Unformatted text preview: Fast Algorithms for Mining Association Rules
Rakesh Agrawal Ramakrishnan Srikant IBM Almaden Research Center 650 Harry Road, San Jose, CA 95120
Abstract We consider the problem of discovering association rules between items in a large database of sales transactions. We present two new algorithms for solving this problem that are fundamentally di erent from the known algorithms. Experiments with synthetic as well as real-life data show that these algorithms outperform the known algorithms by factors ranging from three for small problems to more than an order of magnitude for large problems. We also show how the best features of the two proposed algorithms can be combined into a hybrid algorithm, called AprioriHybrid. Scale-up experiments show that AprioriHybrid scales linearly with the number of transactions. AprioriHybrid also has excellent scale-up properties with respect to the transaction size and the number of items in the database. 1 Introduction
Database mining is motivated by the decision support problem faced by most large retail organizations S+ 93 . Progress in bar-code technology has made it possible for retail organizations to collect and store massive amounts of sales data, referred to as the basket data. A record in such data typically consists of the transaction date and the items bought in the transaction. Successful organizations view such databases as important pieces of the marketing infrastructure Ass92 . They are interested in instituting information-driven marketing processes, managed by database technology, that enable marketers to develop and implement customized marketing programs and strategies Ass90 . The problem of mining association rules over basket data was introduced in AIS93b . An example of such a rule might be that 98 of customers that purchase tires and auto accessories also get automotive services done. Finding all such rules is valuable for cross-marketing and attached mailing applications. Other applications include catalog design, add-on sales, store layout, and customer segmentation based on buying patterns. The databases involved in these applications are very large. It is imperative, therefore, to have fast algorithms for this task. Visiting from the Department of Computer Science, University of Wisconsin, Madison. 1 The following is a formal statement of the problem AIS93b : Let I = fi1; i2; . . . ; img be a set of literals, called items. Let D be a set of transactions, where each transaction T is a set of items such that T I . Associated with each transaction is a unique identi er, called its TID. We say that a transaction T contains X , a set of some items in I , if X T . An association rule is an implication of the form X = Y , where X I , Y I , and X Y = ;. The rule X = Y holds in the transaction set D with con dence c if c of transactions in D that contain X also contain Y . The rule X = Y has support s in the transaction set D if s of transactions in D contain X Y . Our rules are somewhat more general than in AIS93b in that we allow a consequent to have more than one item. Given a set of transactions D, the problem of mining association rules is to generate all association rules that have support and con dence greater than the user-speci ed minimum support called minsup and minimum con dence called minconf respectively. Our discussion is neutral with respect to the representation of D. For example, D could be a data le, a relational table, or the result of a relational expression. An algorithm for nding all association rules, henceforth referred to as the AIS algorithm, was presented in AIS93b . Another algorithm for this task, called the SETM algorithm, has been proposed in HS93 . In this paper, we present two new algorithms, Apriori and AprioriTid, that di er fundamentally from these algorithms. We present experimental results, using both synthetic and real-life data, showing that the proposed algorithms always outperform the earlier algorithms. The performance gap is shown to increase with problem size, and ranges from a factor of three for small problems to more than an order of magnitude for large problems. We then discuss how the best features of Apriori and AprioriTid can be combined into a hybrid algorithm, called AprioriHybrid. Experiments show that the AprioriHybrid has excellent scale-up properties, opening up the feasibility of mining association rules over very large databases. The problem of nding association rules falls within the purview of database mining AIS93a ABN92 HS94 MKKR92 S+ 93 Tsu90 , also called knowledge discovery in databases HCC92 Lub89 PS91b . Related, but not directly applicable, work includes the induction of classi cation rules BFOS84 Cat91 FWD93 HCC92 Qui93 , discovery of causal rules CH92 Pea92 , learning