A Graph-Theoretic Characterization of AND-OR Deadlocks

Transcription

1 A Graph-Theoretc Characterzaton of AND-OR Deadlocks Valmr C Barbosa Maro R F Benevdes Programa de Engenhara de Sstemas e Computação, COPPE Insttuto de Matemátca Unversdade Federal do Ro de Janero Caxa Postal Ro de Janero - RJ, Brazl {valmr, maro}@cosufrjbr Abstract We ntroduce the b-knot as a structure whose exstence n a wat-for graph s necessary and suffcent for the exstence of a deadlock under the AND-OR model Unlke the case of other, more restrcted deadlock models, for the AND-OR model no such graph structure has heretofore been explctly dentfed and characterzed We also show that a well-known asynchronous algorthm for dstrbuted knot detecton can be adapted to yeld an asynchronous dstrbuted algorthm on the wat-for graph for a node to detect whether t s n a b-knot Keywords: AND-OR deadlocks; dstrbuted algorthms; operatng systems 1 Introducton Let N denote a set of processes n a dstrbuted computaton Informally, a deadlock s sad to exst n ths computaton f a subset S N can be dentfed whose members are all blocked for the occurrence of some condton that can only be releved by members of the same subset S A useful abstracton to analyze deadlock stuatons s the wat-for graph G = (N,E), where E s a set of drected edges For n,n j N, an edge exsts n E drected away from n towards n j f n s blocked for some condton that n j may releve G changes dynamcally as the computaton progresses, so whenever we refer to G we mean the wat-for graph that corresponds to a snapshot of the dstrbuted computaton n the usual sense of a consstent global state [1, 4] In general, a necessary condton for the exstence of a deadlock n the dstrbuted computaton s the exstence of a drected cycle n G In order to dscuss more specfc necessary condtons and suffcent condtons, we must frst ntroduce addtonal concepts and notaton For n N, let D denote the set of descendants of n n G (nodes that are reachable from n, ncludng tself) and A denote the set of ancestors of n n G (nodes from whch n s reachable, ncludng tself) Let O D be the set of mmedate descendants of n n G (descendants that are one edge away from n ) and I A ts set of mmedate ancestors n G (ancestors that are one edge away from n ) Nodes n D \ A are called subordnates of n n G 1 A deadlock model for the dstrbuted computaton that underles G s a collecton W 1,,W p of subsets of O for all n N, such that: 1 \ denotes set dfference 1

2 W 1 W p = O ; No two nonempty sets n W 1,,W p are such that one s a subset of the other; In order to ext ts blocked state and proceed wth ts local computaton, a node n for whch O must receve a sgnal from all nodes n at least one of the nonempty sets n W 1,,W p At ths level of generalty, the deadlock model s known as the AND-OR model, reflectng the need for n to be sgnaled by all members of W 1 2 (f nonempty), or all members of W (f nonempty), and so on If at most one of W 1,,W p s nonempty for all n N, then the deadlock model s the AND model Smlarly, f all nonempty sets n W 1,,W p are sngletons for all n N, then the deadlock model s known as the OR model 2 A suffcent condton for the exstence of a deadlock n the AND model s the same as the general necessary condton mentoned earler, that s, that a drected cycle exst n G For the OR model, a necessary and suffcent condton s the exstence of a knot n G A knot s a subset K N wth K > 1 havng the property that, for all n K, D = K For detals on these condtons and related materal, the reader s referred to [6, 10] and the references theren In spte of the exstence of several approaches to the detecton of deadlocks n dstrbuted computatons under the AND-OR model (the approaches n [3, 7, 9] are representatve recent examples), no graph structure appears to have been dentfed that accounts for those deadlocks as a necessary and suffcent condton In ths paper, we descrbe such a structure n Secton 2, and n Secton 3 gve an asynchronous dstrbuted algorthm for a node to detect whether t s n such a structure n G Ths algorthm employs the algorthm of [8] for knot detecton as a frst phase Concludng remarks follow n Secton 4 2 B-knots In ths secton we ntroduce the noton of a b-knot as a structure n G that can be used to characterze deadlocks under the AND-OR