The Graph Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule

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1 The Grah Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule STEFAN D. BRUDA Deartment of Comuter Science Bisho s University Lennoxville, Quebec J1M 1Z7 CANADA bruda@cs.ubishos.ca htt://turing.ubishos.ca Abstract: We comlete the characterization of constant time comutations on arallel models with reconfigurable buses: We have shown reviously that directed reconfigurable multile bus machines (DRMBM) with olynomially bounded resources and running in constant time solve exactly the same roblems as nondeterministic logarithmic sace bounded Turing machines, that write conflict resolution rules such as Priority or even Combining do not add comutational ower over the Collision rule, and that a bus of width 1 (a wire) suffices for any constant time comutation on DRMBM. We now show that the same roerties hold for constant time comutations of directed reconfigurable networks (DRN). We then study the ower of the Collision rule in the general case, linking strongly its universality with the caability of the model to comute the grah accessibility roblem in constant time. Key-words: arallel comutation, real-time comutation, reconfigurable multile bus machine, reconfigurable network, CRCW conflict resolution rules, grah accessibility roblem 1 Introduction Previous work on arallel real-time comutations [4] has roduced a number of incidental results regarding models of arallel comutations with reconfigurable buses. Secifically, a tight characterization of constant time comutations on reconfigurable multile bus machines (RMBMs) was offered: DRMBMs and directed reconfigurable networks (DRNs) with constant running time have been found to have the same comutational ower, which in turn is the same ower as nondeterministic logarithmic sace bounded Turing machines. In addition, it was shown that in the case of constant time RMBM comutations there is no need for such owerful write conflict resolution rules as Priority or Combining as they do not add comutational ower over the easily imlementable Collision rule, that a unitary bus width is enough (i.e., a simle wire as bus will do for all constant time RMBM comutations), and that segmenting buses does not add comutational ower over fusing buses. Motivated by such a result we comlete in this aer the characterization of constant time comutations on models with reconfigurable buses. We find that such roerties (Collision being the most owerful resolution rule and unitary bus width being sufficient) hold for the other model with reconfigurable buses, namely This research was suorted by the Natural Sciences and Engineering Research Council of Canada and by the Fond québécois de recherche sur la nature et les technologies. the RN. We then study whether the Conflict resolution rule is generally (i.e., not only for constant time comutations) universal on DRMBMs and DRNs. We find that this is indeed the case, and we are able to do so in a more general setting. For indeed, the Collision rule is universal on any arallel model of comutation that is caable of comuting the grah accessibility roblem (GAP) in constant time. 2 Preliminaries Results roved elsewhere are introduced henceforth as Proositions, whereas results roved in this work are introduced as Theorems. Intermediate results are all Lemmata. GAP i,j denotes be the following roblem: Given a directed grah G = (V, E), V = {1, 2,..., n} (exressed e.g., by the (boolean) incidence matrix I), determine whether vertex j is accessible from vertex i. We denote by L [NL] the set of languages that are acceted by deterministic [nondeterministic] Turing machines that use at most O(log n) sace (not counting the inut tae) on any inut of length n [6]. For some language L NL there exists a nondeterministic Turing machine M = (K, Σ, δ, s 0 ) that accets L and uses O(log n) working sace. K is the set of states, Σ is the tae alhabet (we consider without loss of generality that Σ = {0, 1}), δ is the transition

