Nearly Optimal Bounds for Distributed Wireless Scheduling in the SINR Model

NEARL Y OPTIMAL BOUNDS F OR DISTRIBUTED WIRELESS SCHEDULING IN THE SINR MODEL MAGN ´ US M. HALLD ´ ORSSON AND PRADIPT A MITRA Abstra ct. W e study the wi reless sc heduling problem in t he SI NR mo del. More sp ecifically , give n a set of n links, each a sender- receiv er pair, w e wish to partition (or sche dule ) the links into the min- im um num ber of slots, each sati sfying interference constrain ts allo wing simultaneo us transmission. In the basic problem, all senders transmit with the same uniform pow er. W e giv e a distributed O (log n )-approximation algorithm for th e scheduling problem, matc hing the b est ratio kn own for central ized algorithms. It holds in arbitrary metric sp ace and for every length-monotone and sublinear p ow er assignment. It is based on an algorithm of Kesselheim and V¨ oc king, w hose analysis we improv e by a logarithmic factor. W e show that every distributed algorithm uses Ω(log n ) slots to schedule certain instances that require only tw o slots, w hich implies that the b est possible absolute p erformance guarantee is logarithmic. 1. Introduction Giv en a set of n wireless links, eac h a send er -r eceiv er pair, what is the minimum n um b er of slots needed to sc hed u le all the links, giv en in terference constrain ts? This is the ca nonical problem of sche duling wireless communicatio n, w hic h we stu dy here in a distribu ted s etting. In a wir eless net w ork, simultaneous transmissions on the same channel inte rfere w ith eac h other. Algorithmic questions f or wireless net w orks dep end crucially on the mod el of interference con- sidered. In this work, we use the physic al, a.k.a. SINR , m o del of int erference, precisely defined in Section 2. It is kn o wn t o capture realit y more fa ithfully than th e graph-based m o dels most common in the theory literature, as sho wn theoretically as well as exp erim entally [21, 23]. Early work on sc heduling in the SINR m o del fo cused on heuristics and/or non-algorithmic a v erag e-case analysis (e.g. [11]). In seminal w ork, Moscibro d a and W attenhofer [22] prop osed the problem of scheduling an arb itrary set of links. Nu merous works on v arious prob lems in the S INR setting ha v e ap p eared since. The sche duling problem has primarily b een studied in a cen tralized setting. In man y realistic scenarios, ho w ev er, it is imp erativ e th at a distributed s olution b e foun d, since a cen tralize d con- troller ma y not exist, and ind ividual no des in the link ma y not b e a w are of the o v erall top ology of the net w ork. F or the sc heduling pr ob lem, the only rigorous result previously kno wn is due to Kesselheim and V¨ oc king [20], w h o show that a s imple and natural d istributed algorithm p ro vides an O (log 2 n )-appro ximation. In this w ork, we adopt the algo rithm of Kesselheim and V¨ oc king, but provide an impr ov ed analysis of an O (log n )-approximati on. T his matc hes the b est upp er b ound kn o wn for cen tralized algorithms. Moreo v er, w e sh o w th is to b e b est p ossible for distr ibuted alg orithms that use no external comm unication infrastructure. 2. Preliminaries and Contribut ions Giv en is a set L = { l 1 , l 2 , . . . , l n } of links, wh er e eac h link l v represent s a communicatio n requ est from a sen der s v to a receiv er r v . Th e distance b etw een t w o p oin ts x and y is d enoted by d ( x, y ). Date : Ma y 21, 2018. Supp orted by grants 90032021 and 120032011 fro m the Icelandic Researc h F un d. Preliminary versi on app eared in ICALP 2011. 1 2 MAGN ´ US M. HALLD ´ ORSSON AND PRADIPT A MITRA The asymmetric distance from link l v to link l w is the distance from v ’s sender to w ’s receiv er, denoted by d vw = d ( s v , r w ). L et ℓ v = d ( r v , s v ) denote the length of link l v . Let P v denote the p o w er assigned to lin k l v , or, in other wo rds, s v transmits with p o w er P v . W e adopt the SINR mo del (a.k.a., physic al mo del ) of inte rference, in whic h a n o de r v successfully receiv es a message fr om a send er s v if and only if the follo wing condition holds: (1) P v /ℓ α v P l w ∈ S \{ l v } P w /d α w v + N ≥ β , where N is a unive rsal constan t d enoting th e ambien t noise, α > 0 denotes the path loss exp onen t, β > 0 denotes the minimum SINR (signal-to-in terference-noise- ratio) required for a message to b e successfully r eceiv ed, an d S is th e set of concur ren tly s c heduled links in the same slot . W e sa y that S is SINR-fe a sible (or s im p ly fe asible ) if (1) is satisfied for eac h link in S . A p ow er assignment P is length-monotone if P v ≥ P w whenev er ℓ v ≥ ℓ w and sub-line ar if P v ℓ α v ≤ P w ℓ α w whenev er ℓ v ≥ ℓ w [20]. Two widely used p ow er assignmen ts in this class are the u niform p o w er assignmen t, where ev ery link transm its with the same p o w er; and th e line ar p o w er assignmen t, where P v is prop ortional to ℓ α v . A third one, me a n p ow er [6, 12 ] has also p r o v ed to b e versatile . Giv en a set of link s L , the sche duling pr oblem is to find a partition of L of minim um size suc h that eac h subset in th e partition is feasible. The size of the partition equals the min im um num