Optimal Oblivious Path Selection on the Mesh

Transcription

1 Optimal Oblivious Path Selection on the Mesh Costas Busch Malik Magon-Ismail Jing Xi Department of Computer Science Rensselaer Polytechnic Institute Troy, NY 280, USA Abstract In the oblivious path selection problem, each packet in the network inepenently chooses a path, which is an important property if the routing algorithm is inepenent. The quality of the paths is etermine by the congestion C, the maximum number of paths crossing an ege, an the ilation D, the maximum path length. So far, the oblivious algorithms stuie in the literature have focuse on minimizing the congestion while ignoring the ilation. An open question is whether C an D can be controle simultaneously. Here, we answer this question for the -imensional mesh. We present an online algorithm for which C an D are both within O( 2 ) of optimal. The algorithm uses ranomization, an we show that the number of ranom bits require per packet is within O() of the minimum number of ranom bits require by any algorithm that obtains near-optimal congestion. For fixe, our algorithm is asymptotically optimal. Introuction Given a set of packet transfer requests in a communication network, a routing algorithm must select the paths that will be traverse in orer to fulfill all the requests. A routing algorithm is oblivious if every path that is selecte for each request is chosen inepenently of every other path. Oblivious algorithms are by their nature istribute an capable of solving online routing problems, where packets continuously arrive in the network. Hence, oblivious routing (path selection) is preferre to non-oblivious routing, since one oes not nee to make assumptions regaring the nature of the traffic. We stuy oblivious algorithms in the context of a synchronous routing moel in which at most one packet traverses any ege uring a time step. Initially, at time zero, a set of packets must simulataneously select their paths. The quality of the paths selecte can be measure

2 by two parameters: the (ege) congestion C, the maximum number of paths that use any ege in the network, an the ilation D, the maximum length of any path. A trivial lower boun for the total time to transfer all the packets along the selecte paths is Ω(C + D), hence C + D is a natural metric by which to measure the quality of the paths output by a routing algorithm. Traitionally, the banwith of the network is usually the major bottleneck, hence optimizing the loa istribution, that is, minimizing the congestion, has been the focus of existing research. In particular, Maggs et al. [9] gave the first oblivious path selection algorithm with optimal ege congestion on the Mesh. Since then, generalizations have been given to oblivious path selection algorithms for arbitrary networks that achieve near optimal congestion [3, 4, 7, ]. However, these algorithms o not control the ilation. For example, a packet path may be stretche by an arbitrary amount a packet that has estination at a neighboring noe may traverse the entire network before reaching its estination. An interesting open problem is to obtain small path stretch in aition to low congestion. Here, we stuy this problem for the -imensional mesh network, for which we show that it is inee possible to obtain near optimal congestion while maintaining small stretch. For general networks, this is not possible. Optimal Oblivious Routing. Given a routing problem (collection of sources an estinations), we efine C to be the optimal congestion attainable by any routing algorithm (oblivious or not). We efine Cobl to be the optimal congestion attainable if the routing algorithm is restricte to being oblivious. For the -imensional Mesh with n noes, Maggs et al. [9] give the lower boun Cobl = log n) in the worst case. We will compare the Ω(C congestion of our algorithm with the lower boun on Cobl. The stretch of a path from a source to a estination is the ratio of the path length to the shortest path length. Clearly, the smallest stretch factor is, since a packet can follow a shortest path; however, the choice of shortest paths may increase the congestion. Our Contributions. We show the surprising result that it is possible to obtain small stretch for any routing instance, using an oblivious algorithm, an without sacrificing on the congestion. Specifically, we give an oblivious routing algorithm for the -imensional mesh with n noes, that achieves congestion O(C log n), an stretch O( 2 ). Consiering the class of oblivious algorithms, our algorithm is within O( 2 ) of optimal for both congestion an ilation, which means that the routing time boun (C + D) is within O( 2 ) of optimal. For fixe, our algorithm is optimal to within constant factors. Our algorithm is base upon a hierarchical ecomposition of the Mesh. Starting at its source noe, a packet constructs its path by ranomly selecting intermeiate points in submeshes of increasing size until the current submesh contains the estination noe. Then ranom intermeiate points are selecte in submeshes of ecreasing size until the estination noe is reache. The key new iea that we introuce is the notion of brige submeshes that make it possible to move from a source to a estination more quickly, without increasing the congestion. These brige submeshes are instrumental in controling the stretch, while 2