2 relation, and s 0 is the initial state. M accets an inut string x iff M halts on x. A configuration of M working on inut x is defined as a tule (s, i, w, j), where s is the state, i and j are the ositions of the heads on inut and working tae, resectively, and w is the content of the working tae. There are oly(n) ossible configurations of M. For two configurations v 1 and v 2, we write v 1 v 2 iff v 2 can be obtained by alying δ exactly once on v 1 [6]. The set of ossible configurations of M working on x forms a directed grah G(M, x) = (V, E) as follows: V contains one vertex for each and every ossible configuration of M working on x, and (v 1, v 2 ) E iff v 1 v 2. It is clear that x L iff some configuration (h, i h, w h, j h ) is accessible in G(M, x) from the initial configuration (s 0, i 0, w 0, j 0 ). For any language L NL and for any x, determining whether x L can be reduced to the roblem of comuting GAP 1, V for G(M, x) = (V, E), where M is some NL Turing machine deciding L. The class of roblems in NL with the addition of (any kind of) real-time constraints is denoted by NL/rt [4]. We denote by rt-proc M (f) the class of those roblems solvable in real time by the arallel model of comutation M that uses f(n) rocessors (and also f(n) buses if alicable) for any inut of size n [3]. The following strongly suorted conjecture is then established. Claim 1 [4] rt-proc CRCW F-DRMBM (oly(n)) = NL/rt. Two main models with reconfigurable buses have been develoed in the literature: the reconfigurable network (or RN for short) [2] and the reconfigurable multile bus machine (or RMBM) [7]. 2.1 The RMBM An RMBM [7] consists of a set of rocessors and b buses. For each rocessor i and bus b there exists a switch controlled by rocessor i. Using these switches, a rocessor has access to the buses by being able to read or write from/to any bus. A rocessor may be able to segment a bus, obtaining thus two indeendent, shorter buses, and it is allowed to fuse any number of buses together by using a fuse line erendicular to and intersecting all the buses. DRMBM, the directed variant of RMBM, is identical to the undirected model, excet for the definition of fuse lines: Each rocessor features two fuse lines (down and u). Each of these fuse lines can be electrically connected to any bus. Assume that, at some given moment, buses i 1, i 2,..., i k are all connected to the down [u] fuse line of some rocessor. Then, a signal laced on bus i j is transmitted in one time unit to all the buses i l such that i l i j [i l i j ]. If some RMBM [DRMBM] is not allowed to segment buses, then this restricted variant is denoted by F-RMBM [F-DRMBM] (for fusing RMBM/DRMBM). The bus width of some RMBM (DRMBM, etc.) denotes the maximum size of a word that may be laced (and read) on (from) any bus in one comutational ste. For CRCW (concurrent read, concurrent write; as oosed to CREW for concurrent read, exclusive write) RMBMs, the most realistic conflict resolution rule is Collision, where two values simultaneously written on a bus result in the lacement of a secial, collision value on that bus. We consider for comleteness other conflict resolution rules such as Common, Arbitrary, Priority, and Combining. However, we find that all of these rules are in fact equivalent to the seemingly less owerful Collision rule (see Proosition 2.1(3)). We restrict only the Combining mode, requiring that the combining oeration be associative and comutable in nondeterministic linear sace. An RMBM (DRMBM, F-DRMBM, etc.) family R = (R n ) n 1 is a set containing one RMBM (DRMBM, F-DRMBM, etc.) construction for each n > 0. A family R solves a roblem P if, for any n, R n solves all inuts for P of size n. We say that some RMBM family R is a uniform RMBM family if there exists a Turing machine M that, given n, roduces the descrition of R n using O(log((n)b(n))) cells on its working tae. We henceforth dro the uniform qualifier, with the understanding that any RMBM family described in this aer is uniform. Assume that some family R = (R n ) solves a roblem P, and that each R n, n > 0, uses (n) rocessors, b(n) buses, and has a running time t(n). We say then that P RMBM((n), b(n), t(n)) (or P F-DRMBM((n), b(n), t(n)), etc.), and that R has size comlexity (n)b(n) and time comlexity t(n). It should be noted that a directed RMBM can simulate a nondirected RMBM by simly keeing all the u and down fuse lines synchronized with each other. 2.2 The Reconfigurable Network An RN [2] is a network of rocessors that can be reresented as a connected grah whose vertices are the rocessors and whose edges reresent fixed connec-

