On Geometric Spanners of Euclidean and Unit Disk Graphs

Symposium on Theoretical Aspects of Computer Science 2008 (Bordeaux), pp. 409-420 www .stacs-conf .org ON GEOMETRIC SP ANNERS O F EUCLIDEAN AND UNIT DISK GRAPHS IY AD A. KANJ AND LJUBOMIR PERK OVI ´ C DePa ul Un ivers ity , Chicago, IL 60604, USA. E-mail addr ess : {ikanj, lperkovic}@ cs.depaul.edu Abstra ct. W e consider the problem of constructing bound ed-degree planar geometric spanners of Euclidean and un it-disk gra p h s. It is well kn own that the Delauna y subgraph is a planar geometric spann er with stretch factor C del ≈ 2 . 42; how ever, its degree may not be bou n ded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretc h factor ρ = 1 + 2 π ( k cos π k ) − 1 and d egree b ounded b y k , for an y integ er parameter k ≥ 14. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean graph with stretch factor ρ · C del and degree bounded by k , for any integer parameter k ≥ 14. Moreov er, the resulting spanner contains a Euclidean Minimum Span n ing T ree (EMST) as a subgraph . Our second contribution lies in d evelo pin g the structural results n ecessary to transfer our analysis and algorithm from Euclidean graphs t o unit disk graphs, the usual mo del for wireless ad-ho c net works. W e obtain a v ery simple distributed, strictly-lo c alize d algori th m that, given a unit d isk graph em b edd ed in the p lane, constru cts a geometric spanner with the ab o ve stretch factor and degree b oun d, and also containing an EMST as a subgraph. The obtained results dramatically impro ve the previous results in all asp ects, as sho wn in the paper. In tro duction Giv en a set of p oints P in the plane, the Eu clidean graph E on P is defined to b e the complete graph whose v ertex-set is P . Eac h edge AB connecting p oin ts A and B is assumed to b e em b edded in the plane as the straigh t line segmen t AB ; w e defin e its cost to b e the E uclidean distance | AB | . W e define the un it disk graph U to b e the sub graph of E consisting of all edges AB with | AB | ≤ 1. Let G b e a subgrap h of E . The cost of a simple path A = M 0 , M 1 , ..., M r = B in G is P r − 1 j =0 | M j M j +1 | . Among all paths b et w een A and B in G , a path with th e sm allest cost is defined to b e a smal lest c ost p a th and we denote its cost as c G ( A, B ). A spanning su bgraph H of G is said to b e a ge ometric sp a nner of G if there is a constan t ρ such that for ev ery 1998 ACM Subje ct Classific ation: C.1.4, F.2.2, G.2. 2. Key wor ds and phr ases: Geometric spanner, euclidean graph, unit disk graph, wirel ess ad- ho c netw orks. The work of the first author w as sup p orted in part by a DePaul Universit y Comp etitive R esearc h Gran t. c  I. Kanj and L. P er kovi ´ c CC  Creative Comm on s Attribution-NoDer ivs License 410 I. KAN J AND L. PERKO VI ´ C t wo p oin ts A, B ∈ G we ha ve: c H ( A, B ) ≤ ρ · c G ( A, B ). The constant ρ is called the str etch factor of H (with r esp ect to the un derlying graph G ). The pr oblem of constructing geometric spanners of Euclidean graphs has recen tly re- ceiv ed a lot of atten tion due to its app licatio n s in computational geo metry , wireless com- puting, and computer graph ics (see, for example, the recen t b o ok [13] for a sur v ey on geometric spanners and their applications in net works). Dobkin et al. [9] sh o w ed that th e Delauna y graph is a p lanar geometric spanner of the Eu clidean graph with stretc h factor (1 + √ 5) π / 2 ≈ 5 . 08 . This ratio was impro ve d by Keil et al [10] to C del = 2 π/ (3 cos ( π / 6)) ≈ 2 . 42, whic h currently stand s as the b est up p er b ound on the stretc h factor of the Delauna y graph. Man y researc hers b elie ve, ho wev er, th at the lo wer b ound of π / 2 sho wn in [7] is also an upp er b ound on the stretc h factor of the Delauna y graph. While Delaunay graphs are go o d planar geometric spanner s of Eu clidean graphs, th ey ma y h a v e u nboun d ed degree. Other geometric (spars e) spanner s w ere also prop osed in th e literature in clud ing the Y ao graphs [16], the Θ-graphs [10 ], and man y others (see [13]). Ho w eve r , most of these prop osed spanners either d o not guarante e planarit y , or do not guaran tee b ounded degree. Bose et al. [2, 3] w ere the fir s t to show h o w to extract a subgraph of the Delauna y graph that is a planar geometric spanner of the Euclidean graph with stretc h f actor ≈ 10 . 