On the metric distortion of nearest-neighbour graphs on random point sets

On the metric distortion of ne arest-neigh b our graphs on random p oin t sets Amitabha Bagc hi ∗ Sohit Bansal ∗ Octob er 30, 2018 Abstract W e study the graph constructed on a Poisson p oint pro cess in d dimensions by connecting each po in t to the k points nea r est to it. This graph a.s. has an infinite cluster if k > k c ( d ) where k c ( d ), known as the critical v alue, dep ends only on the dimension d . This pa per presents an improved upper bound of 1 88 on the v alue o f k c (2). W e also show that if k ≥ 188 the infinite cluster of NN(2 , k ) has an infinite subset of p oints with the prop erty that the distance along the e dg es of the graphs b et w een these po in ts is at most a constant multiplicativ e facto r lar ger than their Euc lidea n distance. Fina lly we discuss in detail the re lev ance of our results to the study of mu lti-hop wir eless sensor netw orks. 1 In tro duction The k -nearest neigh b our grap h of a p oint set S in a metric space is constructed according to the follo wing natural definition: F or eac h p oin t x ∈ S esta blish an edge from x to the k p oin ts of S \ { x } nearest to it. S u c h graphs hav e app licati ons in numerous areas: classification problems of all flav ours, top ology con trol in wireless n et works [6, 22], data co mpression [17, 1] and dimensionalit y reduction [19] and m ulti-agen t systems [10]. W e fo cus on k -n earest neigh b or graphs on random p oin t sets in R d assuming that the distance is the Euclidean distance. F urther w e restrict ourselv es to the case where the ed ges established are undirected. Clearly it is not n ecessary that this graph b e connected for arbitrary k and S or ev en that it ha ve a large c onnected comp onen t. Ho wev er, H¨ aggstr¨ om and Meester [13] ha v e sho wn that if the set S is generated by a P oisson p oint p ro cess then there is a finite v alue k c ( d ) d ep ending only on the dimension su c h that if k > k c ( d ), the k -nearest neigh b or graph has an connected comp onen t whic h is infinite. In th is pap er w e study this setting fu r ther. F ollo wing the notation in [13 ] we will denote this mo del in d dimensions, parametrized b y k as NN( d, k ). In this pap er we sh o w that for NN(2 , k ) that if k ≥ 188 the infi nite cluster 1 has an infin ite su bset of p oin ts with the prop erty the metric distortion b etw een th em is b ounded by a constan t i.e. if there is a pair of p oin ts in this infinite subset the sh ortest d istance b etw een them ac hiev ed along a path in the graph is at most a constan t m ultiplicativ e factor larger than the Euclidean distance b et ween them. In the pro cess of p ro v in g the latter result we improv e the b est kn own b ound for k c (2) to 188 from 213 (due to T eng and Y ao [20]). Our pro of tec h nique generalizes easily to NN( d, k ) f or d ≥ 3. ∗ Dept of Computer Science and Engg, Indian Institute of T echnology , Hauz Khas, New Delhi 110016 . { bagchi,c s503022 } @ cse.iitd. ernet.in. Note: This work h as b een submitted to the IEEE for p ossible publication. Cop yright ma y b e transferred without notice, after which this ve rsion ma y no longer b e accessi ble. 1 W e will use the terms comp onent and cluster in terc hangeably . 1 Organization. T he rest of this section is dev oted to s u rv eying r elated work and introdu cing th e terms and n otatio n we will use. A new b oun d on k c (2) and the resu lt on the metric distortion within the infinite cluster is p r esen ted in Section 2. W e will talk ab out th e applicabilit y of our results to wireless m ulti-hop sensor net w orks in Section 3, concluding with a discussion of some simulat ion results and conjectures arising from them in Section 4. 