3 tions between rocessors. Each edge incident to a rocessor corresonds to a (bidirectional) ort of the rocessor. A rocessor can internally artition its orts such that all the orts in the same block of that artition are electrically connected (or fused) together. Two or more edges that are connected together by a rocessor that fuses some of its orts form a bus which connects orts of various rocessors together. CREW, Common CRCW, Collision CRCW, etc. are defined as for the the RMBM model. The directed RN (DRN for short) is similar to the general RN, excet that the edges are directed. The concet of (uniform) RN family is identical to the concet of RMBM family. The class RN((n), t(n)) [DRN((n), t(n))] is the set of roblems solvable by RN [DRN] uniform families with (n) rocessors ((n) is also called the size comlexity) and t(n) running time. 2.3 RMBM and Small Sace Comutations The characterization of constant time RMBM comutations described in the introduction of this aer can be formally summarized as follows: Proosition 2.1 [4] 1. CRCW DRMBM(oly(n), oly(n), O(1)) = NL = CRCW F-DRMBM(oly(n), oly(n), O(1)) with Collision resolution rule and bus width DRMBM(oly(n), oly(n), O(1)) = DRN(oly(n), O(1)). 3. For any roblem P solvable in constant time by some (directed or nondirected) RMBM family using oly(n) rocessors and oly(n) buses, P CRCW F-DRMBM(oly(n), oly(n), O(1)) with Collision resolution rule and bus width 1. We shall determine a similar result for DRNs. We will adat for this urose two roofs used to derive Proosition 2.1. In order to make this aer self contained we include below these roofs. Lemma 2.2 [4] CRCW DRMBM(oly(n), oly(n), O(1)) NL, for any write conflict resolution rule and any bus width. Proof. Consider some R CRCW DRMBM(oly(n), oly(n), O(1)) erforming ste d of its comutation (d O(1)). We need to find an NL Turing machine M d that generates the descrition of R after ste d using O(log n) sace, and thus [5] an NL Turing machine M d that receives n (the number of rocessors in R) and some i, 1 i n, and oututs the (O(log n) long) descrition for rocessor i instead of the whole descrition. We establish the existence of M d (and thus M d ) by induction over d, and thus we comlete the roof. M 0 exists by the definition of a uniform RMBM family. We assume the existence of M d 1, M d 1 and show how M d is constructed. For each rocessor i and each bus k read by i during ste d, M d erforms (sequentially) the following comutation: M d maintains two words b and ρ, initially emty. For every j, 1 j oly(n), M d determines whether j writes on bus k. This imlies the comutation of GAP j,i (clearly comutable in nondeterministic O(log n) sace since it is NL-comlete [6]). The local configurations of fused and segmented buses at each rocessor (i.e., the edges of the grah for GAP j,i ) are obtained by calls to M d 1. The comutation of GAP j,i is necessary to ensure that we take j into account even when j does not write directly to bus k but instead to another bus that reaches bus k through fused buses. If j writes on bus k, then M d uses M d 1 to determine the value v written by j, and udates b and ρ as follows: (a) If b is emty, then it is set to v ( j is currently the only rocessor that writes to bus k), and ρ is set to j. Otherwise: (b) If R uses the Collision rule, the collision signal is laced in b. (c) If the conflict resolution rule is Priority, ρ and j are comared; if the latter denotes a rocessor with a larger riority, then b is set to v and ρ is set to j, otherwise, neither b nor ρ are modified; the Common and Arbitrary rules are handled similarly. (d) Finally, if R uses the Combining resolution rule with as combining oeration, b is set to the result of b v (since the oeration is associative, the final content of b is indeed the correct combination of all the values written on bus k). Once the content of bus k has been determined, the configuration of i is udated accordingly, b and ρ are reset to the emty word, and the same comutation is erformed for the next bus read by i or for the next rocessor. The whole comutation of M d clearly takes O(log n) sace. Lemma 2.3 [4] Let M = (K, Σ, δ, s 0 ) be an NL Turing machine that accets L NL. Then, given some word x, x = n, there exists a CREW or CRCW F-DRMBM algorithm that comutes G(M, x) (as an incidence matrix I) in O(1) time, and using oly(n) rocessors and oly(n) buses of width 1.

