Near-Optimal Routing Lookups with Bounded Worst Case Performance

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1 Near-Otimal Routing Loous with Bounded Worst Case Performance Panaj Guta Balaji Prabhaar Stehen Boyd Deartments of Electrical Engineering and Comuter Science Stanford University, CA Abstract The roblem of route address loou has received much attention recently and several algorithms and data structures for erforming address loous at high seeds have been roosed. In this aer we consider one such data structure a binary search tree built on the intervals created by the routing table refixes. We wish to exloit the difference in the robabilities with which the various leaves of the tree (where the intervals are stored) are accessed by incoming acets in order to seedu the loou rocess. More recisely, we see an answer to the question How can the search tree be drawn so as to minimize the average acet loou time while eeing the worst-case loou time within a fixed bound? We use ideas from information theory to derive efficient algorithms for comuting near-otimal routing loou trees. Finally, we consider the racticality of our algorithms through analysis and simulation. Keywords Routing loous, Prefix matching, Otimality, Fast routing. I. INTRODUCTION The exlosive growth of the Internet has laced huge demands on its infrastructure. While advances in otical technologies such as DWDM have increased lin seeds to beyond tens of gigabits er second, Internet bacbone routers the rocessing nodes interconnecting these lins, have been lagging behind. One main reason for this is the relatively comlex acet rocessing required at a router for every incoming acet, the router has to loou a forwarding table (also called a routing table) to determine the acet s next ho destination. This rocess of looing u a acet s next ho is aggravated with a steady increase in routing table sizes [1]. As a result, the routing loou roblem has received considerable attention, both in academia and in the industry. The adotion of classless inter-domain routing (CIDR) [2] in 1993 means that a router now erforms a longest refix match to determine the next ho of a acet. A router maintains a set of destination address refixes in a routing table. Given a acet, the loou oeration consists of finding the longest refix in the routing table that matches the first few bits of the destination address of the acet. Several solutions including innovative data structures and algorithms for solving this roblem have aeared in the recent literature (see, for examle, [3], [4], [5], [6], [7], [8]). The early wor [4], [5], [6], [7], [8] focused on the develoment of data structures and algorithms for minimizing the loou time given a routing table and some memory sace constraints. For examle, the wor of Srinivasan and Varghese [8] aims to minimize the worst-case loou time of a acet given memory sace constraints. Also given sace constraints, a more recent aer by Cheung and McCanne [3] considers the frequency with which a certain refix is accessed to imrove the average time taen to loo u an address. Both these aers [3], [8] consider a trie data structure (or its variant) while doing the minimization. This aer also assumes that the frequency with which refixes are accessed is nown. Given this, the aim is to design a routing loou scheme for minimizing the average loou time. In this resect our formulation is similar to that in [3]. However, since our data structure is different (we use a binary search tree instead of a radix trie), our methods and the constraints imosed uon us are different. For examle, redrawing a trie tyically entails comressing it by increasing the degree of some of its internal nodes. This can alter its sace consumtion. In contrast, it is ossible to redraw a binary search tree without changing the amount of sace consumed by it and hence sace consumtion is not a constraint in our formulation. But the use of a binary tree data structure brings u a different roblem: Observe that it is now ossible for the worst-case deth of the binary tree to be very large deending on the distribution of the access robabilities 1. This can lead to rohibitively long loou times for some refixes. It is therefore imortant to constrain the maximum deth of the binary tree to some small resecified number. We aroach the roblem of finding good deth-constrained binary search trees using information theoretic ideas and convex otimization techniques. To this end, we set u the roblem as an average loou time minimization roblem subject to a maximum loou time constraint. There exists an algorithm to solve this roblem due to Larmore and Przytyca [10] with a rerocessing time comlexity of O(nD log n),wherenisthenumber of refixes and D is the worst case number of memory accesses allowed. Desite its otimality the algorithm is comlicated and difficult to imlement. The algorithm obtained in this aer is nearly otimal (to within two memory accesses) and has a rerocessing time comlexity of O(n log n). More imortantly, it is easy to imlement. A useful by-roduct of our aroach is that the resulting data structure is easily arallelizable. That is, if several rocessing engines were available to carry out the loou oeration, then each engine will be sharing the total load almost equally. This load balancing feature when used in arallel hardware designs reduces the average loou time by a multilicative factor. In this aer the routing loou roblem motivates us to find good deth-constrained binary search trees (also called alhabetic trees). Finding good deth-constrained alhabetic and Huffman trees are roblems of indeendent interest, e.g. in comutationally efficient comression. The general aroach of this aer, although develoed for alhabetic trees, turns out to be 1 There are about refixes in a large routing table today [9]. This can lead to a binary search tree with a maximum deth of several hundreds or thousands.