02 and d egree b ound ed by 27. In the con text of unit disk grap h s, Li et al. [11, 12] ga ve a distributed algorithm that constru cts a planar geometric s p anner of a unit disk graph with stretc h factor C del ; how ever, the spanner constructed can ha v e u n b ounded degree. W ang and Li [14, 15] then sh o w ed ho w to co n struct a b ounded-degree planar sp anner of a u nit disk graph with stretc h factor max { π / 2 , 1 + π sin ( α/ 2) } · C del and degree b ounded by 19 + 2 π/α , where 0 < α < 2 π / 3 is a parameter. V ery recen tly , Bose et. al [5] impro ved the earlier result in [2, 3] and show ed ho w to construct a subgraph of the Delauna y graph that is a geometric spann er of the Euclidean graph with stretc h factor: max { π / 2 , 1 + π sin ( α/ 2) } · C del if α < π / 2 and (1 + 2 √ 3 + 3 π / 2 + π sin ( π / 1 2) ) · C del when π / 2 ≤ α ≤ 2 π/ 3, and wh ose d egree is b ounded by 14 + 2 π/α . Bose et al. then applied their construction to obtain a p lanar geometric spanner of a u nit d isk graph with stretc h factor m ax { π / 2 , 1 + π sin ( α/ 2) } · C del and d egree b ounded by 14 + 2 π /α , f or any 0 < α ≤ π / 3. This w as the b est b ound on th e stretc h factor and the degree. W e h a v e t wo new results in this pap er. W e deve lop structural results ab out Delauna y graphs that allo w us to p resen t a very simple lin ear-time algorithm that, give n a Delauna y graph, constructs a subgraph of th e Delauna y graph with stretc h factor 1 + 2 π ( k cos ( π /k )) − 1 (with resp ec t to the Delaunay graph) and degree at most k , for any inte ger parameter k ≥ 14. This result immediately implies an O ( n lg n ) algorithm for co n s tructing a planar geometric spann er of a Eu clidean graph with stretc h factor of (1 + 2 π ( k cos ( π /k )) − 1 ) · C del and degree at most k , for any int eger p arameter k ≥ 14 ( n is the n u m b er of v ertices in the graph). W e then trans late our w ork to unit disk graphs an d present our second result: a v ery simple and strictly-lo c alize d distributed algorithm that, give n a unit-disk graph em b edded in the plane, constructs a planar geometric spann er of the unit disk grap h with stretch factor (1 + 2 π ( k cos ( π /k )) − 1 ) · C del and d egree b ounded by k , for any inte ger parameter k ≥ 14. This efficien t d istributed algorithm exc hanges no more than O ( n ) messages in total, and runs in O (∆ lg ∆) lo cal time at a n o de of degree ∆. W e sh o w that b oth sp anners include a Euclidean Minimum Spanning T ree as a sub graph. Both algorithms significantly improv e pr evious results (describ ed ab o ve) in terms of the stretc h facto r and the d egree b ound. T o sh o w this, w e compare our results with p revious results in more d etail. F or a d egree b ound k = 14, our r esult on Euclidean graphs imply ON GEOM E TR IC SP ANNERS O F EUCLIDEAN A ND UN IT DISK GRAPH S 411 a b oun d of at most 3 . 54 on the stretc h facto r . As the degree b ound k approac hes ∞ , our b ound on the str etch f actor appr oac hes C del ≈ 2 . 42. The very recent results of Bose et al. [5] ac hieve a lo we st d egree b ound of 17, and th at corresp ond s to a b ound on the stretc h factor of at least 23. If Bose et al. [5] allo w the degree b ound to b e arbitrarily large (i.e., to approac h ∞ ), their b oun d on the stretc h f actor appr oac hes ( π / 2) · C del > 3 . 75 . Our stretc h factor and degree b ounds for un it disk graphs are the same as our resu lts for Euclidean graphs. The smallest degree b ound deriv ed b y Bose et al. [5] is 20, and that corresp ond s to a stretch factor of at least 6.19. If Bose et al. [5 ] allo w the degree b oun d to b e arb itrarily large, th en their b ound on the stretch factor app roac hes ( π / 2) · C del > 3 . 75. On the other hand, the smallest d egree b ound derived in W ang et al. [14, 15] is 25, and that corresp on d s to a b ound of 6.19 on th e stretc h factor. If W ang et al. [14, 15] allo w the d egree b oun d to b e arbitrarily large, th en their b ound on the s tr etc h factor approac hes ( π / 2 ) · C del > 3 . 75. Therefore, ev en th e w orst b ound of at most 3.54 on the stretc h f actor corresp ond ing to our lo w est b ound on the degree k = 14, b ea ts the b est b oun d on the s tr etc h factor of at least 3.75 corresp onding to arbitrarily large degree in b oth Bo se et al. [5] and W ang et al. [14, 15]! 