4 Proof. Put n = V (n = oly(n)). The RMBM algorithm uses n + (n 2 n ) rocessors: The first n rocessors i, 1 i n, contain x, i.e., each i contains x i, the i-th symbol of x; i does nothing but writes x i on bus i. We shall refer to the remaining n 2 n rocessors as, 1 i, j n. Each assembles first the configurations corresonding to vertices v i and v j of G(M, x) and then considers the otential edge (v i, v j ) corresonding to I. If such edge exists, then writes T rue to I, and F alse otherwise. There is no interrocessor communication between rocessors, thus any RMBM model is able to carry on this comutation. Clearly, given a configuration v i, can comute in constant time any configuration v l accessible in one ste from v i, as this imlies the comutation of at most a constant number (O(2 k )) of configurations. The whole algorithm runs thus in constant time. 3 The Characterization of Constant Time RN Comutations The generality of the Collision resolution rule is not limited to RMBM comutations. Indeed, the same roerty holds for constant time comutations on RN as well. We also find that a DRN is able to carry out any constant time comutation using only buses of width 1. The first main result of this aer is thus the DRN equivalent of Proosition 2.1, as follows: Theorem 3.1 For any roblem π solvable in constant time on some variant of RN, it holds that π CRCW DRN(oly(n), O(1)) with Collision resolution rule and bus width 1. The roof of Theorem 3.1 is based on the following intermediate results. Lemma 3.2 For any X {CRCW, CREW}, Y {D, ε}, and for any write conflict resolution rule, it holds that X Y RN(oly(n), O(1)) CRCW DRN(oly(n), O(1)) with the Collision resolution rule. Proof. First, note that CRCW DRN(oly(n), O(1)) = NL for the Collision resolution rule [2]. Thus, we comlete the roof by showing that, for any conflict resolution rule, CRCW DRN(oly(n), O(1)) NL. This result is however given by the roof of Lemma 2.2. Indeed, it is immediate that the Turing machines M d and M d, 0 d c for some constant c 1, rovided in the mentioned roof work in the case of a RN R just as well as for the RMBM simulation. The only difference is that buses are not numbered in the RN case. So, we first assign arbitrary (but unambiguous) sequence numbers for the RN buses as follows: There exists an O(log n) sace-bounded Turing machine that generates a descrition of R, since R belongs to a uniform RN family (in fact, such a Turing machine is M 0 ). Then, in order to find bus k, M d uses M 0 to generate the descrition of R until exactly k buses are generated. The descrition is discarded, excet for the last generated bus, which is considered to be bus k. Since M 0 is deterministic, it always generates the descrition in the same order. Thus, it is guaranteed that bus k is different from bus j if and only if k j. The roof of Lemma 2.2 follows then unchanged. The extra sace used in the rocess of generating bus k consists in two counters over the set of buses (one to kee the value k and the other one to count how many buses have been already generated). The counters take O(log n) sace each, since there are at most oly(n) rocessors, and (oly(n)) 2 = oly(n). Thus, the overall sace comlexity remains O(log n), as desired. Lemma 3.3 GAP 1,n CRCW DRN(n 2, O(1)) with Collision resolution rule and bus width 1. Proof. Let R be the DRN solving GAP 1,n instances of size n. Then, R uses n 2 rocessors (referred to as, 1 i, j n), connected in a mesh. That is, there exists a (directional) bus from only to (i+1)j if and only if i+1 n, and to i(j+1) if and only if j+1 n, as shown in Figure 1. As shown in the figure, we also denote by E, S, N, and W the orts of to the buses going to i(j+1), going to (i+1)j, coming from i(j 1), and coming from (i 1)j, resectively. We assume that the inut grah G = (V, E), V = n, is given by its incidence matrix I, and that each rocessor knows the value of I. The DRN R works as follows: Each rocessor, i < j fuses its W and S orts if and only if I = T rue. Analogously, each rocessor, i > j fuses its N and E orts if and only if I = T rue. Finally, each rocessor ii fuses all of its orts. Then, a signal is laced by 11 on both its outgoing buses. If nn receives some signal (either the original one emitted by 11 or the signal corresonding to a collision) the inut is acceted; otherwise, the inut is rejected.