2 Prefix StartPoint EndPoint P 1 * P 2 00* P 3 1* P P 5 001* I I6 TABLE I An examle routing table with 4-bit refixes and endoints of their induced intervals on the number line [0000; 1111]. I1 I2 I4 I5 I1 I2 I3 I4 I5 I6 equally alicable for finding deth-constrained Huffman trees, and comares favorably to recent wor on this toic (see, for examle, Schieber [11] and Mildiu and Laber [12]). The aer is organized as follows: Section 2 describes the roblem of routing loous and its relationshi with Information Theory. Section 3 sets u the otimization roblem and describes our roosed solution. Section 4 rovides simulation results of the roosed algorithms on some large ublicly available routing tables and a large acet trace. And Section 5 concludes. II. ROUTING LOOKUPS A routing table consists of a series of tules of the form hrefix, nexthoi, whererefix reresents the aggregation of several 32-bit destination IP addresses and nextho is the IP address of the corresonding next ho router. Given an incoming acet s destination address, the routing loou roblem is to find the longest (i.e. the most secific) of all the refixes matching the first few bits of the incoming acet s destination address. Each refix can be viewed as an interval on the number line [0; 2 32 ), referred to here as the IP number line. Lamson, Srinivasan and Varghese [6] roose a data structure in which the end oints of this interval, sorted in increasing order after eliminating dulicates to give fp 1 ;P 2 ;::: ;P m g, are stored in the external nodes (leaves) of a binary tree and the corresonding next ho addresses are recomuted for each of the disjoint basic intervals, [P i ;P i+1 ). The internal nodes contain suitably chosen values to guide the search rocess to one of their two children. An examle of a routing table with 4-bit refixes is shown in Table I. The corresonding subdivision of the IP number line and an examle binary search tree is shown in Figure 1. Clearly, with n refixes, there are no more than m =2nsuch endoints and thus the resulting binary search tree has a size which is linear in the number of routing table refixes. While this binary tree structure is very good if all intervals are accessed uniformly, the search time can be imroved considerably by maing use of the frequency with which a certain routing table entry is accessed. We note that today s routers already maintain such er-refix statistics. Hence, minimizing routing loou times by maing use of this information comes at no extra data collection cost. Given this, a natural question to as is What is the best tree data structure given the frequency of access of refix intervals? Viewing it this way, the roblem is readily recognized to be one of minimizing the average P2 P5 P Fig. 1. The binary search tree corresonding to the routing table in Table I. weighted length of a binary tree whose leaves are weighted by the robabilities associated with the intervals reresented by the leaves. The minimization is to be carried over all ossible binary trees with the given intervals and corresonding weights at the leaves. In the language of Information Theory, the roblem stated above translates to: Find a minimum average length alhabetic code (or tree) for an m-letter alhabet (where each letter corresonds to an interval). An alhabetic code is one in which the m letters are ordered lexicograhically on the leaves of the resulting binary tree. That is, if the letter A aears before the letter B in the alhabet then the code word associated with the letter A has a smaller binary value than the code word associated with the letter B. For the examle in Figure 1, the code word associated with Interval I1 is 000 and that associated with Interval I5 is 101, where a bit in the codeword is 0 (res. 1) fortheleft (res. right) branch at the corresonding node. As an examle, if the intervals I1 through I6 are accessed with robabilities 1/2, 1/4, 1/8, 1/16, 1/32 and 1/32 resectively, then the otimal alhabetic tree corresonding to these robabilities (or weights) is shown in Figure 2(a). The codeword for I1 is now 0 andthatofi5is III. ALGORITHMS The average length of a general refix code for a given set of robabilities can be minimized using the well-nown Huffman coding algorithm [13]. However, a Huffman solution is not guaranteed to maintain the alhabetic order of the inut data set. This causes imlementational roblems as simle comarison queries are not ossible at internal nodes to guide the search. Instead, at an internal node of a Huffman tree, one needs to as for membershis in arbitrary subsets of the alhabet to roceed to the next level. Because this is as hard as the original search roblem itself, it is not feasible to use a Huffman solution. Aart from the alhabetic constraint, it is necessary to bound the maximum code word length (or the maximum deth of the tree) to mae the solution useful in ractice. This is because an otimal alhabetic tree for n letters can have a maximum deth of n 1, see for instance, Figure 2(a) (the root is assumed to P3 P4

