Decentralized Search with Random Costs

Decen tralized Searc h with Random Costs Osk ar Sandb erg ∗ No v ember 2 1, 2018 Abstract A decentralized search algorithm is a metho d o f r outing on a ran- dom graph that use s only limited, lo cal, information ab out the r eal- ization of the graph. In some random graph mo dels it is pos sible to define suc h algorithms which pro duce short paths when routing from any v ertex to any other, while for others it is not. W e consider random gr aphs with random costs assigned to the edges. In this situation, w e use the metho ds o f sto chastic dynamic progra mming to crea te a decentralized search metho d which a ttempts to minimize the total cos t, rather than the num b er of steps, of each path. W e show that it succeeds in doing so a mong all decentralized search algo rithms which monotonically appr o ach the destina tion. Our algorithm depends on knowing the expected cos t of routing from every vertex to any other , but we show that this may b e ca lculated itera- tively , and in practice can be ea s ily estimated from the cost of previous routes and compressed in to a small routing table. The metho ds applied here can also b e applied directly in other situations, such as efficien t searching in g raphs with v arying vertex degr ees. 1 In tro duction Jon K leinb erg introdu ced the co ncept of decent ralized searc h algorithms in his celebrated 2000 pap er on the Small-W orld p henomenon [1]. I n particular, he sh o wed that in certain rand om graphs it is p ossible to fin d paths b et ween v ertices of p oly-loga rithmic length ev en when limited to usin g only lo cal kno wledge at eac h step, w hile in o thers it is n ot. Most of the b y now large canon of work in the area (see [2] for a recen t surve y) has b een ded icate d to finding and analyzing algorithms th at route b et w een t w o giv en v ertices in a small n umber of steps. T ypically , the b est ∗ The Department of Mathematical Sciences, Chalmers U nivers ity of T ec hnology and G¨ oteb org U nive rsity . ossa@math.chalme rs.se 1 1.42 12.34 1 2 3 4 5 6 7 8 9 Figure 1: If the goal is to reac h vertex 9 in the f ew est steps, then certainly v ertex 1 should c h oose the long edge to vertex 7. Ho w ev er, if the goal instead is to minimize the total cost, routing to 2 might b e a b etter c hoice. metho d in these situation is greedy: p rogress to the neigh b or w hic h is closest to the destination. I n this p ap er, we consider a generalized situation where the cost of p assing down an edge is not fixed, but ma y b e a random v ariable whose v alue is kn o wn when the c hoice of where to pro ceed to next is made. The goal is then the minim ize the cost of reac hing the destination, w hic h ma y lead to different priorities wh en routing (one m a y not wish to route to a v ertex v ery close to the d estination if the cost of passing down that edge is very high, see Figure 1). This is a pr oblem similar to that applied to time pr ocess in the field of sto c hastic dynamic pr ogramming [3], and we use similar metho ds. The basic id ea is this: If a v ertex kno ws the exp ected cost of eac h of h is neigh b ors r outing to the destination, and also the cost of h im routing to eac h of his neighbors , it makes sense for him to c h oose as the next step that neigh b or which min imizes th e sum of th ese tw o costs. While the exp ected costs are difficult to calculate analyticall y , w e fin d that th ere is little rea- son to do so. When m an y queries are p erformed, one ma y start by usin g an y guess as to these v alues, and th en up d ate these guesses based on past exp erience. W e w ill sho w analytically that this form of search is we ll defined, and that it is optimal among algorithms that monotonically app r oac h the destina- tion, as well as pr esenting some results on the order of the total cost as w ell as approximati on trade-offs. W e cont inue with a discussion and some exp erimen tation on th e p ractica lit y of th e app roac h. Finally , we apply the algorithm with greater generalit y , and see that it p erforms w ell also in these cases. In p articular, we note 1that it can b e applied to routing with non- homogeneous d egree distribu tions. 1.1 Previous W ork The original work on decen tralized searc h was done by Kleinberg in [1] and [4]. Muc h w ork has since b een don e on related problems, in particular further generalization of the results and imp ro vemen ts on the b ounds (see [5] [6] [7] [8] [9] f or some examples). 2 In a b id to improv e the p erformance of F r eenet [10], a decen tralized p eer- to-p eer net work, Clarke [11] prop osed an algorithm in some wa ys sim ilar to that present ed here u nder th e name “NG Routing”. The metho d was im- plemen ted in F reenet at the time, but has n ot b een u sed since the netw ork w as r e-engineered to r oute according to the metho d d escrib ed in [12]. W e b eliev e that the tec hn ical and architec tural problems exp erienced b y previ- ous versions of F reenet we re unrelated to w hat is discussed b elo w, and, in ligh t of our resu lts, that Clark e’s ideas were fu n damen tally soun d . S ¸ im¸ sek and Jens en also p rop osed an algorithm based on the same principles in [13], in tended for routing b ased on b oth verte x degree and s imilarit y . Their “exp ected-v alue na vigation” is b ased on the same idea as our cost- greedy searc h, ho w ev er they use a rough estimate of the exp ected r outing time, whic h cannot b e applied to our problem, to mak e the decisions. They present n o analytic results. F or our tak e on searc h in graphs with v ariable no de d egree, see S ection 6.2 b elo w. F or an int ro duction to the field of sto c hastic dynamic pr ogramming, see the monograph by Ross [3]. 