3 maintaining low congestion. Our algorithm uses ranomization, an we show that ranomization is essential for obtaining low congestion. In a eterministic algorithm, a packet path is fixe, given its source an estination. It can be shown that for any eterministic algorithm, there exists a routing problem with high congestion. The only alternative is to use ranomization: a packet selects a path probabilistically from a choice of κ alternatives, where κ may epen on the source an estination. Such an algorithm requires at least log κ ranom bits per packet. We give a lower boun on the number of bits require, Ω(( D(s,t) ) log ), where D(s, t) is the shortest path istance between the source an estination. The number of ranom bits require by our algorithm is within O() of this lower boun, an hence is near-optimal. Relate Work. Most relate to our work is the original paper by Maggs et al. [9] where they present an oblivious algorithm with congestion O(C log n). However, the stretch factor in that algorithm is unboune. To control stretch, we generalize the hierarchical ecomposition for the mesh enote by an access tree [9] to a more general access graph. In the access tree, only one path exists between a particular source an estination, however, our ecomposition offers several paths, in particular much shorter paths. Our algorithm uses ranomize imension by imension routing, which alone can improve the result in [9] by a factor of to O(C log n). Following the work in [9], there have been extensions to general networks, [3, 4, 7, ], where progressively better oblivious algorithms with near optimal congestion are given. Once again, the stretch is unboune. For general networks, the stretch an congestion cannot inepenently be controle. Non-oblivious approaches to optimizing C + D have receive consierable attention, an near optimal algorithms are iscusse in [, 2, 2, 3]. As alreay mentione, such offline algorithms require knowlege of the traffic istribution a priori an generally o not scale well with the number of packets. We show that for the mesh, istribute an oblivious algorithms are within a logarithmic factor from the optimal offline performance, hence there is no significant benefit from using the offline algorithm. Trae offs between stretch an congestion have been stuie in wireless networks [6]. Lower bouns on the competitive ratio of oblivious routing has been stuie for various types of networks. Maggs et al. [9] give the Ω( C log n) lower boun on the competitive ratio of an oblivious algorithm on the mesh. Valiant an Brebner [4] perform a worst case theoretical analysis on oblivious routing on specific network topologies such as the hypercube. Boroin an Hopcroft [5] an Kaklamanis et al. [8] showe that eterministic oblivious routing algorithms can not approximate the minimal loa on most non-trivial networks, which justifies the necessity for ranomization. Here, we give a lower boun on the amount of ranomization require. Paper Outline. We begin with some preliminary efinitions (Section 2), an continue with our mesh ecomposition algorithm in 2-imensions (Section 3), which we generalize to -imensions in Section 4. We iscuss ranomization requirements in Section 5, an we conclue in Section 6. Several proofs are relegate to the appenix. 3

4 2 Preliminaries The -imensional mesh M is a -imensional gri of noes with sie length m i in imension i. There is a link connecting a noe with each of its 2 neighbors (except at the noes at the bounaries of the mesh). We enote by n the size of M, n = size(m) = i= m i, an by E the number of eges in the network. Each noe has a coorinate with the top-left noe having coorinate (0, 0). We refer to specific submeshes by giving its en points in every imension, for example, [0, 3][2, 5] refers to a 4 4 submesh, with the x coorinate ranging from 0 to 3 an the y coorinate from 2 to 5. The input for the path selection problem is a set of N sources an estinations (i.e. packets), Π = {s i, t i } N i= an the mesh M. The output is a set of paths, P = {p i }, where each path p i P is from noe s i to noe t i. The length of path p, enote p, is the number of eges it uses. We enote the length of the shortest path from s to t by ist(s, t). We will enote by D the maximum shortest istance, max i ist(s i, t i ). The stretch of a path p i, enote stretch(p i ), is the ratio of the path length to the shortest path length between its source an estination, stretch(p i ) = p i /ist(s i, t i ). The stretch factor for the collection of paths P, enote stretch(p), is the maximum stretch of any path in P, stretch(p) = max i stretch(p i ). For a submesh M M, let out(m ) enote the number of eges at the bounary of M, which connect noes in M with noes outsie M. For any routing problem Π, we efine the bounary congestion as follows. Consier some submesh of the network M. Let Π enote the packets (pairs of sources an estinations) in Π which have either their source or estination in M, but not both. All the packets in Π will cross the bounary of M. The paths of these packets will cause congestion at least Π /out(m ). We efine the bounary congestion of M to be B(M, Π) = Π /out(m ). For the routing problem Π, the bounary congestion B is the maximum bounary congestion over all its submeshes, i.e. B = max M M B(M, Π). Clearly, C B. 3 The 2-Dimensional Mesh Here we show how to select the paths in a 2-imensional mesh with equal sie lengths m = 2 k, k 0. The path selection algorithm relies on a ecomposition of the mesh to submeshes, an then constructing an access graph, as we escribe next. 3. Decomposition to Submeshes We ecompose the mesh M into two types of submeshes, type- an type-2, as follows. Type- Submeshes. We efine the type- submeshes recursively. There are k + levels of type- submeshes, l = 0,...,k. The mesh M itself is the only level 0 submesh. Every submesh at level l can be partitione into 4 submeshes by iviing each sie by 2. Each resulting submesh is a type- submesh at level l +. This construction is illustrate in 4