3 I1 1/ I2 1/4 I /8 (a) I4 1/ I I6 1/32 1/32 I1 1/ I2 1/ I3 I4 I5 I6 1/8 1/16 1/32 1/32 Fig. 2. Otimal alhabetic tree corresonding to the tree in Figure 1 with leaf robabilities as shown: (a) unconstrained deth, (b) deth constrained to 4. be at deth 0). This is unaccetable in ractice where the tyical value of n is around 65000, since the access of a dee leaf can cause the router to slow down considerably. Further, any change in the networ toology or in the distribution of incoming acet addresses can lead to a large increase in the access frequency of a dee leaf. It is therefore very desirable to have a small uer bound on the maximum deth of the alhabetic tree. An uer bound on the worst case loou time also simlifies the hardware design of a router. Thus, well-nown algorithms for finding an otimal alhabetic tree such as those in [14], [15], [16] which do not incororate a maximum deth constraint cannot be used in our setting. To understand this last oint better, consider the alhabetic tree of Figure 2(a) which is otimal if the intervals I1 through I6 shown in the binary tree of Figure 1 are accessed with robabilities f1=2; 1=4; 1=8; 1=16; 1=32; 1=32g, resectively. For these robabilities, the average loou time is 1:9375 (equal to 1 1=2+2 1=4+3 1=8+4 1= = =32), while the maximum deth is 5. If we imose a maximum deth constraint of 4, then we need to redraw the tree and obtain another tree as shown in Figure 2(b) where the average loou time has increased to 2. In general, we are interested in the following minimization roblem: minimize fl ig C = P i=n i=1 l i i s.t. l i» D 8i (1) for n intervals with access robabilities i, l i being the number of comarisons required to loou a acet in the i th interval. From the revious discussion, we also require that the l i s obtained as a result of solving the above minimization roblem give rise to an alhabetic tree. Yeung [17] gives a necessary and sufficient condition that the l i s must satisfy in order for the resulting tree to be alhabetic. This condition is stated below as Lemma 1. The smallest ossible value of C is the entroy, H(), of the set of robabilities f i g,whereh() = P i log i. 2 And for 2 All logarithms in this aer are to the base 2. (b) deth constrained alhabetic trees C will be bigger than H() more often than not 3. Finding fast algorithms for comuting otimal deth constrained binary trees (without the alhabetic constraint) is nown to be a hard roblem and good aroximate solutions are aearing only now [18], [12], [19], almost 40 years after the original Huffman algorithm. Imosing the alhabetic constraint renders the roblem even harder [20], [21], [22], [23]. Larmore and Przytyca [10] have roosed an otimum algorithm which runs in time O(nD log n) and finds a deth constrained alhabetic tree. However, their algorithm is very comlicated to imlement. In light of this, we attemt to find a ractical and rovably good solution to the roblem of comuting otimal dethconstrained alhabetic trees. A good aroximate solution to the routing loou roblem is of value because: (1) It is much simler to find than an otimum solution. It is also much simler to imlement. (2) As the robabilities associated with the intervals induced by routing refixes change frequently and are not nown exactly, it does not seem to mae much sense to solve the roblem exactly for an otimum solution. (3) Our aroximate solution can be theoretically roved to be requiring no more than two extra comarisons er loou when comared to the otimum solution. In ractice, the discreancy is very often found to be less than two. A. Algorithm MINDP We first state two results from [17] as lemmas that we will use to develo our algorithm. The first lemma states a necessary and sufficient condition for the existence of an alhabetic code with secified code word lengths, and the second rescribes a method for constructing good, near-otimal trees which are not deth-constrained. Lemma 1: (The Characteristic Inequality): There exists an alhabetic code with code word lengths fl g if and only if s n» 1 where s (L) =c(s 1 (L);2 l )+2 l and c is defined by c(a; b) =da=beb. Proof: For a comlete roof, see [17]. The basic idea is to construct a canonical coding tree, a tree in which the codewords are chosen lexicograhically using the lengths fl i g. For instance, suose that l i = 4 for some i, and in drawing the canonical tree we find the codeword corresonding to the letter i to be Ifl i+1 =4, then the codeword for the letter i +1will be chosen to be 0011; ifl i+1 =3, the codeword for letter i +1is chosen to be 010;andifl i+1 =5, the codeword for letter i+1is chosen to be Clearly, the resulting tree will be alhabetic and Yeung s result verifies that this is ossible if and only if the characteristic inequality defined above is satisfied by the lengths fl i g.2 The next lemma (also from [17]) considers the construction of good, near-otimal codes. (Note that it does not roduce alhabetic trees with rescribed maximum deths. That is the subject of this aer.) Lemma 2: The minimum average length, C min, of an alhabetic code on n letters, where the i th letter occurs with robability i satisfies: H()» C min» H() +2 1 n. 3 The lower bound of entroy is achieved in general when there are no alhabetic or maximum deth constraints.