model Unlke the case of drected cycles and knots, the defnton of a b-knot requres G to be consdered n explct conjuncton wth the deadlock model As wll become apparent shortly, the b n b-knot s an alluson to the fact that, for the defnton of ths structure, edges drected away from a node n must be consdered n bundles that relate closely to the sets W 1,,W p The defnton of a b-knot s based on the defnton of a b-subgraph of G gven the deadlock model If G = (N,E ) s a subgraph of G, then we say that t s a b-subgraph of G f every n N has at most as many mmedate descendants n G as there are nonempty sets n W 1,,W p, and furthermore each of W 1,,W p, f nonempty, ncludes at least one of those mmedate descendants Gven the deadlock model, a subset K N s sad to be a b-knot f K s a knot n some b-subgraph of G whose node set contans K Henceforth, we let the deadlock model be mplctly assumed whenever a b-subgraph or a b-knot of G s mentoned If G s a b-subgraph of G and n s a node of G, then, for 1 k p, let W k nodes contaned n W k be the set of that by defnton appear n G as mmedate descendants of n (f W k s s an OR model for G, so long nonempty), or W k = (otherwse) It follows that W 1,,W p as no two nonempty sets n W 1,,W p are dentcal Illustratons of ths noton of a b-knot are gven n Fgures 1 and 2 In both fgures, part (a) depcts G wth the sets W 1,,W p, whenever nonempty, shown as crcular arcs around n jonng 2 Another deadlock model of nterest s the so-called k-out-of-n model, whch s more general than both the AND model and the OR model, whle beng generalzed by the AND-OR model A generalzaton of the k-out-of-n model s the dsjunctve k-out-of-n model, ths one equvalent to the AND-OR model (see [3], for example, and the references theren, for detals) 2

3 groups of edges that lead to n s mmedate descendants In the case of Fgure 1(a), for example, we have W 1 6 = {n 1,n 2 } and W 2 6 = {n 2,n 5 } The graph of Fgure 1(a) has no b-knots, whch by defnton means that none of ts b-subgraphs has knots Two of these b-subgraphs are shown n Fgures 1(b) and 1(c) In the graph of Fgure 2(a), on the other hand, the set {n 1,n 2,n 6 } s a b-knot Ths same set appears as a knot n the b-subgraph of Fgure 2(b), whle the b-subgraph of Fgure 2(c) has no knots n 1 n 2 n 3 n 1 n 2 n 3 n 1 n 2 n 6 n 5 n 4 n 6 n 5 n 4 n 6 (a) (b) (c) Fgure 1 G wth no b-knots and two b-subgraphs n 1 n 2 n 3 n 1 n 2 n 2 n 3 n 6 n 5 n 4 n 6 n 5 n 6 n 5 n 4 (a) (b) (c) Fgure 2 G wth a b-knot and two b-subgraphs Now suppose that node n s a snk n G (e, n has no descendants n G) It follows that n s not blocked n G, and t makes sense to consder the subgraph H of G that results from the sgnalng by n that all ts mmedate ancestors n G need no longer be blocked as far as t s concerned In H, n s solated (has no ancestors or descendants), and t may happen that one or more of ts mmedate ancestors n G, say node n j, has now become a snk (ths happens f, for some k such that 1 k p j, Wj k = {n }) If n H no wat s superfluous, n a sense that wll become clear shortly, then we say that H s sgnal-reduced from G by n In order to defne ths sgnal-reducton from G by n precsely, let n j be an mmedate ancestor of n n G, and let W j 1,, W p j j be the sets that represents n j s wat condton n H (that s, these sets are part of the deadlock model for H) For 1 k p j, the followng s how the set W j k s obtaned n the process of sgnal-reducton from G by n If there exsts k such that 1 k p j and k k, and furthermore Wj k \ {n } Wj k \ {n }, then W j k = Otherwse, k W j = Wj k \ {n } In addton to beng n consonance wth the defnton of a deadlock model, ths reflects the fact that, n H, t only makes sense for n j to keep on watng for sgnals from nodes n Wj k \ {n } f no other Wj k \ {n } exsts that already ndcates such a wat 3