5 Level, type. Level, type 2. Level 2, type. Level 2, type 2. Figure : Mesh ecomposition for the mesh. Arrows inicate the parents of a submesh. Figure. In general, at level l there are 2 2l submeshes each with sie m l = 2 k l. Note that the level k submeshes are the iniviual noes of the mesh. Type-2 Submeshes. There are k levels of type-2 submeshes, l =,...,k. The type- 2 submeshes at level l are obtaine by first extening the gri of type- meshes by aing one layer of type- meshes along every imension. The resulting gri is then translate by the vector (m l /2, m l /2). In this enlarge an translate gri, some of the resulting translate submeshes are entirely within M. These are the internal type-2 submeshes. For the remaining external type-2 submeshes, we keep only their intersection with M, except that we iscar all the corner submeshes, because they will be inclue in the type- submeshes at the next level. Notice that all the type-2 submeshes have at least sie of length m l noes. Figure illustrates the construction. A submesh of M is regular if it is either type- or type-2. Unless otherwise state, we will refer to regular submeshes. The following lemma follows from the construction of the regular submeshes. Lemma 3. The mesh ecomposition satisfies the following properties. () The type- submeshes at a given level are isjoint, as are the type-2 submeshes. (2) Every regular submesh at level l can be partitione into type- submeshes at level l+. (3) Every regular submesh at level l + is completely containe by a submesh at level l of either type- or type-2, or both. 3.2 Access Graph The access graph G(M), for the mesh M, is a levele graph with k + levels of noes, l = 0,..., k. The noes in the access graph correspon to the istinct regular submeshes. 5

6 Specifically, every level-l submesh (type- or type-2) correspons to a level l noe in G(M). Eges exist only between ajacent levels of the graph. Let u l, u l+ be level l an l + noes of G(M) respectively. The ege (u l, u l+ ) exists if the regular submesh corresponing to u l completely contains the regular submesh corresponing to u l+. We borrow some terminology from trees. We say that u l is a parent of u l+ in G(M); the parent relationship is illustrate in Figure, for the corresponing submeshes. Note that the access graph is not necessarily a tree, since a noe can have two parents (a consequence of Lemma 3., part (3)). The epth of a noe is the same as its level l, an its height is k l. Noes at height 0 have no chilren, an are leaves. The leaves in G(M) correspon to single noes in the mesh. There is a unique root at level 0, which correspons to the whole mesh M. Let p = (u, u 2,..., u k ) be a path in G(M). We say that p is monotonic if every noe is of increasing level (i.e., the level of u i is higher than the level of u i+ ), an the respective submeshes of noes u 2,...,u k are all of type-. If p is monotonic, then we say that u is ancestor of u k. We will use a function g to map noes in the access graph to submeshes. Let u be a noe in the access graph with corresponing submesh M. We efine the function g so that g(u) = M. Denote by g the inverse of function g, that is, g (M ) = u. Using inuction on the height of G(M), an part (2) of Lemma 3., we obtain the following lemma: Lemma 3.2 Let v be any noe of a regular submesh M M, then g (M ) is an ancestor of g (v). Let u an v be two leaves of G(M), an let A be their (not necessarily unique) eepest common ancestor; Note that A exists an in the worst case is g (M) (a consequence of Lemma 3.2). Let p = (u,..., A,...,v), be the concatenation of two monotonic paths, one from A to u an the other from A to v. We will refer to p as the bitonic path between u an v. Submesh g(a) may be type- or type-2, all the other submeshes in p are of type-. We will refer to g(a) as a brige submesh, since it provies the connecting point between two monotonic paths. Note that type-2 submeshes can be use as briges between type- submeshes, when constructing bitonic paths between leaves. Further, only one type-2 submesh is ever neee in a bitonic path. These access graph paths will be use by the path selection algorithm. Suppose that height(a) = h A. The length of a bitonic path from u to v is 2h A. We now show that h A cannot be too large. This will be important in proving that the path selection algorithm gives constant stretch. Lemma 3.3 The eepest common ancestor of two leaves u an v has height at most log ist(g(u), g(v)) + 2. Proof: Let s, t M such that s = g(u) an t = g(v). We show that there is a common ancestor with height at most log ist(s, t) + 2. Assume, for simplicity, that we are on the torus (the same result hols for the mesh, with minor technical etails in the proof ue to ege effects). In this case, all the type-2 meshes are of the same size. We obtain the regular submeshes in the original mesh after truncation of the submeshes at the borers of the torus. log ist(s,t) Let µ = 2 ist(s, t). If 4µ 2 k, then the root, g (M), is a common ancestor with height log ist(s, t) + 2, so assume that 4µ < 2 k. Noe s is containe in 6