1.1.1 A not e on terminology The terminology regarding algorithms for decen tralized path-findin g in ran- dom graphs has not yet settled, and different authors h a ve used differen t terms. W e hav e chosen the term “decen tralized searc h” (follo wing recen t w ork by Klein b erg [2]) bu t others terms hav e b een used to d escrib e the same thing. “Navi gation”, u sed in e.g. [4] [14], is quite common, b ut p erhaps not as d escriptiv e. W e av oid form ally calling our algorithms “routings” as this name has pr eviously b een used in computer s cience literature to d escrib e flo w assignmen ts through a graph with limited edge capacities (for example [15]), a different problem from that currently studied. Ho w ev er, we do use the terms “route” and “to route” follo win g their dictionary d efinitions. 1.2 Organization In S ectio n 2 we set out the b asic defin itions of decent ralized searc h, as w ell as rigorously defin e our new “cost-greedy” searc h algorithm. F ollo wing this, in Section 3 w e prov e the main resu lts regarding cost-greedy search in net works with sufficien t in dep endence. In Sections 4 and 5 we set out the metho ds for applyin g the r esults in p ractice, and, fin ally , in Section 6 we p erform sim ulated exp erimen ts to lo ok at the actual p erformance of the algorithm. 3 2 Definitions A cost graph G = ( V , E , C ) is a graph consisting of a vertex set V , a p ossibly directed set of edges E , and a colle ction of costs, C . F or ea c h elemen t ( x, y ) ∈ E , C ( x, y ) = C (( x, y )) is an i.i.d. p ositiv e random v ariable giving the cost of tra v eling down that edge (time tak en). G ma y b e a random graph, by wh ic h we mean that for eac h x, y ∈ V , there exists a random ind icator v ariable E x,y sa yin g wh ether there is an edge from x to y . These may b e dep endent and differently distrib u ted. Definition 2.1. F or a given c ost gr aph G , a z -searc h for a vertex z ∈ V , is a mapping A : V 7→ V such that: 1. A ( x ) = y only if ( x, y ) ∈ E . 2. A ( z ) = z . 3. F or al l x ∈ V ther e exists is k < ∞ such that A k ( x ) = z . A search of G is a c ol le c tion of z - r outing algorithms for al l z ∈ V . W e call d a distanc e on a set V if d : V × V 7→ R + , if for x, y ∈ V , d ( x, y ) = 0 implies that x = y , an d if for z , y , z ∈ V , d ( x, z ) ≤ d ( x, y ) + d ( y , z ) . A distance is th us a metric without the symmetry requiremen t. In particular, an y connected digraph G implies a distance d G . F or x ∈ V , N ( x ) = { y ∈ V : d G ( x, y ) = 1 } is the set of neigh b ors of x in G . Definition 2.2. A distanc e d is adapted f or searc h in a c onne cte d gr aph G if for every x, z ∈ V , wher e x 6 = z , ther e exists a y ∈ N ( x ) such that d ( y , z ) < d ( x, z ) . A distance function d is th us adapted for searc h if it, in some sen s e, r eflects the structure of G . The most ob vious example is of course d G itself, but d ma y also b e, for instance, graph d istance on any connected sp anning s ub- graph H of G . Another imp ortan t case is that if V is a set of p oin ts in a metric space, then the space’s m etric is adapted for searc h in V ’s Delauna y triangulation (see [9 ]). Definition 2.3. Given a c ost gr aph G , and a vertex z , a decen tralized searc h is a z -se ar ch A , such that for any x ∈ V the r andom variable A ( x ) is me asur able with r esp e ct to: 1. E x,y for al l y ∈ V . 4 2. C ( x, y ) for al l y ∈ N ( x ) . In tuitiv ely , th is means that as w ell as an y information ab out the graph mo del, routing at x m ay u se information ab out w hic h v ertices x do es (and do es n ot) h a ve ed ges to, as we ll as the costs of p assing down those edges. The definition of decen tralized s earch as originally giv en by Kleinb erg w as sligh tly broader than this, allo w in g a route started at a v ertex x to use all the information from A 0 ( x ) , A 1 ( x ) , . . . , A k − 1 ( x ) when taking its k -th step. Because ou r analysis will b e restricted to algorithms meeting the follo wing criteria, excluding this in formation will mak e little difference: Definition 2.4. Given a g r aph G , a distanc e d adapte d for se ar ch in G , and a ve rtex z , a forwa rd searc h is a z -se ar ch F such that for al l x ∈ V \{ z } d ( F ( x ) , z ) < d ( x, z ) . F or a giv en searc h A an d vertex z S z ( x ; A ) = in f  k : A k ( x ) = z  is the num b er of steps it take s to reac h z from x u sing A . Let T z ( x ; A ) = S z ( x ; A ) X i =1 C  A i − 1 ( x ) , A i ( x )  whic h is the c ost . 