1. Definitions and Background W e start with the follo wing w ell known observ ation: Observ ation 1.1. A su bgraph H of graph G has stretc h factor ρ if and only if for ev ery edge X Y ∈ G : the length of a shortest path in H from X to Y is at most ρ · | X Y | . F or three non-collinear p oints X , Y , Z in the plane w e denote by  X Y Z the circum- scrib ed circle of triangle △ X Y Z . A Delaunay triangulation of a set of p oin ts P in the plane is a triangulation of P in wh ich the circumscrib ed circle of ev ery triangle conta ins no p oin t of P in its in terior. It is w ell kno wn th at if the p oints in P are in gener al p osition (i.e., no four p oints in P are co circular) th en th e Delauna y triangulation of P is u nique [8]. In this pap er—as in most pap ers in th e literature—we shall assu me that the p oin ts in P are in general p ositio n; otherwise, the inp ut can b e s ligh tly p ertu r b ed so that this condition is satisfied. Th e Delaunay gr aph of P is d efined as the plane graph w hose p oint-se t is P and whose edges are the edges of the Delaunay triangulation of P . An alte r n ativ e definition that we end up using is: Definition 1.2. An edge X Y is in the Delauna y graph of P if and only if there exists a circle through p oin ts X and Y whose in terior con tains n o p oin t in P . It is well kno wn that the Delauna y graph of a set of p oin ts P is a spannin g subgraph of the E u clidean grap h defined on P (i.e., the complete graph on p oin t-set P ) whose stretc h factor is b ounded b y C del = 4 √ 3 π / 9 ≈ 2 . 4 2 [10]. Giv en integ er parameter k > 6, the Y ao sub gr aph [16] of a plane graph G is constru cted b y p erforming the follo win g Y ao step at ev ery p oin t M of G : place k equally-separated rays out of M (arbitrarily defin ed), thus creating k closed cones of size 2 π /k eac h, and c ho ose the s h ortest edge in G out of M (if an y) in eac h cone. The Y ao subraph consists of edges in G c hosen by either endp oin t. Note that the degree of a p oint in the Y ao sub grap h of G ma y b e un b ounded. Tw o edges M X , M Y incident on a p oint M in a graph G are said to b e c onse cutive if one of the angular sectors determined by M X and M Y con tains n o neigh b ors of M . 412 I. KAN J AND L. PERKO VI ´ C 2. Bounded Degree Spanners of Delauna y Gr aphs Let P b e a set of p oints in the plane and let E b e the complete, Euclidean graph defined on p oint- set P . Let G b e the Delauna y graph of P . This section is dev oted to proving the follo wing theorem: Theorem 2.1. F or every inte ger k ≥ 14 , ther e exists a sub gr aph G ′ of G such that G ′ has maximum de gr e e k and str etch factor 1 + 2 π ( k cos π k ) − 1 . A linear time algorithm that compu tes G ′ from G is the k ey comp onent of our pro of. This v ery simple algorithm essentia lly p erf orms a mo difie d Y ao step (see Section 2.3) and selects up to k edges out of ev ery p oi nt of G . G ′ is simp ly the sp anning subgraph of G consisting of ed ges chosen by b oth endp oin ts. In order to d escrib e the m o dified Y ao step, w e m ust fir st dev elop a b etter understanding of the structur e of the Delauna y graph G . Let C A and C B b e edges in ciden t on p oin t C in G such that ∠ B C A ≤ 2 π /k and C A is th e shortest edge within the angular sector ∠ B C A . W e will s ho w ho w the ab ov e theorem easily follo ws if, for ev ery such pair of edges C A and C B : 1. we show that there exists a p ath p from A to B in G of length | p | , su c h that: | C A | + | p | ≤ (1 + 2 π ( k cos π k ) − 1 ) | C B | , and 2. we mo dify the standard Y ao step to include th e edges of this p ath in G ′ , in addition to including the edges pic k ed by the standard Y ao step but w ith ou t increasing the n umb er of edges c hosen at eac h p oin t b ey ond k . This will ensu re that: for any edge C B ∈ G that is not included in G ′ b y th e mo d ified Y ao step, there is a path from C to B in G ′ , wh ose edges are all includ ed in G ′ b y th e mo dified Y ao step, and whose cost is at most (1 + 2 π ( k cos π k ) − 1 ) | C B | . In the lemma b elo w, w e pr o v e the existence of this p ath and sho w some p rop erties satisfied by edges of this path; we will then m o dify the stand ard Y ao s tep to include edges satisfying these p rop erties. Lemma 2.2. L et k ≥ 14 b e an inte ger, and let C A and C B b e e dges in G such that ∠ B C A ≤ 2 π /k and C A is the short est e dge in the angular se ctor ∠ B C A . Ther e exists a p ath p : A = M 0 , M 1 , ..., M r = B in G such that: (i) | C A | + P r − 1 i =0 | M i M i +1 | ≤ (1 + 2 π ( k cos π k ) − 1 ) | C B | . (ii) Ther e is no e dge in G b etwe en any p air M i and M j lying in the close d r e gion delimite d by C A , C B and the e dges of p , for any i and j satisfying 0 ≤ i < j − 1 ≤ r . (iii) ∠ M i − 1 M i M i +1 > ( k − 2 k ) π , f or i = 1 , · · · , r − 1 . (iv) ∠ C AM 1 ≥ π 2 − π k . W e b r eak do wn th e pro of of the ab ov e lemma into t w o cases: when △ AB C con tains no p oint of G in its in terior, and when there are p oin ts of G in side △ AB C . W e define some additional notation and terminology fir st. W e defin e th e circle ( O ) =  AB C with center O , and set Θ = ∠ B C A . Note that ∠ AO B = 2Θ ≤ 4 π /k . W e will use ⌢ AB to d enote the arc of ( O ) determined by p oints A and B and f acing ∠ AO B . W e w ill make use of the follo wing easily v erified