RELIABILITY OF COMMUNICATION NETWORKS WITH DELAY CONSTRAINTS: COMPUTATIONAL COMPLEXITY AND COMPLETE TOPOLOGIES

Transcription

1 IJMMS 2004:29, PII. S X Hindawi Publishing Cop. RELIABILITY OF COMMUNICATION NETWORKS WITH DELAY CONSTRAINTS: COMPUTATIONAL COMPLEXITY AND COMPLETE TOPOLOGIES H. CANCELA and L. PETINGI Received 22 June 2003 Let G = V,E) be a gaph with a distinguished set of teminal vetices K V. We define the K-diamete of G as the maximum distance between any pai of vetices of K. If the edges fail andomly and independently with known pobabilities vetices ae always opeational), the diamete-constained K-teminal eliability of G, R K G,D), is defined as the pobability that suviving edges span a subgaph whose K-diamete does not exceed D. In geneal, the computational complexity of evaluating R K G,D) is NP-had, as this measue subsumes the classical K-teminal eliability R K G), known to belong to this complexity class. In this note, we show that even though fo two teminal vetices s and t and D = 2, R {s,t} G,D) can be detemined in polynomial time, the poblem of calculating R {s,t} G,D) fofixedvaluesofd, D 3, is NP-had. We also genealize this esult fo any fixed numbe of teminal vetices. Although it is vey unlikely that geneal efficient algoithms exist, we pesent a ecusive fomulation fo the calculation of R {s,t} G,D) that yields a polynomial time evaluation algoithm in the case of complete topologies whee the edge set can be patitioned into at most fou equi-eliable classes Mathematics Subject Classification: 05C99, 90B Intoduction. The components of a communication netwok e.g., nodes, communication links) may be subject to andom failues. Failues may aise fom natual catastophes e.g., huicanes), component weaout, o action of intentional enemies. A communication netwok can be modelled by a gaph G = V,E), whee V and E ae the sets of vetices and edges, espectively, of G. Moeove, the pobabilities of failue of the netwok components can be epesented by assigning pobabilities of failue to the vetices and/o edges of its undelying gaph. A widely used pobabilistic model is the one whee the edges fail andomly and independently with known pobabilities, and whee the vetices ae always opeational. Let G = V,E) be a gaph with a distinguished set K V. We define the K-diamete of G as the maximum distance between any pai of vetices of K. If the edges fail andomly and independently with known pobabilities, in [2, 8], the diamete-constained K-teminal eliability of G, R K G,D), is defined as the pobability that suviving edges span a subgaph whose K-diamete does not exceed D, o equivalently, as the pobability that fo each pai of vetices {u,v} K, thee exists an opeating path between u and v of at most D edges. One paticula application of this measue is when tansmissions between evey two nodes of a teminal set K of a netwok modelled by a gaph G = V,E)) ae equied