4 Therefore, there exists an alhabetic tree on n letters with average code length within 2 bits of the entroy of the robability distribution of the letters. Proof: The lower bound, H(), is obvious. For the uer bound, the code length l of the th letter occurring with robability is chosen to be: l =ρ d log e =1;n d log e +1 2»»n 1 The roof in [17] verifies that these lengths satisfy the characteristic inequality, and shows that a canonical coding tree constructed with these lengths has an average deth satisfying the uer bound.2 We now return to our original roblem of finding otimal deth-constrained alhabetic trees. Since the given set of robabilities f g might be such that min = min 2 D,a direct alication of Lemma 2 could yield a tree where the maximum deth is bigger than D. To wor around this roblem, we transform the given robabilities f g into another set of robabilities fq g such that q min = min q 2 D. This allows us to aly the following variant of the scheme in Lemma 2 to obtain a near-otimal deth-constrained alhabetic tree with leaf robabilities fq g. Given a robability vector fq g such that q min 2 D,we construct a canonical alhabetic coding tree with the codeword length assignment to the th letter given by: linear inequalities. Minimizing convex cost functions with linear constraints is a standard roblem in otimization theory and is easily solved by using Lagrange multilier methods (see, for examle, Section 3.4 of Bertseas [25]). Accordingly, define L(q; ; μ) = i log( i =q i )+ i ( q i )+μ(q i 1) Setting the artial derivatives with resect to q i to zero at q Λ i we =0)qi = i (5) i μ i Putting this bac in L(q; ; μ),wegetthedual G( ; μ) =( i log(μ i )+ i )+(1 μ) i Now minimizing G( ; μ) subject to i 0 and μ i =0) i μ i =08i)= i μ i ρ min(d l Λ log q = e;d) min(d log q e +1;D) =1;n 2»»n 1 (2) which combined with the constraint that i 0 gives us Λ i = max(0;μ i =). Substituting this in Equation (5), we get Each codeword is clearly at most D bits long and the tree thus generated has a maximum deth of D. It remains to be shown that these codeword lengths yield an alhabetic tree. By Lemma 1 it suffices to show that the fl Λ g satisfy the characteristic inequality. We defer this verification to the Aendix. Proceeding, if the codeword lengths are given P by fl Λg,there- sulting alhabetic tree has an average length of l Λ.Now, l Λ» log 1 +2 q = log log +2 q = D(jjq) +H()+2; (3) where D(jjq) is the relative entroy between the distributions and q and H() is the entroy of the distribution. In order to minimize P l Λ, we must therefore choose fqλ i g = fq ig so as to minimize D(jjq). A.1 The minimization roblem We are thus led to the following otimization roblem: minimize DP = D(jjq) = P i ilog( i =q i ) subject to Pi q i =1 and q i =2 D 8i (4) Observe that the cost function D(jjq) is convex in (; q) (see Page 30 of Cover and Thomas [24]). Further, the constraint set is convex and comact. In fact, the constraint set is defined by q Λ i = max( i=μ; ) (6) To finish, we need P to solve Equation (6) for μ = μ Λ under i=n the constraint that i=1 qλ i = 1. fqi Λ g will then be the desired transformed robability distribution. It turns out that we can find a closed form exression for μ Λ, using which we can solveequation (6) by an O(n log n) time and O(n) sacealgorithm. The algorithm first sorts the original robabilities f i g to get f ^ i g such that ^ 1 is the largest and ^ n the smallest robability. Call the transformed (sorted) robability distribution f ^q i Λg. Then the algorithm solves for μ Λ such that F(μ Λ )=0where F(μ) = = i=n ^q i Λ 1 i=1 i= μ i=1 ^ i μ +(n μ) 1 (7) where the second equality follows from Equation 6, and μ is the number of letters with robability greater than μ. Figure 3 shows the relationshi between μ and μ. For all letters to the left of μ in Figure 3, ^q i Λ = and for others, ^q i Λ = ^ i=μ. Lemma 3: F(μ) is a monotonically decreasing function of μ. Proof: First, it is easy to see that if μ increases in the interval [ r+1 ^ =; ^ r =), i.e. such that μ does not change, F(μ) decreases monotonically. Similarly, if μ increases from ^ r = ffl to ^ r = + ffl so that μ decreases by 1, it is easy to verify that F(μ) decreases.2

