Time Complexity of a Path-Formulated Optimal Routing Algorithm

A detailed analysis of convergence rate is presented for an iterative path formulated optimal routing algorithm. The primary objective is to quantify, analytically, how the convergence rate changes as the number of nodes in the underlying graph increases. The analysis is motivated by a particular path formulated gradient projection algorithm that has demonstrated excellent convergence rate -. properties through extensive numerical studies. In particular, the empirical data suggests that the number of iterations required for convergence to within a small fraction of the optimal cost is relatively independent of the number of nodes. Deriving a correspondingly tight analytical bound for the number of iterations required for convergence, as a function of problem size, proves to be a formidable task, primarily because the dimension of the underlying optimization problem (i.e., the total number of paths connecting all origin-destination pairs) generically grows with a super-polynomial function of the number of nodes. The main result of this paper is that the number of iterations for convergence depends on the number of nodes only through the network diameter. I . INTRODUCTION AND PROBLEM FORMULATION A. Basics The primary objective of virtually all routing algorithms is to select routes for those origin destination (OD) pairs that request data communication. A secondary objective is to insure that messages transmitted along the selected routes are delivered to the correct destinations. This latter objective is accomplished by using standard techniques involving protocols and routing tables. In this p q e r the focus is on the former objectivethe route selection problem. It is known that route selection has a substantial impact on the performance of data networks [l-9,12,13,15,16]. Roughly speaking, an optimal routing is a set of routes that yields the "best network performance-based on some quantitative measure. The types of performance measures employed by most optimal routing formulations, estimate, in some sense, the average delay associated with sending a packet of data to a typical destination node. An important issue to consider when implementing a routing algorithm in a large distributed data network is the question of whether the computation should be done in a centralized or distributed manner. Centralized implementations are fairly straightforward: a designated "central" node is sent data (which characterizes the state of the network) from the other network nodes; then, based on this information, the central node solves the optimal routing problem and broadcasts the solution back out to the network. One of the obvious problems with this type of scheme is the associated communication overhead (i.e., bottleneck). In contrast, certain distributed implementations can reduce this communication overhead, by requiring, for example, only nearest neighbor communication. Due to the potentidy high degree of fault tolerance, fast convergence rates, low communication overhead, and other reasons, distributed implementations have received a great deal of attention in the literature over the past decade or so. One fundamental question associated with distributed routing algorithms is that of convergence. Namely, because the order of events in distributed algorithms occur asynchronously (to one degree or another), the question of whether the algorithm will converge becomes a non-trivial one. In references [5] and [6], convergence of a class of distributed optimd routing algorithms is proven under very mild assumptions. The present authors have proven convergence of a class of distributed iterative aggregation algorithms [15], which have applications in optimal routing. While the question of distributed asynchronous convergence has been addressed in the above cited works (and others), the goal in the present paper is to determine the amount of time required for a class of iterative path formulated optimal routing algorithms to converge. The time complexity of routing algorithms is an important practical as well as theoretical issue. In practice, it is imperative that the routing algorithm converge within a certain amount of time, otherwise the eventually arrived upon solution may be of little or no value. In the present paper it is shown how network parameters such as maximum link utilization factors, traffic demand values, link capacity values, and the number of network nodes affect the time required for convergence. In order to achieve meaningful bounds for the convergence rate, a certain price was paid in that the assumed model for computation is essentially synchronous (in terms of the order in which iterations are executed). However, it is believed that the ground-work laid out in this paper should serve well as a guide for future work