b er of s lots requ ired to schedule all links. W e will call this n um b er the sche duling numb er of L , and denote it b y χ ( L ) (or χ w hen clear f rom con text). Distributed algorithms. A comm unication infr astructure for runn ing distributed algorithms is gen- erally assumed to exist in the tr aditional distributed setting. The cur r en t setting, w hic h abstr acts the MA C lay er in net w orks, is differen t, as th e goal actually is to construct s uc h an in frastructure. Th us, our algorithm will w ork w ith very little global kno wledge and m in imal external input. Comm unication is only a v ailable ov er the c hannel. Algorithms op erate in synchronous r ounds with the senders either transmitting or listening in eac h r ound. When transmission is successful, the send er stops transmitting. This necessitates an ackno wledgment from the receiv er, so th at the sender kno ws when h is message h as b een h eard. These ac kno wledgmen ts are sen t o ver the same c hannel as the message; thus, there are no side-c hannels for cont rol messages. W e shall assume this mo del, whic h we call ack- only , in the r est of th e pap er. W e assume that no des ha v e a rough estimate of the net w ork size n and (senders of ) links are assigned a fixed length-monotone, sublinear p o w er function. The p ow er assignment ind irectly requires knowledge of distances and the p ath loss constant α and the tec hnologica l parameters β and N . No inf ormation of lo cations is needed. W e note that th e assu mptions are particularly min im al when using uniform p ow er. The algorithm then needs no kno wledge of distances, the p ath loss constan t α , n or the tec hnological parameters β and N . Only the p olynomial b ound on the num ber n of no des is n eeded. Affectance. W e will u se the notion of affe ctanc e , introd uced in [9, 17] and refined in [20] to the thresholded form used here. Th e affectance a P w ( v ) on link l v fr om another link l w , w ith a giv en p ow er assignment P , is the interference of l w on l v relativ e to the p ow er receiv ed, or a P w ( v ) = min  1 , c v P w /d α w v P v /ℓ α v  , where c v = β / (1 − β N ℓ α v /P v ) d ep ends only on mo del constants and on the length of l v . W e will drop P and assume it to b e an arb itrary length-monotone sub-linear p o w er strategy , unless otherwise stated. Let a v ( v ) = 0. F or a set S of links and a link l v , let a S ( v ) = P l w ∈ S a w ( v ), referred to as in-affe ctanc e , and a v ( S ) = P l w ∈ S a v ( w ), the out-affe ctanc e f rom l v . F or s ets S and R , a R ( S ) = P l v ∈ R P l u ∈ S a v ( u ). Using s u c h n otation, (1) can b e rewritten as (2) a S ( v ) ≤ 1 , DISTRIBUTED SCHEDULING IN THE SINR MODEL 3 whenev er | S | > 2, and th is is the form we will use. 2.1. Related W ork. In the cen traliz ed setting, sc heduling r esu lts ha v e closely follo w ed r esults on the related c ap acity problem, wher e one wa n ts to find the maxim um subset of L that can b e transmitted in a single slot). Go ussevsk aia et al. [10] sho w ed the problem to b e NP-hard for the case of uniform p ow er on the plane and ga v e O (log ∆)-appro ximation result (on the p lane), where ∆ denotes the r atio b et w een the m axim um and m inim um length of a link. Same b ound w as shown b y Andrews and Dinitz [1 ] b ut in comparison with op timum that is allo w ed to c ho ose arbitrary p o w er. Constan t factor appro ximation w as obtained for uniform p o w er, also on the plane, by Goussevsk aia et al. [9], whic h was generalized to all length-monotone, su blinear p o w er assignment s and arbitrary metrics space by Halld´ orsson and Mitra [14]. Kesselheim [18 ] ga v e a constan t-factor app ro ximation for the joint problem of selecting links and assigning them feasible p o w er (see also earlier work of Chafek ar et al. [4]. All the results lead to equ iv alen t b oun ds for the cen tralize d sc heduling problem with O (log n )- factor o v erhead. In particular, O (log n )-appro ximation h olds for scheduling with length-monotone, sublinear p o w er [14] and with arb itrary p o w er con trol [18]. Also, the p roblem remains NP-hard [10]. F or th e resu lts in terms of ∆ on the plane [10, 1], th is ov erh ead can b e av oided (see, e.g., [12]). Sc heduling with arbitrary p o w er cont rol can also b e approxima ted w ithin a factor of O (log n log log ∆) when the algorithm u ses mean p o w er. F or linear p ow er on the p lane, an algo- rithm using O ( χ + log 2 n ) slo ts for in stances with o ptimal sc hedule le ngth χ w as giv en b y F a ngh¨ anel et al. [7]; on the plane, this can b e imp r o v ed to a constan t factor [26]. A b i-directional v ersion w as studied by F angh¨ anel et al. [6] and fu rther treated in [12, 14] and the j oin t m ulti-hop scheduling and routing was treated by C hafek ar et al. [4]. In th e d istributed setting, the capacit y problem w as treated with n o-regret learning b y Dinitz [5] cu lm in ating in a O (1)-appro ximation algorithm for unif orm p o w er of ´ Asgeirsson and Mitra [2]. Ho w ev er, these game-theoretic algorithms take time p olynomial in n to conv erge, and thus can b e view ed more appropriately as determining capacit y in stead of r ealizing it in “real time”. F or distributed