Kranakis, E. (Evangelos)

Vitányi, P.M.B. (Paul)

English

Kranakis, Evangelos

Vitányi, Paul

NARCIS 

Weighted distributed match-making (preliminary version)

CWI's Institutional Repository

Weighted Distributed Match-Making (Preliminary Version) Evangelos Kranakis Centrum voor Wiskunde en Informatica Kruislaan 413 1098 SJ Amsterdam ' ' The Netherlands · Paul M. B. Vitanyi Centrum voor Wiskunde en Informatica and Universiteit van Amsterdam, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands ABSTRACT In many distributed computing environments, processes are con-currently executed by nodes in a store-and-forward network. Distri-buted control issues as diverse as name-server, mutual exclusion and replicated data management, involve making matches between processes. The generic paradigm is a formal problem called "distri-buted match-making''. The applications require solutions to weighted versions of the problem. We define new multi-dimensional and weighted versions, and the relations between the two, and develop a very general method to prove lower bounds on the complexity as a trade-off between number of messages and "distributedness". The resulting lower bounds are tight in all cases we have examined. 1. Introduction A distributed system consists of computers (nodes) connected by a communication network. Each node can communicate with each other node through the network. There is no other communication between nodes. Distributed computation entails the concurrent execution of more than one process, each process being identified with the execution of a program on a computing node. Communication networks come in two types: broadcast networks and store-and-forward networks. In a broadcast network a message by the sender is broadcasted and received by all nodes, including the addressee. In such networks the communication medium is usually suited for this, like ether for radio. An example is Ethernet. Here we are interested in the latter type, store-and-forward networks, where a message is routed from node to node to its destination. Such networks occur in the form of wide area networks like Arpa net, but also as the communication network of a single multi-computer. The necessary coordination of the separate processes in various ways constitutes distributed control. The situation gets more complicated by assuming that processes can migrate from host to host, e.g., to balance the load in the system. 362 We focus on a common aspect of seemingly unrelated issues in this area, such as name server, mutual exclusion and replicated data management. Namely, processes residing in different nodes need to find each other, without knowing the host addresses of each other in advance. E.g., in a name-server a client process wants to know the host address of a server process providing a particular service; in distri-buted mutual exclusion a process that wants to enter the critical section needs to know whether some other process wants to do so as well (see [7] for a general over-view). This aspect is formalized in [4] as the paradigm "Distributed Match-Making." Roughly speaking, the problem consists in associating with each node v in the network two sets of network nodes, P(v) and Q(v), such that the intersection P(v)nQ(v') for each ordered node pair (v,v') is nonempty. We want to minimize the average of IP(v)I + jQ(v')I, the average taken over all pairs (v,v'). This average is related to the amount of communication (number of messages) involved in implementations of the distributed control issues mentioned. In the application to name-server: vis a server that posts its whereabouts in all nodes P(v), and v' is a client that looks for a particular service (as provided by v) in all nodes in the query set Q(v'). Nodes in P(v)nQ(v') can establish contact between v and v' by e.g. sending a message to v with the address of v'. In distributed mutual exclusion the interpretation