2.1 Greedy and W eigh ted Greedy Searc h In th e follo win g, we fix z . Gr e e dy se ar ch is given by A G ( x ) = argmin y ∈ N ( x ) ( d ( y , z )) . (1) This is alw ays a d ecentrali zed searc h, and if d is adapted, than it is a forwa rd searc h (if d is not adapted, it ma y not b e we ll defi n ed to b egin with). Standard greedy searc h d o es n ot tak e the costs C in to accoun t. A v ariant that do es, is weighte d gr e e dy se ar ch . F or a giv en z , let w z ( x ) for x ∈ V b e a collectio n of we igh ts, where w z ( z ) = 0. These weig hts sp ecify a searc h algorithm A w ( x ) = argmin y ∈ N ( x ) ( C ( x, y ) + w z ( x )) (2) whic h may or may n ot b e we ll defined. I f w e restrict this to b eing a forward searc h , we get F w ( x ) = argmin y ∈ N ( x ) : d ( y ,z ) d ( x, z ) and d ( v , z ) > d ( u, z ), y ❀ x is indep enden t of v ❀ u if y 6 = v . Examples of s u c h grap h s are addin g outgoing edges from eac h ve rtex with destinations chosen indep endently (as in Klein b erg’s w ork [1 ]) or allo wing eac h edge, either seen as directed or undirected, to exist ind ep enden tly of all others (lik e in classical random graphs and long-range p ercolation [16]). W e call a grap h constructed in this manner e dge indep endent . Theorem 3.1. F or an e dge indep endent r andom gr aph G , ther e exists a solution ˜ w to e quation (4) so that c ost-gr e e dy se ar ch is wel l define d. ˜ w is a glob al ly attr active fix-p oint of the iter ation given b y w i +1 = E [ T z ( x ; F w i )] . (5) 6 Pr o of. Define the rank of x with resp ect to z , r z ( x ), as the p osition of x when all the elemen ts in V \{ z } are ordered by in creasing d istance f r om z , using s ome d eterministic tie-breaking ru le. Let r z ( z ) = 0. W e will pro ceed by ind uction on r z ( x ). Let w 0 b e an y we igh ting. Let x ∈ V such th at r z ( x ) = 1. F or an y forwa rd searc h F , F ( x ) = z , wh ence E [ T z ( x ; F )] d oes not dep end on the algorithm. In particular E [ T z ( x ; F w )] is do es not d ep end on w , whence w i ( x ) is constant for i ≥ 1. Let r z ( x ) = k , and assume that for all y ∈ V suc h that r z ( y ) < r z ( x ), w i ( y ) tak es the same v alue for all i ≥ k − 1. This means that E [ T z ( x ; F w i )] tak es the same v alue for all i ≥ k . It follo ws that for all x ∈ V w i ( x ) is fixed for i ≥ r z ( x ). Hence w k = w k +1 for all k ≥ n − 1 , and ˜ w = w n is a solution to (4). Theorem 3.2. L et G b e an e dge indep endent g r aph, and F a forwar d se ar ch for a vertex z ∈ V . Then for al l x ∈ V E [ T z ( x ; F ˜ w )] ≤ E [ T z ( x ; F )] wher e F ˜ w is c ost-gr e e dy se ar ch for z as in Definition 2.5. Pr o of. Use th e same definition of r z ( x ) as in the pro of of Theorem 3.1. Like there, w e will use indu ction on r z ( x ). If r z ( x ) ≤ 1, th en all forward searc hes from x are the same, and there is nothing to prov e. Let F b e an y forward search. Giv en x ∈ V , assume that for all y ∈ V suc h that r z ( y ) < r z ( x ), E [ T z ( y ; F ˜ w )] ≤ E [ T z ( y ; F )]. Let v = F ( x ) and v ∗ = F T ( x ), the places th e resp ective algorithms c h oose as th e next step. Belo w, w e mean by “lo cal kno wledge” that which decen- tralized algorithm may u s e, as giv en in 2.3. W e note, crucially , th at b ecause of our assumptions, T z ( v ; F ) is in d ep enden t of lo cal knowledge at x , while C ( x, y ) is measurable with resp ect to it. E [ T z ( x ; F )] = E [ E [ T z ( x ; F ) | lo cal kn o wledge at x ] ] = E [ D ( x, v ) + E [ T z ( v ; F )] ] ≥ E [ D ( x, v ) + E [ T z ( v ; F ˜ w )] ] ≥ E [ D ( x, v ∗ ) + E [ T z ( v ∗ ; F ˜ w )] ] = E [ T z ( x ; F ˜ w )] where the first inequalit y follo ws by induction sin ce r z ( v ) < r z ( x ) and the last b ecause the expression inside the first exp ectation is what v ∗ minimizes. 7 3.1 The Small-W orld Graph A p articular example graphs meeting th e criteria of the last chapter are the small-w orld augmenta tions first int ro duced b y Kleinber g [1]. This constru c- tion starts with a fixed fin ite grid H , letting d = d H , and creating G by adding a rand om outgoing directed edge f rom eac h v ertex x to destination a y with p r obabilit y P ( x ❀ y ) ∝ 1 /d G ( x, y ) α (6) d is naturally adapted for routing in G . F or simplicit y , w e let H b e a ring of n v ertices (Kleinb erg originally u s ed a t wo-dimensional squ are lattice, but the pro ofs are iden tical). Let G ( n, α ) b e the family of rand om graphs so constructed. Using previous results ab out greedy routing on suc h graph s, w e can calculate the cost order of cost-greedy searc h . In particular, we can see the order in n cannot b e differen t from greedy r outing. Theorem 3.3. (Kleinb er g) If G ∈ G ( n, α ) with α = 1 ther e exists N 1 such that f or n ≥ N 1 , E [ S z ( x ; A G )] ≤ k 1 log n log d ( x, z ) wher e k 1 is a c onstant indep endent of x , z , and n . F urther results ab out su c h graphs, prov ed in [5] and [17] r esp ectiv ely are Theorem 3.4. (Barr ier e et al.) If G ∈ G ( n, α ) with α = 1 , then ther e exists N 2 such that for n ≥ N 2 , E [ S z ( x ; A G )] ≥ k 2 log n log d ( x, z ) wher e k 2 is a c onstant indep endent of x , z , and n . Theorem 3.5. (Singh Manku) If G ∈ G ( n, α ) , with α ≥ 0 , then E [ S z ( x ; A G )] ≤ E [ S z ( x ; A )] for any x, z ∈ V ( G ) and de c entr alize