7 some type- submesh of sie length 4µ. Without loss of generality assume that this submesh is M = [0, 4µ ] 2. If M also contains t, then we are one, since by Lemma 3.2, g (M ) is a common ancestor at height log ist(s, t) + 2. So suppose that t is containe in some other (ajacent) type- submesh M 2. Without loss of generality, there are two possibilities for M 2. (i) M 2 = [4µ, 8µ ][4µ, 8µ ]. Since ist(s, t) µ, s [3µ, 4µ ] 2 an t [4µ, 5µ ] 2, an so the type-2 submesh [2µ, 6µ ] 2 contains both s an t. (ii) M 2 = [0, 4µ ][4µ, 8µ ]. There are four cases: (a) s [0, 2µ ][3µ, 4µ ] an t [0, 2µ ][4µ, 5µ ], in which case the type-2 submesh [ 2µ, 2µ ][2µ, 6µ ] contains s, t; (b) s [2µ, 4µ ][3µ, 4µ ] an t [2µ, 4µ ][4µ, 5µ ], in which case the type-2 submesh [2µ, 6µ ] 2 contains s, t; (c) s [µ, 2µ ][3µ, 4µ ] an t [2µ, 3µ ][4µ, 5µ ], in which case the type-2 submesh [µ, 3µ ][3µ, 5µ ] at height log ist(s, t) + contains s, t; () s [2µ, 3µ ][3µ, 4µ ] an t [µ, 2µ ][4µ, 5µ ], which is similar to (c). In all cases, s, t are containe in a submesh of height at most log ist(s, t) Path Selection Given the access graph, the proceure to etermine a path from a given source s to a estination t is summarize in the following algorithm. : Input: Source s an estination t in the mesh M; 2: Output: Path p(s, t) from s to t in M; 3: Let (u 0,...,u l ) enote a bitonic path in G(M) from g (s) to g (t); 4: for i = 0 to l o 5: Select a noe v i in g(u i ) uniformly at ranom; {v 0 = s an v l = t} 6: if i l then 7: Construct subpath r i from v i to v i by picking a imension by imension shortest path (an at most one-ben path), accoring to a ranom orering of the imensions; 8: The path p(s, t) is obtaine by concatenating the subpaths r i, p(s, t) = r 0 r r l ; Note that the algorithm is oblivious an local, since each source-estination pair can obtain a path inepenently of the other paths. We will now show that our algorithm with the generalize access graph, in aition to obtaining optimal congestion, also controls the stretch. First we show the constant stretch property of the selecte paths. Theorem 3.4 For any two istinct noes s an t of the mesh, stretch(p(s, t)) 64. 7

8 Proof: Let h be the height of the eepest common ancestor of s an t. Then p(s, t) is the concatenation of paths constructe by the imension to imension paths in meshes of sies 2,...,2 h, 2 h, 2 h,...,2. By aing the length of the paths we have that p(s, t) 2( h + 2 h h) which implies that p(s, t) 2 h+3 4h. Since s an t are istinct, h. By Lemma 3.3, h log ist(s, t) + 3, an the theorem follows. We now relate the congestion of the paths selecte to the optimal congestion C. Let e enote an ege in M. Let C(e) enote the loa on e, i.e., the number of times that ege e is use by the paths of all the packets. We will get an upper boun on E[C(e)], an then using a Chernoff boun we will obtain a concentration result. We start by bouning the probability that some particular subpath forme by the path selection algorithm uses ege e. Consier the formation of a subpath r i from a submesh M to a submesh M 2, such that M 2 completely contains M, an e is a member of M 2. Accoring to the path selection algorithm, mesh M is of type-, thus all of its sies are equal to m l, where l is the level of M. We show the following lemma. Lemma 3.5 Subpath r i uses ege e with probability at most 2/m l. Proof: For subpath r i, let v be the starting noe in M an v 2 the ening noe in M 2 for subpath r i. Suppose e = (v 3, v 4 ). Without loss of generality, suppose e is vertical. Since the subpath is a one-ben path, ege e is use either when v or v 2 have the same x coorinate with e. This event occurs with probability at most 2/m l. Let P be the set of paths that go from M to M 2 or vice-versa. Let C (e) enote the congestion that the packets P cause on e. We show: Lemma 3.6 E[C (e)] 2 P /m l. Proof: We can write P = P P 2, where P is the set of subpaths from M to M 2, an P 2 is the subpaths from M 2 to M. Then, from Lemma 3.5, the expecte congestion at ege e ue to the subpaths in P is boune by 2 P /m l. With a similar analysis, we have that the expecte congestion at e is boune by 2 P 2 /m l. Since P an P 2 are isjoint, we obtain E[C (e)] 2( P + P 2 )/m l = 2 P /m l. From the efinition of the bounary congestion, we have that B B(M, Π) P /out(m ). Therefore, C P /out(m ). Since each sie of M has m l noes, we have that out(m ) 4m l. From Lemmas 3.6 we obtain: Lemma 3.7 E[C (e)] 8C. We charge this congestion to submesh M 2. By Lemma 3.3, only submeshes up to height h < log D + 3 can contribute to the congestion on ege e (submeshes of type-). By summing the congestions ue to these at most 2(log D + 3) submeshes (a type- an a type-2 submesh at each level), an by using Lemma 3.7, we arrive at an upper boun for the expecte congestion at ege e: 8