4 Note that the absence of a b-knot n G mples, by the defnton of a b-knot, that G has at least one snk, and therefore there exsts a graph that s sgnal-reduced from G by each of G s snks Note also that, n the OR model, the process of sgnal-reducton preserves all knots exstng n the orgnal graph whle creatng no new knots Ths s what happens, then, when that orgnal graph s a b-subgraph of G and ts deadlock model s derved from that of G as we dscussed earler An llustraton s provded n Fgure 3 of ths process of sgnal-reducton from a snk The graph shown n Fgure 3(a) s sgnal-reduced from the one of Fgure 1(a) by n 1 Note that not only does n 1 become solated, but also the edge drected from n 6 to n 5 need no longer exst (once an unblockng sgnal s receved by n 6 from n 1, the only further sgnal that t needs s from n 2, as a sgnal from n 5 s rrelevant to ts wat condton) The remanng graphs n Fgure 3, those n parts (b) and (c), are both b-subgraphs of the graph of Fgure 3(a) The one n Fgure 3(b) s sgnal-reduced from the graph n Fgure 1(c) by n 1 (ths one a b-subgraph of the graph n Fgure 1(a)), whle the one n Fgure 3(c) s not However, all t takes for the graph of Fgure 3(c) to be sgnal-reduced from that same graph by n 1 s the addton of the solated n 1 to t n 1 n 2 n 3 n 1 n 2 n 2 n 6 n 5 n 4 n 6 n 6 (a) (b) (c) Fgure 3 Sgnal-reducton by n 1 We now demonstrate, after statng and provng a supportng lemma, that the exstence of a b-knot n G s closely related to the exstence of a deadlock n the AND-OR model Lemma 1 If G has no b-knots and H s sgnal-reduced from G by one of G s snks, then H has no b-knots Proof: Let n be the snk n G such that H s sgnal-reduced from G by n, and let H be a b-subgraph of H If H s also a b-subgraph of G, then by hypothess H has no knots If H s not a b-subgraph of G, then H ncludes an mmedate ancestor n j of n n G for whch Wj k and W j k =, where 1 k p j In other words, the nonempty Wj k n G became an empty W j k n H through the sgnal-reducton by n In order for ths to have happened, there has to exst k such that 1 k p j, k k, and Wj k {n } such that Wj k \ {n } Wj k \ {n } Now consder a b-subgraph G of G that ncludes n and n j, and n addton ncludes an edge drected from n j to n and another drected from n j to any member of Wj k \ W j k (ths set s nonempty, by defnton of a deadlock model) If n s a node of H, then there exsts such a G from whch H s obtaned va a sgnal-reducton by n If n s not a node of H, then the result of ths sgnal-reducton s H enlarged by the solated n In ether case, any knots n H must also be knots n G These, however, are ruled out by hypothess, so n ths case too H has no knots It follows that H has no b-knots Theorem 2 There exsts a deadlock n the AND-OR model f and only f G has a b-knot Proof: If G has a b-knot, then let G be a b-subgraph of G n whch a knot exsts A node n ths knot s blocked for the recept of a sgnal from at least one of ts mmedate descendants n G, but 4

5 the exstence of the knot means that such a sgnal wll never be sent As a consequence, that node s permanently deprved of any progress n the dstrbuted computaton, that s, a deadlock exsts Conversely, suppose that G does not have a b-knot In order to prove that n ths case no deadlock exsts, we must show that, f G can only evolve by the removal of edges as sgnals are sent to unblock watng nodes, then eventually all wats are elmnated and G stablzes as a graph wth no edges But ths s guaranteed drectly by Lemma 1, thence the theorem If at most one of W 1,,W p s nonempty for all n N (ths s the AND model), then n every b-subgraph of G every node has at most one mmedate descendant A knot n such a subgraph s a drected cycle, so the condton that Theorem 2 asserts for the AND-OR model becomes the known condton for the exstence of a deadlock n the AND model Smlarly, f all nonempty sets n W 1,,W p are sngletons for all n N (the OR model), then n every b-subgraph node n has as many mmedate descendants as t has n G A knot n such a subgraph s then a knot n G as well, and the condton for the AND-OR model gven by Theorem 2 s reduced to the condton for the exstence of a deadlock n the OR model 3 Checkng membershp dstrbutedly In ths secton we descrbe an asynchronous dstrbuted algorthm for a node, say n 1 N, to detect whether t s n a b-knot n G The algorthm we gve employs a smplfed verson of the algorthm of [8] as an ntal phase That algorthm has been gven for the detecton by n 1 of whether t belongs to a knot n G We dscuss that algorthm frst 31 Knot detecton The algorthm of [8] (and hence the one we gve n ths secton) s descrbed for the followng model of computaton A node n G s dentfed wth a process that can only compute reactvely to the recepton of messages from other nodes Upon recevng