is about the same, except that there is no difference between client and server, i.e. P(v)=Q(v), see e.g. [3], [4]. For application to replicated data management see [4], final version. We make the simplifying assumption that the involved processes do not migrate during execution of a match-making instance. Previously, for instance in name servers in distributed operating systems, only ad hoe solutions were proposed, e.g, [5] and references in [4]. Lack of any theoreti-cal foundation necessarily entailed that comparisons of the relative merit of different solutions could only be conducted on a haphazard basis. The question about how to distribute the name-server in a distributed operating system that is currently being implemented [6], prompted our initial investigation in distributed match-making [4]. Our analysis leads to a natural quantification of the distributed-ness of a match-making algorithm, and trade-offs between number of messages and distributedness. Thus, the complexity results hold for the full range from central-ized via hierarchical to totally distributed algorithms for match-making. As pointed out in [4], in many applications we are actually interested in weighted versions, i.e., we want to minimize the average of jP(v') I +a(v',v) I Q(v) I- It turns out that to do so we have to look at multi-dimensional versions first. We develop a very general argument to obtain lower bounds on both versions that include as special case the ones in [ 4]. The structure of the paper is as follows. First we formally define the multidimensional version and the weighted version of the problem. In the next sec-tion we derive the lower bound trade-off (Theorem 1) on the multidimensional case. We then show that the lower bound is tight for the binary n-cube topology and projective n-space topology, by exhibiting distributed algorithms that match the lower bound. In the final section, we derive the promised lower bound on the weighted version of distributed match-making (Theorem 2). This development 363 enhances applicability of the theory of distributed match-making in practical situa-tions. 1.1. Formal Framework To simplify notation from now on let the set N of network nodes be equal to {1, ... ,n }. Let P = (Pa, ... ,Ps) be a communication strategy in a given network as follows. (For convenience, with some abuse of notation, we use letters a through s to denote both node variables and the numbers 1 through s.) For each j = a, ... ,s, P1: N ~ 2N is a total function, and for each s-tuple (a',b', ... ,s') of nodes Pa(a')nPb(b')n ... nPs(s')=f:.0. For any s-tuple (a',b', ... ,s') of nodes let m[P](a', ... ,s') = IP a(a') I + ... + I Ps(s') I be the number of messages required for the match-making instance (a, ... ,s) following strategy P. The average number M[P] of point-to-point messages necessary for match-making is (deleting here and elsewhere [P] because P is understood): M - -s""' (I ') - n ""m a , ... ,s , (1) with the sum taken over (a', ... , s') E Ns. Let us interpret the case s = 2 in terms of the name-server, in order to give the intuitive background for considering weighted versions. Since a server i posts its whereabouts at all the nodes in P(i), by sending messages to all these nodes, and a client j queries each node in Q(j), we have P=(P,Q). The number m(i,j) of point-to-point messages in the match-making instance (i,j) must be at least I P(i) I + I QU) I . Another more general situation arises when the average call for a service i by a client j occurs a(i,j)-times more often than the average posting of a service available at i. Here one wants to minim-ize (1), with m(i,j) = jP(i)I + a(i,J)IQ(J)I. A similar case arises when in the match-making instance (i,j) the server i is allowed to post p (i,j)-many times to the nodes in P(i) and the customer j is allowed to query q(i,j)-many times the nodes in Q(j) in order to increase reliability of the network. In this case the number m(i,j) of point-to-point messages is equal top (i,j) I P (i) I + q (i,j) IQ (j) I · In contrast to the post-query case (s =2), which is best visualized in two dimensions, the more general case (s > 2) is best visualized in s dimensions. (Each axis is marked with a node from 1, .. .,n and at the vertex (a, ... ,s) a point of the intersection nPr is located.) To obtain lower bounds on the complexity of the weighted versions and the versions with retransmission, it turns out that it is advan-tageous to analyse the generals-dimensional case first. 