1.1 Related work The study of random graphs obtained by applying connection ru les on stationary p oin t p ro cesses is kno wn as contin uum p ercolation. Meester and Ro y’s m onograph on the sub j ect pro vides an excellen t view of the deep theory that has b een develo p ed aroun d this general setting [16]. Th e NN( d, k ) m o d el w as in tro duced by H¨ aggstr¨ om a nd Meester [13]. They sh o wed that ther e w as a fin ite critical v alue, k c ( d ) for all d ≥ 2 suc h th at an infinite cluster exists in this mo d el. They pro v ed that the infi nite cluster was unique and that there w as a v alue d 0 suc h that k c ( d ) = 2 for all d > d 0 . T eng and Y ao ga v e an u pp er b ound of 213 for k c ( d ) [20]. k -nearest neigh b or graphs on random p oin t sets con tained in side a finite regio n h a ve been extensiv ely studied. The ma jor concern, d ifferen t from ours, h as b een to ensure that al l the p oints within the region are connecte d within the same cluster. Ballister, Bollob´ as, S ark ar and W alters [5] sho w ed that the smallest v alue of k that will ensur e connectivit y lies b et ween 0 . 3043 log n and 0 . 5139 log n , impr oving earlier results of Xu e and Ku mar [22]. Ballister et. al. also studied th e problem of co v ering the region with the discs con taining the k -nearest neighbour s of the p oin ts. W e refer the reader to [5] for an in teresting discussion relating this setting to earlier wo rk by P enrose and others. Eppstein, P aterson a nd Y ao [9] stud ied k -nearest neigh b our graphs on random p oint sets in t w o dimensions in some detail and pr o ved in teresting b ound s showing that the n um b er of p oin ts in a comp onen t of depth D w as p olynomial in D when k w as 1 and exp onen tial in D when it w as 2 or greater. Their pr imary int erest was in obtaining lo w dilation em b eddings of nearest-neigh b or graphs. Algorithms for searc hing for n earest n eighb ors (see e.g. [8, 21]) and constr u cting n earest neigh b or graphs efficien tly ha v e also receiv ed a lot of attent ion (see e.g. [18]). Ho wev er these are not directly related so w e do n ot su rv ey this literature in d etail. 1.2 Definitions an d N otation P oisson p oin t pro cesses. Our r andom p oin t sets are generated by homogenous P oisson p oin t p ro- cesses of in tensit y λ in R d where d ≥ 1. Under this mo del the num b er of p oin ts in a region is a random v ariable that dep end s only on its d -dimensional volume i.e. the num b er of p oints in a b ounded, mea- surable set A is Po isson distr ibuted with mean λV ( A ) where V ( A ) is the d -dimensional v olume of A. F ur th er, the r andom v ariables asso ciated with the num b er of p oin ts in disj oint sets are indep endent . Site p ercolation. Consider an infin ite graph defin ed on the v ertex set Z d with edges b et w een p oin ts x and y such that k x − y k 1 = 1. Site p ercolation is a probabilistic pro cess on this graph. Eac h p oin t of Z d is tak en to b e op en with probabilit y p and close d with probabilit y 1 − p . The p ro duct of all the measures for individ ual p oin ts f orm s a measure for the space of p ossible configurations. An edge b et w een t wo op en v ertices is considered op en. All ot her edges are considered closed. A co mp onent in which op en v ertices are connected throu gh p aths of op en edges is k n o wn as an op en cluster. It is kno wn that there is a v alue p c suc h th at for all p > p c the graph obtained h as an infin ite op en cluster. This v alue is known as the cr itical probabilit y . When p > p c then eac h p oint of Z d has some non-zero probabilit y of b eing part of an infinite cluster. The r eader is referr ed to [12] for a f ull treatmen t of p ercolation and to [7] for a recen t up date on some new directions in this area. 2 2 An infinite s u bset of C ∞ has constan t metric distortion The graph distance b etw een p airs of p oin ts in a k -nearest neighb