5 n n-1 µ +1 µ Fig. 3. Showing the osition of μ and μ Lemma 3 imlies that we can do a binary search for finding a suitable value of r such that μ 2 [ r =; r 1 =) and F( r =) 0 and F( r 1 =) 0. This will tae us only O(log n) time. Once we now that μ belongs to this halfclosed interval, we now the exact value of μ = K and we can P then directly solve for μ Λ using Equation (7) to get μ Λ i=k = ( i=1 ^ i)=(1 (n K)). Putting this value of μ Λ in Equation (6) will then give us the transformed set of robabilities f ^q i Λg. 4 Given such fq ^ i g, the algorithm then constructs a canonical alhabetic coding tree as in [17] with the codeword lengths l Λ as chosen in Equation (2). This tree then clearly has a maximum deth of no more than D, and its average weighted length is worse than the otimum algorithm by no more than 2 bits. To see this, let us refer to P the code lengths in the otimum tree as fl ot g.thenc ot = l = H() +D(jj2 lot ). As we have chosen q Λ to be such that D(jjq Λ )» D(jjq) for all robability distributions q, it follows from Equation (3) that C mindq» C ot +2. We have thus roved the following main theorem of the aer: Theorem 1: Given a set of n robabilities f i g in a secified order, an alhabetic tree with a deth constraint D can be constructed in O(n log n) time and O(n) sace such that the average codeword length is at most 2 bits away from the otimum deth-constrained alhabetic tree. Further, if the robabilities are given in sorted order, such a tree can be constructed in linear time. B. Deth-Constrained Weight Balanced Tree (DCWBT) In this subsection, we resent a heuristic algorithm to generate near-otimal deth constrained alhabetic trees. This heuristic is similar to the weight balancing heuristic roosed by Horibe [26] with the modification that the maximum deth constraint is never violated. The trees generated by this heuristic algorithm have been observed to have lower average weighted length than those generated by algorithm MINDP. Also, imlementation of this algorithm turns out to be even simler. Desite its simlicity, it is unfortunately hard to rove any otimality roerties of this algorithm. We roceed to describe the normal weight balancing heuristic and then describe the modification needed to incororate the constraint of maximum deth. In a tree, suose the leaves of a articular subtree corresond to letters numbered r through t we say that the weight of the subtree as well as of its root is 4 Note that we will have to send O(n) time in the calculation of P i=μ i=1 ^ i anyway, so if we want we can simly imlement a linear search instead of a binary search to find the interval [ r=; r 1=). µ 2 1 P i=t i=r i. The root node of this subtree is said to reresent the robabilities f i g i=t i=r. Thus, the root node of an alhabetic tree has weight 1 and reresents the robability distribution f i g i=n. i=1 In the normal weight balancing heuristic of Horibe [26], one constructs a tree such that the weight of the root node is slit into two arts reresenting the weights of its two children in the most balanced manner ossible. The weights of the two children nodes are then slit recursively in a similar manner. In general, at an internal node reresenting the robabilities f r ::: t g,we tae the left and right children as reresenting the robabilities f r ::: s gand f s+1 ::: t gif s is such that (r;t) = fi i=s i=t i i=r i=s+1 = min 8u:r»utfi i=u i i=t i fi i=r i=u+1 i fi ; r» st This to-down algorithm clearly roduces an alhabetic tree. As an examle, the weight-balanced tree corresonding to Figure 1 is the tree shown in Figure 2(a) for the shown robabilities. Horibe roves that the average deth of such a weight-balanced tree is greater than the entroy of the underlying robability distribution f i g by no more than 2 (n +2) min,where min is the minimum robability in the distribution. Again this simle weight balancing heuristic can roduce a tree which has an unbounded maximum deth. For instance, a distribution f i g such that n = 2 (n 1) and i = 2 i 81» i» n 1, will roduce a highly sewed tree of maximum deth (n 1). Figure 2(a) is an instance of a tree on such a distribution and so is highly sewed. Here is a simle modification we roose to resect the deth constraint we follow Horibe s weight balancing heuristic constructing the tree in the normal to-down weight balancing manner until we reach a node such that if we were to slit the weight of the node further in the most balanced manner, the deth constraint would be violated. Instead, we slit the node maintaining as much balance as we can while resecting the deth constraint. If this haens at a node at deth d reresenting the robabilities f r ::: t g we tae the left and right children as reresenting the robabilities f r ::: s gand f s+1 ::: t gif s is such that (r;t) = fi i=s i=t i i=r i=s+1 i=r i=u = min 8u:a»ubfi i i fi ; a» sb i=t i=u+1 where a = t 2 D d 1 and b = r+2 D d 1. Therefore, the idea is to use the weight balancing heuristic as far down into the tree as ossible. Intuitively, any node where we are unable to use the heuristic is exected to be very dee down in the tree. This would mean that the total weight of this node is small enough so that aroximating the weight balancing heuristic does not cause any substantial effect to the average ath length. For instance, Figure 4 shows the deth-constrained weight balanced tree for a maximum deth constraint of 4 for the tree in Figure 1(a). i fi