2. The s-Dimensional Lower Bounds In this section the main lower bound results are derived. In order to be able to prove the most general results possible it will be necessary to formulate the required concepts with a higher level of abstraction than in the introduction. The motivation however is derived from the previous section, and the results are neces-sary to resolve weighted match-making in the next section. 364 Let N,N0 , ••• ,N5 be nonempty sets, and n=INI, na=INal.···· ns=INsl· For convenience we set N = { 1, ... ,n }. lt is important to note that, in this general setting, N 0 , ••• , Ns are arbitrary finite sets (of integers), in particular, they can have more elements than N. Consider a strategy P = {P0 (a'), ... ,Ps(s'): a'EN0 , ••• ,s'EN5 }, with total mappings Pr:N,~2N, and Pr(x) = IP,(x)I, for r E{a, ... ,s}. Let Kibe the set of s-tuples (a', ... ,s') such that i EP0 (a')n · · · nPs(s') and let ki = I K; I · (lt is clear that if each of these intersections is nonempty then k 1 + · · · + kn ;;;;;:: na · · · n9 , and equality holds if all intersections are singleton sets.) For the given strategy P define the product II and the sum M associated with P by the following formulas: II = (n 0 • • • n9 )- 1 Da(a') · · · p5 (s') , M = (n0 • • • n5)- 1 ~[p0(a')+ · · · +ps(s')], with the sums taken over (d, ... ,s')EN0 X ... XN5 • Further, for r E{a, ... ,s} define M, = n; 1 »,(r') (with summation over r' ENr), so that II = Ma · · · M5 and M = Ma+ · · · +Ms . The main result of the section is the following Theorem 1. For any strategy P the following inequalities hold: (2) II;;;;;:: (na · · · n5 )- 1 [ ~ k} 15] 5 and M;;;;;:: s(na · · · n5 )- 115 [ ~ k} 15 ]. i EN i EN Remark If na = · · · = n5 = n then II;;;;;:: [n- 1 _,f k} 15] 5 and M;;;;;:: sn- 1 [.fkf 15 ]. r=l 1=! Additionally considering the symmetric case where all k;'s are equal, viz., k; =n 5 - 1, i = 1, ... ,n. Then Theorem 1 specializes to the important "truly distri-buted" case: II ;;;;;:: n 5 - 1 and M ;;;;;:: sn (s - I) / s. We will find matching upper bounds below. Remark. M equals the right-hand side of the inequality in which it occurs, exactly when Ma '= · · · = M5 , i.e. the strategy P is optimal exactly when the average number of messages is equally balanced in all directions. Proof: The following inequality, also known as inequali~y of the arithmetic and geometric means, holds for s-many nonnegative real numbers a, ... , o, a + · · · + o;;;;;:: s(a · · · 0) 115. (3) In fact, equality holds exactly when all the surnmands are equal [2]. Thus, the ine-quality in the Theorem concerning the sum M follows immediately from the 365 inequality concerning product II, identities (2), and inequality (3). It is only left to prove the inequality concerning II. For each rE{a, ... ,s} and each i EN, define the set Hr,i r;;,.Nr such that r' EHr,i iff for some s-tuple (a', .. /, .,s') holds i EPa(a') n ... nPr(r') n ... n Ps(s'). Set hr,i = I Hr,i I· Clearly, for all i = 1, ... ,n, h .... h . = IH x XH I a,1 s,1 a,i · · · s,i ;;.. I {(a', ... ,s'): i EPa(a')n ... nPs(s')} I = k;. (4) Now, for all rE{a, ... ,s}, ~ hr,i ~ L I {r' : i EPr(r')} I iEN iEN = L L I {(i,r') : i EPr(r')} I i EN r'EN, = 2: j{i: iEPr(r')}I r'EN, = L Pr(r') = nr Mr . r'EN, To obtain the lower bound on II, we now proceed as follows. II = Mi · ··Ms (by (2)) · "' (na · · · n,)- 1 [.