or graph is clearly at least the Eu clidean distance b et ween them. The qu estion arises if the distance is arbitrarily larger th an the Euclidean. Clearly , for p oin ts in differen t clusters the distance this question makes no sense. W e also ignore f or no w the question of what h ap p ens inside finite clusters, fo cussing for no w on the infin ite cluster in th e sup ercritical p hase of NN ( d, k ). W e conjecture th at it is p ossib le to sh o w that the d istance b et w een any pair of p oin ts in the infinite cluster is only a small factor larger than th e Eu clidean distance b et w een them. In this pap er we prov e a w eak er result: the in fi nite cluster con tains an infinite sub set of p oin ts whose pairwise distances are not distorted by more than a co nstan t factor. In order to do this w e first presen t a construction that allo ws u s to couple NN(2 , k ) with a site p ercolatio n pro cess in Z 2 . This constru ction also imp r o ves the b est kn o wn u pp er b oun d for k c (2). Then we s h o w how to use the algorithm of Angel et. al. [2] for routing on a p er colated mesh to fi n d a short path b et w een a pair of v ertices in NN( d, k ). 2.1 Coupling NN (2 , k ) to site p ercolation in Z 2 Lik e H¨ aggstr¨ om and Meester’s pr oof for th e existence of a cr itical v alue [13] and T eng and Y ao’s pro of for the w eaker of their tw o upp er b ound s on k c (2) [20], w e pro ceed by constructing a coupling w ith a site p ercolation pro cess on Z 2 . Ho wev er, our construction giv es a b etter up p er b ound than T eng and Y ao’s impr ov ement of th eir own result (also in [20]) to k c (2) ≥ 213 whic h u ses a coupling to a mixed p ercolation pro cess. W e are able to imp ro ve this result to sho w k c (2) ≥ 188. Note that b oth pap ers, the one by H¨ aggstr¨ om and Meester and the one by T eng and Y ao, rep orted that simulat ions seemed to indicate that the v alue of k c (2) app ears to b e aroun d 3. Our s im u lations bac ked up this fin ding. Let us no w p ro ceed to a formal statemen t of the main theorem of this section and it’s pr o of. Theorem 2.1 F or the k -ne ar est neighb our mo del in a Poisson p oint pr o c ess setting k c (2) ≤ 188 . C x C b C l E l C 0 E b z E r C r C t E t x C z Figure 1: A tile t and it’s 9 relev ant regions. Note that the region E r lies wholly within all discs of the form C x and C z cen tr ed at p oin ts on the b oundary of the discs C 0 and C r . Pro of. In order to prov e the theorem w e couple a site p ercolation pro cess on Z 2 with th e k -nearest neigh b our graph as follo ws. W e d ivide R 2 in to square tiles of side 10 a where a is a parameter whose 3 v alue will b e fixed later. W e create a bijection, φ , b et w een these tiles in R 2 and p oints in Z 2 suc h that neigh b ouring tiles in R 2 corresp ond to neigh b ouring p oin ts in Z 2 . W e couple the pro cesses b y sa ying that a giv en p oint x in Z 2 is op en only if the tile t = φ − 1 ( x ) a certain even t A t o ccurs. W e no w d efine this ev ent A t . Let u s lo ok at a tile cen tred at (0 , 0) with b ottom left co rner ( − 5 a, − 5 a ) and top righ t corner (5 a, 5 a ). F or conv enience w e will refer to the tiles surrou n ding the tile t as, couuntercloc kwise starting from the righ t t r , t t , t l and t b . W e consider five circles of r adius a : C 0 cen tr ed at (0 , 0), C l cen tr ed at ( − 4 a, 0), C r cen tr ed at (4 a, 0), C t cen tr ed at (0 , 4 a ) and C b cen tr ed at (0 , − 4 a ). There are four other region which are named E l , E r , E t and E b in the figur e. E r is d efi ned as follo ws. Consider the largest circle centred at any p oint in C 0 or C r that lies wholly within th e tw o tiles t and t r . Two su c h circles, C x and C z , are depicted in Figure 1. E r is the lo cus of the p oint s con tained in all such circles. The regions E l , E t and E b