a message, a node may compute and send messages to any other nodes that are drectly connected to t n G, regardless of the drectons of the edges Only node n 1 can compute (and possbly send messages) wthout beng trggered by the arrval of a message, but t must do so only once and behave reactvely lke the others thereafter Such a dstrbuted computaton s referred to as a dffusng computaton ntated by n 1 As n the standard model for asynchronous dstrbuted computatons [1], every node has an ndependent tme bass, and messages are guaranteed to be delvered wth fnte (though unpredctable) delays Fnally, we note that the algorthms dscussed n ths secton adopt the same vew of [2] for the dstrbuted detecton of stable propertes, that s, the vew that G stands for the (unchangng) wat-for graph at some consstent global state of the underlyng dstrbuted computaton The departng pont of the algorthm n [8] s that n 1 s n a knot n G f and only f D 1 \A 1 = What the algorthm does s to compute the cardnalty of D 1 \A 1 In order to acheve ths, messages of three types, called desc, anc, and ack, are employed, along wth the followng sute of varables for node n descendant : Boolean varable ndcatng whether n D 1 (ntally set to true f = 1, false otherwse); ancestor : Boolean varable ndcatng whether n A 1 (ntally set to true f = 1, false otherwse); subordnate : Integer varable havng value 1 f n D 1 \ A 1, 0 otherwse (ntally set to 0); cs : Integer varable contanng the sum of subordnate k over some nodes n k N (ntally set to 0) The algorthm seeks to establsh cs 1 = n N subordnate (1) 5

6 = D 1 \ A 1 at global termnaton Towards ths goal, t proceeds as follows Node n 1 starts by sendng desc to ts mmedate descendants and anc to ts mmedate ancestors, and reples at once wth an ack upon recevng any messages of these types Another node n forwards the frst desc t receves to ts mmedate descendants and the frst anc t receves to ts mmedate ancestors The ack that corresponds to a desc or anc receved when n does not expect to receve any ack s tself s wthheld and only sent when n once agan comes to expect no further ack s All other desc s or anc s receved by n are repled to wth an ack mmedately The frst desc and anc that n receves cause descendant and ancestor, respectvely, to be set to true As one readly recognzes, ths algorthm s an nstance of the algorthm of [5] (see also [1] for another descrpton) for n 1 to detect the global termnaton of dffusng computatons ntated by tself Ths occurs when n 1 has receved ack s from all of ts mmedate descendants and ancestors In order to guarantee that (1) holds at ths moment, the algorthm of [8] employs the followng addtonal rules (a) Upon recevng a desc or an anc, node n n 1 does { 1, f descendant and not ancestor cs := cs subordnate + ; 0, otherwse; { 1, f descendant and not ancestor subordnate := ; 0, otherwse; (b) Every ack that node n 1 sends s sent as ack(0); (c) Every ack that node n n 1 sends s sent as ack(cs ), and then cs s reset to 0; (d) Upon recevng an ack(c), node n does cs := cs + c Now consder a consstent global state of the dffusng computaton ntated by n 1, and let C be the sum, over all ack(c) s that are n transt n that global state, of the parameters c The followng s an easy consequence of (a) through (d) At all consstent global states of the dffusng computaton ntated by n 1, n N cs + C = n N subordnate (2) In partcular, at all consstent global states at whch global termnaton holds, we have C = 0 and, by (c), cs = 0 for all nodes n n 1 By (2), t follows that the equalty of (1) s acheved by n 1 upon detectng global termnaton 32 B-knot detecton Let us now turn to the detecton by node n 1 of whether t s n a b-knot n G By defnton, what n 1 must detect s whether t partcpates n a knot n some b-subgraph of G For n N, let S 1,,Sq be the subsets of O havng at least one node from each nonempty set n W 1,,W p and at most as many nodes as there are nonempty sets n W 1,,W p For one of the q 1 subsets S1, 1,S q 1 1, an equvalent condton for n 1 to detect s whether there exsts a b-subgraph of G that ncludes n 1 and that subset, but no subordnate of n 1 Node n 1 must be n a b-knot f and only f ths condton holds for at least one of S1, 1,S q 1 1 Our algorthm comprses two phases The frst phase s a smplfcaton of the algorthm of [8] that removes most of the actons descrbed under (a) through (d) What the smplfed verson 6