under more relaxed (i.e., asynchronous) assumptions. The main time complexity results are for a class of path-formulated gradient projection-based algorithms. B. Fornulation of the Optimal Routing Problem The following formulation uses the same notation and is based on the same approximating assumptions as set forth by Bertsekas and Gallager in reference [2]. Delay Models Queuing theory is the primary methodological framework for analyzing network performance. Oftentimes its use requires simplifying assumptions for the sake of mathematical tractability. Due to the complexity of realistic networks, it is typically impossible to obtain accurate quantitative delay predictions, however, the models used often provide valuable qualitative results and insights [2]. Perhaps the simplest queuing model is the so-called M/M/l queuing system that consists of a single queuing station and a single server. It is assumed that customers (i.e., packets of data) arrive according to a Poisson process with rate F , and the probability distribution of the service rate is exponential with mean C. By applying Little's Theorem, the average delay for a packet to traverse link (i, j) is given by where Cij and Fij denote the service rate and arrival rate respectively, associated with link (i, j). Jackson's Theorem states that in a network of single server queues in which customers arrive from outside the network at each queue according to independent Poisson processes, the average number of outstanding packets in the (steady-state) system can be derived as if each queue in the network is an M/M/l queue. So, for the purpose of measuring network performance, modeling the entire network with simple M/M/ 1 queues is justified. Based on Jackson's Theorem and Equatioa -. (I), the cost function is defined as a weighted sum of all link delays: where & is the set of all links and links having more traffic flow are given higher relative weightings, i.e., Equation (1) is multipled by F;;. Note that each term in the sum represents the average size of the queue associated with link (i, j). Therefore, D(F) is an estimate of the total number of outstanding packets in the network. For the purposes of this paper, determining routes that minimize D(F), for a given set of OD traffic demands, will constitute the notion of an optimal routing. Preliminary Notation The following notation is needed in order to formally state the optimal routing problem. Throughout the paper, script fonts such as W and P are used exclusively to denote sets. W : The set of OD pairs requesting communication. w : A generic OD pair in W. r, : The arrival rate (tr&c demand) measured in packets/sec, for the OD pair w. Pw : For the OD pair w, this is the set of all logical paths connecting the origin node to the destination node. p : A generic path in P,. x, : The flow rate on the logical path p. Constraint Equations The following constraint equations arise naturally due to conservation of flow. and all pashs p containing (i J ) C xp = r , for all w E W Note that the cost function being minimized, see Equation (2), can be expressed in terms of the path vector x, defined as x = [ x ~ J ~ ~ P ~ . By combining constraint Equation w E W (3) with the definition of the path vector, the cost function of (2) can be written as where Kij is a row vector with components equal to either zero or one. Specifically, the pth component of K;j is one if link ( 2 , j) is on path p, otherwise, the pt" component of Kij is zero. The Path-Formulated Optimal Routing Problem Given r,, for each w E W, minimize {D(x)) , such that Equations (4) and (5) are satisfied. 11. THE PATH FORMULATED GRADIENT PROJECTION ( P F G P ) ALGORITHM It can be shown that the path-formulated optimal routing problem can be transformed into an equivalent box-constrained problem, see [15J for more details. Also, the function D(x) is a differentiable convex function of the path vector x. Therefore, the pathformulated optimal routing problem can be solved numerically by using well established techniques from nonlinear programming; the focus here is on the gradient projection method. The main idea of the gradient projection technique is that after a step is made in the direction of the negative gradient, the result is orthogonally projected onto the positive orthant. The iteration equation that results from applying the gradient projection method to the path-formulated optimal routing problem necessitates the definition of the first derivative length (FDL) of a path. The FDL of path p, denoted dp, is defined by where . on path p .Next, the minimum FDL (MFDL) paths, denoted as p, for each w E W, are defined by dFw = min {d,) , for all w E W. PEP, (11) Note that for any particular w, there may be more than one MFDL path. In case of such an event, 25, is an arbitrarily chosen MDFL path. The iteration equation associated with