sc heduling, the only w ork that we are a w are of is the groun dbreaking pap er of Kesselheim and V¨ oc king [20], who giv e a d istributed O (log 2 n )-appro ximation alg orithm for the sc heduling pr oblem with fi xed length-monotone and sublinear p o w er assignmen t. Ou r r esults constitute a Ω (log n )-factor improv ement. Kesselheim and V¨ oc king also extend their r esults to m ulti-hop sc h eduling, with the same appro ximation facto r, for whic h our impro v emen ts do n ot apply , and to routing, with an extra logarithmic factor. A v ersatile m easure introd u ced in [20 ] is the maxim um a v erage affectance A of a link set L , defined as A ( L ) := max R ⊆ L a vg l ∈ R a R ( l ) = max R ⊆ L a R ( R ) | R | . They then sh ow tw o results that combined yield the O (log 2 n )-appro ximation factor. On the one hand, they sho w that A ( L ) = O ( χ ( L ) log n ). On the other hand , they pr esen t a natural algorithm (whic h w e also use in this work) that s chedules links in O ( A ( L ) log n ) slots. W e show th at b oth of these b ounds are tigh t. T h us, it is not p ossible to obtain impro v ed appro ximation using the measure A . F ollo wing the original pu blication of this w ork, the r esults hav e b een applied to d istributed connectivit y and aggregat ion [15, 3]. A different approac h for distributed capacit y was prop osed b y P ei and Kumar [24], w ith complexit y that is a fu nction of the link lengths. In a recen t follo w-up w ork, Hall d´ orsson et al. [13] ha v e sho wn that A ( L ) = O ( χ ) for all su blinear, lengt h-monotone p o w er assignmen ts other than uniform p o w er. 2.2. Our C on tributions. W e achiev e the follo wing results: 4 MAGN ´ US M. HALLD ´ ORSSON AND PRADIPT A MITRA Theorem 1. Ther e is a O (log n ) -appr oximate distribute d algorithm for the sche duling pr oblem, in arbitr ary metric sp ac e and for al l length-monotone subline ar p ower assignments. Theorem 2. F or every n , ther e is an instanc e L n of links on the r e al line that c an b e sche dule d in two slots but for which every eve ry distribute d algorithm uses Ω(log n ) -slots (w.h.p). Thus, Θ(log n ) is the b est absolute appr oximation factor for a distribute d sche duling algorithm. As in [20], our upp er b ound results hold in arbitrary d istance metrics (and do not require the common assump tion that α > 2). W e also show that the results hold in d ep end ent of the ambien t noise term N , extending [20]. The lo w er b ound result necessarily h olds indep enden t of p ow er assignmen t strategy and for all settings of the tec hnolog ical constants α , β and N . One of our main tec hnical in s igh ts is to devise a different measure that inv olv es me dian rather than a v erag e affectance. The measure Λ = Λ( L ) is giv en by Λ( L ) := max R ⊆ L median( A ( R )) , where A ( R ) = { a R ( l ) : l ∈ R } is the multi-set of in -affectance v alues of links in the subs et R , and median( X ) den otes the median of a multi-set X . Since w e only insist that half of the given sub set R of lin k s h a v e affectance b ound ed b y Λ, the v alue of Λ ma y b e m uc h sm aller than A . I ndeed, we sho w that Λ = O ( χ ) and that the algorithm sc hedules all link s in time O (Λ log n ), ac hieving the claimed ap p ro ximation factor. The other main tec h n ical cont ribution of the p ap er is the int ro duction of the concept of anti- fe asibility . A set S of lin k s is anti-fea sible 1 if a v ( S ) ≤ 2, for ev ery l v in S ; i.e., if th e out-going affectance from eac h link is small. A set is bi - fe a sible if it is b oth f easible and an ti-fea sible. W e observ e in t his pap er that every feasible set con tains a large bi-feasible set and that c ertain analyses are easie r on b i-feasible sets. This h as pro v ed usefu l i n late r w orks, e.g., in giving simp lified a nalysis of capacit y appr o ximation algorithms [19, 16]. In the next section, we giv e the impr o v ed analysis of a O (log n )-factor for d istributed sc heduling, via the measure Λ; th e treatmen t of ac kno wledgmen ts is giv en in Section 3.2. W e sh o w in Section 4 that this logarithmic factor is b est p ossible, and giv e a construction in Section 5 that sho ws that this r esult cannot b e obtained in terms of the measur e A . 3. O (log n ) -Appro xima te Distributed Schedul ing Algorithm The algorithm from [20], listed b elow as Distributed , is a natural bac k off sc heme, in the tra- dition of ALOHA [25]. It is ru n s ync hronously , but indep endent ly , on eac h sender of a lin k . T he algorithm, and all the r esults in this section, work for an arbitrary fixed sublinear length-monotone p ow er assignment. The algorithm is mostly self-descriptive. The constan t c 1 is to b e c hosen to satisfy the h igh probabilit y b ound desired. One p oin t to note is that Line 2 necessitates some sort of ac kno w ledg- men t mec hanism f or the distr ib uted algorithm to stop. F or simplicit y , we will defer the issue of ac kno wledgmen ts to Section 3.2 and simply assume their existence f or n o w. Thm. 3 b elo w implies our main p ositiv e result. Let Λ = Λ( L ). Theorem 3. If al l links of a set L of n links run Distribute d , then L i s ful ly sche dule d in O (Λ log n ) slots, with high pr ob ability. T o pro v e Thm. 3, we claim the follo wing. Lemma 3.1. Consider a subset R ⊆ L of links and a p articular time slot t in which e ach sender of R tr ansmits with pr ob ability q ≤ 1 2Λ . Then, the exp e cte d numb e r of suc c essful tr ansmissions is at le ast q ·| R | 4 . 