9 Lemma 3.8 E[C(e)] 6C (log D + 3). Note that without increasing the expecte congestion, we can always remove any cycles in a path, so without loss of generality, we will assume that the paths obtaine are acyclic. We now obtain a concentration result on the congestion C obtaine by our algorithm, using the fact that every packet selects its path inepenently of every other packet. Theorem 3.9 C = O(C log n) with high probability. Proof: Let X i = if path p i uses ege e, an 0 otherwise. Then E[C(e)] = E[ i X i] 6C (log D + 3). Let E be the number of eges in the mesh. Asymptotically in E, E[C(e)] 6C log( E D ). Let κ > 2e, then applying a Chernoff boun [0], an using the fact that C we fin that P[C(e) > 6κC log( E D )] < ( E D ) 6κ. Taking a union boun over all the eges, we obtain P[max e E C(e) > 6κC log( E D )] < ( E D ) 6κ. Using the fact that D = O( E ), E = O(n 2 ), an choosing κ = 2e +, we get C = O(C log n) with high probability. 4 The -Dimensional Mesh The 2-imensional ecomposition can be irectly generalize to a -imensional mesh with equal sie lengths (2 k, k 0) at each imension. However, the stretch becomes O(2 ), which is excessively high for large. In orer to alleviate the problem, we present an alternative ecomposition for which the path selection algorithm has congestion O( 2 C log n), an stretch O( 2 ). For fixe, these are constant factors from optimal. 4. Decomposition As in 2 imensions, we have the type- meshes, an the translate meshes. However, we now introuce Θ() types of translate meshes at each level. To be specific, consier the level l type- mesh with sie length m l. Set λ = max{, m l /2 log(+) }. We shift the type- submeshes by (j )λ noes in each imension to get the type-j submeshes, for j >. Notice that the number of ifferent types of submeshes at any level is at most 2( + ). If m l +, then there are at least ( + ) ifferent types of submeshes. Thus, an ege is containe in O() submeshes at every level. Figure 2 shows an example for = 3, m l = 4, an λ = m l =. The access graph can be constructe using these submeshes. The path 4 selection is similar to the 2-imensional case, where the brige is some type-j submesh, while all the other submeshes in the bitonic path are of type-. In particular, suppose we are given two noes s an t in the mesh, we show that in the access graph there is some regular submesh (of some type-j) that completely contains s an 9

10 Type. Type 2. Type 3. Type 4. Figure 2: Mesh ecomposition for the 3 imensional mesh. Only 2 of the 3 imensions are epicte t an has sie length O( ist(s, t)). As we i for = 2, assume, for simplicity, that we are on the torus. Suppose that s = (s,...,s ) an t = (t,...,t ), with t i s i ist(s, t). Let R = [a, b ] [a, b ] be a region of the mesh efine by s an t, such that [a i, b i ] = [min(s i, t i ), max(s i, t i )]. Note that R contains s an t. We will fin a regular submesh which completely contains R. Consier now the eepest level that has submeshes of sie at least 2( + ) ist(s, t). Let h be the height of this level. The sie length of submeshes at height h is m h = 2 h, where 4( + ) ist(s, t) m h 2( + ) ist(s, t), an let the shift amount at this height be λ h = m h /2 log(+) ist(s, t). We have: Lemma 4. For every height h, R is completely containe by some regular submesh. Proof: First consier height h. The anchor noe of a submesh is the noe with the smallest coorinate in each imension. Consier imension i. Let i = [a i, b i ], be the sie of R in imension i. By construction of the ifferent types of submeshes, we have that i contains the ith imension of anchors of at most one type of submesh at height h. This is because in each imension the istance between anchors of ifferent mesh types is at least i. Consiering now all the imensions, since we have imensions an at least + ifferent types of submeshes, by the pigeonhole principle, there exists at least one type of submesh, say type-ζ, such that R oes not contain any anchors of any type-ζ submesh in any imension. This implies that R is completely containe by a type-ζ submesh M at height h. The same argument also hols for heights > h. From the proof of Lemma 4., it follows that there is some regular submesh M 2 at level h + that contains R. On the access graph, the bitonic path is constructe so that it goes from s to t through the brige submesh at height h +. Expect possibly for the brige, the other submeshes on the path are of type-. By construction, it is guarantee that M 2 is ecompose into type- submeshes at height h = log ist(s, t). Let M be the type- submesh at height h that contains s, an M 3 the submesh that contains t. The path is forme by first using submeshes of type- from s up to submesh M, then to brige M 2, own to M 3, an then own to t by using type- submeshes. The reason of using the height h+ for briges instea of height h, is ue to technical reasons explaine in the appenix. 0