~}'·· l · · · [,~}'·'] =(na···n5 )-l 2: ha,0t.···hs,<J 0t., .. .,<JEN (5) (by (5)) Set S(a, ... ,p,a) =ha.a··· hr,phs,a· By cyclically rotating the indices o:, ... ,p,a of S(a,. . .,p,a) one obtains the followings-many summands: a 1 = S(a, ... ,p,a) = ha a · · · h5 (J ' ' a1 = S(/3,. .. ,a,a) = ha,{3 · · · hs,0t. (6) as = S(a,. . .,7T,p) = ha,<J · · · hs,p. Using inequalities (3) and (4) and regrouping terms in the resulting product a 1 · · · a5 it is easy to see that a1 + · · · + a5 ~s(a1 · ··a5 ) 115 ~s(ka ···k0 ) 115 . After adding the s-many summands of (6), each one summed with respect to a,. .. ,a, dividing again by s to eliminates-multiple copies, and taking into account the last inequality, we obtain: 0t., ... ,aEN 0t., ... ,<JEN [ 2: k} Is] s iEN 366 This completes the proof of the lower bound of II, and hence the proof of the theorem is complete. 0 Corollary. Both propositions 1 and 2 of [4] are immediate consequences of Theorem 1. 3. Optimality We show that Theorem I is optimal in some special cases (which are of sufficient generality), by exhibiting matching strategies. (Multidimensional Cube Network) Let the number of nodes be n = 2d and suppose that s is a divisor of d. Addresses of nodes consist of d bits, like u 1 u 2 · · · ud. Nodes are connected by an edge exactly when they differ by a single bit. Let P = (P 1, ... ,P5 ) be a strategy, and, for each r E{l, ... ,s}, let Pr(u1 · · · ud) be the set {x1 ···x(r-J)d/sU(r-l)d!s+l ···urd/sXrdls.,..1 ···xd:x;E{0,1}}. Clearly, each of the above sets has size 2(s - 1)d1 s and k; = 2Cs -- 1 )d = n s - 1 . Th us, one easily obtains that M:,.;; sn(s-l)ls, i.e. the average number of point-to-point message transmissions is at most sn (s - I)/ s. In view of Theorem I this strategy is also optimal. (Multidimensional Projective Plane). Consider generalized mutual exclusion in a distributed setting, where s -1 processors are allowed to be in the critical section simultaneously, but not s or more processors. For background and nondistributed solutions we refer to [1]. In [3], Maekawa considers the distributed version of mutual exclusion for s =2, the commonly studied variant. In our terminology, for mutual exclusion with s = 2 we can set P 1(i)=P2 (i), which is some sort of sym-metry condition. Each instance of mutual exclusion contains a match-making instance [4]. For the truly distributed case, with k 1 = ... =kn =n and s =2 we find that on the average each match-making instance takes at least 2Vn messages [4]. Maekawa obtains a similar lower bound, and exhibits an algorithm that achieves 5 Vn [3]. Theorem l gives a lower bound of sn (s - I)/ s for the generalized version. We exhibit an algorithm that achieves this. The s-dimensional projective plane PG (s, k) has k5 + ks - l + · · · + 1 = n nodes, each node is incident to k 5 - I + k 5 - 2 + · · · + 1 hyperplanes, and each hyperplane contains k 5 - I + k5 - 2 + · · · + 1 nodes. Each s-element set of hyperplanes intersects in precisely one node. Let P=(P 1,. .. ,P5 ) be a symmetric strategy with each query set S (i) = P 1 (i) = · · · = Ps(i) of a node i consists of the set of e- 1 + ks - 2 + · · · + 1 nodes incident to a hyperplane containing node i. It does not matter which hyperplane we pick, because any s hyperplanes intersect in a single node. The average cost M of point-to-point messages associated with a par-ticular mutual exclusion instance is therefore (generalizing Maekawa's method for s =2 [3]) O(s(k5- 1 + e- 2 + · · · + 1)) ~ O(snCs-l)ls). In view of Theorem 1 this strategy is also optimal. 