Concise Papers. Main Memory Indexing: The Case for BD-Tree 1 INTRODUCTION 3 COST ANALYSIS 2 THE MEMORY-BASED BD-TREE

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1 870 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 16, NO. 7, JULY 2004 Concise Papers Main Meory Indexing: The Case for BD-Tree BinCui,BengChinOoi,Meber, IEEE, Jianwen Su, Senior Meber, IEEE, and Kian-Lee Tan, Meber, IEEE Coputer Society Abstract In this paper, we adapt and optiize the BD-tree for ain eory data processing. We copare the eory-based BD-tree against the B þ -tree and CSB þ -tree. We present cost odes for exact atch uery for these indexes, incuding L2 cache and transation ookahead buffer (TLB) iss ode and execution tie ode. We aso ipeented these structures and conducted experienta study. Our anaytica and experienta resuts show that a we-tuned BD-tree is superior in ost cases. Index Ters Main eory databases, BD-tree, B þ -tree, CSB þ -tree. 1 INTRODUCTION æ THE ipact of advances in hardware and ain eory capacity is beginning to be fet by the software counity [3], [4], [6], [7], [12]. Not ony is it possibe to store the entirety of a database in eory now, any agoriths that were designed in the past need to be reexained. For exape, the eory-based index structure T -tree [10] is no onger attractive because of its ow fanout, poor cache behavior, and excessive utiization of pointers. In this paper, we reexaine the indexing issue in ain eory databases. To iprove perforance, an indexing structure ust faciitate effective use of the L2 caches and CPU. One proising structure is the CSB þ -tree [12] that reduces the cache isses by storing a the chid nodes of any given node contiguousy, and keeping ony the address of the first chid in each node, thereby increasing the utiization of the cache ine. However, whie the CSB þ -tree is efficient for range ueries, it does not perfor we for exact atch ueries (copared to hash-based schees). This is because utipe nodes need to be accessed during the tree traversa. This propted us to search for a structure that can cobine the benefits of hashing and tree indexing. In this paper, we present our soution to faciitate efficient uery processing in ain eory environent. We adapted and optiized the Bounded Disorder (BD) ethod introduced by Litwin and Loet [11] for ain eory processing. The BD-tree is essentiay a B þ -tree where the eaf nodes are arge-sized partitions, and each eaf node is organized by hash tabes. We evauated the optiized eory-based BD-tree against the CSB þ -tree and B þ -tree anayticay and epiricay. Our resuts show that a we-tuned BD-tree outperfors the other two indexes in ost cases. 2 THE MEMORY-BASED BD-TREE Fig. 1 shows the structure of the BD-tree [11], which coprises two tiers. In the first tier, a hierarchica utiway tree structure (e.g.,. B. Cui, B.C. Ooi, and K.-L. Tan are with the Departent of Coputer Science, Nationa University of Singapore, 3 Science Drive 2, Singapore E-ai: {cuibin, ooibc, tank}@cop.nus.edu.sg.. J. Su is with the Departent of Coputer Science, University of Caifornia at Santa Barbara, Santa Barbara, CA E-ai: su@cs.ucsb.edu. Manuscript received 1 Oct. 2002; revised 11 May 2003; accepted 28 Juy For inforation on obtaining reprints of this artice, pease send e-ai to: tkde@coputer.org, and reference IEEECS Log Nuber /04/$20.00 ß 2004 IEEE Pubished by the IEEE Coputer Society B þ -tree) range partitions the data. The second tier begins at the eaf eve of the tree, which organizes the data in each eaf node using a hashing ethod. Records within a eaf are not ordered. The BD-tree is efficient for eory-based uery processing. First, the eaf node size of a BD-tree is uch arger than that of a B þ -tree. This eans fewer interna nodes are needed, i.e., a shorter tree. Therefore, fewer cache and TLB 1 isses are needed during tree traversa. Moreover, since eaf nodes are organized as hash buckets, ony the necessary buckets need to be searched. This ceary benefits exact atch ueries. Second, the tree structure essentiay acts as a dynaic range partitioning echanis, aowing each eaf node to have different range size. As such, ony the necessary eaves need to be exained for range ueries. To