2 1552 H. CANCELA AND L. PETINGI to expeience a maximum delay DT, whee T is the delay expeienced at a single node o edge. The pobability that afte andom failues of the communication links the suviving netwok meets the delay DT is R K G,D). As eal netwoks ae subject to failues, the diamete-constained eliability can be useful in diffeent contexts. Fo example, this measue gives an indicato of the suitability of an existing netwok topology to suppot good-quality voice ove IP applications between a pai of teminals. In the case of a videoconfeence, we take K to be the set of the paticipating nodes, and the diamete-constained eliability gives the pobability that we can find shot enough paths between all of them. Anothe potential case of inteest is a numbe of potocols which, in ode to avoid congestion by looping data, assign a timeout date o a maximum numbe of hops to each data packet, to contol infomation; that is, the case fo some pee-to-pee P2P) netwoks, such as Feenet, Gnutella, o othes [3, 7], in which a fixed maximum numbe of hops ae allowed fo communication between nodes. In these cases, the diamete-constained uneliability i.e., one minus the eliability) gives the pobability that, due to failed links, thee ae some nodes of the netwok, which ae not mutually eachable using these potocols. In Section 2, we intoduce some basic notation and definitions that will be used in the following sections, and we pesent R K G,D) as a genealization of the classical eliability measue R K G), allowing us to conclude that, in geneal, the complexity of evaluating R K G,D) is NP-had. In Section 3, we show that even though R K G,D), fo K =2andD = 2, can be calculated efficiently, this poblem is NP-had fo fixed values of D, D 3. In Section 4, we genealize this esult fo any fixed numbe of teminal vetices. In Section 5, we pesent a ecusive fomulation fo the calculation of R {s,t} G,D) that yields a polynomial-time evaluation algoithm fo the case of complete topologies, whose edge set can be patitioned into at most fou equi-eliable classes. Finally, in Section 6, we pesent some open poblems and final emaks. The notation in this pape follows that of Haay [5], unless othewise noted. 2. Peliminaies. The following notation and definitions will be used in the sequel. i) Let G = V,E, E)) be a pobabilistic gaph with a distinguished set K V, K 2, and D Z +,with1 D n 1, whee n = V and whee : E [0,1] ae the opeational pobabilities of the set of edges E. Fo ease of notation, we epesent the opeational pobability of an edge e E as pe) = 1 qe) qe) is the pobability of failue). ii) Let the sample space Ω epesent the set of all possible subsets of E coesponding to sets of opeational links i.e., Ω = 2 E ). iii) Unde the assumption of independent edge failues, each H Ω has occuence pobability PH)= pe) qe). 2.1) e H e H iv) H Ω is a pathset o opeating state if H spans a subgaph whose K-diamete is at most D. v) Let D K E) ={H Ω : H is a pathset}.

3 RELIABILITY OF COMMUNICATION NETWORKS vi) An opeating state H of D K E) is called a minpath if H {e i} D K E), fo all e i H. vii) H Ω is a failue state if H spans a subgaph whose K-diamete is geate than D if H spans a subgaph whee two vetices of K belong to diffeent connected components, then its K-diamete is infinite). viii) Let D K E) ={H Ω : H is a failue state}. Fom the definition of R K G,D) and definition v), one gets R K G,D) = P D K E) ) = H D K E) e H pe) e H qe). 2.2) Similaly, 2.2) can be ewitten in tems of the failue states: R K G,D) = 1 Q K G,D) = 1 qe). 2.3) H D K E) e H pe) e H A widely used pobabilistic measue see [1, 4, 11, 12]) is the classical K-teminal eliability ofagaphg, R K G), defined on the same pobabilistic model i.e., edges fail andomly and independently with known pobabilities and the vetices ae always opeational). The measue R K G) is the pobability that fo evey pai of vetices u,v K, thee exists an opeating path between u and v. In this case, thee ae no length estictions of the paths joining the vetices of K, and by noting that the maximum length of a path joining a pai of vetices is of at most n 1 edges, whee n is the numbe of vetices of G, then R K G) = R K G,n 1). 2.4) This genealization of the classical eliability paamete allows us to eflect moe stingent pefomance objectives by esticting the maximum length of a path in a netwok. Let G = V,E) and let K be a set of teminal vetices of G. Fo the classical eliability measue, computations of the K-teminal eliability see [10]) and the specific cases when K =2see[13]) and K = V see [6, 9]) wee shown to be NP-had. Fom these esults and the fact that R K G) = R K G,n 1), R {s,t} G) = R {s,t} G,n 1), andr V G) = R V G,n 1), whee n is the numbe of vetices of the gaph G, and by esticting D = n 1, we have the following theoem. Theoem 2.1. Fo the diamete-constained eliability, the computational complexity of computing R K G,D), R {s,t} G,D), and R V G,D) is NP-had. Even though it is vey unlikely that R K G,D) can be evaluated efficiently, we cannot peclude that it is the case when fixed values of the diamete paamete D ae unde consideation. We addess this question in Sections 3 and Evaluating R K G,D) when K =2 and when D is a constant value. In this section, we establish the computational complexity of computing R K G,D) when K is composed of two teminal vetices s and t, and fo fixed values of diamete paamete D.