d se ar ch A . T ogether, these allo w us to p ro ve the observ ation that Prop osition 3.6. If G ∈ G ( n, α ) and 0 < E [ C ( x, y )] < ∞ then E [ T z ( x ; F ˜ w )] = Θ(log n log d ( x, z )) 8 Pr o of. Th e upp er b oun d comes directly f rom Theorems 3.2 and 3.3, since E [ T z ( x ; F ˜ w )] ≤ E [ T z ( x ; A G )] = E [ C ( x, y )] E [ S z ( x ; A G )] ≤ E [ C ( x, y )] k 1 log n log d ( x, y ) . where the middle equalit y follo w s from the fact A G routes indep end en tly of the costs, and the simple form of W ald’s equation. T o pro v e th e up p er b ound, consider all edges in the graph as directed, letting the edges of the H b e denoted by doub le d irected edges. S ince a forw ard searc h can only ev er tra v erse an edge in one dir ectio n, this d o es n ot affect its cost. Now let R z ( x ; F ) = S z ( x ; A ) X i =1 C min ( F ( x ) i − 1 ) (7) where C min ( x ) = min { C ( x, y ) : y ∈ N ( x ) } . This count s, at eac h step, the minim um cost of an y outgoing edge, rather than the cost of the edge whic h w as actually used. Since the degree of eac h vertex is fixed, C min ( x ) is i.i.d. for all x . Let S i b e the σ -algebra generated b y all th e information seen in steps 1 , 2 , . . . , i of th e searc h (as listed in Definition 2.3). Note: • C min ( F i +1 ( x )) is ind ep endent of S i . • S z ( x ; A ) is a Stopp ing Time w ith resp ect to the filtration { S i } ∞ i =1 . Th us we ma y use W ald’s Equ ation to conclude that E [ R z ( x ; F )] = E [ C min ( x )] E [ S z ( x ; F )] . (8) W e n o w use the immediate fact that R z ( x ; F ) ≤ T z ( x ; F ), follo w ed by (8) and T heorems 3.5 and 3.4, to conclude E [ T z ( x ; F )] ≥ E [ R z ( x ; F )] = E [ C min ( x )] E [ S z ( x ; F )] ≥ E [ C min ( x )] E [ S z ( x ; A G )] ≥ E [ C min ( x )] k 2 log n log d ( x, z ) for suffi cien tly large n . Since this holds f or an y forward-searc h F , it holds in p articular for F ˜ w . Prop osition 3.6 tells u s that in this mo del, the order of cost-greedy routing will not b e different from that of greedy rou tin g. The pro of of the lo wer b ound assumes, how ev er, that E [ C ( x, y )] < ∞ and that the degree of eac h v ertex is b ounded as n gro ws. Neither of these things, and particularly not the latter, necessarily h old in applications. 9 3.2 Appro ximated W eigh t s W e consid er the situation when the solution ˜ w to (5) is not kno wn exactly but appro ximated b y an other set of we igh ts. Prop osition 3.7. If ˜ w is the solution to (5) and w another set of p ositive weights such that max x ∈ V | w ( x ) − ˜ w ( x ) | ≤ ǫ then for any e dge-indep endent gr aph of size n E [ T z ( x ; F w )] − E [ T z ( x ; F ˜ w )] ≤ 2 nǫ (9) and mor e gener al ly, for any k ≥ 0 E [ T z ( x ; F w )] − E [ T z ( x ; F ˜ w )] ≤ 2 ǫ ( k + n P ( S z ( x ; F w ) > k )) . (10) Pr o of. Let err w ( x ) = E [ T z ( x ; F w )] − E [ T z ( x ; F ˜ w )]. It follo ws th at E [ T z ( x ; F w )] = E [ C ( x, F w ( x )) + E [ T z ( F w ( x ); F w )]] = E [ C ( x, F w ( x )) + E [ T z ( F w ( x ); F ˜ w )]] + err w ( F w ( x )) = E [ C ( x, F w ( x )) + ˜ w ( F w ( x ))] + err w ( F w ( x )) . No w, since b y the definition of a wei gh ted greedy searc h C ( x, F w ( x )) + w ( F w ( x )) ≤ C ( x, y ) + w ( y ) for all y ∈ N ( x ) ≤ E [ C ( x, F ˜ w ( x )) + w ( F ˜ w ( x )) − w ( F w ( x )) + ˜ w ( F w ( x ))] + err w ( F w ( x )) ≤ E [ C ( x, F ˜ w ) + ˜ w ( F ˜ w ( x ))] + E | w ( F ˜ w ( x )) − ˜ w ( F ˜ w ( x )) | + E | ˜ w ( F w ( x )) − w ( F w ( x )) | + err w ( F w ( x )) ≤ E [ T z ( x ; F ˜ w )] + 2 ǫ + err w ( F w ( x )) It follo ws that for any k ≥ 0 err w ( x ) ≤ 2 ǫk + err w ( F k w ( x )) . (11) If k > S z ( x ; F w ) then F k w ( x ) = z and err w ( F k w ( x )) = 0, so (9) follo ws s ince n > S z ( x ; A ) for all searc h es. T o prov e (10), n ote that by th e same reasoning err w ( F k w ( x )) = err w ( F k w ( x ) | F k w ( x ) 6 = z ) P ( S z ( x ; F w ) > k ) . Since the graph is edge ind ep enden t err w ( F k w ( x ) | F k w ( x ) 6 = z ) is s imply the error f r om some p oint wh ic h is not z , bu t where (9 ) s till applies. 10 What the p rop osition, and in particular (11) sa y s is that if an appro ximation w of ˜ w is off by ǫ , then eac h step in the routing add s at most 2 ǫ to the optimal routing time. Th is is intuitiv ely clear, sin ce w hile F w ma y c ho ose the wron g v ertex in a given step, it can on ly do so wh en the total (actual) cost of routing via that v ertex is within 2 ǫ of the cost of routing via the real one. F or the same reason, it is unlik ely that a b etter b oun d can b e ac h iev ed without fu rther assumptions on the graph an d the cost distribution. 