optiize the BD-tree for ain eory processing, we adapt the hash function with ow coputation cost to reduce the CPU overhead of searching the eaf nodes. Since the use of tree structure essentiay acts as a dynaic range partitioning echanis and the difference between the nubers of keys in the eaf nodes is bounded, the average perforance can be accuratey predicted under the unifority assuption. Moreover, since data ay be skewed or the hash functions ay not distribute the data unifory within each eaf, we aow each bucket to have at ost k chains (for soe integer k>¼ 1). In other words, a eaf wi be spit ony when a new record is to be inserted into a bucket whose k chained buckets are aready fu. There are exceptions where the dupicated keys (hash keys) overfow the chained buckets and no spitting wi hep. In these cases, we aow ore than k chains. For sipicity, we ignore such specia cases in our discussion. Since the fanout of a ain eory tree structure is typicay sa copared to a diskbased index, we can reduce the height of BD-tree significanty by setting a reativey arge eaf size. Within the eaf, we set the bucket size to correspond to the cache ine size. Note that, athough we can keep the eaf nodes of a B þ -tree=csb þ -tree arger than its interna nodes, a arger eaf node size does not resut in better search perforance. Because a arge eaf node size aso eans ore cache isses when the node is accessed, soe noncontiguous segents of the eaf need to be oaded into cache, even if we use binary search. For exact atch ueries in the BD-tree, we ony need one cache iss to access each chain bucket and at ost k buckets to be retrieved in eaf eve. For range ueries, because the eaf partitions are ordered, we just scan the eaves that overap the desired uery ranges. To speed up the range uery, we store the chained buckets contiguous with the priary buckets of the eaf in the eory and, hence, reduce the TLB isses of scanning. 3 COST ANALYSIS To have an insight into the perforance of the B þ -tree, CSB þ -tree, and BD-tree, we present anaytica odes to evauate the cost of exact atch uery, i.e., a cache and TLB isses ode and an execution tie ode. We ode the cost of exact atch uery as a function of three variabes: nuber of L2 cache isses, TLB isses, and instructions. There are other factors that affect perforance, such as branch ispredictions, instruction cache isses, and L1 data cache isses. However, these factors do not pay a significant roe in deterining the overa cost [3]. The representative paraeters used for the anaysis are shown in Tabe 1. Soe of the vaues were produced by easuring the perforance of indexes in the experient. And, to sipify the anaysis, we assue that the buckets in the eaf nodes of the BD-tree have no overfow chains. 1. The TLB is an essentia part of odern processors, which keeps the apping fro ogica eory address to physica eory address.

2 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 16, NO. 7, JULY Fig. 1. The BD-tree. 3.1 Cache and TLB Miss Mode The L2 cache isses and TLB isses pay an iportant roe in the tree operations. For each node access, there ay be severa cache isses and one TLB iss for various node sizes (< virtua eory page size). The node size and tree height are two ain paraeters in this cost ode. A arger node size ipies a shorter tree, and vice versa. To deterine the height of the tree, we first present the forua for fanout, f. 8 j k < u jnj jj þ 1 : CSB þ -tree f ¼ j k : u jnj jj þjpj þ 1 : B þ -tree=bd-tree: Aso, with the given size of data set, the nuber of eaf node, L, is: 8 c >< : B þ -tree=csb þ -tree u jnj jj L ¼ >: c ud jbj : BD-tree: Therefore, the height of the tree, h, is given as foows: h ¼dog f L þ 1e: For an exact atch uery, we ust traverse the tree fro the root to the eaf to get the answer. At each eve, a chid node is deterined by a binary search in the parent node, so the nuber of L2 cache ines accessed of a node is dðog 2 ðu n c Þþ0:5Þe, where n c is the nuber of cache ines spanned by one node (n c ¼d jnj jj e). Because we aways need to access the first cache bock of the node, we add 0.5 cache iss for each node access. On the first uery of the index, the nuber of cache isses is the cache ines