11 4.2 Stretch an Congestion We now compute the stretch of the path selection algorithm in the imensions. Theorem 4.2 (Stretch for Dimensions) For any two istinct noes s an t of the mesh, stretch(p(s, t)) = O( 2 ). Proof: Let p enote the path from s to t. Let r, r 2, an r 3, enote the respective subpaths from s to M, from M to M 2 to M 3, an from M 3 to t. First we compute r. Subpath r consists of h subpaths where the subpath from height i to height i has length at most (2 i ). Therefore, r 2 h i=0 (2i ) = 2(2 2 h h ). Since 2 h ist(s, t), we have r 2(2 ist(s, t) h) = O( ist(s, t)). The path length of r 2 is no more than 2(2 h+ ). Since 2 h+ 8( + ) ist(s, t), we have that r 2 2(8( + ) ist(s, t) + ) = O( 2 ist(s, t)). Further, r 3 = r. Thus, p = r + r 2 + r 3 = O( 2 ist(s, t)). With an analysis similar to the 2-imensional case, we can show that the expecte congestion cause on an ege e from subpaths between a mesh of a lower height an a mesh of a higher level, is E[C (e)] 4C ) (the analysis can be foun in the appenix). We charge this congestion to the higher height mesh. Since paths use heights up to O(log(D )), an each height has O() ifferent types of submeshes, at most O( log(d )) ifferent submeshes can contribute to the congestion on e. Thus E[C(e)] = O(C log(d )). As with the 2-imensional analysis, we get a concentration result by applying a Chernoff bouning argument an the facts that D = O( E ), E = O(n), an = O(n): Theorem 4.3 (Congestion for Dimensions) C = O(C log n) with high probability. 5 Ranom Choices Here we show that ranomization is unavoiable for oblivious algorithms, if they are to obtain near optimal congestion: such algorithms nee access to a substantial number of ranom bits, an we show that our algorithm uses a near minimal number of ranom bits. Consier the -imensional mesh, with equal sie lengths (2 k, k 0) in each imension. We say that a path selection algorithm A is a κ-choice algorithm, if for every sourceestination pair (s, t), A chooses the resulting path from κ possible ifferent paths from s to t. The path choice is ranomize accoring to some probability istribution which may be specific to each source/estination pair. For the case in which κ =, the algorithm is eterministic. A κ-choice algorithm requires log κ bits of ranomization per packet to select any particular path. 5. Path Choices an Congestion Consier a κ-choice algorithm A. We now construct a routing problem Π A which gives a lower boun on κ in terms of the congestion. The routing problem construction uses algorithm A

12 itself. Suppose that l = 2 s, for some s 0. Each noe in the network is the source of one packet an the estination of one packet (permutation). The istance of every packet to its estination is l. Such a problem can be constructe by iving the network into submeshes of sie length l, an then forming pairs of submeshes which exchange their packets at the respective noes. However, the problem construction has not finishe yet. Every packet uses the path from its source to its estination so that it is the path with maximum probability among the κ possible paths provie by algorithm A for the particular source an estination. This way, we obtain paths for all the packets in the network. The average number of paths crossing an ege is at least m l = l. Therefore, there is an ege e which is crosse by at m least l/ packets. Let Π A enote a set consisting of l/ packets crossing e. This conclues the construction of Π A. Note that Π A keeps only a subset of the packets in the original permutation problem. We have the following result that relates path choices an congestion. Lemma 5. For any κ-choice algorithm A an corresponing routing problem Π A with D = l, the expecte congestion Y is at least l κ. Proof: By construction of Π A, there is an ege e which is use by each packet in Π A with probability at least /κ. The expecte congestion Y e on ege e is boune by Y e Π A = l. κ κ The expecte congestion Y = E[max ei C(e i )] E[C(e)] = Y e, since max ei C(e i ) C(e). 5.2 Lower Boun on Ranomization Here, we establish a lower boun on the number of bits per packet require by any algorithm that achieves congestion comparable to our algorithm. Let algorithm H enote our - imensional algorithm presente in Section 4. Consier an arbitrary κ-choice algorithm A an its corresponing routing problem Π A as escribe in the previous section. We will give an upper boun on the performance of H for routing problem Π A (notice that H an Π A are inepenent unless H = A). Let C H enote the expecte congestion of H for Π A. We will give an upper on C H in terms of l an. ( ( Lemma 5.2 C H = O l ) ) log n. Proof: From the analysis in Sections 3 an 4, C H = O(B log n) = O(C log n), where B is the bounary congestion. We give an upper boun on B in terms of l an. For any arbitrary submesh M, we give an upper boun on B(M, Π A ), which is also an upper boun for B. Assume M is a m m with n noes. It can be shown that out(m ) n (see a proof in the appenix). Let Π enote the packets that have to cross the borer of M (have source outsie an estination insie M or vice versa). Clearly, Π min{ l, n }, since Π Π A = l, an each noe is the source/estination of at most one packet. By efinition of the bounary congestion, B(M, Π A ) = Π. There are two out(m ) cases: () l n : Π out(m ) l/ n ( l ) 2 = l l (+ ); 2