1 F or a tec hnical rea son w e use a differen t constan t here than f or feasibili ty; the signal-strengthening result of [1 7] implies that this only affects constants in th e approximatio n factors. DISTRIBUTED SCHEDULING IN THE SINR MODEL 5 Algorithm 1 Distrib uted 1: k ← 0 2: lo op 3: q = 1 4 · 2 k 4: for 4 q c 1 ln n slots do 5: transmit with i.i.d. probability q 6: if successful (and ac kno wledged) then 7: halt 8: end if 9: end for 10: k ← k + 1 11: end lo op Pr o of. Define M = M Λ ( R ) = { l u ∈ R : a R ( u ) ≤ Λ } . By the defi n ition of Λ, | M | ≥ | R | / 2. Thus, it suffices then to sh o w that at least q | M | / 2 transmissions in slot t are successful in exp ectation. In tuitiv ely , the success p robabilit y of a lin k is prop ortional to its in-affectance. The links in M are the ones with lo w in-affectance, so as long as the tr an s mission pr obabilit y q is less than 1 / (2Λ), they will s u cceed with probabilit y 1 / 2 if tr an s mitting. F or l u ∈ R , recall that T u = T u ( t ) is the indicator r andom v a riable that link l u transmits, and let S u = S u ( t ) b e the indicator random v ariable that l u succeeds. W e shall make use of a few ele men tary facts ab out probabilities. F or a (Be rnoulli) in d icator random v ariable X , E ( X ) = Pr( X ). F or random v ariables X 1 , X 2 , . . . , it holds by the lin earit y of exp ectation that P i E ( X i ) = E ( P i X i ). An d, for a random v ariable X that assumes non-negativ e v alues, P ( X > 1) ≤ E ( X ). Armed w ith these facts, w e can no w b ound the probabilit y that a transmitting link l u ∈ M is unsuccessful: P ( S u = 0 | T u = 1) = P   X l v ∈ R a v ( u ) T v > 1   ≤ E   X l v ∈ R a v ( u ) T v   = X l v ∈ R a v ( u ) E ( T v ) = q X l v ∈ R a v ( u ) ≤ q · Λ , where the fi rst equalit y uses (2), and the last inequ ality u ses th e defi n ition of M . Th us, wh en q ≤ 1 2Λ , P ( S u = 0 | T u = 1) ≤ 1 / 2 , whic h allo w s us to b ound the probabilit y of link l u transmitting in the time slot b y E ( S u ) = P ( S u = 1) = P ( T u = 1) P ( S u = 1 | T u = 1) = q (1 − P ( S u = 0 | T u = 1)) ≥ q / 2 . 6 MAGN ´ US M. HALLD ´ ORSSON AND PRADIPT A MITRA The exp ected num ber of s uccessful links in the time s lot is then E   X l u ∈ R S u   = X l u ∈ R E ( S u ) ≥ X l u ∈ M E ( S u ) ≥ | M | · q / 2 ≥ | R | · q / 4 , implying the lemma.  of Thm. 3 . Giv e n Lemma 3.1, the th eorem follo ws essentiall y from the argu m en ts in Thms. 2 and 3 of [20]. Let ˆ q = 2 − (1+ ⌈ lg Λ ⌉ ) , i.e., th e unique p o w er of tw o satisfying 1 4Λ ≤ ˆ q ≤ 1 2Λ . W e fir st b ound the probability th at n ot all links are scheduled during the iteration of the outer lo op w hen q in Line 3 equals ˆ q . Let ˆ t b e the first time slot w here q ≤ ˆ q . Let n t b e the rand om v ariable indicating the num b er of links that did not su ccessfully transmit in the first t time slots. Lemma 3.1 imp lies that for any giv en v alue s and time slot t ≥ ˆ t , E ( n t | n t − 1 = s ) ≤ s − ˆ q 4 s , and th us E ( n t ) ≤ ∞ X s =0 P ( n t − 1 = s ) · (1 − ˆ q / 4) s = (1 − ˆ q / 4) E ( n t − 1 ) . Noting that n 0 = n , this yields that E ( n t ) ≤ (1 − ˆ q / 4) t n . No w, after ˆ t + 4 c 1 ln n/ ˆ q time slots, the exp ected n umber of remaining requests is E ( n ˆ t +4 c 1 ln n/ ˆ q ) ≤ (1 − ˆ q / 4) 4 c 1 ln n/ ˆ q E ( n ˆ t ) ≤  1 e  c 1 ln n n = n 1 − c 1 . By Mark o v’s inequalit y , P ( n ˆ t +4 c 1 ln n/ ˆ q 6 = 0) = P ( ˆ t + n 4 c 1 ln n/ ˆ q ≥ 1) ≤ E ( n ˆ t +4 c 1 ln n/ ˆ q ) ≤ n 1 − c 1 . Th us, with high probabilit y all the links are scheduled wh ile q ≥ ˆ q . Finally , to b ound the total run ning time of the algorithm, w e su m up the sp ent for v alues of q smaller than ˆ q , b oun ding t 0 . This is a geometric series giv en b y t 0 = lg(1 / ˆ q ) X i =2 8 c 1 ln n 2 − i = 8 c 1 ln n lg(1 / ˆ q ) X i =2 2 i ≤ 8 c 1 ln n · 2 lg (1 / ˆ q )+1 = 8 c 1 ln n · 2 ˆ q ≤ 64 c 1 Λ ln n , establishing the time complexit y .  DISTRIBUTED SCHEDULING IN THE SINR MODEL 7 3.1. Bounding the Measure. W e n eed the f ollo wing lemma to get a h andle on affectances. Recall that w e assumed that the imp licit p o w er assignmen t is length-monotone and s u blinear. Lemma 3.2 (Lemma 7, [20]) . L et L b e a fe asible set and l u 6∈ L b e link with ℓ u ≤ ℓ v for al l l v ∈ L . Then, a L ( u ) = O (1) . W e no w pro v e the follo wing complemen tary result. It can b e con trasted w ith Lemm a 9 of [20], whic h without th e ant i-feasibilit y condition can only give a v ( L ) = O (log n ). The second part of the lemma essen tially follo w s Lemma 11 of [2] (whic h had the unnecessary assumption that L is feasible). W e first need the follo wing result. Lemma 3.3 ([12]) . L et l u , l v b e links with min( a u ( v ) , a v ( u )) ≤ 1 / q . Then, d uv · d vu ≥ q 2 · ℓ u ℓ v . Lemma 3.4. L et L b e an anti-f e asible se t with length-monotone and subline ar p ow er and let l v 6∈ L b e a link with ℓ v ≤ ℓ u , for every l u ∈ L . Then, a v ( L ) = O (1) . Pr o of. W e fir st use a v ariatio n of the signal strengthening technique of [17], giv en as Thm . 