6 I1 1/ I2 1/ Deth constraint would have been violated here 1101 tree constructed by the MINDP algorithm, the two subtrees of an internal node are almost equally balanced. This imlies that such a tree data structure can be efficiently arallelized. Suose that we had two searate loou engines for traversing a binary tree. Then we can assign the left-subtree of the root node to one of these engines and the right-subtree to the other engine. Since, the wor load is exected to be balanced among the two engines, we can get twice the loou rate that is ossible with one engine. Such an architecture is attractive in arallelizable designs. This near-erfect load-balancing thus hels achieve linear seedu with the number of loou engines. This is a nice scalability roerty for instance, if the routing tables were to increase 8 times in size, the same loou rate could be achieved by having 8 subtrees, each being traversed by a searate loou engine, as shown in Figure 5. I3 I4 I5 I6 1/8 1/16 1/32 1/32 Fig. 4. Weight balanced tree for Figure 1 with a deth constraint of 4. T1 T2 T3 T4 T5 T6 T7 T8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 Fig. 5. Showing 8-way arallelism in the constructed alhabetic tree. As mentioned above, we have been unable to come u with a rovably good bound on the distance of this heuristic from the otimum solution, but its concetual and imlementational simlicity along with the simulation results (see next section) suggest its usefulness. Lemma 4: A deth constrained weight balanced tree (DCWBT) for n leaves can be constructed in O(n log n) time and O(n) sace. Proof: At an internal node, the signed difference in the weights between its two subtrees is a monotonically increasing function of the difference in the number of nodes in the left and right subtrees. Thus a suitable slit may be found by binary search in O(log n) time at every internal node 5. Since there are n 1 internal nodes in a binary tree with n leaves, the total time comlexity is O(n log n). The sace comlexity is the comlexity of storing the binary tree and is thus linear.2 C. Load Balancing From the manner in which the DCWBT is constructed, it is easy to see that the two children subtrees of any internal node are nearly as weight-balanced as ossible. Even in the alhabetic 5 Note that we may need P access to P i=s i=r i; 81» r;s» n. This can be i=s obtained by recomuting i=1 i; 81» s» n in linear time and sace. IV. SIMULATION RESULTS A lot at the CAIDA web site [27] shows that the amount of traffic er refix length is very non-uniformly distributed. This rovides some real-life evidence of the ossible benefits to be gained by otimizing the routing table loou data structure based on the access frequency of the table entries. For the urose of detailed simulation to further demonstrate this claim, we too two large default-free routing tables ublicly available at IPMA [9] and another smaller table at VBNS [28]. To mae an accurate evaluation of the advantages of the otimization algorithms roosed in this aer, a nowledge of the access robabilities of the routing table entries is crucial. However, there are no ublicly available acet traffic traces with unencryted destination addresses that access these tables. Fortunately, we were able to find one trace of about 2.14 million acet destination addresses at NLANR [29] this trace has been taen from Fix-West and thus does not access the same routing tables as obtained from IPMA. Still, as the default-free routing tables should not be too different from each other, the use of this trace should give us valuable insights into the advantages of the roosed algorithms. In addition, we also consider the uniform distribution in our simulations, where the robability of accessing a articular refix is roortional to its size, i.e. an 8-bit long refix has a robability of access twice that of a 9-bit long refix. Table II shows the sizes of the three routing tables considered in our simulation, along with the entroy values of the uniform robability distribution and the robability distribution obtained from the trace. Also shown is the number of memory accesses required in an unotimized binary search, which simly is dlog(#intervals)e. In Figures 6 and 7, we lot the average loou query time (measured in terms of the number of memory accesses) versus the maximum deth constraint value for the two different robability distributions. These figures show that as the maximum deth constraint is relaxed from dlog 2 ne to higher values, the average loou time quicly aroaches the entroy of the corresonding distribution (see Table II). An interesting observation from the lots is that the simle weight-balancing heuristic DCWBT almost always erforms better than the near-otimal MINDP algorithm, esecially at higher values of maximum deth constraint.