of logical de nitions MF92 Qui90 , tting of functions to data LSBZ87 Sch90 , and clustering ANB92 C+ 88 Fis87 . The closest work in the machine learning literature is the KID3 algorithm presented in PS91a . If used for nding all association rules, this algorithm will make as many passes over the data as the number of combinations of items in the antecedent, which is exponentially large. Related work in the database literature is the work on inferring functional dependencies from data Bit92 MR87 . Functional dependencies are rules requiring strict satisfaction. Consequently, having determined a dependency X ! A, the algorithms in Bit92 MR87 2 consider any other dependency of the form X + Y ! A redundant and do not generate it. The association rules we consider are probabilistic in nature. The presence of a rule X ! A does not necessarily mean that X + Y ! A also holds because the latter may not have minimum support. Similarly, the presence of rules X ! Y and Y ! Z does not necessarily mean that X ! Z holds because the latter may not have minimum con dence. There has been work on quantifying the usefulness" or interestingness" of a rule PS91a . What is useful or interesting is often application-dependent. The need for a human in the loop and providing tools to allow human guidance of the rule discovery process has been articulated, for example, in B+ 93 KI91 Tsu90 . We do not discuss these issues in this paper, except to point out that these are necessary features of a rule discovery system that may use our algorithms as the engine of the discovery process. 1.1 Problem Decomposition and Paper Organization
The problem of discovering all association rules can be decomposed into two subproblems AIS93b : 1. Find all sets of items itemsets that have transaction support above minimum support. The support for an itemset is the number of transactions that contain the itemset. Itemsets with minimum support are called large itemsets, and all others small itemsets. In Section 2, we give new algorithms, Apriori and AprioriTid, for solving this problem. 2. Use the large itemsets to generate the desired rules. We give algorithms for this problem in Section 3. The general idea is that if, say, ABCD and AB are large itemsets, then we can determine if the rule AB = CD holds by computing the ratio conf = supportABCD supportAB . If conf minconf, then the rule holds. The rule will surely have minimum support because ABCD is large. Unlike AIS93b , where rules were limited to only one item in the consequent, we allow multiple items in the consequent. An example of such a rule might be that in 58 of the cases, a person who orders a comforter also orders a at sheet, a tted sheet, a pillow case, and a ru e. The algorithms in Section 3 generate such multi-consequent rules. In Section 4, we show the relative performance of the proposed Apriori and AprioriTid algorithms against the AIS AIS93b and SETM HS93 algorithms. To make the paper self-contained, we include an overview of the AIS and SETM algorithms in this section. We also describe how the Apriori and AprioriTid algorithms can be combined into a hybrid algorithm, AprioriHybrid, and demonstrate the scale-up properties of this algorithm. We conclude by pointing out some related open problems in Section 5. 3 2 Discovering Large Itemsets
Algorithms for discovering large itemsets make multiple passes over the data. In the rst pass, we count the support of individual items and determine which of them are large, i.e. have minimum support. In each subsequent pass, we start with a seed set of itemsets found to be large in the previous pass. We use this seed set for generating new potentially large itemsets, called candidate itemsets, and count the actual support for these candidate itemsets during the pass over the data. At the end of the pass, we determine which of the candidate itemsets are actually large, and they become the seed for the next pass. This process continues until no new large itemsets are found. The Apriori and AprioriTid algorithms we propose di er fundamentally from the AIS AIS93b and SETM HS93 algorithms in terms of which candidate itemsets are counted in a pass and in the way that those candidates are generated. In both the AIS and SETM algorithms see Sections 4.1 and 4.2 for a review, candidate itemsets are generated on-the- y during the pass as data is being read. Speci cally, after reading a transaction, it is determined which of the itemsets found large in the previous pass are present in the transaction. New candidate itemsets are generated by extending these large itemsets with other items in the transaction. However, as we will see, the disadvantage is that this results in unnecessarily generating and counting too many candidate itemsets that turn out to be small. The