4 Calculating the W eigh ts Theorem 3.1 pro vides us with a metho d of calculating the weig ht s ˜ w f or cost-greedy searc h. One can start b y assigning any initial weigh ting w 0 , and then calculate w 1 , w 2 , . . . u s ing (5). A closed analytic form for E [ T z ( x ; F w )] as a function of the vec tor w is probably v ery d ifficult to find , ev en in the most simp le situations. On e can note h o wev er that it can b e wr itten recursive ly as E [ T z ( x ; F w )] = X ( E [ T z ( x ; F w )] + E [ C ( x, y ) | F w ( x ) = y ]) P ( F w ( x ) = y ) where the sum is o ver all y ∈ V suc h that d ( y , z ) < d ( x, z ). In the v ery s im p lest cases (suc h as a directed lo op with one augmented outgo- ing shortcut c hord p er verte x) it is p ossible to calculate P ( F w ( x ) = y ) and E [ C ( x, y ) | F w ( x ) = y ] analytically , in w hic h case E [ T z ( x ; F w )] can b e cal- culated numericall y by recursion. Because this is complicated, and u nlik ely to b e of muc h in terest in pr actice , we do not linger on it. A m uc h more rew arding strategy is to calculat e the weig hts empirically . That is, start b y simula ting a large n um b er of searc h es from randomly c hosen p oin ts using F w 0 . While this is b eing done, s amp le the a v erage routing cost to z fr om eac h vertex (due to the Mark o vian n atur e of forward searc h on an edge indep enden t graph, a ve rtex ma y take a sample eve ry time a qu ery passes through it). After a suffi cien t num b er of queries, the av erage should b e an estimate at E [ T z ( x ; F w 0 )] by the law of large n umbers. One ma y then tak e the av erage costs from eac h p oint as w 1 , and contin ue in this manner. Prop osition 3.7 ind icates h o w close an approximat ion is n eeded, but u nfortunately it is n ot strong enough to derive a rigorous b ou n d us ing a p olynomial num b er of s amples. F urther, we note tw o things ab out the sampling implemen ted. Firstly , one needs to b e careful ab out the wa y th e rep eated queries are done. Since we w an t th e edge costs to b e r andom, C ( x, y ) must b e p ic ke d anew, indep en- den tly , for eac h query samp led. If the graph is r andom, th e edges may b e redra wn, bu t must n ot – it sim p ly d ep ends on whether they are to b e seen as r andom or fi x ed edges in the G ab o v e. 11 Secondly , the p ro of of Theorem 3.1 guaran tees con vergence in n steps, mean- ing that an optimal routing is ac hiev ed once w n has b een calculated (if an empirical metho d is used, the resulting w eigh ting may still suffer i naccuracies due to the sampling). Th is is an unfortun ately large num b er of iterations, esp ecially giv en that eac h ma y require sim ulating a large num b er of qu eries, but we find th at in practice, muc h fewer iterations (t ypically t w o or three, ev en for very large net w orks) are needed, see Section 6 b elo w. 5 Practicalit y and Decen tralization W e p ro ceed to discuss actual app licatio ns of Definition 2.5. O n the face of it, the routin g metho d describ ed do es n ot seem particularly p ractica l. Even if we can calculate th e wei gh ting ˜ w , this giv es a routing table of size of n , and suc h a table is needed for eac h z we w ish to route for. The complete table of w eigh ts needed to route b et ween an y t wo v ertices is th us of size n 2 . Sev eral assu mptions can help here ho w ev er. Cen t ralization: T ranslation In v ariance If we assum e that the graph is translation inv arian t, then E [ T z ( x ; F )] = E [ T 0 ( x − z ; F )] so x needs on ly kno w the routing cost fr om eac h starting ver- tex to a distinguished v ertex 0. In fact, in man y cases (such as the common case of au gmenting sin gle cycle with rand om outgoing edges) E [ T z ( x ; F T )] ma y b e exactly , or at least approxi mately , a function of d ( x, z ), in whic h case x need only kno w the exp ected cost of r outing a given d istance. Th is kno wledge is the same for all x , so may b e calculated as a single, global, v ector. Decen t ralization If one wish es for a completely decentral ized searc h system, as, for instance in p eer-to-p eer systems suc h as [10], then one cannot store a global v ector of weigh ts. Instead, eac h vertex m ust store the weig hts needed to route to ev ery other v ertex. In particular, eac h vertex x n eeds to b e able to calculate w z ( y ) for eac h y ∈ N ( x ) and z ∈ V . If one assumes translation in v ariance, x need only store one such weigh t vec tor, and can translate it to apply to its n eigh b ors . Without such in v ariance, it needs to store | N ( x ) | v ectors. W eigh t V ector Compression In b oth cases ab ov e, ho we v er, w e are still left with a r outing table size of at least n , whic h is defi n itely n ot desirable. The heart of what mak es our 12 Figure 2: Rather than storing ev ery v alue of w v ector (the d otted line) as a routing table, we store v alues at exp onent ially increasing p ositions, and use these to appr o ximate the v alues b etw een them. metho d practically usefu l co mes from the fact that the previous theory ab out decen tralized searc h mak es compression to a logarithmic s ize p ossib le. If we consider graphs of t yp e G ( n, α ) d escrib ed in Section 3.1, we kno w from Prop osition 3.6 that E [ T z ( x, F ˜ w )] ≈ c E [ C ] log ( n ) log( d ( x, z )) . The utilit y of this is that if we k n o w that E [ T z ( x, F T )] gro ws logarithmically with d ( x, z ) (as indicated b y Prop osition 3.6), w e are motiv ated to assume that it, and th us the w eigh ts ˜ w in Definition 2.5, can b e appro ximated b y assuming ˜ w ( x ) and ˜ w ( y ) hav e similar v alues if x and y are su ch that log( d ( x, z )) ≈ log ( d ( y , z )). In particular if 0 ≤ d ( y , z ) − d ( x, z ) ≤ r we get | ˜ w ( y ) − ˜ w ( x ) | ≈ c 1 E [ C ] log( n ) r d ( x, z ) . (12) It is easy to pr o ve , using the same metho ds as in the pr o of of T heorem 3.3 that f or greedy routing in G ( n, α ) P ( S z ( x ; A G ) ≥ log 3 n ) ≤ c 2 log n n . Assuming that a similar b ound holds for F ˜ w , equation (10) with k = log 3 n in Pr op osition 3.7, gives that E [ T z ( x ; F w )] − E [ T z ( x ; F ˜ w )] ≤ 4 max x ∈ V | ˜ w ( x ) − w ( x ) | log 3 n so if r < ǫd ( x, z ) / 4 log 3 n in (12), then E [ T z ( x ; F w )] − E [ T z ( x ; F ˜ w )] < ǫ. Th us, the weig ht w ( x ) can b e s u bstituted by the w eigh t of a v ertex ǫd ( x, z ) / log 3 n steps fr om x . T o do this, we d ivide the routing d istances into zones of size 2 i for i = 0 , 1 , 2 , . . . , and record only the w eigh ts of log 3 n/ǫ eve nly spaced v ertices within eac h zone (Figure 2). The routing table th u s conta ins a 13 3 4 5 6 7 8 9 10 11 100 1000 10000 100000 1e+06 Steps Size Greedy Cost-Greedy Figure 3: Th e steps take n by greedy and cost-gree dy searc h w hen all edge costs are fi xed to 1. In theory , the latter should conv erge to th e former (whic h is optimal) but due to the inaccuracy of the estimates and the routing-table compression cost-greedy here p erforms ab out 5% worse than the optimum. p olylogarithmic num b er of ent ries ( O (log 4 n )), and y et by using th e clos- est recorded w eigh t as a su bstitute for w ( x ), w e incur only an ǫ -error on the total routing cost. Proving this r igorously , ho w ev er, dep ends on tight er b ounds then Prop osition 3.6 or eve n Th eorems 3.3 and 3.4 pr o vide. W e w ill see exp erimenta lly in S ectio n 6 that a routing table of size around E [ S z ( x ; A G )] w orks w ell in practice, b oth when using a single v ector and in a d ecen tralized system. While it may seem lik e a limitation that this will only work on graphs where routing in a logarithmic num b er of s teps is p ossible, th ose are lik ely to accoun t for most situations where decen tralized searc h is of in terest. Bey ond Klein b erg’s small-wo rld mo del, other cases where decen tralized routing is exp ected to tak e a logarithmic n umber of steps are hyp ercub es (where the hamming d istance is adapted for routing), and Chord net w orks [18] (where the circular distance is). 6 Exp erimen ts 6.1 Direct Applications W e start by simula ting the algorithm und er the most b asic conditions. W e let G consist of a single dir ected cycle of n ve rtices, augmen ted w ith log n outgoing shortcuts fr om eac h v ertex, according to K leinb erg’s small-w orld mo del. That is, eac h shortcut from x is to an indep end en tly chosen v ertex selected with according to (6) with α = 1, wh ic h in this case translates to 1 /h n d ( x, y ), where d is d istance in H , and h n ≈ log n is a norm alize r. 14 0 5 10 15 20 1000 10000 100000 Cost/Steps Size Greedy - Cost/Steps Cost-Greedy - Cost Cost-Greedy - Steps Figure 4: The cost and steps tak en by cost-greedy and greedy searc h on net w orks with Exp(1) distributed costs along eac h edge, plotted against the size of the net w ork. Net works consist of directed rin gs with log 2 n directed shortcuts p er v ertex. 0 5 10 15 20 25 30 35 40 45 50 55 1 2 3 4 5 6 7 8 9 10 Cost/Steps Round Greedy - Cost/Steps Cost-Greedy - Cost Cost-Greedy - Steps Figure 5: The cost and s teps tak en by cost-greedy and greedy r outing on a net w ork of size 262144 with Exp(1) distribu ted costs along eac h edge. Cost- greedy p erformance is plotted against iteration of the system in Theorem 3.1, s tarting with all zero weig hts. 15 W e s tart b y assigning w 0 ( x ) = 0 f or all x ∈ V , and calculate th e exp ected routing times by simulating 20 n queries b etw een randomly c h osen p oint s (this n umber of iterations is probably excessiv e). W e re-sample the costs for eac h query , but th e graph is kept the same. Ho wev er, b ecause only one s ample vec tor of the exp ected routing times ov er eac h d istance is ke pt, w e still end up marginalizing o ver the s h ortcuts. W e use the logarithmic compression of w describ ed in Section 5 (in p ractice , we fi nd th is outp erf orms using a full w vect or except when an extremely large num b er of qu eries is sim ulated), and u s e w 10 as an estimate of th e fin al v alue. The difference b et ween cost-greedy and stand ard greedy searc h in term s of query cost dep ends crucially on th e distrib ution of C ( x, y ). Quantitat iv ely , it is p ossible to mak e the b enefit of cost greedy as large (or small) as one wishes by a strategic c h oice of this distribu tion. F or example, if C ( x, y ) = ( 2 with probability 1 2 0 otherwise. then cost-greedy searc h will most often incu r zero cost assuming the v ertex degree is large en ou gh (as w ill a simple lo w est cost routing). It would thus b e dishon est of us to claim that our metho ds are motiv ated based on the p erformance ac h iev ed with d ela ys c hosen by us. Th e exp eriments in this section are thus meant to v erify that cost-greedy searc h b eha ves as exp ected, rather than to illustrate its b enefi t: the p oten tial b enefit of the algorithm m ust b e ev aluated for ev ery particular situation where it ma y b e