7 Routing #Prefixes #Intervals Entroy Entroy Unot srch Table (Uniform) (Trace) VBNS [28] MAE WEST [9] MAE EAST [9] TABLE II Routing tables considered in simulations. Unot srch is the number of memory accesses required in a naive, unotimized binary search tree VBNS_DCWBT VBNS_DP MAE_WEST_DCWBT MAE_WEST_DP MAE_EAST_DCWBT MAE_EAST_DP ObservedTrace Distribution Equal Distribution Average Tree Deth Probability e Maximum Tree Deth 1e Interval# Fig. 6. Showing how the average loou time decreases when the worst case deth constraint is relaxed, for the uniform robability distribution. Y in the legend means that the lot relates to algorithm Y when alied to the routing table. Fig. 8. Showing the robability distribution as derived from the acet trace on the the MAE EAST routing table. Note that the Equal Distribution corresonds to a horizontal line at y=1.5e-5. Average Tree Deth VBNS_DCWBT VBNS_DP MAE_WEST_DCWBT MAE_WEST_DP MAE_EAST_DCWBT MAE_EAST_DP Maximum Tree Deth Fig. 7. Showing how the average loou time decreases when the worst case deth constraint is relaxed, for the robability distribution derived by the two million acet trace available from NLANR. Y in the legend means that the lot relates to algorithm Y when alied to the routing table. A. Tree Reconfigurability Because the routing tables and the refix access atterns are not static, the data-structure build time is an imortant consideration. This is the amount of time required to comute the otimized tree data structure. Our simulations show that even for the biggest routing table at MAE EAST, the MINDP algorithm taes about 0.96 seconds to comute a new tree, while the DCWBT algorithm taes about 0.40 seconds. 6 The build times 6 These simulations were carried out by imlementing the algorithms in C and running the simulation as a user-level rocess under Linux on a 333MHz Pentium-II rocessor with 96MB of memory. for the smaller VBNS routing table are only and seconds for the MINDP and DCWBT algorithms resectively. Comutation of a new tree could be needed because of two reasons: (1) change in the routing table, or (2) change in the access attern of the routing table entries. The average frequency of routing udates in the Internet today is of the order of a few udates er second, even though the ea value can be u to a few hundred udates er second. We can simly batch several udates to the routing table and run the tree comutation algorithm eriodically. Changes in the routing table structure can therefore be easily managed. The change in access atterns is harder to redict, but there is no reason to believe that it should haen at a very high rate. Indeed, if it does, there is no benefit to otimizing the tree anyway. In reality, we exect the global access attern to not change a lot while a small change in the robability distribution is exected over shorter time scales. Hence, an obvious way for udating the tree would be to ee trac of the current average loou time as measured by the last few acet loous in the router and do a new tree comutation whenever this differs from the tree s average weighted length (which is the exected value of the average loou time if the acets were obeying the robability distribution) by more than some settable threshold amount. The tree could also be recomuted at fixed intervals regardless of the changes. In summary, we believe that the tree build times are small enough to mae the algorithms of ractical use. To investigate tree reconfigurability in more detail, we simulated the acet trace with the MAE EAST routing table. For simlicity, we divided the 2.14 million acet destination ad-

8 Probability Uniform Distribution PtNum luwst luavg lusd Entroy WtLen 0-0.5M M M M M e-05 1e Interval# Fig. 9. Showing the uniform robability distribution as derived from the MAE EAST routing table. In this distribution, the robability of accessing an interval is roortional to its length. TABLE III Statistics for the MINDP tree constructed at the end of every 0.5 million acets in the 2.14 million acet trace for the MAE EAST routing table. The maximum loou time is constrained to be 22 memory accesses. All times/lengths are secified in terms of the number of memory accesses to reach the leaf of the tree storing the interval. luwst is the worst case loou time, luavg is the average loo u time and lusd is the standard deviation. WtLen is the average weighted length of the tree. dresses in the trace into grous, each grou consisting of 0.5M acets. We fed the addresses one at a time to the simulation and simulated the effects of udating the tree after seeing the last acet in every grou. We started out with the equal distribution, i.e. every tree leaf, which corresonds to a refix interval, is equally liely to be accessed by an incoming acet. Thus our initial tree is simly the comlete tree of deth dlog 2 ne. The simulation showed that once the first tree udate (at the end of the last acet of the first grou) is done, the average loou time decreases significantly and the other subsequent tree udates do not considerably alter this loou time. In other words, the access attern changes only slightly across grous. Figure 