are defin ed similary by C 0 alongwith C l , C t and C b resp ectiv ely and th e tiles t l , t t and t b resp ectiv ely . No w, for a tile t , the even t A t is said to o ccur if 1. the n u m b er of p oint s inside t is at most k / 2 and 2. the nine r egions C 0 , C r , C t , C l , C b , E r , E t , E l and E b con tain at least one p oint eac h. If A t o ccurs we call the p oint con tained in C 0 the r epr e sentative p oint of the tile t , den oted rep( t ). In order to relate th e pro cess on Z 2 defined via th ese ev en ts A t to the NN( d, k ) mo del, we claim that the existence of an edge in Z 2 implies the existence of a path from the representati v e p oin ts of the tw o tiles corresp ond ing to the tw o end p oin ts of the edge. W e state this formally , including an observ ation ab out th e metric distortion of th e length of th e path b et ween the t w o representa tiv e p oint s. Claim 2.2 If an e dge exists in the p er c olate d mesh Z 2 b etwe en two p oints x and y then 1. Ther e is a p ath b etwe en the r epr esentative p oints r ep ( φ − 1 ( x )) and r ep ( φ − 1 ( y )) of the tiles c orr e- sp onding to x and y in NN (2 , k ) and 2. ther e is a c onstant c tiles such that d k ( r ep ( φ − 1 ( x )) , r ep ( φ − 1 ( y ))) ≤ c · d ( r ep ( φ − 1 ( x )) , r ep ( φ − 1 ( y ))) . Figure 2: A p ath b etw een tw o r epresen tativ e p oints of tiles for b oth of whic h the eve n t A t has o ccured. Pro of of Claim 2.2: The pro of of the claim is depicted in Figure 2 Clearly any circle dra wn from rep( t ) th at sta ys within t con tains all of E r in it b y the defin ition of E r . Since there are at most k / 2 4 p oin ts in ev er y tile for whic h A t has o ccured, hen ce there is an edge from rep( t ) to the p oin t guarant eed to b e con tained in E r , let’s call it x r , b y the definition of A t . W e do not make any claims on where th e edges established b y x r to its neighbou r s lie, observin g only that an y p oin t that lies in C r m ust ha ve an edge to x r , again by the d efi nition of E r . Ho wev er, an y disc cen tred at a p oin t in C r that remains within t and t r m ust con tain the left d isc of its n eigh b oring tile. Hence, if A t and A t r o ccur then a path f rom rep( t ) to rep( t r ) occur s. The second part of the claim is obvio usly true. T he constant can easily b e calculated using calculus. ⊓ ⊔ F rom Claim 2. 2, it is easy to dedu ce that if an infinite comp on ent exists in the site p ercolati on setting, then an infinte comp onent exists in NN(2 , k ). Hence w e need to determine for what settings of our p arameters a and, more imp ortant ly , k , the site p ercolation pro cess is sup ercritical. The critical probabilit y for site p ercolation is 0.59 (see e.g. [15 ]). Numerical calculations sh o wed that the smallest v alue of k for whic h the probability of A t exceeds this v alue is 188, and the v alue of a for w hic h th is happ ens is 0.893. ⊓ ⊔ 2.2 A subset with constant metric distortion W e no w sh o w that th er e is a set of p oints in C ∞ and constan t α suc h that for eac h pair of p oints x, y in this s et D k ( x, y ) ≤ α · D ( x, y ) . W e will pr ov e the follo wing theorem Theorem 2.3 F or NN (2 , k ) wh er e k > 188 , ther e i s a set of p oints S ⊆ C ∞ such that | S | = ∞ with the fol lowing pr op erty: L et x, y ∈ S b e two p oints with Euclide an distanc e D ( x, y ) b etwe e n them whose k -NN distanc e is D k ( x, y ) . F or some α, c dep ending only on k P ( D k ( x, y ) > α · D ( x, y )) < e − c · D ( x,y ) . Pro of. W e identi fy S to b e the set of represen tativ e p oints lying in the in finite cluster of NN(2 , k ) of th e construction describ ed in the pr o of of Theorem 2 .1 as the sub set that w e will claim has this prop erty . W e u s e th e coup ling w ith site p ercolation in Z 2 in tro duced in that pro of to help us fin d s