applied. Our first exp erimen t, sho wn in Figure 3, is th us to see what happ ens if w e fix the costs to 1 for all the edges. In th is case cost and steps are the same, and since it is kno wn (Theorem 3.5 ) that greedy searc h is optimal in the exp ected n umber of steps, the theory tells us that cost-greedy should, ideally , giv e the same v alue. In fact w e fi nd th at it un der-p erforms by ab out 5% in all the sizes tested – presum ab ly due to the emp irical estimate of the exp ected v alue, an d the losses d ue to the logarithmic compression of the w eigh t v ector. Figure 4 sho ws the p erformance of cost-greedy w hen the costs are exp onen- tially distrib uted as a fun ction of the graph size. W e choose an exp onential distribution simply b ecause it is a common mo del for w aiting times, and the mean of 1 means that the cost and steps of a route are of the same scale. W e see, as exp ected, th at cost-greedy searc h is able to pr o duce r outes that cost less b y taking more steps than normal greedy searc h d oes. In Figure 5 we p lot p erformance for a single n et wo rk size against the iterations of (5) when starting with all zeros. W e see that ev en in a netw ork of hundreds of thousands of v er tices, no measurable p erform an ce is gained after the fourth round - su pp orting our hyp othesis that conv ergence is a lot f aster than the b ound giv en ab ov e. 16 0 5 10 15 20 1000 10000 100000 Cost Size Greedy Cost-Greedy SJ Method Figure 6: Example of a net w ork with t wo different v ertex t yp es - one ten th of the ve rtices are augmented b y 55 outgoing edges, wh ereas the rest get just 5. All edge costs are exactly one, and the v alues are a v eraged ov er four sim ulations to decrease v ariance. “SJ Metho d” is the metho d of S ¸ im¸ sek an d Jensen describ ed in Section 6.2. 6.2 Out-degree D istribution Another question th at has b een asked ab out na vigabilit y is h ow to route in a n etw ork if the v ertices h a ve v ariable d egree, and if the degree as well as the p osition of the n eigh b ors is kno wn when the r outing decision is made. This pr oblem is motiv ated b y the nature of so cial n et works, w hic h app ear to b e na vigable, b u t where it is kno wn that the vertex degree follo ws a h ea vy- tailed p ow er-la w . This problem is in man y w a ys similar to that whic h we discuss ab o ve: lik e with the edge costs, d egree distribu tions ma y incen tivize a wa y fr om a pur e greedy strategy , and in stead call f or a trade-off b et ween getting close to th e destination, and other factors (in this case, wan ting to route a ve rtex with high degree). S ¸ im¸ sek and Jensen [13] ha ve stud ied this pr oblem b y sim ulation. T heir metho d is fu ndamen tally similar to ours: they also seek to choose the neigh- b or w hic h minimizes the exp ected num b er of steps to the destination (as w e do if the costs are fixed to a unit v alue). Ho wev er, rather th an attempting to calculate the fix -p oin t of the weigh ts, they mak e a rough app ro ximation of the v alue usin g the right-hand sid e of th e inequalit y E [ S z ( x )] ≥ P ( S z ( x ) > 1) = P ( z / ∈ N ( x )) (13) to estimate the left. T o apply our metho ds ab o ve to the prob lem, we let the we igh ts b e a function not only of the d istance to the destination, but also of the d egree of the v ertex. Since th is question related only to the num b er of steps, we fi x all the costs to 1. W e exp ect the w eigh t for any particular distance to b e smaller 17 0 2 4 6 8 10 12 1000 10000 100000 Cost Size Greedy Cost-Greedy (Centralized, 10 Rounds) Cost-Greedy (Decentralized. 20 Rounds) Cost-Greedy (Decentralized. 10 Rounds) Figure 7: Th e same situations as in Figure 4, b ut n ow including the results when using a separate w eigh t v ector at every v ertex in the manner describ ed in S ectio n 6.3. A round is 20 × n simulate d queries as ab o v e. for ve rtices with higher degree (since the amoun t of ground gained in the first step sh ou ld b e b etter). Figure 6 sh o ws a simple example of this. In that case we hav e exactly t wo p ossible out-degrees: a few (0 . 1 n ) of the vertice s ha v e 55 shortcuts, wh ile the r est hav e just 5. W e compare the cost-gree dy searc h as used ab ov e with regular greedy search and the metho d of S ¸ im¸ sek and Jensen. Th e results seems to vind icate th e appro ximation used in the latter m etho d , with this distribution it slightly out-p erform s cost-gree dy searc h, meaning that th e n umerical losses in estimating the true w eigh ts are greater than th e analytic loss of th e approxi mation. T his seems to b e the case for most sensible such distributions, we fi nd that cost greedy search only tak es a sligh t lead when the p opu lar v ertices hav e more than a hundred times the degree of un p opular ones. If w e presume that cost-greedy searc h can come within 5% of b eing optimal also h ere, w e are forced to conclude that so do es th e SJ metho d. One adv ant age that cost-greedy searc h has ov er the metho d of S ¸ im¸ sek and Jensen, is that their m ethod r equires detailed kn o wledge ab out the mo d el in order to calculat e the r igh t-hand side of (13), which cost-greedy searc h do es