7 acheves, upon global termnaton, s the correct computaton of subordnate for all n N Ths frst phase employs the same messages and varables we have consdered so far, except for the parameters carred by the ack s and the cs varables The second phase s also ntated by node n 1 and comprses two sub-phases, one nested nto the other It employs messages desc, ack d, anc, and ack a, n addton to the followng varable (among others, to be ntroduced later) for node n s : Array [1 q ] of Booleans, each ndcatng whether all b-subgraphs of G that nclude n and the correspondng set n S 1,,Sq also nclude a subordnate of n 1 (ntally set to false) The goal of the second phase of our algorthm s to compute s 1 [k] for all k such that 1 k q 1 Clearly, f ths s acheved at global termnaton, then n 1 s n a b-knot f and only f 1 k q 1 s 1 [k] = false (3) We start by provdng an nformal descrpton of the second phase Node n 1 ntates the second phase by sendng desc to all ts mmedate descendants It then reples mmedately wth an ack d to any desc t receves When n 1 has receved an ack d for every desc t sent, global termnaton of the second phase has occurred A node n n 1, upon recevng the frst desc, wthholds the correspondng ack d and then checks whether t s a subordnate of n 1 If t s not, then t forwards desc to all ts mmedate descendants and wats for no ack d s to be any longer expected n order to send the ack d t wthheld If t s a subordnate of n 1 and has never receved an anc message, then t ntates another dffusng computaton, whose termnaton wll sgnal that t may send the ack d t wthheld Upon ntatng ths dffusng computaton, n sets s [k] to true for all k such that 1 k q Every further desc receved by n s repled to wth an ack d mmedately The dffusng computaton that a subordnate n of n 1 ntates proceeds as follows Frst n sends anc to all of ts mmedate ancestors that are also descendants of n 1 (ths can be recorded locally at n durng the propagaton of desc messages n the frst phase) Node n reples mmedately wth an ack a to any anc that t receves, and detects global termnaton of the dffusng computaton t ntated when t receves as many ack a s as t sent anc s Upon recevng anc from an mmedate descendant n l, a node n j n sets s j [k] to true for all k such that n l Sj k, and sends anc to all of ts mmedate ancestors that are also descendants of n 1 f s j [1] = = s j [q j ] = true The ack a that corresponds to the anc t receved s wthheld n the affrmatve case, otherwse t s sent at once If wthheld, t s sent when n j no longer expects to receve any ack a s A more detaled descrpton of the second phase s gven next as Algorthm Compute(s), n whch the followng addtonal varables are used at node n descendant k : Boolean varable ndcatng whether n k D 1 for n k I (ths varable s assumed to have been set durng the frst phase as desc messages are receved); subordnate : Integer varable havng value 1 f n s a subordnate of n 1, 0 otherwse (ths varable s assumed to have been set durng the frst phase); reached d : Boolean varable ndcatng whether n has receved at least one desc (ntally set to true f = 1, false otherwse); expected d : Integer varable contanng the number of ack d s n expects to receve (ntally set to 0); parent d : Varable used to pont to a specal mmedate ancestor of n (ntally set to nl); ntator : Boolean varable ndcatng whether n s one of the nodes that ntate the sendng of anc s (ntally set to false); 7

8 expected a : Integer varable contanng the number of ack a s n expects to receve (ntally set to 0); parent a : Varable used to pont to a specal mmedate descendant of n (ntally set to nl) Algorthm Compute(s) comprses actons (4) through (8), respectvely for ntaton by node n 1 and for n N to respond to the recepton of a desc, an ack d, an anc, and an ack a Algorthm Compute(s): Intal acton by n 1 : Send desc to all n k O ; Set expected d accordngly (4) Acton upon recept by n of desc from n j I : (5) f reached d then Send ack d to n j (51) begn reached d := true; parent d := n j ; f subordnate = 0 then begn (52) Send desc to all n k O ; Set expected d accordngly; f expected d = 0 then Send ack d to parent d end f s [1] = = s [q ] = false then begn (53) ntator := true; s [k] := true for all k such that 1 k q ; Send anc to all n k I such that descendant k ; Set expected a accordngly end end Acton upon recept by n of ack d from n j O : (6) expected d := expected d 1; f expected d = 0 then f = 1 then Global termnaton has occurred (61) Send ack d to parent d (62) 8