hort paths b et w een pairs of p oin ts in S . Let us consid er an y t w o tiles t 1 and t 2 whose repr esentati v e p oin ts rep( t 1 ) and r ep( t 2 ) lie in C ∞ . W e den ote d istance b et ween tw o p oin ts a, b in Z 2 is denoted D latt ( a, b ). First w e relate the distance in the (u np ercolated) lattice to the euclidean d istance b et w een these t wo p oin ts b y observing a simple fact. F act 2.4 Given that c tiles is the c onstant define d in Claim 2.2 then for two tiles t 1 , t 2 D latt ( φ ( r ep ( t 1 )) , φ ( r ep ( t 2 ))) ≤ √ 2 · D ( r ep ( t 1 ) , r ep ( t 2 )) c tiles . When the lattic e undergo es p ercolation, the simple op en path from φ (rep( t 1 )) to φ (rep( t 2 )) ma y b e brok en at several p oin ts. Anta l and Pisztora studied this setting and pro v ed a p o werful theorem whic h helps us here [3, Theorems 1.1 and 1.2]. W e u se it as a lemma h ere, adopting the restatemen t of Angel et. al. [2, Lemma 8]. Lemma 2.5 [3, 2 ] F or any p > p c and any x, y c onne cte d thr ough an op en p ath in a cub e M d of the infinite lattic e, let D p latt ( x, y ) b e the distanc e b etwe en the two p oints in the p er c olate d lattic e. F or some ρ, c 2 > 0 dep ending only on the dimension and p and for any a > ρ · D latt ( x, y ) pr ( D p latt ( x, y ) > a )) < e − c 2 a . 5 Figure 3: The path b etw een t w o representa tiv e p oin ts mimics th e path in Z 2 . T o fi nd a path b et w een rep( t 1 ) and rep( t 2 ) w e simply tak e the path in the p ercolated lattice b et w een φ ( t 1 ) and φ ( t 2 ) and mimic it R 2 as depicted in Figure 3 and the result f ollo ws by combining F act 2.4 and Lemma 2.5. ⊓ ⊔ W e note here that our claim that the constan t in the statemen t of Theorem 2.3 d ep ends only on the v alue of k follo ws from the fact that the constan ts in Lemma 2.5 dep end only on p , sin ce in our construction the s ize of a tile and the probabilit y of A t o ccuring for a tile c h anges when we change k . W e also note that An tal and Pisztora [3] pr o ve their theorem for b ond p ercolation but note that their metho ds can easily b e extended to site p ercolation. It is p ossible to extend Theorem 2.3 easily for d > 2. The constant s c h an ge and their dep endence on d has to b e hand led carefully bu t the pro of remains basically the same. 3 Applications to m ulti-hop wireless sensor net w orks. Multi-hop sensor net w orks, wh ere n o d es act not on ly to sense but also to rela y information, ha v e prov en adv an tages in terms of energy efficiency ov er single hop s ensor netw orks [14] and are useful necessary tasks lik e time synchronizatio n [11]. And for collab orativ e tasks like target trac king [23] sensor-to- sensor comm u nication is essentia l. But the total connectivit y sough t to b e ac h iev ed in [22, 5] b etw een all the p oin ts of a p oin t p ro cess is n ot necessary for these n et works. It m a y b e the correct mo del for general ad ho c w ireless net works where all n o d es need to b e connected, bu t for a sensor net w ork we argue the p resence of large connected comp onen t is enough. Sensor net w orks seek to ac hiev e cov erage of a target area. When th e lo cations of sensors are mo delled by p oint pro cesses achievi ng most co v erage measures (whether it is single p oin t co verage or k -co ve rage or barrier co verag e) has found that there is a critical densit y of t he point pro cess a b o v e whic h the particular measure is satisfactory . F or example [4] estimates the critical density r equired for barrier cov erage in str ip -lik e regions, a n otion of co v er age where an ob j ect must b e sensed if it tr ies to 6 n k a vg. max. v alue p ercen tage 500 3 1.727 15.180 96 . 96% 500 4 1.364 7.543 97 . 96% 500 5 1.204 5.874 99 . 38% 1000 3 1.660 22 .64 97 . 08% 1000 4 1.333 8.39 98 . 92% 1000 5 1.172 4. 385 99 . 82% 1500 4 1.322 7. 