not. Algorithmically , an y vertex ma y implement cost greedy searc h for its queries, and it needs only ha v e the abilit y to to measure th e cost of the queries it send s to its neighbors, n othing more. 6.3 A D ec entralized and Generalized Implemen tation T o lo ok at the practical viabilit y of the algorithms describ ed ab ov e, we also sim ulate a completely distribu ted v arian t. In the decentrali zed v arian t, we equip eac h v ertex with its o wn w eigh t vec tor, measuring the mean cost of routing from it to destinations at v arying distances. Lik e b efore w e use a 18 0 2 4 6 8 10 12 1000 10000 100000 Cost Size Greedy Cost-Greedy SJ Method Figure 8: Using a separate wei gh t ve ctor at every v ertex in the manner describ ed in S ection 6.3. Costs are Exp(1) distributed, and out-degrees are distributed according to a p ow er-la w with tail-exp on ent 2. “SJ Metho d” is the metho d of S ¸ im¸ sek and Jensen d escrib ed in Section 6.2. log 2 n compr ession of the weig hts – coalescing all d istances b et we en 2 k and 2 k +1 in to the same entry – but unlike ab ov e we do not calculate eac h w eigh t b y sampling o ve r a fi xed num b er of queries. Instead, w e let the weigh t v ector at eac h ve rtex x b e calculated as th e mean of the entries in a FIFO buffer, sho wing the cost of the last m quer ies x h as routed destined for vertices of every distance category . As b efore we do n ot c hange the edges of the graph b et we en the queries from which th e w eigh ts are estimated. Because no marginalizing is o ccurr ing here, the graph mo del is actually one fixe d realizatio n f or eac h size – the exp ectation is actually tak en only o v er the costs. T o route a query , x uses the w eigh t v ectors of eac h of the v ertices in N ( x ) to minimize (2 ). In a real wo rld implemen tation, these v alues could b e p eri- o dically copied b et we en neighbors. One p roblem we find with this metho d is that if x initially has m queries in a certain distance category that incur a very high cost, he will not attract more qu eries in that category from his neigh b ors (wh o see it as v ery costly to send suc h q u eries to him), meaning it tak es a long time to clear the err ant v alues from the FIF O b uffer. Ev en- tually the buffer will b e r eplaced, if not otherwise then by the cost of the queries initiated at x itself, b ut in our sim ulations we fi nd that th is slo ws the con v ergence. T o alleviate this, w e keep a count of the num b er of queries x r eceiv es for eac h distance categ ory dur ing an in terv al. The theory sa ys that these should b e equal, so if one of the counts has fallen a lot b ehind (is less than a quarter of the queries x receiv es for itself ) w e s et all the v alues in the b uffer to 0. Ev en with this metho d, th e con vergence is, as exp ected, slow er than in the cen tr alize d version. Figure 7 sho ws the equiv alen t of Figure 4 b ut u sing 19 lo cal weig ht ve ctors at eac h v ertex. Here we let m = 20. W e can note three thin gs: there is an ab s olute p er f ormance cost of the decen tralized v ersion, the cost seems to get worse for larger sizes, but it still consid erably outp erforms greedy searc h. The first is p r obably du e to eac h estimate of the exp ected r outing time b eing based on far fewer v alues, w h ile the second is due to us not simulating enough qu eries for full conv ergence at the large sizes, as seen b y the increasing difference b et w een the v alues as 10 and 20 rounds. W e n ote that ev en 20 roun ds is actually only 400 queries initiated at eac h vertex – a large num b er when w e m ust s im u late it for a quarter of a million ve rtices, b ut v ery little compared to the num b er of queries one w ould exp ect in most DHT’s or other dep lo yments of d istributed net w orks. Finally , in Figure 8 w e us e th e d ecen tralized metho d to r oute in a situation when w e b oth h a ve exp onentia l edge costs, and v ertices of v arying out-degree (in this case a p ow er-la w with P ( | N ( x ) | > t ) ≈ t − 2 ). Decen tralized cost- greedy searc h can optimize b oth for v arying costs and vertex degrees at the same time. 7 Conclusion W e h a ve p resen ted a m ethod for d ecen tralized searc h that tak es v arying costs of routing do wn different edges int o accoun t. W e ha v e show ed that this metho d is optimal among all s uc h algorithms th at monotonically appr oac h the destination of the qu ery , and that the necessary we igh ts can b e calculated iterativ ely . On small-w orld graphs, w e can calculate the order of costs, and sa y something ab out the appro ximation cost. Bey ond these analytic facts, ha v e presen ted a n um b er of tec hniqu es whic h mak e the algorithm pr actic al, and exp erimen ted with actual implementa tions u sing simula tion. It wo uld b e v ery d esir ab le to b e able to b etter m otiv ate our app ro xima- tions rigorously . T o do r equires strengthening prop ositions 3.6 and 3.7, and p erhaps a lot of work b eyond that. In the sh ort term, pr oving that any p olylogarithmic r outing table, and an y p olynomial num b er of samples, is sufficien t w ould b e a big improv emen t. 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