5 11 21 n W S N E... Figure 1: A mesh of n n rocessors 1n nn It is immediate that R solves GAP 1,n, by an argument similar to the one for RMBM [4] (also note that a similar construction is resented and roved correct elsewhere [8]). In addition, the content of the signal received by nn is clearly immaterial, so a bus of width 1 suffices. Recall now that the grah G(M, x) is the grah of configurations of the Turing machine M working on inut x. Lemma 3.4 For any language L NL (with the associated NL Turing machine M acceting L), and given some word x, x = n, there exists a constant time CREW (and thus CRCW) DRN algorithm using oly(n) rocessors and buses of width 1 that comutes G(M, x) (as an incidence matrix I). Proof. This fact is obtained by the same argument as the one resented in the roof of Lemma 2.3. Indeed, excet for the distribution of inut x to rocessors, there is no interrocessor communication; as such, any arallel machine will do. Thus, the comutation of G(M, x) = (V, E) will be erformed by the same mesh of rocessors R deicted in Figure 1, this time of size n n (where n = V ). In addition, the desired inut distribution will be accomlished by n additional meshes identical to R. We will denote these meshes by R i, 1 i n. For any 1 i, j n and 1 k n, the rocessor at row i, column j in mesh R k [R], will be denoted by k [n+1 ]. Each rocessor k has two new orts U and D. There exists a bus connecting ort D of k to ort U of k+1 for any 1 k n. The n + 1 meshes and their interconnection are shown in Figure 2. At the beginning of the comutation, x k, the kth symbol of inut x, is stored in a register of rocessor k 11, 1 k n. We note from the roof of Lemma 2.3 that each rocessor n+1 of R is resonsible for checking the existence of a single edge (i, j) of G(M, x). In order to accomlish this, it needs only one symbol x h from x, namely the symbol scanned by the head of the inut tae in configuration i. We assume that all the rocessors k, 1 k n, know the configuration i (and thus the value of h ). It remains therefore to show now how x h reaches rocessor n+1 in constant time, for indeed, after this distribution is achieved, R is able to comute the incidence matrix I exactly as shown in the roof of Lemma 2.3. The set of n + 1 meshes erforms the following comutation: For all 1 k n and 1 i, j n, 1. Each k 11 broadcasts x k to all the rocessors in R k. To do this, all rocessors k fuse together their N, S, E, and W orts, and then k 11 laces x k on its outgoing buses. 2. Each k comares k and h, and writes T rue in one of its registers d if they are equal and F alse otherwise. 3. Each k fuses its U and D orts, thus forming i j vertical buses. 4. Each k for which d = T rue laces x k on its ort D. 5. Finally, each n+1 stores the value it receives on its U ort. This is the value of x h it needs in order to comute the element I of the incidence matrix. It is immediate that the above rocessing takes constant time. In addition, it is also immediate that exactly one rocessor writes on each vertical bus, and thus no concurrent write takes lace. Indeed, there exists exactly one rocessor k, 1 k n, such that k = h. Therefore, we realized the inut distribution. I is then comuted by rocessor n+1 without further communication, as shown in the roof of Lemma 2.3. The construction of the DRN algorithm that comutes I is therefore comlete. Clearly, buses of width 1 are enough for the whole rocessing, since x is a word over an alhabet with 2 symbols. Given Lemmata 3.3, 3.4, and 3.2 we can now rove our first main result.

6 R 1 R k 1 R k R k+1 R k+1 1 k 1 k U D n+1 Figure 2: A collection of n meshes connected together Proof of Theorem 3.1. That the Collision resolution rule is the most owerful follows from Lemma 3.2. It remains to be shown only that a bus width 1 suffices. Given some language L NL, let M be the (NL) Turing machine acceting L. For any inut x, the DRN algorithm that accets L works as follows: Using Lemma 3.4, it obtains the grah G(M, x) of the configurations of M working on x. Then, it alies the algorithm from Lemma 3.3 in order to determine whether vertex n (halting/acceting state) is accessible from vertex 1 (initial state) in G(M, x), and accets or rejects x, accordingly. In addition, note that the values I comuted by (and stored at) n+1 in the algorithm from Lemma 3.4 are in the right lace as inut for in the algorithm from Lemma 3.3 (that uses only the mesh R). It is immediate given the aforementioned lemmata that the resulting algorithm accets L and uses no more than oly(n) rocessors, and unitary width for all the buses. The roof is now comlete, since all the roblems solvable in constant time on RN are included in NL. 4 GAP and the Universality of Collision As far a constant time comutation is concerned, we note an aarent contrast between the ower of conflict resolution rules for models with reconfigurable buses (RMBM and RN) on one hand, and for shared memory models (PRAM) on the other hand. According to our results, Collision is the most owerful rule on RMBM and RN. By contrast, it is widely believed that the Combining CRCW PRAM is more owerful than the CRCW PRAM using the equivalent of a Collision resolution rule. To our knowledge however no roof on the matter exists to date. We believe that an investigation in this direction is an interesting ursuit. We also believe that the contrast between RN and PRAM is not only aarent (that is, we believe that the Combining CRCW PRAM is indeed more owerful than the Collision CRCW PRAM). The reason for our belief is that the ability of some model to comute GAP in constant time is central to the constant time universality of the Collision rule, and also determines that exactly all DRMBM and DRN comutations are in NL; we also note that GAP is NLcomlete [6]. In light of this motivation, consider the classes M <GAP, M GAP, and M >GAP of arallel models of comutations using olynomially bounded resources (rocessors and, if alicable, buses), such that: M <GAP contains exactly all the models that cannot comute GAP in constant time, and cannot comute in constant time any roblem not in NL. An examle of such a model is the Common CRCW PRAM [9] (and thus the less owerful PRAM variants). contains exactly all the models that can comute GAP in constant time, but cannot comute in constant time any roblem not in NL. This class includes the RMBM and RN. contains exactly all the models that can comute GAP in constant time and can comute in constant time at least one roblem not in NL. To our knowledge, no model has been roved to ertain to such a class, but a ossible candidate is the broadcast with selective reduction model [1]. M GAP M >GAP Our second main result is then formulated as follows: Theorem 4.1 The Collision resolution rule is universal on any model M M GAP M >GAP, in the following sense: For any R M with any write conflict resolution rule and t(n) running time for inut size n there exists an R M that erforms the same comutation as R in O(t(n)) time using the Collision resolution rule.