accessed. However, there is a higher probabiity of finding soe nodes in the cache for subseuent ueries, e.g., the nodes in the upper eves of the tree are accessed freuenty and ay aways reside in the cache. Reaccessing of these nodes wi resut in fewer cache isses. To ode cache isses ore efficienty, we use the Cardenas s forua [5]: Xðn; Þ ¼n ð1 ð1 1=nÞ Þ; where X is the nuber of uniue nodes accessed, n is the nuber of nodes avaiabe, and is the nuber of ueries. Using this forua, the average nuber of uniuey accessed nodes for a uery, i.e. Xðn;Þ, reduces when we increase the nuber of ueries. We cobine the Cardenas s forua with the tree structure to deterine the nuber of uniue nodes that are accessed when we traverse the tree. The nuber of cache isses in the eaf node of BD-tree is 1, as we assue no chained buckets in the ode. Thus, we can ode the nuber of cache isses per uery as: TABLE 1 Paraeters for Anaysis

3 872 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 16, NO. 7, JULY 2004 Fig. 2. The nuber of (a) cache isses and (b) TLB isses for a singe uery. 8 P h >< XðnðiÞ;Þdðog i¼1 2ðun cþþ0:5þe : B þ -tree=csb þ -tree C ¼ P h 1 >: XðnðiÞ;Þdðog i¼1 2ðun cþþ0:5þeþxðnðhþ; Þ : BD-tree; where nðiþ is the nuber of nodes at eve i in the tree (nðiþ ¼f i 1 ). The ode gives the copusory cache isses, but not the capacity cache isses [9]. Because the cache size is finite, eventuay, soe highy accessed cache ines can be repaced after running any ueries. However, with arge cache size, these effects are not significant for a first-order approxiation. For an individua uery, the nuber of TLB isses can be eua to the height of the tree, whie the TLB isses ay be satisfied fro the L2 cache hits and top eves of the tree ay stay cached for subseuent ueries. Therefore, the nuber of the TLB isses, TLB, can be uch saer than the tree height and aso saer than C because of arger TLB page size. TLB is represented by the foowing euation: TLB ¼ ( C og 2 ðun cþþ0:5 : B þ -tree=csb þ -tree C 1 og 2 ðun þ 1 cþþ0:5 : BD-tree: We show the average nuber of TLB and cache isses for 1,000 exact atch ueries using the above odes in Fig. 2. The B þ -tree perfors worst aong these trees. Copared with the CSB þ -tree, the B þ -tree has a saer branching factor than the CSB þ -tree since it needs to store ore chid pointers. Hence, the B þ -tree is taer than the CSB þ -tree, and incurs ore TLB and cache isses. Not surprisingy, the BD-tree is best because its eaf node (partition) has ore keys, thus the BD-tree is shorter. When we increase the node size, both the nuber of cache and TLB isses reduce. The nuber of cache isses is uch saer than the nuber of cache ines accessed, because soe of the freuenty accessed nodes can reside in the cache. The nuber of TLB isses is fewer than that of cache isses when the node size is arger than 64 bytes, because one node access ony incurs one TLB iss, but severa cache isses in this case. For the CSB þ -tree, when the node size is 64 bytes (cache ine size), the nuber of cache isses is saest. For the B þ -tree and BD-tree, the optia node size is 128 bytes. In a cases, the cache isses do not increase uch for arger node sizes. Because the tree is shorter, ore proportiona accessed nodes can be resident in the cache, athough ore cache ines need to be accessed. The nuber of cache isses is a coproise of these factors. 3.2 Execution Tie Mode The tie to execute a singe uery (T Q ) incudes coputation tie (T C ), L2 cache isses tie (T CM ), and TLB isses tie (T TLBM ). Thus, we can estiate the execution tie T Q as: T Q ¼ T C þ T CM þ T TLBM : The above euation ignores the effects of out-of-order execution, which can hide a portion of cache iss atency by continuing to execute the instructions out-of-order on the processor [3]. It is easy for us to get T CM and T TLBM because we know the nuber of TLB and cache isses fro the TLB and cache iss ode. We obtain the cost of each TLB and cache iss using the Caibrator Too [1], e.g., a L2 cache iss atency c is 150 cyces and a TLB iss atency t is 100 cyces for SUNFire The coputation cost is the tie taken to execute a the instructions for a uery. The nuber of instructions executed at each eve is ainy the binary