7 in the App endix. T his allo ws us to decomp ose the s et L int o ⌈ 4 · 3 α ⌉ 2 sets, wh ere eac h s et S satisfies a w ( S ) ≤ 1 3 α , for all l w ∈ S . W e shall pro v e the claim for S ; the claim will then hold for L by summing ov er the ⌈ 4 · 3 α ⌉ 2 sets. Let l u = ( s u , r u ) ( l w = ( s w , r w )) b e th e lin k in S whose sender (receiv er) is closest to s v , i.e., d ( s v , s u ) ≤ min l x ∈ S d ( s v , s x ) ( d ( s v , r w ) ≤ min l x ∈ S d ( s v , r x )), r esp ectiv ely . L et h = d ( s v , s u ). W e claim th at for all lin ks l x in S , l x 6 = l w , it holds that (3) d ( s v , r x ) ≥ 1 2 h . T o p ro v e this, assume, for contradictio n, that d ( s v , r x ) < 1 2 h . Then, b y the definition of l w , d ( s v , r w ) < 1 2 h , and by the definition of l u , d ( s v , s x ) ≥ d ( s v , s u ) ≥ h and d ( s v , s w ) ≥ h . Th us, ℓ w ≥ d ( s v , s w ) − d ( s v , r w ) > h 2 and similarly ℓ x > h 2 . O n the other hand, b y the triangular inequalit y and th e assumed inequalit y , d ( r w , r x ) ≤ d ( r w , s v ) + d ( s v , r x ) < h 2 + h 2 < h . No w, d w x · d xw ≤ ( ℓ w + d ( r w , r x ))( ℓ x + d ( r w , r x )) < ( ℓ w + h )( ℓ x + h ) < 9 ℓ w ℓ x , con tradicting Lemma 3.3. This establishes (3). No w, by the triangular in equalit y , the definition of h and (3), d ux = d ( s u , r x ) ≤ d ( s u , s v ) + d ( s v , r x ) ≤ 3 d ( s v , r x ) = 3 d vx . W e observe that P v ≤ P u holds b y length-monotonicit y . Also, note that since the maxim um affectance b et w een links in S is 1 3 α , the thr esholding in th e affectance definition do es not tak e effect, implying th at a u ( x ) = c x P u d α ux ℓ α x P x . Th us, a v ( x ) = c x P v d α vx ℓ α x P x ≤ c x 3 α P u d α ux ℓ α x P x = 3 α a u ( x ) . 8 MAGN ´ US M. HALLD ´ ORSSON AND PRADIPT A MITRA Finally , sum ming o v er all links in S , a v ( L ) = a v ( w ) + X l x ∈ S \{ l w } a v ( x ) ≤ 1 + 3 α X l x ∈ S \{ l w } a u ( x ) ≤ 1 + 3 α · 2 = O (1) , using an ti-feasibilit y in the last inequalit y . T h e lemma follo ws.  W e can no w deriv e the needed b ound on th e m easure. Theorem 4. L et L b e a set of links. Then, Λ( L ) = O ( χ ( L )) . Pr o of. Let χ = χ ( L ) and let R b e an arb itrary subset R ⊆ L . T o pr o v e th e theorem, it suffices to sho w th at at least h alf of the links in R h a v e in-affectance at O ( χ ( L )). Consider a partition of R into χ feasible subsets S 1 , S 2 , . . . , S χ , and defin e S ′ i = { l v ∈ S i : a v ( S i ) ≤ 3 } . W e claim that S ′ i con tains at least tw o thir ds of the links in S i . Claim 3.5. F or al l i , | S ′ i | ≥ 2 | S i | 3 . Pr o of. Since S i is fea sible, it follo ws from (2) that a S i ( v ) ≤ 1, for every l ink l v ∈ S i . Let ˆ S i = S i \ S ′ i . No w, a ˆ S i ( S i ) ≤ X l v ∈ S i a S i ( v ) ≤ X l v ∈ S i 1 ≤ | S i | . But, a ˆ S i ( S i ) = P l v ∈ ˆ S i a v ( S i ) ≥ 3 · | ˆ S i | , by th e d efinition of ˆ S i . Thus, | ˆ S i | ≤ 2 | S i | / 3, proving the claim.  Let R ′ = ∪ i S ′ i . By the ab ov e claim, 3 | R ′ | / 4 ≥ | R | / 2. W e next sho w the follo win g. Let c 2 ( c 3 ) b e the constan t implicit in the big-oh n otation in Lemma 3.2 (Lemma 3.4), resp ectiv ely . Claim 3.6. a R ( R ′ ) ≤ ( c 2 + c 3 ) | R | · χ. Pr o of. W e fi rst observ e that for every i, j , a S j ( S ′ i ) = X l u ∈ S ′ i X l v ∈ S j a v ( u ) ≤ X l u ∈ S ′ i X l v ∈ S j ℓ v ≥ ℓ u a v ( u ) + X l u ∈ S ′ i X l v ∈ S j ℓ v ≤ ℓ u a v ( u ) ≤ X l u ∈ S ′ i c 2 + X l v ∈ S j X l u ∈ S ′ i ℓ u ≥ ℓ v a v ( u ) ≤ c 2 | S ′ i | + X l v ∈ S j c 3 ≤ c 2 | S i | + c 3 | S j | , (4) DISTRIBUTED SCHEDULING IN THE SINR MODEL 9 using Lemma 3.2 and rearrangemen t in th e second inequalit y , and Lemma 3.4 in the third inequalit y . W e then obtain that a R ( R ′ ) = χ X i =1 χ X j =1 a S j ( S ′ i ) ≤ χ X i,j =1 c 2 | S i | + c 3 | S j | (By (4)) = χ X i,j =1 ( c 2 + c 3 ) | S i | ) (By symmetry) = ( c 2 + c 3 ) χ χ X i =1 | S i | = ( c 2 + c 3 ) χ | R | (Defn. of S i )  It follo ws th at the av erage in-affectance a R ( l ′ ) o v er the link s l ′ ∈ R ′ is at most a R ( R ′ ) | R ′ | ≤ ( c 3 + c 4 ) | R | · χ | R ′ | ≤ µ := 3( c 2 + c 3 ) 2 χ . Recall that M 4 µ ( R ) = { l ∈ R : a R ( l ) ≤ 4 µ } is the set of links in R of in-affectance at m ost four times the a v erage . By Marko v’s inequalit y , at least three fourth s of the links hav e in-affectance at most four times the av erage; namely , | M 4 µ ( R ) | ≥ | M 4 µ ( R ′ ) | ≥ 3 | R ′ | / 4 ≥ | R | / 2 . That is, at least half the links in R hav e in-affectance at most 4 µ . Hence, the median in-affectance of links in R is b ound ed ab o v e b y median( A ( R )) ≤ 4 µ = O ( χ ) . Since th is holds for ev ery giv en R , the theorem follo ws.  3.2. Ac kno wledgmen ts. In the pr eceding exp osition, we ignored th e iss u e of send ing ac kno wledg- men ts from receiv ers to senders. W e can treat ac kno wledgmen ts in a fashion similar to Kesselheim and V¨ oc king [20]. W e outline their appr oac h briefly , but direct the reader to their pap er for the details. A sp ecial slot for ackno wledgments is inserted b et w een the time slots used b y Algorithm 1. A no d e that successfully receiv ed a pac k et will transmit an ac kno wledgmen t with p robabilit