Delauna y graph p rop erty: Prop osition 2.3. If C A and C B ar e e dges of G then the r e gion inside ( O ) subtende d by chor d C A and away fr om B and the r e g ion inside ( O ) subtende d by chor d C B and away fr om A c ontain no p oints. ON GEOM E TR IC SP ANNERS O F EUCLIDEAN A ND UN IT DISK GRAPH S 413 2.1. The Outw a rd P ath W e consider fir st the case when no p oin ts of G are inside △ AB C . S in ce b oth C A and C B are edges in G and by Prop osition 2.3 , th e region of ( O ) subtended by c h ord AB closer to C has no p oints of G in its interior. Keil an d Gutwin [10] sho we d that, in this case, there exists a path b et w een A and B in G in side the region of ( O ) su btended by chord AB a wa y from C , whose length is b ounded by the length of ⌢ AB (see Lemma 1 in [10]). W e find it con v enient to use a recursive definition of their p ath (for more details, we refer the reader to [10]): 1. Base case: If AB ∈ G , the path consists of edge AB . 2. Recursiv e step: Otherwise, a p oin t must reside in the region of ( O ) subtended b y c hord AB and a wa y from C . Let T b e such a p oin t with th e p rop erty that the region of  AT B subtend ed by c hord AB closer to T is empty . W e call T an interme diate p oint with resp ect to the pair of p oin ts ( A, B ). Let ( O 1 ) b e the circle passing through A and T whose cen ter O 1 lies on segmen t AO and let ( O 2 ) b e the circle passing through B and T whose cen ter O 2 lies on segmen t B O . Then b oth ( O 1 ) and ( O 2 ) lie inside ( O ), and ∠ AO 1 T and ∠ T O 2 B are b oth less than ∠ AO B ≤ 4 π /k . Moreo v er, the region of ( O 1 ) subtended by c hord AT that con tains O 1 is empt y , and the r egion of ( O 2 ) subtend ed b y c h ord B T and con taining O 2 is empty . Th er efore, w e can r ecur siv ely constru ct a path from A to T and a path from T to B , and then concatenate them to obtain a path f rom A to B . Definition 2.4. W e call the path constructed ab o ve the outwar d p ath b et ween A and B . Keil and Gut win [10], from this p oin t on, use a purely geometric argumen t (with no use of Delaunay graph pr op erties) to s ho w that the length of the obtained path A = M 0 , M 1 , · · · , M r = B (where eac h p oin t M p , f or p = 1 , · · · , r − 1, is an in termediate p oin t with resp ect to a pair ( M i , M j ), wh ere 0 ≤ i < p < j ≤ r ) is smaller than the length of ⌢ AB . Figure 1 illustr ates an out ward p ath b et ween A and B . C B = M 3 A = M 0 M 1 M 2 ≤ 2 π /k Figure 1: Illustration of an ou tw ard path. Prop osition 2.5. In e v ery r e cursive step of the outwar d p ath c onstruction describ e d ab ove, if M p is an interme diate p oint with r esp e ct to a p air of p oints ( M i , M j ) , then: (a) ther e is a cir cle p assing thr ough C and M p that c ontains no p oint of G , and (b) cir cles  C M i M p and  C M j M p c ontain no p oints of G exc ept, p ossibly, in the r e gion subtende d by chor ds M i M p and M p M j , r esp e ctively, away fr om C . 414 I. KAN J AND L. PERKO VI ´ C Pr o of. W e assume, by induction, that there are circles ( O M i ) and ( O M j ) passing through C and M i , and C and M j , resp ectiv ely , con taining n o p oin ts of G , and that the circle ( O ) =  C M i M j con tains n o p oin t of G in the inte rior of the region R ′ subtended by c hord M i M j closer to C . (This is certainly true in the b ase case b ec ause C A, C B ∈ G , by Prop osition 2.3, and by our initial assumptions). Since M i M j is not an edge in G , the p oin t M p c hosen in the construction is th e p oin t with the p rop erty that th e region R of  M i M p M j subtended by chord M i M j a w ay from C , con tains no p oin t of G . Then the circle passing th rough C and M p and tangen t to  M i M p M j at M p is completely inside ( O M i ) ∪ ( O M j ) ∪ R ∪ R ′ , and therefore dev oid of p oints of G . This p ro ve s part (a). The region of  C M i M p subtended by c h ord M i M p and conta ining C is insid e ( O M i ) ∪ R ∪ R ′ , and therefore conta ins no p oin t of G in its interior. The same is tru e for the region of  C M j M p subtended b y c hord M j M p and con taining C , and part (b) holds as well. W e are now r eady to pro ve Lemma 2.2 in the case wh en no p oin t of G lies inside △ AB C . In this case we define the path in Lemma 2.2 to b e the out w ard path b et w een A and B . Pr o of of L emma 2.2 for the c ase of outwar d p ath. ( i ) With Θ = ∠ B C A , we ha ve | ⌢ AB | = 2Θ · | O A | and sin Θ = | AB | / (2 | O A | ) . W e note that | C A | + | ⌢ AB | is largest when | C A | = | C B | , i.e. when C A and C B are symmetrical w ith resp ect to the diameter of  C AB p assin g through C ; this follo ws from th e fact that th e p erimeter of a conv ex