search inside the node, so we can give the euation to cacuate the nuber of tota instructions I Q as foows: I Q ¼ 8 I b ðh 1Þ og 2 u jnj jj þjpj og 2 u jnj jj þ 1 >< I b ðh 1Þ og 2 u jnj jj og 2 u jnj jj þ 1 I b ðh 1Þ og 2 u jnj jj >: og 2 u jbj þ 1 : B þ -tree : CSB þ -tree : BD-tree; where I b is the nuber of instructions reuired to evauate a key and seect the next position in a binary search. The Perfon too [2] can be used to count the nuber of instructions/cyces of an evauation. Hence, we can get the nuber of instructions for an exact atch uery using the above euation. The CPU cyces per instruction (CPI) used in the ode is eua to 2. The nuber of instructions and tota CPU cyces for a uery are shown in Fig. 3. Fig. 3a shows the nuber of instructions executed versus node size per uery. The nuber of instructions is uite stabe when the node size is arger than 64 bytes because we execute binary search in each node and the overa coputation cost is siiar. More instructions are incurred when the node size is eua to 32 bytes because the binary search is not efficient when the node size is too

4 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 16, NO. 7, JULY Fig. 3. The nuber of instructions and cyces for a singe uery. (a) Instructions and (b) CPU cyces. sa. The tota execution tie is the cobination of TLB and cache isses and CPU cost. When the node size is eua to the cache ine size, the CSB þ -tree perfors best, whie 128 bytes is an optia node size for B þ -tree and BD-tree. Ceary, the BD-tree perfors best aong these trees, foowed by the CSB þ -tree and B þ -tree. 4 APERFORMANCE STUDY In this section, we present an experienta evauation on SUNFire 4800, which has 750MHz CPU, 16 GB RAM, and 8M L2 cache. The data set we used consists of 10 iion uniue integers. The data in the structures consists of < key; pointer > pairs, both 4 bytes ong, and the keys are unifory distributed. We aso conducted sensitivity anaysis on the techniues by varying distributions of data sets, cardinarity of data sets, and so on. For a fu set of of the experienta study, readers are referred to [8]. The BD-tree has two extra paraeters, the nuber of buckets in eaf and the bucket size. Our preiinary study to tune these paraeters shows that the exact atch uery perforance of BDtree iproves with a arge nuber of buckets. On the other hand, the arge eaf size ay reduce the perforance of range ueries with sa uery range. We found that the perforance of tree is near optia when the eaf has 64 buckets, each of size 64 bytes, and we use it as the defaut vaues in our study here. Moreover, we aow at ost a chain of two buckets (i.e., k ¼ 2). A the buckets in a eaf node are aocated in the contiguous eory space to iniize TLB isses. 4.1 On Exact Match Query In this experient, we study the effect of node size on the perforance of exact atch uery of the various schees by varying the node size fro 32 bytes to 256 bytes. This aso serves to tune the various schees for optia settings of node sizes. We conducted 1,000 operations of exact atch ueries, and recorded their perforance. The resuts are shown in Fig. 4, and it is cear that the perforance of the three indexes is consistent with the anaytica resuts. Fig. 4a shows the overa eapsed tie of exact atch ueries. For a tree-based schees, when the node size is too sa (e.g., 32 bytes), the fanout becoes so sa that it resuts in a very ta tree. This eads to ore TLB and cache isses as the tree is traversed. When the node size increases, the nuber of TLB and cache isses decreases. For the B þ -tree, its optia perforance for exact atch ueries occurs when the node size is eua to 128 bytes. For the CSB þ -tree, we observe that it is optia when the node size is 64 bytes which is the cache ine size of L2. Siiar to the B þ -tree, the BD-tree yieds optia perforance when the interna node size is eua to 128 bytes. Coparing these various B þ -tree-based structures, we see that the CSB þ -tree is better than the B þ -tree in a cases, but the difference is not significant when the node size is arger than 128 bytes. This resut is consistent with Fig. 4. Perforance of exact atch uery. (a) Eapsed tie of exact atch and (b) cache isses of exact atch.