7 Proof. Suose that some machine R M comutes GAP i,j for a grah G with n vertices in constant time and using oly(n) rocessors and oly(n) buses (if alicable); R exists by definition. Given the log n restriction to the size of the registers of the articiating rocessors, a rocessor cannot hold but a constant number of edge descritions. We then assume without loss of generality that m rocessors hold information about the m edges in G. Since the whole comutation erformed by R comletes in constant time, it follows that the m rocessors holding the edge information are made to communicate with each other in constant time by using extra oly(m) resources. This aarently irrelevant roerty can be ut in a more interesting way: given some R M with m rocessors, there exists an R M that includes R such that (a) all the original rocessors from R communicate with each other in constant time in R, and (b) R uses at most oly(m) more resources than R. Call this roerty the constant-gap-constantcommunication roerty. Let now R M be some CRCW machine with olynomially bounded resources that uses the Combining resolution rule to erform its comutation. We then relace R with a variant that uses the Collision resolution rule and then we slit each ste i of the comutation into the following constant number of stes: 1. Each rocessor of R reads the content of external resources (buses, memory locations, etc.) as required and then erforms the rescribed comutation for ste i, excet that whenever wants to write to external resource k it also writes the same value into a dedicated resource k (there is one such a resource for each rocessor). Note that the Collision value might be laced in k. 2. A machine R g then finds which of the original resources hold a Collision value, and for each such a resource: (a) determines based on the configurations of the original rocessors which of these rocessors write into the resective resource, (b) comutes the resulting value to be laced into the resective resource (instead of the Collision signal), and (c) writes the comuted value into the resource. Clearly the running time of the combination R, R g, and the new resources k ut together is of the same order as the running time of the original R. Indeed, Ste (2.a) might imly reeated comutations of some GAP i,j, whilst ste (2.b) is comutable in logarithmic sace given that the Combining oeration is associative and comutable in linear sace. The whole rocess erformed by R g is thus achievable in nondeterministic logarithmic sace. On the other hand GAP i,j is NL-comlete, so R g can be a imlemented as a machine in M running in constant time. It is also immediate that, if R uses oly(n) rocessors and resources, then the combination also uses oly(n) resources. In addition to all of these, the rocessors from R g should be able to communicate in constant time with all the rocessors in R (to insect their configurations); this can be accomlished however with a oly(n) increase in resources according to the constant-gap-constant-communication roerty stated at the beginning of the roof. The result is then established for the Combining rule. The other conflict resolution rules are considered similarly in an immediate manner, and the theorem obviously holds for any machine that does not use a conflict resolution rule (i.e., a CREW machine). Given that both DRMBMs and DRNs are in M GAP, the following is an immediate consequence of Theorem 4.1 and comletes the characterization of models with reconfigurable buses: Corollary 4.2 The Collision resolution rule is universal on models with reconfigurable buses. That is: For any X {CRCW, CREW}, Y {D, ε}, Z {RN(oly(n), ), RMBM(oly(n), oly(n), )}, t : IN IN, and for any write conflict resolution rule, it holds that X Y Z(t(n)) X DZ(O(t(n))) with the Collision resolution rule. 4.1 GAP and Real-Time Comutations Comare the revious discussion on GAP with the following immediate generalization of Claim 1: Theorem 4.3 For any models of comutation M 1, M 2, and M 3 such that M 1 M <GAP, M 2 M GAP, and M 3 M <GAP, it holds that rt-proc M 1 (oly(n)) NL/rt (1) rt-proc M 2 (oly(n)) = NL/rt (2) rt-proc M 3 (oly(n)) NL/rt (3) Proof. Minor variations of the arguments used in [4] show that those comutations which can be erformed in constant time on M i, 1 i 3, can be erformed in the resence of however tight time constraints (and thus in real time in general). Then, Relations (1) and (3) follow immediately from Claim 1.