the PFGP algorithm [5,20] can now be stated: (*+I) = rnax{O, X r ) a(') ('1 -1 ('1 x~ (Hpp ) (d, q)}, for all w E W , P E Pw, P # Pw, (12) where k is the iteration count. The term a(') denotes the step size and the term H;:) is a scaling factor that is related to the second derivative length of path p. It is easy to verify that the term a(')(~$:))-l (df) 4:) 2 0, for all p E Pw, and therefore, the above iteration equation need not be applied to those paths for which xg) = 0. Thus, the set of active paths at iteration k, denoted by @ik), is defined as as So, a more efficient version of the PFGP algorithm (as described originally in [20]) is the following: ('+I) = max{O, 2: ) a(*)(H('))-'(dr) 4))) for all w E W, p E pik), p # pw, x~ PP W ' (13) (k+l) = rw xPW C xf), for all w E W, (14) P E * ~ , P # F , PLk+l) = { p E 1 xf) a (k ) (~ (k ) ) l (d (k ) P P P 4;) > 0) LJ {A} , for all w E W. (15) The PFGP algorithm of Equations (13) through (15) has been efficiently implemented as a serial FORTRAN code, see reference [I]. This code uses a constant step size and the value of H: ! ) is an approximation of the pth diagonal element of the Hessian matrix. The set of active paths for each w E W are iniwized with a single (randomly selected) shortest hop path. A. Basics The overall time complexity of the PFGP algorithm is given by the product of the complexity of each iteration and the complexity of the number of iterations required for convergence to an acceptably small neighborhood of the optimal solution. From Equations (13) through (15), the complexity of each iteration is clearly dependent on the following three quantities: ( i ) the number of active paths: 1 pLk)l; (ii) the number of OD pairs: 1 WI; and (iii) the number of nodes in the network: n. The dependence on \Pkk)l is due to the fact that the flow on each (non-MFDL) active path must be updated according to Equation (13). The dependence on I W ( comes from the fact that a MFDL path must be determined for each w E W. Finally, the dependence on the number of nodes, denoted by n, is due to the fact that solving shortest path problems (i.e., finding the MFDL paths) generically depends on the size of the graph. The complexity of each iteration (El) is therefore denoted as TEI(k, I W 1, n), where k actually denotes the dependence on l@ik)(. TEI k, 1 W 1, n) is fairly straightforward to e s t ima te the only difficulty comes in estimating the maximum number of active paths used in any single iteration. The following is an obvious upper bound for I Pik)(: because at each iteration at most one new active path is added to each set pik). In contrast to the fairly straightforward task associated with estimating the complexity of each iteration (described above), the main concern in this paper is to estimate the complexity associated with the number of iterations, say Nr, required for the PFGP algorithm to converge. Most of the classical results related to convergence rates of numerical optimization algorithms depend on the values of the largest and smallest eigenvalues of the Hessian. For example, it is shown in [lo, p. 338-3401 that by using a special step size rule, the convergence rate for gradient projection algorithms is bounded by .where D(*) denotes the value of the cost function at iteration k, and B and b are, respectively, the largest and smallest eigenvalues (in magnitude) of the Hessian. From Equation (17) it is easy to see that if the difference B b is large (or b -+ 0), then [(B b)/(B + b)I2 -, 1. On the other hand, if b -+ B, then [(B b)/(B + b)12 -t 0. Clearly, the smaller the value of [(B b)/(B + b)12, the faster the convergence rate, which implies fewer iterations are required for convergence to within a fixed neighborhood of the optimal solution. Unfortunately, the convergence rate of Equation (17) has some practical problems when considering the application of the gradient projection technique to the optimal routing problem. First, the assumed step size rule used to derive Equation (17) is based on a type of line minimization technique which would be difficult to implement in a large distributed network-in practice a constant (or simple) step size rule is used. Second, it is difficult to determine a meaningful lower bound for b, primarily because the number of active paths can (potentially) grow according to a super-polynomial function of n. In particular, lpLk)l is bounded above by (P,$')l, where ( P ~ ) J denotes the total number of (potential) paths that interconnect the OD pair w. For all but the sparsest of graphs, I P?) 1 grows as a super-polynomial function of n. (Consider, for example, the fact that there exists O(2") distinct paths that interconnect various OD pairs in a simple n-node planar mesh.) In estimating the number of iterations for convergence for the PFGP algorithm, one of the most crucial issues is the assumed