5 874 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 16, NO. 7, JULY 2004 Fig. 5. Perforance of range uery and insertion. (a) Eapsed tie of range uery and (b) eapsed tie of insertion. the findings in the anaytica study. The BD-tree perfors best and is about 40 percent better because of the shorter tree traversa and the efficient bucket access in the eaf eve. In Fig. 4b, we show the nubers of average cache isses of exact atch uery, which were obtained by Perfon [2]. Because we did not fush the L2 cache after each uery and the L2 cache can be fairy arge on odern achines, running any ueries on the index wi eventuay resut in fewer cache isses per uery, e.g., soe highy accessed cache ines can aways reside in the L2 cache. Note that there is no index node residing in the cache before the operations. So, saer index size can benefit ore fro this echanis as the upper eves of the tree can aways reside in the cache. Based on the resuts of the previous experient, we pick the node size for the BD-tree and B þ -tree to be 128 bytes, the node size for the CSB þ -tree to be 64 bytes. These wi be used as the defaut settings for the subseuent experients. 4.2 On Range Query In this experient, we study the various schees perforance on range ueries. We vary the seectivities of the ueries ranges fro 0.01 percent to 10 percent of the tota nuber of keys. The resut is shown in Fig. 5a. The cost of range ueries fas into two parts, tree traversa to ocate the eaf node containing the first record and scanning a eaf nodes that contain the answers. The CSB þ -tree is aways better than the B þ -tree because the CSB þ -tree can ocate the eaf faster and fewer TLB isses are incurred for scan, as the eaf nodes with the sae parent node are stored seuentiay in eory. Athough the BD-tree can ocate the eaf node uch faster, the disorder within the eaf incurs soe additiona cost. Therefore, when the range seectivity is sa, e.g., 0.01 percent seectivity, the BD-tree shows poorer resuts because it needs to access the whoe eaf. On the other hand, the BD-tree perfors as we as the CSB þ -tree and better than the B þ -tree when the seectivity is arger than 0.1 percent in our experient. Each eaf node of the BD-tree has uch ore keys than the other two trees and, hence, the scan of eaf nodes causes fewest TLB isses. 4.3 On Insertion We aso study the insertion perforance of the various schees. Fig. 5b shows the resut of 1,000 insertion for various node sizes. In a cases, we observe siiar perforance: the BD-tree is the ost efficient, the CSB þ -tree is the worst and the B þ -tree is sighty better than the CSB þ -tree. The CSB þ -tree is worse than the B þ -tree for the foowing reasons. The insertion cost has three parts: search cost, sort cost, and the spit cost. The spit cost of the CSB þ -tree incudes copying a copete node group, whereas a node spit of the B þ -tree invoves creating a new node. The BD-tree perfors better than the B þ -tree because the eaf partitions of the BD-tree have uch ore keys than those of the CSB þ -tree and B þ -tree, thus the tree is shorter; at the sae tie, inserting a key within a eaf is aso very efficient, as we ony need to cacuate the hash vaue and insert the key into the target bucket. 5 CONCLUSION In this paper, we have revisited the probe of accessing data in ain eory databases. We adapted and optiized the BD-tree [11] to faciitate fast search in ain eory environent. The BD-tree taps on the strengths of a hierarchica tree structure to eveny distribute and range partition the keys (faciitating efficient range ueries processing) and the efficiency of hash-based ethods for processing exact atch ueries. We studied the BD-tree against the B þ -tree and CSB þ -tree anayticay and epiricay. Our resuts showed that the BD-tree is a proising index structure for eory-based processing. REFERENCES [1] The Caibrator Too, [2] The Perfon Too, [3] A. Aiaaki, D.J. DeWitt, M.D. Hi, and D.A. Wood, DBMS on a Modern Processor: Where Does Tie Go, Proc. 25th Very Large Databases Conf., pp , [4] P. Bohannon, P. Mcroy, and R. Rastogi, Main-Meory Index Structures with Fixed-Size Partia Keys, Proc. ACM SIGMOD, pp , [5] A. Cardenas, Anaysis and Perforance of Inverted Database Structures, Co. ACM, vo. 18, pp , May [6] S. Chen, P.B. Gibbons, and T.C. Mowry, Iproving Index Perforance through Prefetching, Proc. ACM SIGMOD, pp , [7] S. Chen, P.B. Gibbons, T.C. Mowry, and G. Vaentin, Fracta Prefeching B+-Tree: Optiizing Both Cache and Disk Perforance, Proc. ACM SIGMOD, pp , [8] B. Cui, B.C. Ooi, J.W. Su, and K.L. Tan, Using BD-Tree for Main Meory Processing, technica report, Schoo of Coputing, Nat Univ. of Singapore, [9] J. Hennessy and D. Patterson, Coputer Architecture: A Quantitative Approach. Morgan Kauffan, [10] T. Lehan and M. Carey, A Study of Index Structures for Main Meory Database Manageent Systes, Proc. 12th Very Large Databases Conf., pp , [11] W. Litwin and D. Loet, The Bounded Disorder Access Method, Proc. 17th Int Conf. Data Eng., pp , [12] J. Rao and K. Ross, Making B+-Trees Cache Conscious in Main Meory, Proc. ACM SIGMOD, pp , For ore inforation on this or any other coputing topic, pease visit our Digita Library at