13 (2) l > n : Π n out(m ) n In both cases, B(M, Π A ) ( ( C H = O l ) ) log n. = n / < ( l ) = l (+ ). l (+ ). Since M was arbitrary, B l (+ ), consequently Suppose that algorithm A, is at least as goo as H for arbitrary routing problems with respect to congestion. Let C A enote the expecte congestion of algorithm A on a routing problem. Then C A C H for routing problem Π A. From Lemmas 5.2, an 5., algorithm ) log n ) ( A requires κ = Ω( l path choices, or equivalently Ω ( ( ) log l ranom bits per packet. Since D = l we have the following result. log log n) Lemma 5.3 There is a routing problem with D = Ω(log n) for which any algorithm A with C A = O(C H ) requires Ω ( ) ( log D ) ranom bits per packet. i.e., there exist routing problems for which significant ranomization is unavoiable for any algorithm with congestion at least as goo as algorithm H. 5.3 Upper Boun on Ranomization Here we compute an upper boun on the number of ranom bits per packet require by our -imensional algorithm H. Consier a path formation from submesh M to a submesh M 2 (we will refer to this as a step in the algorithm). The algorithm makes the following ranom selections: i. A ranom orering of the imensions from the! possible orerings, requiring O( log ) ranom bits each time. ii. A ranom noe in M 2. The largest submesh in the bitonic path has size O((D ) ) (Lemma 4.), so the number of ranom bits require each time is O( log(d )). Consequently, in a single step of the algorithm, the number of bits require per step is O( log(d )). Since, the number of steps to select a path is at most 2 log D +, the total number of ranom bits neee per packet is O( log 2 (D )). We can ecrease the number of ranom bits neee by a factor of log(d ), as follows: i. We select the ranom orer of imensions only once at the beginning of the path creation an we use the same orer at the subsequent steps of the algorithm for the same path. ii. We compute two ranom noes, namely v an v 2, at the largest submesh in the bitonic path. For the path formation we only use bits from v an v 2 to compute the (ranom) noes in the submeshes of the bitonic path. We use as many bits as necessary from v or v 2, epening on the size of the submesh consiere each time. We alternate the use of noes v an v 2 in the path formation, so that if the path is forme on submeshes M,...,M k, we use bits from v for M, M 3,..., (o ineces) an from v 2 for M 2, M 4,... (even ineces). 3

14 Therefore, only O(O( log )) ranom bits are neee for the ranom orer of imensions, an O( log(d )) ranom bits for the ranomization of v an v 2. This gives us in total O( log(d )) bits. Lemma 5.4 For any routing problem, algorithm H requires O( log(d )) ranom bits. From Lemma 5.3 we have that there exist routing problems with D = Ω(log n) such that any oblivious routing algorithm which is at least as goo as H in expecte congestion, requires Ω(( D ) log ) ranom bits. The same result hols for H too. Therefore, when D = Ω( + log n), algorithm H uses a number of bits which is only a factor O() from optimal. We have the following theorem: Theorem 5.5 (Approximation of Ranom Bits) The number of ranom bits use by Algorithm H is within O() of optimal. 6 Discussion We have shown that for the mesh, one can simultaneously control both the stretch an the congestion. As alreay mentione, this cannot be one for general networks. Interesting open issues that remain are to evelop similar algorithms for other specific networks, an to evelop oblivious algorithms that minimize C + D for general networks. References [] J. Aspens, Y. Azar, A. Fiat, S. Plotkin, an O. Waarts. Online loa balancing with applications to machine scheuling an virtual circuit routing. In Proceeings of the 25th ACM Symposium on Theory of Computing, pages , 993. [2] B. Awerbuch an Y. Azar. Local optimization of global objectives: competitive istribute ealock resolution an resource allocation. In Proceeings of 35th Annual Symposium on Founations of Computer Science, pages , Santa Fe, New Mexico, 994. [3] Y. Azar, E. Cohen, A. Fiat, H. Kaplan, an H. Racke. Optimal oblivious routing in polynomial time. In Proceeings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pages , San Diego, CA, June ACM Press. [4] Marcin Bienkowski, Miroslaw Korzeniowski, an Haral Räcke. A practical algrorithm for constructing oblivious routing schemes. In Proceeings of the 5th Annual ACM Symposium on Parallelism in Algorithms an Architectures, pages 24 33, Jun [5] A. Boroin an J. E. Hopcroft. Routing, merging, an sorting on parallel moels of computation. Journal of Computer an System Science, 30:30 45, 985. [6] Jie Gao an Li Zhang. Traeoff between stretch factor an loa balancing ratio in wireless network routing. In Proceeings of th Symposium on Principles of Distribute Computing (PODC), page to appear,