y p = 1 / 8. The p ow er P ∗ v used for the ackno wledgment on lin k l v is c hosen to b e p r op ortional to P ∗ v = ℓ α /P v (using the right scaling factor). Kesselheim and V¨ oc king show that at least half of these ac kno wledgmen ts are su ccessful in exp ectati on. Th at implies that w e can mo d ify Lemma 3.1 to claim th at the exp ected num b er of successfully ac kno wledged tr ansmissions is at least p · q | R | / 4 = q | R | / 32, losing only a constant f actor. The r est of the argumen ts are then iden tical. The only catc h is that th ey do assume in their analysis th at ther e are no w eak links in the instance; a link l v is said to b e we ak iff c v > C β , for an app ropriately c hosen constant C (whose v alue affects the c hoice of p ). W e sho w here ho w to extend the approac h to deal with w eak link s . F or simplicit y of exp osition, we illustrate it for th e case of uniform p o w er and assume that w eak links satisfy c v > 3 max( β , 1). 10 MAGN ´ US M. HALLD ´ ORSSON AND PRADIPT A MITRA The original transmissions, using Algorithm 1, are u nc hanged, bu t w e allo cate a separate time slot for the ac kno wledgmen ts of wea k links. Eac h receiv er of a successfully tr ansmitting w eak link sends an ac kno wledgmen t in that time slot with prob ab ility p ′ (to b e c hosen). The key observ ation in the follo wing lemma is that weak links must b e spatially w ell-separated. This implies that differences b et w een the p ositions of the sender and receiv er of a link are min or, allo w ing us to r elate the su ccess prob ab ility for an ac kno wledgmen t in terms of the observe d s uccess of the original transmission. Lemma 3.7. Assume the use of uniform p ower. L et l v b e a we ak link that tr ansmits suc c essful ly in a given time slot t of Algorith m 1. Then, the tr ansmission is suc c essful ly acknow le dge d with pr ob ability p ′ / 2 when p ′ ≤ (1 + 2 ln 3 · α ) − α . Pr o of. Let l u b e another weak link that successfully transmitted at time t . Since b oth were suc- cessful, (2) is satisfied in b oth directions, whic h implies that (5) d α uv ≥ c v ℓ α v ≥ 3 ℓ α v , and d α vu ≥ 3 ℓ α u . By the triangular inequalit y , d uv ≤ d vu + ℓ u + ℓ v , wh ic h by (5) implies that d uv  1 − 1 3 1 /α  ≤ d uv − ℓ v ≤ d vu + ℓ u ≤  1 + 1 3 1 /α  d vu . No w, obser ve that  1 + 1 3 1 /α   1 − 1 3 1 /α  ≤ 1 + 2 e (ln 3) /α − 1 ≤ 1 + 2 α ln 3 . Th us, (6) d α uv ≤  1 + 2 α ln 3  α d α vu . No w, let l ∗ v = ( s ∗ v , r ∗ v ) = ( r v , s v ) b e the dual link of l v , with the r oles of sender and receiv er rev ersed. A transmission on l v is ac kno wledged on l ∗ v . W e u se (6) to b ound the in-affectances of a dual link l ∗ v from another d ual link l ∗ u : a u ∗ ( v ∗ ) = c v  ℓ v d vu  α ≤  1 + 2 α ln 3  α c v  ℓ v d uv  α =  1 + 2 α ln 3  α a u ( v ) . Let S b e the set of wea k links that successfully tran s mitted in slot t and S ∗ the set of the cor- resp ond ing dual links. Sup p ose eac h link in S transmits an ac kno wledgmen t with probabilit y p ′ . DISTRIBUTED SCHEDULING IN THE SINR MODEL 11 Then, the exp ected in-affectance of a link l ∗ v that transmits an ac k is b ound ed b y E   X l ∗ u ∈ S p ′ · a u ∗ ( v ∗ )   ≤ X l ∗ u ∈ S ∗ E ( p ′ a u ∗ ( v ∗ )) = p ′ X l ∗ u ∈ S ∗ a u ∗ ( v ∗ ) ≤ p ′  1 + 2 α ln 3  α X l ∗ u ∈ S ∗ a u ( v ) ≤ 1 2 a S ( v ) ≤ 1 2 , using th e feasibilit y of S . Hence, the p r obabilit y that a link receiv es less than t wice the exp ected in-affectance is at least 1 / 2, i.e., a du al link that d o es attempt to transmit an ac kno wledgmen t has at least 50% c hance of success. The probability that a giv en link b oth attempts to send an ac kno wledgmen t and th at the transmission is su ccessful, is then at least p ′ / 2.  4. Ω(log n ) -F a ctor Lower Bound for Distributed Sched uling W e construct a set of 2 n unit length links on the line that can b e scheduled in t w o slots w hile no d istributed algorithm can sc hedule the set in less than Ω(log n ) slots. W e assume that all s enders start at the same time in the same state and use the s ame (random- ized) algorithm. Not e that the algorithm pr esen ted op erates u nder th ese assump tions. F or simplicit y , we assume the noise N = 0, bu t note that the construction can b e mo dified to hold for differen t v alues of N . W e allo w α and β to b e arbitrary p ositive v alues. W e start with a gadget F w ith t w o identica l links of lengt h 1, in a yin -yang p osition, i.e., with the sender of one link in the same p osition as the receiv er of the other (it suffi ces that they b e separated by at most ( P max / ( β P min )) 1 /α , w here P max ( P min ) is the maxim um (minim um) p o w er that can b e used, resp ectiv ely). Le t x = (2 β n ) 1 /α . The construction consists of n such gadgets F i , i = 1 , 2 , . . . n , placed on the line as follo ws : The sender of one link and the r eceiv er of the other link in F i are placed at p oint i ( x + 1) and the other tw o no des of F i are placed at i ( x + 1) + 1. The construction ensu r es that a link suc c essful ly tr a nsmits only if the other link in the gadget do es not tr an smit . This holds indep en d en t of the p ow e r used on these links. On the other hand, when using un if orm p o w er, the affectance fr om