858 99 . 12% 2000 4 1.285 9. 512 99 . 4% T able 1: Metric distortion in NN(2 , k ). The last column sho ws th e p ercen tage of pairs d istorted by a factor of 2 or less. cross a p articular region. 2 Our results sho w that it is p ossib le to fin d an infin ite comp onen t with wh ic h has an infinite su bset of no des wh ose graph d istance is a constan t times their euclidean distance. O ur constr u ction for the pro ofs of T heorems 2.1 and 2.3, tak en along with the f act that for an y p oint in Z 2 there is a non- zero probability of b eing part of the infinite comp onent in the sup ercritical p hase imply the f ollo wing theorem Theorem 3.1 F or any λ , ther e is a λ ′ such that NN (2 , k ) bu i lt on a p oint p r o c ess of density λ ′ with k > 188 , has an infinite c omp onent with the pr op erty th at an infinite subset of p oints with d ensity at le ast λ has the pr op erty that that gr aph distanc e b etwe en them is at most a c onstant times the Euclide an distanc e b etwe e n them. Mor e over ther e is a c onstant c such that λ ′ < cλ . Clearly the existence of suc h a subset can fulfil th e sen sing requiremen t while not compr omising on the sensor-to-sensor data transfer capabilit y . The v alue 188 seems pr ohibitiv e for most pr actica l purp oses. But it is our hop e that this upp er b ound will b e impr o ved down to a reasonable v alue closer to the 2 conjectured by H¨ aggstr¨ om and Meester [13] and T en g and Y ao [20] and that it w ill b e p ossib le to pro v e Th eorem 2.3 for this imp ro ved b ound as well . W e omit the pr o of of this theorem here since it do es not add any n ew insight ov er th e pro ofs already seen in this pap er. 4 Conclusion W e conclud e by presen ting some conjectures ab out the relationship of the metric d istortion in NN(2 , k ) to the parameter k . These conjectures come fr om simulatio ns we ran. The exp eriments had to be carried out on a fin ite b o x i n R 2 , b ut to negate b oundary effects we sim ulated a p oint pro cess in a large b o x and lo ok ed at th e largest comp onent formed within a smaller b o x cont ained w ell with in this finite b o x . W e placed a num b er of p oin ts rand omly within the larger area (thereby ac hieving a target density). In T able 4 th e fir st column has the num b er of p oints placed. The table s h o ws the a v erage distortion for differen t v alues of k , m aximum v alue of the distortion and the p ercen tage of p oin ts having distortion less than t wo times the a v erage. This table also indicates that there the distortion is indep endent of the num b er of p oin ts u nder consideration but dep end s on the v alue of k . T o show relationship b etw een k and av erage distortion w e plotted a v erage r atio w ith k 2 for a r ange of v alue of k fr om 3 to 13 f or t w o rand om p oin t sets. Figures 4 and 4 sh o w plots for t wo such sets 2 See [14, Chap 13.2] for a succinct summary of the issues inv olv ed in co vera ge. 7 1 1.2 1.4 1.6 1.8 2 0 20 40 60 80 100 120 140 160 Average Distortion k 2 Figure 4: Ave rage metric distortion on the y -axis and k 2 on the x axi s. The curv e plot ted is 1 + 4.62/ k 2 . 1 1.2 1.4 1.6 1.8 2 0 20 40 60 80 100 120 140 160 Average Distortion k 2 Figure 5: Ave rage metric distortion on the y -axis and k 2 on the x axi s. The curv e plot ted is 1 + 5.03/ k 2 . 8 along with a function f ( k ) = 1 + a/k 2 where a is determined b y least square fitting fun ctions. These findings lead us to conjecture that: Conjecture 4.1 F or the NN (2 , k ) mo del at a v alue k > k c (2) 1. The metric distortion of the p oints of C ∞ is at most 2 with pr ob ability tending to 1 and 2. ther e is a c onstant such that th e exp e cte d metric distortion of the p oints of C ∞ is of the form 1 + a k 2 . References [1] M. Adler and M. Mitzenmac h er. 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