4 1554 H. CANCELA AND L. PETINGI In [8], an efficient fomulation was given fo the evaluation of diamete-constained two-teminal eliability of a netwok when teminals s and t should be connected by opeating paths of at most two edges i.e., D = 2). Theoem 3.1. Let G = V,E) be a simple gaph whee each edge u,v) E opeates independently with pobability p u,v), and let {s,t} ={u 1,u 2,...,u l } be the common neighbohood of teminal nodes s and t; then whee 1 R :s,t) E, R {s,t} G,2) = 1 ) 1 p s,t) R :s,t) E, R = l i=1 3.1) ) 1 ps,ui )p ui,t). 3.2) Even though R {s,t} G,2) can be computed in time linealy on the numbe of nodes of G, we next show that the complexity of evaluating R {s,t} G,D), fo fixed values of D, is NP-had. Fo ease of notation, instead of epesenting a state opeational o failue) as a set of edges of a gaph G, we epesent it as a subgaph of G spanned by this set. Theoem 3.2. Evaluating R {s,t} G,D), fo fixed D 3, is NP-had. Poof. We show that counting the numbe of vetex coves of a bipatite gaph is polynomial-time Tuing educible to counting the numbe of failue states of an undiected gaph with teminal vetices s and t and fixed diamete paamete D 3. An instance of the bipatite vetex cove consists of a bipatite gaph G = V,E); let X and Y be the classes in the bipatition of V. A vetex cove is a set of vetices C = C X C Y, C X X, and C Y Y such that evey edge of E has at least one endpoint in C. The poblem of counting the numbe of vetex coves of a bipatite gaph was shown to be # -complete by Povan and Ball [9]. Let D = 3 + d and let P = V P),EP)) be a path on d + 1 vetices, whee VP) = {s,s 1,...,s d } and EP) ={s,s 1 ),s 1,s 2 ),...,s d 1,s d )} it is undestood that if d = 0, P is composed of an isolated vetex s d = s). We constuct a pobabilistic gaph G = V,E, E )) fom G. The vetex set V consists of V, VP), and t. The edge set E consists of E, EP), the edge sets E X ={s d,x) : x X} and E Y ={t,y) : y Y } see Figue 3.1). The edges opeational pobabilities ae pe) = 1ife E, pe) = 1if e EP), and pe) = 1/2 othewise. The teminal set is K ={s,t}. It is clea that this tansfomation is polynomial on the size of the input of the bipatite gaph, since D is a constant value. Fom this constuction, a state H of G consists of the bipatite gaph G, thepathp, the vetex s d, possibly joined to some vetices of X, and the vetex t, possibly joined to some vetices of Y of the bipatite gaph G. We then constuct a one-to-one coespondence Z : W D {s,t}e ) fom the set of vetex coves W of the bipatite gaph to the failue states of G as follows: if w = S T) W, whee S X and T Y, is a vetex cove of the bipatite gaph, then

5 RELIABILITY OF COMMUNICATION NETWORKS s s 1 s d t Path on d + 1 vetices Bipatite gaph Figue 3.1. Gaph G constucted fom bipatite G and constant d. s s d t Bipatite Figue 3.2. Failue state constucted fom a vetex cove of bipatite gaph black vetices epesent a vetex cove). Zw) = V,E EP) {s d,x): x X S} {t,y) : y Y T }) is a failue state of G see Figue 3.2). To show that this is a one-to-one coespondence, let w = S T) be a vetex cove of the bipatite gaph G, and suppose x X S and y Y T. Clealy, x,y) is not an edge in Zw), othewise if x,y) is an edge, this edge is not coveed by w. Thus, thee ae no paths of length D joining s and t in Zw). Theefoe, Zw) is a failue state of G. Convesely, conside a failue state H which, as emaked peviously, must include all the edges of path P, all the edges of set E, and possibly some edges joining vetex s d to vetices in X and joining vetex t to some vetices in Y.AsH is a failue state, the vetices of X and Y adjacent to s d and t must fom an independence set if that was not the case, thee would exist a path of length D joining s and t). Theefoe, the emaining vetices of the bipatite gaph fom a vetex cove which is Z 1 H). Fom 2.3), we obtain R {s,t} G,D ) X + Y = 1 = 1 i=0 F i G,{s,t},D ) 1 2 X + Y i=0 F i G,{s,t},D ) 2 X + Y, ) i ) 1 X + Y i 1) E + EP) 2 3.3)

