, L. Cn, K. N, Y. Lu, n G.. nw: Grnt ounry tton or t srs snpsot onstruton n snsor ntwors. IEEE Trnstons on Pr n Dstrut Systs, 18(10): , J. Jn n M. Woroys. Dttn s topoo ns n snsor ntwors y o rton. CM GIS, Irvn, C, US, J. Jn n M. Woroys. Evnt-s topooy or yn pnr r ots. Intrnton Journ o Gorp Inorton Sn, 23(1):33-60, I. Mu, K. Hornsy, n I.D. sop. Mon ospt vnts n pts trou quttv n. In T. rowsy t., s.: Spt Conton V, Vou 4387 o Ltur Nots n rt Intn, p Sprnr, rn, D.J. Puqut n N. Dun. n vnt-s sptotpor t o (ESTDM) or tpor nyss o orp t. Intrnton Journ o Gorp Inorton Systs, 9:7-24, D.J. Puqut. Mn Sp or T: Issus n Sp-T Dt Rprsntton. Gonort. 5(1):11-32, F.P. Prprt n M.I. Sos. Coputton Gotry: n Introuton. Sprnr, Nw Yor, M.J. Sq n M. Du. Et o noroo on n-ntwor prossn n snsor ntwors. In T.J. Cov t., s.: GISn 2008, LNCS, vou 5266, ps Sprnr, Hr, C.E. Snnon. Counton n t prsn o nos. In Prons o t Insttut o Ro Ennrs 37(1), ps 10-21, P. Sr, Q. Fn,. Nuyn, n L. Gus. Swps ovr wrss snsor ntwors. In IPSN 06: Prons o t 5t Intrnton Conrn on Inorton Prossn n Snsor Ntwors. ps , T. Wr, P. Cor, P. S, L. Kn, Y. Guo, C. Crossn, P. Vn, D. Swn, n G. sop-hury. Trnsorn rutur trou Prvsv Wrss Snsor Ntwors. IEEE Prvsv Coputn, 6:2(50-57), M. Woroys. Evnt-ornt ppros to orp pnon. Intrnton Journ o Gorp Inorton Sn, 19(1):128, M. Woroys n M. Du. Montorn quttv sptotpor n or osnsor ntwors. Intrnton Journ o Gorp Inorton Sn, 20(10): , 2006.