b o dy is not smaller than the p erimeter of a conv ex b od y con taining it (see p age 42 in [1 ]). If | C A | = | C B | , s in Θ 2 = | AB | 2 | C B | . Using element ary trigonometry , it follo ws from the ab o ve facts an d f rom | C A | ≤ | C B | that: | C A | + | ⌢ AB | ≤ | C B | + 2Θ · | O A | = | C B | + ( Θ sin Θ ) · | AB | = | C B | + ( Θ cos Θ 2 ) · | C B | ≤ (1 + 2 π ( k cos π k ) − 1 ) | C B | . The last inequalit y follo ws from Θ ≤ 2 π/k and k > 2. ( ii ) If M i M j w as an ed ge in G th en , for ev ery p b etw een i and j , the circle  M i M p M j w ould not con tain C . This, h ow ev er, cont r adicts part (a) of Prop ositio n 2.5. ( iii ) If the outw ard path con tains a single in termediate p oint M 1 , then since M 1 lies inside ( O ) =  C AB , ∠ AM 1 B ≥ π − ∠ AO B / 2 ≥ π − 2 π /k = ( k − 2) π /k (note that ∠ AO B = 2 · ∠ AC B ), as desired. No w the statement follo ws by indu ction on the n umb er of steps tak en to constru ct the out w ard path b et w een A and B , using the fact (pro v ed in [10]) th at ea ch angle ∠ M i − 1 O i M i +1 at the cent er of the circle ( O i ) defining the intermediate p oin t M i , is b ounded b y ∠ AO B . ( iv ) Th is follo ws from the fact that ∠ C AM 1 ≥ ∠ C AB ≥ π / 2 − π /k . The last inequalit y is true b ecause | C A | ≤ | C B | and ∠ B C A ≤ 2 π/k in △ C AB . 2.2. The In ward P ath W e consider now the case when th e interior of △ AB C conta ins p oin ts of G . Let S b e the s et of p oint s consisting of p oi nts A and B p lus all the p oin ts in terior to △ AB C (n ote ON GEOM E TR IC SP ANNERS O F EUCLIDEAN A ND UN IT DISK GRAPH S 415 that C / ∈ S ). Let C H ( S ) b e the p oin ts on the con vex h u ll of S . Then C H ( S ) consists of p oints N 0 = A and N s = B , and p oints N 1 , ..., N s − 1 of G in terior to △ AB C . W e ha ve the follo wing pr op osition: Prop osition 2.6. F or every i = 1 , · · · , s − 1 : (a) C N i ∈ G , (b) | C N i | ≤ | C N i +1 | , and (c) ∠ N i − 1 N i N i +1 ≥ π , wher e ∠ N i − 1 N i N i +1 is the angle facing p oint C . Pr o of. These f ollo w follo w from the follo wing facts: C A and C B are edges in G , C A is the shortest edge in its cone, and hence | C A | ≤ | C N i | , for i = 0 , · · · , s , and p oin ts N 0 , · · · , N s are on C H ( S ) in the listed order. Since | C N i | ≤ | C N i +1 | and no p oin t of G lie s inside △ N i C N i +1 ( N i and N i +1 are on C H ( S )), C N i is the sh ortest edge in the angular sector ∠ N i C N i +1 . Since ∠ N i C N i +1 ≤ ∠ B C A ≤ 2 π /k , by Lemma 2.2 th er e exists an outw ard path P i b et ween N i and N i +1 , for ev- ery i = 0 , 1 , · · · , s − 1, satisfying all the prop erties of Lemma 2.2. Let A = M 0 , M 1 , · · · , M r = B b e the concatenat ion of the paths P i , for i = 0 , · · · , r − 1. Definition 2.7. W e call the path A = M 0 , M 1 , · · · , M r = B constructed ab o ve the inwar d p ath b et ween A and B . Figure 2 illustr ates an inw ard path b et w een A and B . C B = N 3 A = N 0 N 1 N 2 ≤ 2 π /k Figure 2: Illustration of an inw ard path. W e no w pr ov e Lemma 2.2 in the case when there are p oi nts of G interior to △ AB C . In this case we define the path in Lemma 2.2 to b e the in ward path b et ween A and B . Pr o of of L emma 2.2 for the c ase of inwar d p ath. ( i ) Define A ′′ to b e a p oin t on the half-line [ C A s u c h that | C A ′′ | = | C B | , and let ( O ′′ ) =  C A ′′ B . Denote b y α ′′ the length of the arc of  C A ′′ B su b tended by c hord A ′′ B and facing ∠ A ′′ C B . F or ev ery i = 0 , 1 , · · · , s − 1, w e d efi ne arc α i to b e the arc of  C N i N i +1 subtended b y c hord N i N i +1 and f acing ∠ N i C N i +1 . F or ev ery i = 0 , 1 , ..., s − 1, w e d efine N ′ i to b e the p oin t on the h alf-line [ C N i suc h that | C N ′ i | = | C N i +1 | , ( O i ) to b e the circle  C N ′ i N i +1 , and α ′ i to b e the arc of ( O i ) subtended b y c hord N ′ i N i +1 and facing ∠ N ′ i C N i +1 . Finally , for ev ery i = 0 , · · · , s − 1, w e define N ′′ i to b e th e p oin t of intersecti on of the half-line [ C N i and circle ( O ′′ ), and α ′′ i to b e the arc of ( O ′′ ) sub tended b y c hord N ′′ i N ′′ i +1 and facing ∠ N ′′ i C N ′′ i +1 . As s ho wn in section 2.1, the length of the outw ard path P i b et ween N i and N i +1 is b ound ed by the length of α i . Since the con ve x b o dy C 1 delimited b y C N i , C N i +1 and α i is con tained insid e th e con v ex b ody C 2 delimited b y C N ′ i , C N i +1 and α ′ i , 416 I. KAN J AND L. PERKO VI ´ C b y [1 ], the p er im eter of C 1 is n ot larger than that of C 2 . Denoting by | P i | the length of path P i , w e get: | P i | ≤ | N i N ′ i | + α ′ i , i = 1 , · · · , s − 1 . (2.1) Since ( O i ) and ( O ′′ ) are concen