6 1556 H. CANCELA AND L. PETINGI whee F i G,{s,t},D) is the numbe of failue states of G with E + EP) +i edges. But F i G,{s,t},D) is equal to the numbe of vetex coves of the bipatite gaph G, thus the esult follows. In the next section, we genealize this esult fo any fixed numbe of teminal vetices. 4. Evaluating R K G,D) fo any fixed numbe of teminal vetices K and when D is a constant value, D 3. In this section, we show that complexity of calculating R K G,D) is NP-had when any fixed numbe of teminal vetices ae unde consideation and when D is a constant value, D 3. Theoem 4.1. Evaluating R K G,D) fo a fixed numbe of teminal vetices and fo fixed D, D 3, is NP-had. Poof. Let k = K with k>2, and D = 3 + d. We constuct a pobabilistic G = V,E, E )) obtained fom the pobabilistic gaph G = V,E, E )) mentioned in the poof of Theoem 3.2 as follows: a) fo each j, 1 j k 2, let P j = V j ={u j 1,uj 2,...,uj D}, E j ={u j 1,uj 2 ),...,uj D 1, u j D )}) be a path on D vetices; b) let the set of edges E s ={s,u j 1 ) :1 j k 2} i.e., we make the teminal vetex s adjacent to each vetex u j 1 of P j,1 j k 2); c) let the set of edges E t ={t,u j D ) :1 j k 2} i.e., we make the teminal vetex t adjacent to each vetex u j D of P j,1 j k 2); d) let G = V k 2 j=1 V j ),E k 2 E s Et j=1 E j ), E )) see Figue 4.1). Fo the opeational pobabilities of G,ife is an edge in which one of the endpoints is s d o t and the othe endpoint is a vetex of the bipatite gaph G, thenpe) = 1/2, othewise let pe) = 1. Let also the teminal set K ={s,t} k 2 j=1{u j 1 }). This constuction is also polynomial on the size of the input of the bipatite G, since both K and D ae constant values. If s and t ought to be connected by a path of at most D edges, then this path must compise the path P of G see the poof of Theoem 3.2). In a failue state H of G, evey pai of teminal vetices is connected by paths of at most D edges, with the exception of s and t; thus H = V H),EH) ) is a failue state of G H = VH) k 2 k 2 V j,eh) Es Et E j is a failue state of G. j=1 j=1 4.1) Also, it follows fom 4.1) that the occuence pobability see Section 2, definition iii)) of the edge set EH ) of a failue state H of G is equal to the occuence pobability oftheedgeseteh) of its coesponding failue state H of G as P E H )) = P EH) ) 1 Es 1 E t 1 k 2 j=1 E j = P EH) ). 4.2)

7 RELIABILITY OF COMMUNICATION NETWORKS u 1 1 s s 1 s d t u k 2 1 Bipatite gaph Path of length D Figue 4.1. Gaph G constucted fom bipatite gaph G. Theefoe, it follows fom 2.3), 3.3), 4.1), and 4.2) that R K G,D ) = R {s,t} G,D ) = 1 X + Y i=0 F i G,{s,t},D ) 2 X + Y. 4.3) But, as was shown in the poof of Theoem 3.2, F i G,{s,t},D) is equal to the numbe of vetex coves of the bipatite gaph G, thus the esult follows. 5. Two-teminal eliability of complete topologies with at most fou edge eliability values. In this section, we define a class of complete gaphs, whose edges can be patitioned into fou elementay eliability values. We will denote these gaphs by G A n,q st,q s,q t,q), whee n is the numbe of intemediate nodes between the teminals s and t, q st = 1 st is the uneliability of the edge connecting s and t, q s = 1 s is the uneliability of the edges connecting s to the othe n intemediate nodes, q t = 1 t is the uneliability of the edges connecting t to the n intemediate nodes, and q = 1 is the uneliability of the edges whose endpoints ae intemediate nodes see Figue 5.1a)). The eliability fo this netwok, R {s,t} G A n,q st,q s,q t,q),d), is a multinomial in q st, q s, q t, q. We will expess the eliability of netwok G A by taking into account the independence between the edge s,t) and the othe paths, and by conditioning on the numbe of opeational edges between s and the intemediate nodes exploiting the symmety between those nodes). If D = 1, the only feasible path is the edge s,t); then tivially R {s,t} G A n,q st,q s,q t,q),1) = 1 q st. We can now look at the case whee D>1. As a fist step, we use the fact that s,t) is a feasible path and that it is independent fom all othe simple paths fom s to t. Thenwecanwite R {s,t} GA n,qst,q s,q t,q ),D ) = 1 q st ) +qst R {s,t} G A n,qs,q t,q ),D ), 5.1)