367 4. Weighted Distributed Match-Making We can now examine weighted distributed match-making. This can be formulated as communication strategies with multiple transmissions allowed. We use Theorem l to derive significant lower bounds on the average number of message transmis-sions in distributed networks when multiple transmissions are allowed. Consider a strategy P=(Pai ... ,P5 ), with all parameters as above, and define a weighted version of m. I.e., define the number of messages for the match-making instance S =(a', ... ,s') as m[P](S) = la(S)pa(a') + · · · + l5(S)ps(s'), where each la(S), ... ,ls(S) is a positive integer. Then, with S as above, define Nr,r'' for all r E { a, .. ,s} and r' ENr so that it satisfies: SEN0 X ... XN, ~ [ ~ / 0 (S)]pa(a') + · · · + ~ [ ~ ls(S)lp5 (s') dE~ SE~ ~E~ SE~ = ~ Na,a'Pa(a') + · · · + ~ Ns,s'Ps(s') , (7) s'EN, where Na,d = "LsES)a(S), etc. Define N'r=~r'EN N,,r. Consider the following r related strategy Q for the set of nodes N. Q = {Q0 (a'), ... ,Q5 (s'): a'EN'a, ... ,s1 EN15 }, such that, for each r E{a, ... ,s} and each y ENr there are Nr,y distinct x's, with Qr(x)=Pr(y). I.e., Q is formed from the strategy P by repeating each set Piy), Nr,y·times. Let qr(x) = I Qr(x) I· Note that we have chosen the definitions such that ~ q,(r') = ~ Nr,r'Pr(r') , r'EN', r'EN, for all r from a through s. Then we can relate M[P] with II[Q]: M[P] = (na · · · n5 )- 1 [ ~ qa(a') + · · · + ~ %(s')] (by (7)) a'EN'0 s'EN', ;,,, s(na · · · n5 )- 1 [ [ }: %(a')] · · · [ }: q5 (s')] ] 11 s (by (3)) dEN'0 s'EN', = s(na · · · n5 )- 1(N'a · · · N'5 ) 115II[Q] 115 (by definition) ~ s(na · · · n5 )- 1 ~ k;[Q] 115 . (by Theorem 1) iEN It remains to compare the quantities k;[P], k;[Q]. This can be done by comparing the sizes of the sets K;[P], K;[Q]. Now, for each s-tuple (a', ... ,s') such that 368 i EPa(a')n ... nPs(s') there are at least Na,d · · · Ns,s' s-tuples (a", ... ,s") such that i EQa(a") n ... n Qs(s"). Namely, there are N,,r copies of P,(r'), for r,r' from a,a' through s,s', in Q. Therefore, each (a', ... ,s') E K;[P] corresponds to a disjoint sub-set of at least Na a' · · · Ns rmany s-tuples in the set K;[Q]'s. Hence, it has been , , proved that -1 [ . ' ' Jl/s M[P] ;;;;-. s(n1 · · ·ns) ~ ~{Na,a'-··Ns,s'. (a, ... ,s)EK;[P]} . iEN (8) In particular, with some computation we can specialize the general result (8) to: Theorem 2. For a~y strategy P, if there are positive integers A.0 , ••• ,A.5 such that for all (a', ... ,s') holds m(a', ... ,s') = A.0p0 (a') + · · · + AsPs(s'), thm s(A.1 ... A )1 Is n M ;;;,-. s 2: kJ1s. n ; =I (9) Moreover, the quantity M equals the right-hand side of the inequality above, exact{y when AaM a = · · · = As Ms. D Corollary. Routine calculation shows that Theorem 2 also holds for rational A's. (Hint: for A.,=p,I q, apply Theorem 2 for µ,,=cA., with c =%···% (r E{a, ... ,s}). This gives an inequality for cM. Substituting the A.'s for the µ,'s, we can cancel con both sides of the inequality.) References [1] Fischer, M.J., Lynch, N.A., Bums, J.E., and Borodin, A., Distributed FIFO allocation of identical resources using small shared space, Massachusetts Institute of Technology, Cambridge, Mass., Report MIT/LCS/TM-290, October 1985. [2] Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities, Cambridge Univer-sity Press, 1934. [3] Maekawa, M., A VN Algorithmfor Mutual Exclusion in Decentralized Systems, ACM Transactions on Computer Systems 3 (1985), pp. 145-159. [4a] Mullender, S. J. and Vitanyi, P. M. B., Distributed Match-Making for Processes in Computer Networks, Preliminary Version, Proceedings of the 4th annual ACM Symposium on Principles of Distributed Computing, 1985, pp. 261-271. [4b] Mullender, S. J. and Vitanyi, P. M. B., Distributed Match-Making, Algorith-mica, (1988). [5] Powell, M.L. and Miller, B. P., Process Migration in DEMOSl.MP, Proceed-ings of the 9th ACM Symposium on Operating Systems Principles, 1983, pp. 110-119. [6] Tanenbaum, A.S. and S.J. Mullender, The Design of a Capabili~y Based Distri-buted Operating System, The Computer Journal, 29(1986). [7] Tanenbaum, A.S., Computer Networks, Prentice-Hall, 1981. 

Weighted distributed match-making (preliminary version)

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