links of other gadgets is negligible. T o see this, consider the affectance on a link l u ∈ F i from all links of other gadgets, i.e., from all links l v ∈ ˆ F := ∪ j 6 = i F j . There are 2 n − 2 links in ˆ F . The d istance d vu ≥ x . Therefore, P ˆ F a v ( u ) ≤ (2 n − 2) β x α < 1. Thus, b ehavio r of links in other gadgets is immaterial to the success of a link. This also implies that the sc heduling num b er of this set of links is 2. Note that since the constru ction uses equi-length links, the only p ossible oblivious p ow er assignment is the un if orm one. T o pro v e the lo w er b ound , w e sa y th at gadget F i is active at time t if neither link of F i has succeeded by time t − 1, and d en ote the ev en t by A i ( t ). Let T u ( t ) denote the indicator random v ariable that link l u transmits at time t . Lemma 4.1. L et F i b e a gadget and t ≥ 0 b e a time. The tr ansmission pr ob abilitie s of the two links in F i at time t ar e identic al and indep endent, c on ditione d on F i b eing active at time t . Pr o of. Let l u and l v b e the links in gadget F i . Let T u = T u ( t ) and T v = T v ( t ), f or short. By sym- metry , the distribu tions of T u and T v are identica l, th us w e need only to pro v e their in dep end ence. W e can m o del the r andomness used by the algorithms as an i.i.d. random c hoice ov er a set F of f unctions. E ach f ∈ F is a function th at tak es a history of past transmissions and r eceptions 12 MAGN ´ US M. HALLD ´ ORSSON AND PRADIPT A MITRA o v er previous slots, and return s a b inary transmission decision. Note that if A i ( t ) o ccurs then the histories of l u and l v o v er the p revious t − 1 slots are identica l. The differen t histories that can result in A i ( t ) o ccurring are disj oin t; th us, it is enough to pro v e indep endence f or a fi xed history H . Let f u and f v denote the fun ctions chosen b y l u and l v , and allo w them also to represent the ev en t that they get c hosen. Once again, b y symmetry , there is some F ′ ⊆ F su c h that H happ ens iff f u ∈ F ′ and f v ∈ F ′ . W e will use the Iv erson brac k et [ X ] to denote the v alue 1 if X is true and 0 otherwise. Then, for fixed Bo olean ou tcomes a and b , P ( T u = a, T v = b | H ) = X f u ∈F ′ ,f v ∈F ′ P ( f u f v )[ f u ( H ) = a ][ f v ( H ) = b ] = X f u ∈F ′ ,f v ∈F ′ P ( f u ) P ( f v )[ f u ( H ) = a ][ f v ( H ) = b ] = X f u ∈F ′ P ( f u )[ f u ( H ) = a ] · X f v ∈F ′ P ( f v )[ f v ( H ) = b ] = P ( T u = a | H ) P ( T v = b | H ) , thereb y pr o ving indep endence. W e hav e used that P ( f u f v ) = P ( f u ) P ( f v ) in the second equalit y , whic h follo w s f r om the fact that f u and f v are chosen a priori and indep endently .   Let p t denote the i.i.d. prob ab ility that some link in a given gadget F i transmits at time t . No w P ( A i ( t + 1) | A i ( t )) = p 2 t + (1 − p t ) 2 , whic h is minimized for p t = 1 2 with v alue 1 2 . Th us, (7) P ( A i ( t + 1) | A i ( t )) ≥ 1 2 . In tuitiv ely , on av erage, at most half of the activ e gadgets b ecome inactiv e in an y give n round, and th us it tak es lg n rounds for all gadgets to b ecome inactiv e. Theorem 5. L e t z ( n ) b e a r andom variable whose value is the smal lest time t at which none of the g adgets ar e active. Then, E ( z ( n )) = Ω(log n ) . Pr o of. Consider gadget F i . Note that for ev ery t > 0, A i (1) ∩ A i (2) ∩ · · · ∩ A i ( k ) = A i ( k ) and P ( A i (0)) = 1. Let t 0 = ⌈ lg n ⌉ . Then , for ev ery t ′ ≥ t 0 , P ( A i ( t ′ )) = P ( A i (0)) t ′ Y t =2 P ( A i ( t ) | ∩ j 1 n , b y (7). Let Q t ′ = ∩ i A i ( t ′ ) b e the ev en t that none of the n gadgets are activ e at time t ′ . S ince ev en ts of d ifferen t gadgets are indep end en t, it holds for an y t ′ ≥ t 0 that P ( Q t ′ ) = n Y i =1 (1 − P ( A i ( t ′ )) ≤  1 − 1 n  n ≤ e − 1 . Then, b y definition of exp ectatio n, E ( z ( n )) = ∞ X t =1 Pr( Q t ) ≥ t 0 · P r( Q t 0 ) ≥ (1 − e − 1 ) lg n . DISTRIBUTED SCHEDULING IN THE SINR MODEL 13  Note that b ounding E ( z ( n )) su ffices to low er b ound the exp ected time b efore all links successfully transmit, since by definition a link cannot succeed as long as the corresp onding gadget is activ e. 5. Tight Boun d on Anal ys is via A W e ac hiev e d a O (log n )-appr o ximation by a v oiding the measure A in our analysis. In con trast, the O (log 2 n ) b oun d in [20] is ac hiev ed by p ro ving t w o sep arate b oun ds inv olvi ng A : first ALG = O ( A log n ), and second A = O ( χ log n ), where ALG is the exp ected time tak en b y the algorithm. The tight ness of the b ound on ALG u nder any oblivious p o w er assignmen t follo ws from Section 4, as it is easy to verify that A = Θ(1) in that construction. W e giv e a construction b elo w f or whic h the second b oun d is tight. Thus, going through A is not sufficien t to obtain improv ed b oun ds, and differen t analysis is required. Our construction uses u niform p o w er. T his is necessary , since for other oblivious p o w er assign- men ts A = O ( χ ), by recen t results of [13]. Theorem 6. F or every numb ers ˆ n and every numb er t , ther e i s a set ˆ L of ˆ n links with χ ( ˆ L ) = Θ( t ) and A ( ˆ L ) = Ω( χ ( ˆ L ) log ˆ n ) under u niform p ower. This lemma shows, p erhaps surprisingly , th at there