8 1558 H. CANCELA AND L. PETINGI st s t s t s t s t s t s t K n K n a) G A b) G A Figue 5.1. G A n,q st,q s,q t,q) and G A n,q s,q t,q) netwoks. whee G A n,q s,q t,q) is the pobabilistic netwok obtained fom G A n,q st,q s,q t,q),by deleting the edge s, t) see Figue 5.1b)). As this netwok is completely symmetic with espect to the edges between s and the intemediate nodes all nodes except s and t), we will define a patition of the pobability state space fo the netwok based on the numbe of those edges that fail o ae opeational. We define A k to be the event whee k edges fom s to intemediate nodes fail and the emaining n k ae opeating; its pobability is P { A k } = n k) qs ) k 1 qs ) n k. 5.2) The set {A k :0 k n} is a patition of the pobability space, as the events ae paiwise-disjoint and thei union has pobability one. Applying the total pobability theoem, we then have R {s,t} G A n,qs,q t,q ),D ) = P { opeational path of length D fom s to t in G A } n = P { 5.3) opeational path of length D fom s to t in G A } { } Ai P Ai. i=0 We must now find an expession fo the geneal tem with k n. The leftmost netwok shown in Figue 5.2 coesponds to this event, whee k edges between s and intemediate nodes fail i.e., can be emoved fom the netwok) and the emaining n k ae opeational, which ae pesented with a bolde tace. Finding an opeational path of length less than o equal to D in this netwok coesponds to finding a path of length less than o equal to D 1 in the netwok shown at the cente of Figue 5.2, which is obtained by identifying s and the intemediate nodes to which it is unconditionally

9 RELIABILITY OF COMMUNICATION NETWORKS K n k t t 1 1 t ) n k s t t t t s t t t t s 1 1 ) n k t t 1 1 ) n k t K k K k K k Figue 5.2. G A n,q s,q t,q) when n k edges between s and intemediate nodes wok and the est fail. connected into one node. In addition, as edges fail independently, the opeational pobability of a bank of paallel edges is the complement to one) of the poducts of thei failue pobabilities. Thus, the set of paallel edges between s and t shown at the cente of Figue 5.2 can be eplaced by a single edge with eliability 1 qt n k. Similaly, the set of paallel edges between s and an intemediate node of K k can be eplaced by a single edge with eliability 1 q n k. The netwok esulting fom this last opeation is shown at the ight of Figue 5.2, and coesponds to a topology, whee the opeational paths must have length at most D 1. Fom 5.1), 5.2), and 5.3) and the latest fact, we can expess the eliability of the oiginal netwok G A by the following fomula: R {s,t} GA n,qst,q s,q t,q ),D ) = ) n ) n 1 qs ) n kq k 1 q st +qst s R {s,t} GA k,q n k t,q n k,q t,q ),D 1 ) 5.4). k k=0 The ecusive application of this fomula gives a multinomial on q st, q s, q t, q. It can be obseved that the diect application of the ecusive fomula leads to evaluating the multinomial many times fo values of k n and d<d, but with diffeent values to be substituted fo the paametes q st, q s, q t, and q. This obsevation leads us to the following altenative fomulation. We define a subclass of the class of netwoks G A. These netwoks will be denoted by G B m,h,q t,q),withg B m,h,q t,q) = G A m,q h t,q h,q t,q). This means that these netwoks have the same complete gaph topology, but that the eliabilities of the fou-edge classes ae defined by only two intege and two eal paametes. It is tivial to obseve that R {s,t} GB m,h,qt,q ),D ) = R {s,t} GA m,q h t,q h,q t,q ),D ). 5.5) Substituting in 5.4), we have that R {s,t} GA n,qst,q s,q t,q ),D ) = 1 q st ) +qst n k=0 ) n 1 qs ) n kq k s R {s,t} GB k,n k,qt,q ),D 1 ) 5.6) k