8 By the same argument, rt-proc M 2 (oly(n)) rt-proc CRCW F-DRMBM (oly(n)) holds as well. The equality (and thus Relation (2)) is given by the arguments that suort Claim 1 [4]. Thus, the characterization of real-time comutations established by Claim 1 does hold in fact for any machines that are able to comute GAP in constant time. The characterization resented in Theorem 4.3 emhasizes in fact the strength of Claim 1. Indeed, as noted above, no model more owerful than the RMBM is known to exist. That is, according to the current body of knowledge, M >GAP =. Unless this relation is found to be false, it follows from Claim 1 that no roblem outside NL can be solved in real time in general, not only as far as RMBM comutations are concerned. 5 Conclusions We found that there exists a very strong similarity between the two models with reconfigurable buses, the RN and the RMBM: Not only they solve the same roblems (namely, exactly all the roblems in NL), but in both cases (a) the smallest ossible bus width is enough for all roblems, and (b) the Collision resolution rule is the most owerful (even as owerful as the Combining rule). Furthermore, we showed that Collision is the most owerful on DRNs and DRMBMs for any running time. According to our results, the discussion regarding the ractical feasibility of rules like Priority or Combining on satially distributed resources such as buses is no longer of interest. Indeed, such rules are not only of questionable feasibility, but also not necessary! We also noted the central role of the grah accessibility roblem (GAP) for the DRN and DRMBM results obtained here and also reviously [4]. In fact, the universality of the Collision rule on models with reconfigurable buses was established as a consequence of our more general result that the Collision rule is universal on any model of arallel comutation that is able to comute GAP in constant time. Having found that Collision is universal on all the models from M GAP and given that GAP is not comutable in constant time on models from M <GAP, we believe that the Collision resolution rule is not universal on any model from M <GAP. We also exect that a lesser conflict resolution rule is universal on models in M >GAP (if any). Showing (or disroving) this is an intriguing oen roblem. References: [1] S. G. AKL AND G. R. GUENTHER, Broadcasting with selective reduction, in Proceedings of the IFIP Congress, 1989, [2] Y. BEN-ASHER, K.-J. LANGE, D. PELEG, AND A. SCHUSTER, The comlexity of reconfiguring network models, Information and Comutation, 121 (1995), [3] S. D. BRUDA AND S. G. AKL, Pursuit and evasion on a ring: An infinite hierarchy for arallel real-time systems, Theory of Comuting Systems, 34 (2001), For an extended version see htt://turing.ubishos.ca/home/bruda/- aers/ursuit. [4], On the relation between arallel real-time comutations and logarithmic sace, in Proceedings of the IASTED International Conference on Parallel and Distributed Comuting and Systems, Cambridge, MA, Nov. 2002, For an extended version see htt://turing.ubishos.ca/- home/bruda/aers/nlogsace. [5] R. GREENLAW, H. J. HOOVER, AND W. L. RUZO, Limits to Parallel Comutation: P- Comleteness Theory, Oxford University Press, New York, NY, [6] A. SZEPIETOWSKI, Turing Machines with Sublogarithmic Sace, Sringer Lecture Notes in Comuter Science 843, [7] J. L. TRAHAN, R. VAIDYANATHAN, AND R. K. THIRUCHELVAN, On the ower of segmenting and fusing buses, Journal of Parallel and Distributed Comuting, 34 (1996), [8] B.-F. WANG AND G.-H. CHEN, Constant time algorithms for the transitive closure and some related grah roblems on rocessor arrays with reconfigurable bus systems, IEEE Transactions on Parallel and Distributed Systems, 1 (1990), [9], Two-dimensional rocessor array with a reconfigurable bus system is at least as owerful as CRCW model, Information Processing Letters, 36 (1990),