tric circles (of cen ter C ), and the r ad iu s of ( O i ) is not larger than that of ( O ′′ ), we ha ve α ′ i ≤ α ′′ i , for i = 0 , · · · , s − 1. It follo ws f rom Inequalit y (2.1) that: | P i | ≤ | N i N ′ i | + α ′′ i , i = 1 , · · · , s − 1 . (2.2) Using Inequalities (2.1 ) and (2.2) we get: | C A | + s − 1 X i =0 | P i | ≤ | C A | + s − 1 X i =0 | N i N ′ i | + s − 1 X i =0 α ′′ i . (2.3) Noting that P s − 1 i =0 | N i N ′ i | = | C B | − | C A | , that P r − 1 i =0 α ′′ i = α ′′ , and using the same argumen t as in part ( i ) of Lemma 2.2) completes the pr o of. ( ii ) Since C N p ∈ G f or p = 1 , · · · , s − 1 by part ( a ) of Prop ositi on 2.6, by planarit y of G , if suc h an edge b et w een t wo p oin ts M i and M j exists, th en M i and M j m ust b elong to an out wa rd path b et we en t wo p oints N p and N p +1 of C H ( S ). But this con tradicts part ( ii ) of Lemma 2.2 for the case of the out wa r d path applied to N p and N p +1 . ( iii ) F or eac h i = 0 , · · · , r , either M i = N j ∈ C H ( S ), or M i is an inte rm ediate p oint on the outw ard path b et ween tw o p oin ts N p and N q in C H ( S ). In th e former case ∠ M i − 1 M i M i +1 ≥ ∠ N j − 1 M i N j +1 ≥ π ≥ ( k − 2) π /k f or k ≥ 14 ( N j − 1 and N j are p oints b efore and after M i = N j on C H ( S )), by part (c) of Prop osition 2.6. In the latter case ∠ M i − 1 M i M i +1 ≥ ( k − 2) π /k b y the p ro of of p art ( iii ) of Lemma 2.2 applied to the outw ard path b et ween N p and N q . ( iv ) Th is follo ws from | C A | = | C M 0 | ≤ | C M 1 | and ∠ AC M 1 ≤ ∠ AC B ≤ 2 π/k , in triangle △ C AM 1 . 2.3. The Mo dified Y ao Step W e no w augmen t the Y ao step so edges formin g the paths describ ed in Lemma 2.2 are included in G ′ , in addition to the edges chosen in the standard Y ao step. Lemm a 2.2 says that consecutiv e edges on su c h paths f orm mo derately large angles. The mo dified Y ao step will ens ure that consecutiv e edges f orming large angles are includ ed in G ′ . The algorithm is describ ed in Figure 3. Since the algorithm selects at most k edges incident on an y p oint M and s ince only edges chosen by b oth endp oin ts are included in G ′ , eac h p oint has d egree at most k in G ′ . Before w e complete the pro of of Theorem 2.1, w e sh o w that the runn ing time of the algorithm is linear. Note first that all edges inciden t on p oin t M of degree ∆ can b e mapp ed to the k cones around M in linear time in ∆. Then, the s hortest edge in ev ery cone ca n b e f ound in time O (∆) (step 2. in the algorithm). Since k is a constan t, selec ting the ℓ / 2 edges cloc kwise (or coun terclockwise) from a sequence of el l < k empty cones around M (step 3.1.) can b e done in O (∆) time. Noting that the total num b er of edges in G is linear in th e num b er of v ertices completes the analysis. T o complete the p ro of of Theorem 2.1, all we need to do is sh o w: ON GEOM E TR IC SP ANNERS O F EUCLIDEAN A ND UN IT DISK GRAPH S 417 Algorithm Mo dified Y ao step Input: A Del aunay gr aph G ; inte ger k ≥ 14 Output: A sub gr aph G ′ of G of maximum de gr e e k 1. d efine k disjoin t cones of size 2 π/k around every p oin t M in G ; 2. in every n on-empty cone, select the shortest edge incident on M in this cone; 3. for every maximal sequence of ℓ ≥ 1 consecutive empt y cones: 3.1. if ℓ > 1 then select the first ⌊ ℓ/ 2 ⌋ unselected incident edges on M clo ckwi se from the sequence of empt y cones and the first ⌈ ℓ/ 2 ⌉ unselected edges inciden t on M counterclockwise from the sequence of empt y cones; 3.2. el se (i.e., ℓ = 1) let M X and M Y b e the inciden t edges on M clockwise and countercl o ckwise, respectively , from the empt y cone; if either M X or M Y is selected then sele ct th e other edge (in case it has not b een selected); otherwise select the shorter edge betw een M X and M Y breaking ties arbitrarily; 4. G ′ is the spanning subgraph of G consisting of edges selected by b oth endp oin ts. Figure 3: The mo dified Y ao Step. Lemma 2.8. If e dge C B is not sele cte d by the algorithm, let C A b e the shortest e dge in the c one out of C to which C B b elongs. Then the e dges of the p ath describ e d in L e mma 2.2 ar e include d in G ′ by the algorithm. Pr o of. F or brevity , instead of saying that the algorithm Mo dified Y ao Step selects an edge M X out of a p oin t M , w e will sa y that M selects edge M X . T o get started, it is ob vious that C will select edge C A . By p art ( iv ) of Lemma 2.2 , the angle ∠ C AM 1 ≥ π / 2 − π /k ≥ 6 π /k for k ≥ 14. Therefore, at least t wo empt y cones m u st fall within the sector ∠ C AM 1 determined b y the t wo consecutive edges C A and AM 1 , and edges AC and AM 1 will b oth b e selected by A . Since edge C A is also