10 1560 H. CANCELA AND L. PETINGI and that if D>1, R {s,t} GB m,h,qt,q ),D ) = 1 qt h ) m +q h t k=0 ) m 1 q h ) m k q hk R {s,t} GB k,m k,qt,q ),D 1 ). k 5.7) If D = 1, then R st GB m,h,qt,q ),D ) = 1 q h t. 5.8) Moeove, it is also noted that the last tem i.e., k=m)of5.7) is null since G B m,0,q t,q) epesents a netwok with m + 2 nodes in which the edges fom s to the intemediate nodes fail, and whee s is adjacent to the teminal node t by an edge whose eliability is 0 i.e., 1 q 0 t ). The next theoem states the space and time complexity of evaluating R {s,t} G A n,q st, q s,q t,q),d). Theoem 5.1. Evaluating R {s,t} G A n,q st,q s,q t,q),d) takes On 3 ) and ODn 3 ) space and time complexity, espectively. Poof. We fist stat with the standad assumption that the edges eliabilities ae ational numbes p/q, whee both p and q ae of length On), and that aithmetic opeations on such ational numbes take constant time i.e., O1)) see[4, page 4]). We next note, fom the ight-hand side of 5.7), that evaluation of R {s,t} G B m,h,q t, q),d) fo specific values of m and h depends on powes of the edges uneliabilities q t and q. Since the possible values fo m and h ae in [0,n], we fist poceed to evaluate and stoe the values qt i,fo0 i n, and qi,fo0 i n. This pocedue equies On 2 ) space complexity n values to be stoed, each having length On)) andon 2 ) time complexity. We then calculate and stoe all possible values q ij,fo0 i,j n.this can be accomplished by utilizing an n n two-dimensional aay and values aleady stoed fo powes of q. This pocedue takes On 3 ) space and time complexity. Values fo 1 q i ) j ae stoed in a simila fashion. Finally, we poceed to evaluate R {s,t} G B m,h,q t,q),d) fo specific values of m, h, and d. In ode to execute this last step, we note, fom 5.7), that an evaluation of R {s,t} G B m,h,q t,q),d) equies at most n values i.e., n tems in the summation) of R {s,t} G B m,h,q t,q),d 1). An efficient evaluation is, fo example, to fist evaluate and stoe R {s,t} G B m,h,q t,q),d),fod = 1, and fo all values of m with 0 m n) and h with 0 h n) i.e.,on 3 ) space and On 2 ) time complexity). Next we detemine and stoe the values of R {s,t} G B m,h,q t,q),d) fo d = 2 and fo each pai m,h), by utilizing aleady stoed values; by the pevious assumptions, this step will take On) execution time. We sequentially epeat this last pocedue fo all possible values of d, 1 d<d, yielding an ODn 3 ) space and time complexity. We can also impove the stoage to On 3 ) by noting that to evaluate R {s,t} G B m,h,q t,q),d), it is only necessay to stoe values fo d 1.

11 RELIABILITY OF COMMUNICATION NETWORKS Table 5.1. Execution times fo evaluating complete gaph eliability. n D Execution time CPUs) Iteative method Recusive method 10 2 < < < < < < < > > > > > We implemented this iteative pocedue as well as the ecusive method which is obtained by diectly applying 5.4). Table 5.1 shows the execution times obtained on an Intel Celeon PC) fo both the iteative and the ecusive methods, fo diffeent values of n and D. Some enties ae maked > 36000, coesponding to uns which exceeded 10 CPU hous seconds) and wee aboted. It is easy to see that the iteative method has a much bette behavio than the ecusive fomulation, leading to smalle execution times especially fo high values of D). 6. Conclusions and futue wok. In this pape, we investigated the computational complexity of the diamete-constained K-teminal eliability. In paticula, this measue subsumes the classical K-teminal eliability measue when thee is no estiction on the length of the paths connecting the teminal vetices. This equivalence between these two eliability measues allows us to conclude that, in geneal, the computational complexity of the diamete-constained K-teminal eliability is NP-had. Nevetheless, the poblem of detemining the computational complexity of R K G,D), fo specific cases of K and D, was open. We showed that even though fo two teminal vetices s and t and D = 2, R {s,t} G,D) can be detemined in polynomial time, the poblem of calculating R {s,t} G,D) fo fixed values of D, D 3, is NP-had. Moeove, we genealized this esult fo any abitay numbe of teminal vetices. As a consequence, this esult justifies the implementation of appoximation methods fo the evaluation, and the detemination of bounds fo R K G,D). Anothe elevant open poblem is to detemine subclasses of gaphs fo which polynomial-time evaluation algoithms exist. In this pape, we showed that fo the case of complete topologies whee the edge set can be patitioned into at most fou equi-eliable classes, R {s,t} G,D) can be computed efficiently.