8 shows the robability distribution derived from the trace, and also lots the equal distribution (which is just a straight line arallel to the x-axis). Also shown for comarison is the uniform distribution in Figure 9. We found the distribution derived from the trace to be relatively unchanging from one grou to another, and therefore only one of the grous is shown in Figure 8. This is also borne out in the tree statistics for the MINDP trees comuted for every grou as shown in Table III for (an arbitrarily chosen) maximum loou time constraint of 22 memory accesses. The table shows how comuting a new tree at the end of the first grou brings down the average loou time from to 9.26 memory accesses roviding a nearly 42% imrovement in the loou rate. This imrovement is exected to be greater if the deth constraint were to be relaxed further. While it is not ossible to mae a direct comarison with the other roosed schemes [3] [8] for otimizing the routing table data structures because of the different nature of roblems being solved; we can mae a comarison of the comlexity of comutation of the otimal data structures. The comlexity of the data structure in [8] is stated to be O(HnB) where H = 32, n = number of refixes, B = constant believed to be around 3 [8]. This is thus around 96n. The comlexity of the algorithm in [3] is stated to be O(HnB) where B is a constant around 10, which maes it about 320n. In contrast, both the MINDP and the DCWBT algorithms are of comlexity O(n log n) for n refixes which, strictly seaing, is worse than O(n). However, including the constants in calculations, these algorithms have comlexity Cnlog n, wherec is a constant no more than 3. Thus even for very large values of n, say2 16 = 64K, the comlexity of these algorithms is no worse than aroximately 48n. V. CONCLUSIONS This aer roosed two near-otimal algorithms for doing route loous using a binary search tree data structure with a constraint on the maximum deth. The comlexity, erformance and otimality roerties of the algorithms were exlored. Simulations erformed on data taen from routing tables in the bacbone of the Internet show that the algorithms rovide a significant gain in erformance. REFERENCES [1] BGP Table, htt://telstra.net/os/bgtable.html. [2] Y. Rehter and T. 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9 [16] T. C. Hu and A. C. Tucer, Otimal comuter search trees and variable length alhabetic codes, SIAM Journal on Alied Mathematics, vol. 21, , [17] R. W. Yeung, Alhabetic codes revisited, IEEE Transactions on Information Theory, vol. 37, no. 3, , [18] R. L. Mildiu and E. S. Laber, Imroved bounds on the inefficiency of length-restricted refix codes, unublished. [19] R. L. Milidiu, A. A. Pessoa, and E. S. Laber, Efficient imlementation of the WARM-UP algorithm for the construction of length-restricted refix codes, in Proceedings of the ALENE, Baltimore, Maryland, USA, January 1999, vol of Lecture Notes in Comuter Science, Sringer. [20] M. R. Garey, Otimal binary search trees with restricted maximal deth, SIAM Journal on Comuting, vol. 3, , [21] E. N. Gilbert, Codes based on inaccurate source robabilities, IEEE Transactions on Information Theory, vol. 17, , [22] A. Itai, Otimal alhabetic trees, SIAM Journal on Comuting, vol. 5,. 9 18, [23] R. L. Wessner, Otimal alhabetic search trees with restricted maximal height, Information Processing Letters, vol. 4, , [24] T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley Series in Telecommunications, [25] D. P. Bertseas, Nonlinear Programming, Athena Scientific, [26] Y. Horibe, An imroved bound for weight-balanced tree, Information and Control, vol. 34, , [27] KC Claffy, Caida Internet Measurement Presentation, htt:// [28] All vbns routes snashot, htt:// [29] NLANR Networ Analysis Infrastructure, htt://moat.nlanr.net. Therefore, s n 1» P i=n 1 i=1 q i =1 q n»1 2 ln.and s n = 2 ln + c(s n 1 ; 2 ln )» c(1 2 ln ; 2 ln )+2 ln = 1 2 ln +2 ln =1. This comletes the roof that these lengths satisfy the characteristic inequality.2 APPENDI Lemma 5: For a deth constrained alhabetic tree with maximum deth D, the lengths of n letters when chosen as follows satisfy the characteristic inequality of Lemma 1, i.e. s n» 1 where s = c(s 1 ; 2 l )+2 l, s 0 =0and c is defined by c(a; b) =da=beb; when the code length l of the th letter occurring with robability q (q 2 D 8) is given by: l =ρ min(d log q e;d) =1;n min(d log q e +1;D) 2»»n 1 Proof: We first rove by induction that s i» i q 81» i» n 1 For the base case, s 1 =2 l1»q 1 by the definition of l 1.For the induction ste, assume the hyothesis is true for i 1. By definition, s i =2 li +c(s i 1 ;2 li ). Now there are two ossible cases: 1. d log q i e +1»D, and therefore 2 (li 1)» q i.usingthe fact that da=be a=b +1or c(a; b) a+bfor all nonzero real numbers a and b, we get that s i» 2 li + s i 1 +2 li. Now using the inductive hyothesis, we get s i» 2 (li 1) + =i 1 q» q i + =i 1 q = =i 2. d log q i e +1 D. Thisimliesthatl i = D and hence q i 2 li. Also, as s j is an integral multile of 2 D 8j, c(s i 1 ; 2 li )=s i 1» P i 1 q and thus =i 1 =i s i =2 li +c(s i 1 ;2 li )»q i + q = q q