selected by p oint C , edge AC ∈ G ′ . By p art ( iii ) of Lemma 2.2, for ev ery i = 1 , 2 , · · · , r − 1, the angle ∠ M i − 1 M i M i +1 ≥ ( k − 2) π /k ≥ 10 π /k for k ≥ 12, and h ence at least four cones fall within the angular sector ∠ M i − 1 M i M i +1 . S ince by part (ii) of Lemma 2.2 M i C is the only p ossible ed ge inside the angular sector ∠ M i − 1 M i M i +1 , it is easy to see that r egardless of the p osition of th ese four cones with resp ect to edge M i C , M i ends up selecting all edges M i M i − 1 , M i M i +1 and M i C in steps 2 and/or 3 of the algorithm. Since we show ed ab o ve th at A selects edge AM 1 , this sho ws that all edges M i M i +1 , for i = 0 , · · · , r − 2, are selected b y b oth th eir endp oin ts, and hence m ust b e in G ′ . Moreo ve r , edge M r − 1 M r = M r − 1 B is selected by p oint M r − 1 . W e no w argue that edge B M r − 1 will b e selected by B . First, observe that | B M r − 1 | ≤ | ⌢ AB | < | C B | . Let C D b e the other consecutiv e edge to C B in G (other than C M r − 1 ). Because C do es n ot select B , it follo ws that ∠ M r − 1 C D ≤ 6 π /k . O therwise, since C M r − 1 and C B are in the same cone, t wo empt y cones would fall within the sector ∠ B C D and C w ould select B . S ince C B is an ed ge in G , by the c haracterizatio n of Delaunay edges [8], ∠ C M r − 1 B + ∠ C D B ≤ π . By considering the quadr ilateral C DB M r − 1 , we hav e ∠ M r − 1 C D + ∠ D B M r − 1 ≥ π . T h is, together with the fact that ∠ M r − 1 C D ≤ 6 π /k , imply that ∠ D B M r − 1 ≥ ( k − 6) π /k ≥ 8 π /k , for k ≥ 14. Therefore, ∠ D B M r − 1 con tains at least 418 I. KAN J AND L. PERKO VI ´ C three cones of size 2 π /k out of B . If one of these cones falls within the angular sector ∠ C B M r − 1 then, since | M r − 1 B | < | C B | , B M r − 1 m ust ha ve b een selected out of B . Supp ose no w that ∠ C B M r − 1 con tains n o cone inside and hence ∠ C B M r − 1 < 4 π /k . If one of these three cones within sector ∠ D B M r − 1 con tains edge C B , then the remaining t wo cones must fall within ∠ D B C an d B M r − 1 will get selected out of B when considering the sequence of at least t wo empt y cones con tained within ∠ C B D . Supp ose no w that all three empt y cones fall w ith in ∠ C B D . T hen w e ha v e ∠ C B D ≥ 6 π /k . If ∠ M r − 1 C D ≥ 4 π /k , then since M r − 1 C and C B b elong to the same cone, the sector ∠ B C D m ust con tain an emp t y cone. Because D is exterior to  C B M r − 1 , ∠ C B M r − 1 < 4 π /k , and ∠ M r − 1 C B ≤ 2 π /k , it follo ws th at ∠ C DB < ∠ M r − 1 C B + ∠ C B M r − 1 < 6 π/k < ∠ D B C . Therefore, by considerin g the triangle △ C D B , we n ote that | C B | < | C D | . But then edge C B w ould ha v e b een selected by C in step 3 since the sector ∠ B C D con tains an empt y cone, a con tradiction. It follo ws that ∠ M r − 1 C D ≤ 4 π /k , and therefore ∠ M r − 1 B D ≥ ( k − 4) π/k ≥ 10 π/k for k ≥ 14. This means th at at least four cones are conta in ed inside sector ∠ D B M r − 1 . It is easy to chec k no w that regardless of the placement of the edge B C with resp ect to these cones, edge B M r − 1 is alwa ys selected out of B by the algorithm. This completes the pro of. Corollary 2.9. A Eu clide an Minimum Sp anning T r e e (EM ST) on P is a su b gr aph of G ′ . Pr o of. It is well known that a Delauna y graph ( G ) cont ains a E MS T. If an edge C B is not in G ′ , then, b y Lemma 2.8, a path from C to B is includ ed in G ′ . All edges on this path are no longer than C B , so there is a EMST n ot includ in g C B . Since a Delauna y graph of a Euclidean graph of n p oin ts can b e computed in time O ( n lg n ) [8] and has stretc h factor C del ≈ 2 . 42 , w e hav e the follo wing theorem. Theorem 2.10. Ther e exists an algorithm that, given a set P of n p oints in the plane, c omputes a plane ge ometric sp anner of the Euclide an gr aph on P that c ontains a E MST, has maximum de gr e e k , and has str e tch factor (1 + 2 π ( k cos π k ) − 1 ) · C del , wher e k ≥ 14 is an inte ger. M or e over, the algorithm runs in time O ( n lg n ) . 