can b e a h uge difference b et w een th e in- affectance and out-affec tance of a link in a f easible set, thereby illustrating the need for the bi- feasibilit y concept. Lemma 5.1. F or every n , ther e is a set L of n links on the line and a link l 0 ∈ L , such that under uniform p ower, L i s fe asible while a 0 ( L ) = Ω(log n ) . Pr o of. W e form the set L = { l 0 , l 1 , . . . , l n − 1 } as follo w s. T he sender s i of link l i is p ositioned at co ordinate d ( s 0 , s i ) = c · i 1 /α · 2 i , w here c > 1 is a constan t to b e determined. The length of th e link l i is ℓ i = 2 i and the r eceiv er r i is p ositioned at r i = s i + ℓ i = ( c · i 1 /α + 1)2 i . Then, a 0 ( L ) = n − 1 X i =1  ℓ i d 0 i  α = n − 1 X i =1  2 i ( c · i 1 /α + 1)2 i  α < 1 (2 c ) α n − 1 X i =1 1 i = Ω(log n ) . T o sho w feasibilit y , we first b ound distances b et w een links by: d i − 1 ,i = d 0 i − d ( s 0 , s i − 1 ) = ( c · i 1 /α + 1)2 i − c ( i − 1) 1 /α 2 i − 1 > c · i 1 /α 2 i − 1 , 14 MAGN ´ US M. HALLD ´ ORSSON AND PRADIPT A MITRA and for m > 0, d i + m,i = d ( s 0 , s i + m ) − d ( s 0 , r i ) > c ( i + m ) 1 /α 2 i + m − ( c · i 1 /α + 1)2 i > ci 1 /α 2 i + m − ( c · i 1 /α + 1)2 i + m − 1 = (2 c · i 1 /α − ( c · i 1 /α + 1))2 i + m − 1 ≥ ( c − 1)2 i + m − 1 . W e then b ound the in-affectance of eac h link by a L ( i ) = X k ,k i a k ( i ) ≤ i · a i − 1 ( i ) + n − i − 1 X m =1  ℓ i d i + m,i  α ≤ i ·  ℓ i d i − 1 ,i  α + n − i X m =1  2 i ( c − 1) · 2 i + m − 1  α ≤ i ·  2 i c · i 1 /α · 2 i − 1  α + 1 ( c − 1) α X m =0  1 2 α  m = 2 c α + 1 ( c − 1) α · 2 α 2 α − 1 . Th us, when c ≥ 1 +  3  1 + 1 2 α − 1  1 /α , it holds that a L ( i ) ≤ 1 for eac h link l i , i.e., L is f easible.   W e no w turn to pr o ving Thm. 6. W e construct the set ˆ L that satisfies the claim of the theorem. Let L b e th e set feasible u nder uniform p o w er and the link l 0 = ( s 0 , r 0 ) ∈ L with a L ( l 0 ) = Ω(log n ), promised by Lemma 5.1 . Let L 1 denote t isometric copies of L with links in the s ame p osition as L . W e next tak e an arbitrary s et S on n links that is feasible under u niform p o w er, and s cale its distances so that maximum p airwise distance is o (1). F or ins tance, w e can let S = { l ′ 1 , l ′ 2 , . . . , l ′ n } b e ˆ L scaled b y a factor of 4 − n so that the length of l ′ i = ( s ′ i , r ′ i ) is 2 i − 2 n , s ′ i = ci 1 /α 2 i − 2 n , and r ′ i = s ′ i + 2 i − 2 n . By the same argum ent as Lemma 6, S is feasible; observ e also that pairwise distances of p oin ts within S are O ( n/ 2 n ) = o (1). Let L 2 denote t copies of S , with the same co ordinates; th us, the n o des of L 2 are all close to the n o de s 0 in L 1 . Finally , we form the com bined instance ˆ L = L 1 ∪ L 2 with a total of ˆ n = 2 tn links. Observe that for ev ery l j ∈ L 2 and l i ∈ L 1 ( i > 0), that d j i = d 0 i (1 + o (1) ). Since we use uniform p ow er, it h olds for eac h of the tn links l j ∈ L 2 that a j ( L 1 ) = Θ( a 0 ( L 1 )) = Θ( t · a 0 ( L )) = Ω( t log n ) . Th us, a ˆ L ( ˆ L ) ≥ a L 2 ( L 1 ) = | L 2 | Ω( t log n ) , implying that A ≥ 1 | ˆ L | a ˆ L ( ˆ L ) = Ω( t log ( ˆ n/t )) . On the other han d , the set ˆ L clearly h as a s cheduling num b er of 2 t , as it is formed by 2 t feasible sets. Hence, the th eorem. DISTRIBUTED SCHEDULING IN THE SINR MODEL 15 6. Conclusions W e hav e giv en a distribu ted sc heduling algorithm that is O (log n )-appr o ximate in th e sc heduling mo del, and sho wn this facto r cannot b e impro v ed in general. Our low er b ound construction, ho w ev er, applies only to in s tances w ith small sc heduling num b er. A similar randomized scheduling algorithm was shown by F angh¨ anel et al. [7] to yield an asymp - totic constan t-factor ap p ro ximation for the case of linear p o w er assignment. One key difference is that in the case of linear p ow er, all links ha v e lo w affectance ( O ( χ )), while for general sublinear length-monotone p ow er assignments this only holds on a v erage. It remains an imp ortant and in triguing op en question wh ether a b etter asymptotic appro ximation ratio can b e obtained. 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Then, S c an b e p artitione d into t =  4 p  2 sets S 1 , S 2 , . . . , S t , e ach satisfying a v ( S i ) ≤ p , for every l v ∈ S i . Pr o of. W e fir st partition S in to a sequence T 1 , T 2 , . . . of sets as follo w s. Ord er the links in S in decreasing order. F or eac h link l v , assign l v to the firs t set T j for which a v ( T j ) ≤ p/ 2, i.e. the accum ulated affectance of l v on the p r evious, longer links in T j is at m ost p/ 2. Since eac h link l v originally had out-affecta nce at most 2, then b y the additivit y of affectance, the n umber of sets used is at most ⌈ 2 p/ 2 ⌉ = ⌈ 4 p ⌉ . W e then rep eat the same approac h on eac h of the sets T i , pro cessing the links this time in increasing ord er. The num b er of sets is again ⌈ 4 p ⌉ for eac h T i , or ⌈ 4 p ⌉ 2 in total. In eac h fin al slot (set), th e affectance of a link on the sh orter links in the same slot is at most p/ 2. In tota l, then, the out-affectance of eac h lin k is at most 2 · p/ 2 = p .  ICE-TCS, School of C omputer Science, Reykja vik Unive rsity, Iceland. E-mail addr ess : mmh@ru.is, ppmitra@gmai l.com