12 1562 H. CANCELA AND L. PETINGI Refeences [1] F. T. Boesch, X. Li, and C. Suffel, On the existence of unifomly optimally eliable netwoks, Netwoks ), no. 2, [2] H. Cancela and L. Petingi, Diamete constained netwok eliability: exact evaluation by factoization and bounds, Poceedings of the Intenational Confeence on Industial Logistics Japan), 2001, pp [3] I. Clake, O. Sandbeg, B. Wiley, and T. W. Hong, Feenet: a distibuted anonymous infomation stoage and etieval system, Intenational Wokshop on Design Issues in Anonymity and Unobsevability, The Intenational Compute Science Institute, Califonia, 2000, pp [4] C. J. Colboun, The Combinatoics of Netwok Reliability, Intenational Seies of Monogaphs on Compute Science, The Claendon Pess, Oxfod Univesity Pess, New Yok, [5] F. Haay, Gaph Theoy, Addison-Wesley, Massachusetts, [6] M. Jeum, On the complexity of evaluating multivaiate polynomials, Ph.D. thesis, Depatment of Compute Science, Univesity of Edinbugh, Edinbugh, [7] G. Panduangan, P. Raghavan, and E. Upfal, Building low-diamete P2P netwoks, 42nd IEEE Symposium on Foundations of Compute Science Las Vegas, Nev, 2001), IEEE Compute Society, Califonia, 2001, pp [8] L. Petingi and J. Rodiguez, Reliability of netwoks with delay constaints, Cong. Nume ), [9] J. S. Povan and M. O. Ball, The complexity of counting cuts and of computing the pobability that a gaph is connected, SIAMJ. Comput ), no. 4, [10] A. Rosenthal, A compute scientist looks at eliability computations, Reliability and Fault Tee Analysis Conf., Univ. Califonia, Bekeley, Calif, 1974), Society fo Industeial and Applied Mathematics, Pennsylvania, 1975, pp [11] A. Satyanaayana and J. N. Hagstom, A new algoithm fo the eliability analysis of multiteminal netwoks, IEEE Tans. Reliab ), [12] D. R. Shie, Netwok Reliability and Algebaic Stuctues, Oxfod Science Publications, The Claendon Pess, Oxfod Univesity Pess, New Yok, [13] L. G. Valiant, The complexity of enumeation and eliability poblems, SIAMJ. Comput ), no. 3, H. Cancela: Instituto de Computación, Facultad de Ingenieía, Univesidad de la República, J. Heea y Reissig 565, Montevideo, Uuguay addess: cancela@fing.edu.uy L. Petingi: Compute Science Depatment, College of Staten Island, City Univesity of New Yok, 2800 Victoy Boulevad, Staten Island, NY 10314, USA addess: petingi@postbox.csi.cuny.edu

13 Advances in Opeations Reseach Advances in Decision Sciences Jounal of Applied Mathematics Algeba Jounal of Pobability and Statistics The Scientific Wold Jounal Intenational Jounal of Diffeential Equations Submit you manuscipts at Intenational Jounal of Advances in Combinatoics Mathematical Physics Jounal of Complex Analysis Intenational Jounal of Mathematics and Mathematical Sciences Mathematical Poblems in Engineeing Jounal of Mathematics Discete Mathematics Jounal of Discete Dynamics in Natue and Society Jounal of Function Spaces Abstact and Applied Analysis Intenational Jounal of Jounal of Stochastic Analysis Optimization