3. Geometric Spanners of Unit Disk Graphs In this section we generalize our planar geometric spanner algorithm to u n it d isk graphs . Unit disk graphs mo del wireless ad-ho c and sensor net works and, for pac ket routing and other applicatio n s , a b ounded-degree p lanar geomet r ic spanner of the wir eless net work is often desired. Due to th e limited computational p ow er of the netw ork devices and th e requirement that th e n et wo rk b e robust with resp ect to device joining and lea ving the net wo rk , the construction/alg orithm should id eally b e strictly-lo c alize d : the computation p erformed at a p oint dep ends solely on the information a v ailable at the p oint and its d -hop neigh b ors, for some constant d (in our case d = 2). In particular, no global pr opagation of information should take place in the net wo rk. The results in the previous section do not carry o ver to unit disk graphs b ecause not all Delauna y graph edges on a p oin t-set P are unit d isk edges. Ho w ever, if U is the u n it disk graph on p oints in P and U D el ( U ) is the subgraph of the Delauna y graph on P ob tained b y deleting edges of length greater th an one unit, then U D el ( U ) is a connected, planar, spanning subgraph of U with stretc h factor b ounded by C del (see [11, 4]). Th erefore, if we ON GEOM E TR IC SP ANNERS O F EUCLIDEAN A ND UN IT DISK GRAPH S 419 apply the results from the previous section to U D el ( U ) and observe that all edges on th e path defined in Lemma 2.2 m ust b e u nit disk edges (giv en that edges C A and C B are), it is easy to see that Theorem 2.1 and Th eorem 2.10 carry o ve r to unit disk graph s . The only problem, ho wev er, is that the construction of U D el ( U ) cannot b e done in a strictly-l o calized manner. T o solv e this problem, W ang et al. [11, 12] in tro d u ced a su bgraph of U denoted LD el (2) ( U ). It was sho wn in [11, 12] th at LD el (2) ( U ) is a planar su p ergraph of U D el ( U ), and hence also has stretc h factor b ounded by C del . Moreo ve r, the resu lts in [6, 15] sho w ho w LD el (2) ( U ) can b e compu ted with a str ictly-lo calized distributed algorithm exc h an g- ing n o more than O ( n ) messages in total ( n is the n umb er of p oin ts in U ), and having a lo cal pro cessing time of O (∆ lg ∆) = O ( n lg n ) at a p oin t of d egree ∆. I n a style similar to Definition 1.2, L D el (2) ( U ) can b e defined as follo ws: Definition 3.1. An ed ge X Y of U is in LD el (2) ( U ) if and only if there exists a circle through p oints X and Y whose in terior contai n s no p oin t of U that is a 2-hop neighb or of X or Y . W e will use G = LD el (2) ( U ) as th e u nderlying subgraph of U to r eplace the Delauna y graph G used in the p revious section. W e note that G is p lanar, is a sup ergraph of U D el ( U ), and hence has stretc h factor C del . T o translate our resu lts to unit disk graph s, w e n eed to sho w that the in ward and out ward paths are still w ell defin ed in G . In particular, w e need to sho w that Lemma 2.2 holds f or G = LD el (2) ( U ). W e outline the general app roac h and omit th e d etails for lac k of sp ace. The f ollo wing is equiv alent to Prop ositi on 2.3: Lemma 3.2. If C A and C B ar e e dges of G then the r e gion of ( O ) =  AB C sub tende d by chor d C A and away fr om B and the r e gion of ( O ) subtende d by chor d C B and away fr om A c ontain no p oints that ar e two hop neighb ors of A , B and C . Pr o of. By symmetry it is enough to prov e the lemma for the regio n of ( O ) sub tended by c hord C A and a wa y from B . By Definition 3.1, there is a circle ( O C A ) passing through C and A wh ose int erior is empty of an y p oint w ithin tw o hops of C or A . The region of ( O ) subtended by chord C A and a w ay from B is in s ide this circle, so w e only n eed to argue that it d o esn’t con tain t wo hop neigh b ors of B either. If it did, sa y p oint X , then an y neigh b or of X and B w ould hav e to b e a n eigh b or of C or A as well, a con tradiction. With th is lemma in hand, the recursive construction of th e out w ard path giv en in Subsection 2.1 can b e applied to the graph G = LD el (2) ( U ). The follo wing p rop osition f or G = LD el (2) ( U ) corresp onds to Prop osition 2.5 for Dela u n a y graphs and is p ro ven in an equiv alent mann er : Prop osition 3.3. In every r e cursive step of the outwar d p ath c onstruction, if M p is an interme diate p oint with r e sp e ct to a p air of p oints ( M i , M j ) , then: (a) ther e is a cir cle p assing thr ough C and M p that c ontains no p oint of G that i s a two-hop neighb or of C or M p , and (b) cir cles  C M i M p and  C M j M p c ontain no p oints of G that ar e two -hop neighb ors of C , M i and M p and C , M j , and M p , r esp e ctively, exc e pt, p ossibly, in the r e gion subtende d by chor ds M i M p and M p M j , r esp e ctively, away fr om C . With this prop osition, we can show that Lemma 2.2 h olds tru e for G = LD el (2) ( U ) for out wa rd paths. It holds for in ward paths as w ell, using th e same argument as in Section 2.2. 420 I. KAN J AND L. PERKO VI ´ C Finally , it is ob vious h ow the Mo dified Y ao Step algorithm in Section 2.3 can b e easily describ ed as a strictly-lo calized algorithm. W e can show, therefore, the f ollo win g theorem: Theorem 3.4. 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