9 Acton upon recept by n of anc from n j O : (7) f ntator then Send ack a to n j (71) begn s [k] := true for all k such that n j S k ; (72) f s [1] s [q ] then begn (73) parent a := n j ; Send anc to all n k I such that descendant k ; Set expected a accordngly; f expected a = 0 then Send ack a to parent a end Send ack a to n j (74) end Acton upon recept by n of ack a from n j I : (8) expected a := expected a 1; f expected a = 0 then f ntator then Send ack d to parent d (81) Send ack a to parent a (82) 33 Correctness and complexty In ths secton we frst concentrate on establshng the correctness of Algorthm Compute(s) and after that analyze the complexty of the overall algorthm, comprsng Algorthm Compute(s) and the frst phase that precedes t Throughout ths secton, we let node n be called an ntator f, durng the executon of Algorthm Compute(s), n ever executes (53), thereby settng ntator to true Note that, f n s an ntator, then there exsts n G a drected path startng at n 1 on whch n s the frst subordnate of n 1 to appear Ths s so because the absence of such a path would requre n to be reached by a desc message through another subordnate of n 1 frst, whch, by (53), never happens Algorthm Compute(s) s a combnaton of several dffusng computatons, each ncludng the termnaton-detecton mechansm of [5] The frst dffusng computaton occurs as a sngle nstance and s ntated by n 1 It uses desc and ack d messages, and proceeds accordng to (4) and (52) for the sendng of desc s, and accordng to (51), (52), (62), and (81) to send ack d s Each of the other dffusng computatons s ntated by an ntator, and employs anc messages (sent accordng to (53) and (73)) and ack a messages (sent accordng to (71), (73), (74), and (82)) The followng s how the varous dffusng computatons are combned At an ntator, by (53) the computaton ntated by n 1 s suspended and a new computaton, based on anc and ack a messages, s ntated Upon termnaton of ths computaton, the computaton ntated by n 1 s resumed by (81) It s an mmedate consequence of the results n [5] that all these computatons do ndeed termnate correctly, that s, the detecton of global termnaton by n 1 9

10 n (61) s correct We then concentrate on argung that, upon global termnaton of Algorthm Compute(s), s 1 has been computed correctly for use n checkng whether (3) holds Theorem 3 Let k be such that 1 k q 1 Upon global termnaton of Algorthm Compute(s), s 1 [k] = true f and only f all b-subgraphs of G that nclude n 1 and S k 1 also nclude a subordnate of n 1 Proof: If s 1 [k] = true when Algorthm Compute(s) termnates globally, then by (72) n 1 must have receved an anc from an mmedate descendant n S1 k Let n 1 and the mmedate descendants from whch n 1 receved an anc be marked, and proceed wth the markng of other nodes as follows If node n s marked and s not an ntator, then mark the nodes from whch n receved an anc By (53) and (73), at least one node n each of the sets S 1,,Sp gets marked, and the markng process halts at ntators or at nodes already marked Now consder any subgraph of G that ncludes n 1, S1 k, and for every n that s ncluded, one of the sets S 1,,Sp (unless n s not marked or s an ntator, such a set ncludes a marked node) It follows that ths subgraph necessarly ncludes an ntator, and s therefore a b-subgraph of G that ncludes n 1, S1 k, and a subordnate of n 1 If s 1 [k] = false upon global termnaton of Algorthm Compute(s), then by (72) no anc ever arrved at n 1 from an mmedate descendant n S1 k We do the markng process agan, startng at n 1 and all nodes n S1 k For a marked n, we mark every node from whch n receved no anc Once agan by (53) and (73), all nodes n at least one of the sets S 1,,Sp get marked, and the markng stops at nodes already marked wthout ever reachng an ntator Now consder any subgraph of G that ncludes n 1, S1, k and for every n that s ncluded, one of the sets S 1,,Sp whose nodes are all marked Clearly, ths subgraph s a b-subgraph