15 [7] Chris Harrelson, Kristen Hilrum, an Satish Rao. A polynomial-time tree ecomposition to minize congestion. In Proceeings of the 5th Annual ACM Symposium on Parallelism in Algorithms an Architectures, pages 34 43, Jun [8] Christos Kaklamanis, Danny Krizanc, an Thanasis Tsantilas. Tight bouns for oblivious routing in the hypercube. In Proceeings of 2n IEEE Symposium on Parallel an Distribute Processing (2n SPAA 90), pages 3 36, Crete, Greece, July 990. [9] B. M. Maggs, F. Meyer auf er Heie, B. Vöcking, an M. Westerman. Exploiting locality in ata management in systems of limite banwith. In Proceeings of the 38th Annual Symposium on the Founations of Computer Science, pages , 997. [0] Rajeev Motwani an Prabhakar Raghavan. Ranomize Algorithms. Cambrige University Press, Cambrige, UK, [] Haral Räcke. Minimizing congestion in general networks. In Proceeings of the 43r Annual Symposium on the Founations of Computer Science, pages 43 52, Nov [2] P. Raghavan an C. D. Thompson. Ranomize rouning: A technique for provably goo algorithms an algorithmic proofs. Combinatorica, 7: , 987. [3] A. Srinivasan an C-P. Teo. A constant factor approximation algorithm for packet routing, an balancing local vs. global criteria. In Proceeings of the ACM Symposium on the Theory of Computing (STOC), pages , 997. [4] L. G. Valiant an G. J. Brebner. Universal schemes for parallel communication. In Proceeings of the 3th Annual ACM Symposium on Theory of Computing, pages , May 98. A Appenix A. The -Dimensional Mesh Here, we consier the -imensional path selection algorithm of Section 4. We compute an upper boun on the expecte congestion on an ege. Let M an M 2 be two submeshes with respective sie lengths a,...,a an b,...,b, such that: (i) M is of type-, (ii) M 2 completely contains M, (iii) each sie of M 2 is at least twice the sie of M, b i 2a i, for all i. The -imensional algorithm preserves conitions (i)-(iii), for every pair of consecutive submeshes of the bitonic path (conition (iii) is preserve even when M 2 is a brige, since the height of the brige is h + ). Consier the formation of a subpath r from a submesh M to M 2, where r is forme by following a shortest path from a ranomly chosen noe v in M to a ranomly chosen noe v 2 in M 2. The path is forme by following a imension by imension path, with a ranom orering of imensions. Since M is of type-, it hols that all of its sies are equal, a = a 2 = = a = a. Let e be an ege of M 2. We show the following result: 2 2 This result cannot be use for the 2-imensional analysis of Section 3.3, since the coniction (iii) liste above oes not hol for that algorithm. 5

16 Lemma A. Subpath r uses ege e with probability at most 2 a. Proof: Without loss of generality, suppose that e is an ege in M 2 along imension l, from (x,...,x l,...,x ) to (x,...,x l +,...,x ). Suppose v = (y,...,y ) an v 2 = (z,..., z ). Let ρ be a ranom permutation which etermines the orer of imensions. Suppose that imension l occurs in position i, i.e., ρ(i) = l. Then the probability P[e] that ege e is use by r is boune from above by the probability that, z ρ() = x ρ(),...,z ρ(i ) = x ρ(i ), an, y ρ(i+) = x ρ(i+),...,y ρ() = x ρ(). We obtain: P[e] = b ρ() b ρ(i ) a ρ(i+) a ρ() 2a ρ() 2a ρ(i ) a ρ(i+) a ρ() = 2 i a. Since ρ is a ranom permutation, the probability that ρ(i) = l for any i is /, so multiplying by /, summing, an using the fact that i= 2, we obtain: 2 i P[e] i= 2 i a = a i= 2 2 i a. Let P be the set of paths that go from M to M 2 or vice-versa. Let C (e) enote the congestion that the packets P cause on e. We show: Lemma A.2 E[C (e)] 2 P a. Proof: We can write P = P P 2, where P is the set of subpaths from M to M 2, an P 2 is the subpaths from M 2 to M. Then, from Lemma A., the expecte congestion at ege e ue to the subpaths in P is boune by 2 P /(a ). With a similar analysis, we have that the expecte congestion at e is boune by 2 P 2 /(a ). Since P an P 2 are isjoint, we obtain E[C (e)] 2( P + P 2 )/(a ) = 2 P /(a ). We have that C B(M, Π) P /out(m ). Since each sie of M has a noes, we have that out(m ) 2a. From Lemma A.2, we obtain: Lemma A.3 E[C (e)] 4C. A.2 Lower Boun on Ranom Choices Here we show that any submesh with a particular number of noes must have a lower boun on its outgoing eges. Lemma A.4 For any submesh M with n noes, out(m ) n. 6

17 Proof: Let m i i be the sies of M. Let z i enote the prouct m m i m i+ m. Each z i represents a surface of M. At most ifferent surfaces coul be at the borer of M an thus may not be use at out(m ). Therefore, out(m ) i z i. We want to fin which m i minimize i z i, given that i m i = n. This will give us a lower boun on out(m ). Consier the Lagrangian L = i z i λ( i m i n ). Then, i z i is minimize when L m i = 0, for all i. We have, L m = ((m 3 m ) + (m 2 m 4 m ) + + (m 2 m )) λ L m 2 = ((m 3 m ) + (m m 4 m ) + + (m m 3 m )) λm which gives, m i = 0, an i=2 m i = 0, i=3 λ = m 2 + m m, an λ = m + m m. Therefore, m = m 2. By computing the rest of the partial erivatives, we can show m i = m i for 2 i. Therefore, i z i is minimize when m i = m j, for all i, j. Consequently, i z i m ( ) i = (n ) ( ) = n. Which implies that out(m ) n. 7