bound for 1 ~ ~ ~ 1 . If one uses the fact that (PL~) ( 5 I P ~ ) ( , then the resulting analysis indicates that the number of iterations for convergence is bounded by a function that depends on IP,$ ')( , which can result in an overall bound that grows with a super-polynomial function of n. This assumption and the resulting convergence rate result are apparently too loose because the empirical data suggests that the number of iterations for convergence actually grows (at most) slowly as the number of nodes in the graph is increased. In the paper [19], it is shown that if one assumes that Ipt)( 5 lK,,J, where (%,I is a constant (independent of both k and n), then the number of iterations for convergence is indeed bounded by a slowly increasing function of n. In [19], the assumption that 1@kk)( < IP,axJ is argued to be reasonable because numerous simulation studies indicated that ) P L ~ ) ~ rarely exceeded ten, regardless of the size of the network. Also, the main thrust in [19] is not in getting a necessarily tight bound for the convergence rate but rather in showing that the convergence rate assuming a Jacobi-type updating rule is the roughly the same as the convergence rate assoc&ted with a Gauss-Seidel updating rule. In the present paper, we allow J@Lk)l to grow according to k + J P ~ ) I (i.e., no uniform bound is assumed) and show that the number of iterations for convergence is still bounded by (at most) a slowly increasing function of n (and thus, from this result, we confirm indeed that only a small number of active paths are required to achieve convergence). Because no uniform bound is assumed for (pLk)l in the present paper, the analysis techniques are significantly different from those used in reference [19]. B. Serial Versus Distributed Time Complexities Thus far a distinction has not been made between the time complexity of the PFGP algorithm relative to serial and distributed implementations. In a serial (single processor) implementation, the value for TEl(k, 1 W 1 , n) is the sum of the time required to solve the shortest path problem for each w E W and the time required to update all active path flows in @ik). In a distributed implementation, a distributed shortest path algorithm could be employed (such as the distributed Bellman-Ford algorithm [2,14]) and each node i could assume the responsibility of updating all active path flows originating at node i. Of course one of the main difficulties associated with distributed algorithms (in general) is the asynchronous nature of the communication overhead. For the purposes of this paper it shall be assumed that iteration k + 1 is executed only after iteration k is completed. Under this simplifying assumption, the complexity for the number of iterations is the same for both the serial and distributed implementations of the PFGP algorithm. In terms of the distributed implementation, this assumption implicitly assumes the existence of a uniform upper bound for the communication time complexity of each iteration. This type of assumption results in what is typically called a partially asynchronous distributed algorithm. It has been proven that the distributed PFGP algorithm will actually converge (eventually) in a virtually totally asynchronous computing environment, see [5,6]. However, with such mild restrictions on the ordering of events it becomes very difficult to tightly bound the rate of convergence. The analysis technique introduced in the present paper for estimating the convergence rate (for the partially asynchronous case) serves a twofold purpose. First, the resulting bound may serve as a nominal estimate of the convergence rates associated with a more asynchronous model. Second, the analysis techniques developed in deriving the bound for the partially asynchronous case may serve as a guide for deriving similar results under more relaxed assumptions. .. IV. THE COMPLEXITY OF THE CONVERGENCE RATE In this section upper bound results for the convergence rate of the PFGP algorithm are derived. First, all necessary notation for stating the main results is introduced. A . Notation From this point on, the superscript "(n)" is placed on those variables or sets that are explicitly dependent on the number of nodes in the network. Likewise, the superscript "(k)" is used to indicate dependence on the iteration count, k. Variables with neither a "(n)" nor a "(k)" superscript are assumed to be constants, independent of both n and k. One particularly important yet subtle point is that the set of all logical paths associated with the OD pair w is denoted by PC) , while the set of active paths at iteration k associated with OD pair w is denoted by @ik). w(") : The set of OD pairs requesting communication. w : A generic OD pair in ~ ( " 1 . r, : The arrival rate for the OD pair w E ~ ( " 1 . r: The minimum arrival rate, for all w E ~ ( " 1 :