of G that ncludes n 1 and S1 k, but no subordnate of n 1 We now turn to an analyss of the complexty of our algorthm to detect membershp of n 1 n a b-knot n G To ths end, we employ the standard measures of tme and communcaton complexty for asynchronous dstrbuted algorthms [1] Also, we let E1 a be the set of edges that le on paths drected towards n 1 n G, and E1 d the set of edges lyng on paths drected away from n 1 n G Note, ntally, by the results n [8], that the tme requred by the frst phase of our algorthm s no larger than 2 ( A 1 D 1 ) In addton, the frst phase requres exactly 2 ( E1 a + E1 d ) messages to be sent Of these, exactly one desc and one ack are sent on each member of E1 d, and exactly one anc and one ack on each member of E1 a To analyze the complexty of the second phase (Algorthm Compute(s)), note, by (4), (52), (53), and (73), that messages are only sent on edges that connect two descendants of n 1 So the tme taken by Algorthm Compute(s) s at most 4 D 1 In order to assess the number of messages nvolved n the algorthm, note that a descendant of n 1 only receves a desc message, and therefore starts partcpatng n the computaton, f a path exsts drected from n 1 to t wth no ntermedate subordnates of n 1 As a consequence, the number of messages sent by Algorthm Compute(s) s at most 4 E1 d Of these, at most one desc/ack d par flows on each member of Ed 1, and lkewse at most one anc/ack a par 4 Concludng remarks In ths paper we have dentfed b-knots as graph structures that account for deadlocks n the AND- OR model as a necessary and suffcent condton Unlke cycles and knots, b-knots are defned n explct conjuncton wth the deadlock model We have also gven an asynchronous dstrbuted algorthm for node n 1 to check whether t s n a b-knot n G Ths algorthm extends the algorthm of [8], whch s essentally a procedure to count the subordnates of n 1 Our algorthm employs a smplfcaton of ths procedure as a frst 10

11 phase (t smply dentfes the subordnates of n 1 ), and then a second phase n whch each node that s a descendant of n 1, and from whch a subordnate of n 1 s reachable, can detect whether a subordnate of n 1 s reachable from t n all b-subgraphs of G n whch t partcpates Detecton of whether n 1 s n a b-knot follows easly Ongong work ncludes nvestgatng how exstng algorthms to detect AND-OR deadlocks (those n [3, 7, 9], for example) relate to the presence of a b-knot n the wat-for graph, and how they can be mproved, f at all, by explctly consderng the b-knot whose presence we have demonstrated to be necessary and suffcent for an AND-OR deadlock to exst We also pont out that the algorthm of Secton 3 has been gven as an extenson of a well-known algorthm for knot detecton only The queston of how to explot the defnton of a b-knot more closely n order to mprove that algorthm (and then carry out an all-encompassng comparatve study of the algorthms that detect AND-OR deadlocks) s subject of ongong research as well Acknowledgments We acknowledge partal support from CNPq, CAPES, the PRONEX ntatve of Brazl s MCT under contract , the LOCUS Project of ProTeM/CNPq, and a FAPERJ BBP grant References 1 V C Barbosa, An Introducton to Dstrbuted Algorthms, The MIT Press, Cambrdge, MA, G Bracha and S Toueg, Dstrbuted deadlock detecton, Dstrbuted Computng 2 (1987), J Brzeznsk, J-M Hélary, M Raynal, and M Snghal, Deadlock models and a general algorthm for dstrbuted deadlock detecton, J of Parallel and Dstrbuted Computng 31 (1995), K M Chandy and L Lamport, Dstrbuted snapshots: determnng global states of dstrbuted systems, ACM Trans on Computer Systems 3 (1985), E W Djkstra and C S Scholten, Termnaton detecton for dffusng computatons, Informaton Processng Letters 11 (1980), E Knapp, Deadlock detecton n dstrbuted databases, ACM Computng Surveys 19 (1987), A D Kshemkalyan and M Snghal, Effcent detecton and resoluton of generalzed dstrbuted deadlocks, IEEE Trans on Software Engneerng 20 (1994), J Msra and K M Chandy, A dstrbuted graph algorthm: knot detecton, ACM Trans on Programmng Languages and Systems 4 (1982), D-S Ryang and K H Park, A two-level dstrbuted detecton algorthm of AND/OR deadlocks, J of Parallel and Dstrbuted Computng 28 (1995), M Snghal, Deadlock detecton n dstrbuted systems, IEEE Computer 22 (1989),