Apriori and AprioriTid algorithms generate the candidate itemsets to be counted in a pass by using only the itemsets found large in the previous pass without considering the transactions in the database. The basic intuition is that any subset of a large itemset must be large. Therefore, the candidate itemsets having k items can be generated by joining large itemsets having k,1 items, and deleting those that contain any subset that is not large. This procedure results in generation of a much smaller number of candidate itemsets. The AprioriTid algorithm has the additional property that the database is not used at all for counting the support of candidate itemsets after the rst pass. Rather, an encoding of the candidate itemsets used in the previous pass is employed for this purpose. In later passes, the size of this encoding can become much smaller than the database, thus saving much reading e ort. We will explain these points in more detail when we describe the algorithms. and each database record is a TID, item pair, where TID is the identi er of the corresponding transaction. We call the number of items in an itemset its size, and call an itemset of size k a k-itemset. Items within an itemset are kept in lexicographic order. We use the notation c 1 c 2 . . . c k 4 Notation We assume that items in each transaction are kept sorted in their lexicographic order. It is straightforward to adapt these algorithms to the case where the database D is kept normalized Table 1: Notation
k-itemset An itemset having k items. Set of large k-itemsets those with minimum support. Lk Each member of this set has two elds: i itemset and ii support count. Set of candidate k-itemsets potentially large itemsets. Ck Each member of this set has two elds: i itemset and ii support count. Set of candidate k-itemsets when the TIDs of the generating transactions Ck are kept associated with the candidates. to represent a k-itemset c consisting of items c 1 ; c 2 ; . . . c k , where c 1 c 2 . . . c k . If c = X Y and Y is an m-itemset, we also call Y an m-extension of X . Associated with each itemset is a count eld to store the support for this itemset. The count eld is initialized to zero when the itemset is rst created. We summarize in Table 1 the notation used in the algorithms. The set C k is used by AprioriTid and will be further discussed when we describe this algorithm. 2.1 Algorithm Apriori
Figure 1 gives the Apriori algorithm. The rst pass of the algorithm simply counts item occurrences to determine the large 1-itemsets. A subsequent pass, say pass k, consists of two phases. First, the large itemsets Lk,1 found in the k,1th pass are used to generate the candidate itemsets Ck , using the apriori-gen function described in Section 2.1.1. Next, the database is scanned and the support of candidates in Ck is counted. For fast counting, we need to e ciently determine the candidates in Ck that are contained in a given transaction t. Section 2.1.2 describes the subset function used for this purpose. Section 2.1.3 discusses bu er management.
1 L1 = flarge 1-itemsetsg; 2 for k = 2; Lk,1 6= ;; k++ do begin 3 Ck = apriori-genLk,1; New candidates see Section 2.1.1 4 forall transactions t 2 D do begin 5 Ct = subsetCk , t; Candidates contained in t see Section 2.1.2 6 forall candidates c 2 Ct do 7 c:count++; 8 end 9 Lk = fc 2 Ck j c:count minsupg 10 end S 11 Answer = k Lk ; Figure 1: Algorithm Apriori 5 2.1.1 Apriori Candidate Generation
The apriori-gen function takes as argument Lk,1 , the set of all large k , 1-itemsets. It returns a superset of the set of all large k-itemsets. The function works as follows. 1 First, in the join step, we join Lk,1 with Lk,1 : insert into Ck select p.item1, p.item2, ..., p.itemk,1, q.itemk,1 from Lk,1 p, Lk,1 q where p.item1 = q.item1, . . ., p.itemk,2 = q.itemk,2, p.itemk,1 q.itemk,1; Next, in the prune step, we delete all itemsets c 2 Ck such that some k , 1-subset of c is not in Lk,1: forall itemsets c 2 Ck do forall k ,1-subsets s of c do if s 62 Lk,1 then delete c from Ck; Contrast this candidate generation with the one used in the AIS and SETM algorithms. In pass k of these algorithms see Section 4 for details, a database transaction t is read and it is determined which of the large itemsets in Lk,1 are present in t. Each of these large itemsets l is then extended with all those large items that are present in t and occur later in the lexicographic ordering than any of the items in l. Continuing with the previous example, consider a transaction f1 2 3 4 5g. In the fourth pass, AIS and SETM will generate two candidates, f1 2 3 4g and f1 2 3 5g, by extending the large itemset f1 2 3g. Similarly, an additional three candidate itemsets will be generated by extending the other large itemsets in L3 , leading to a total of 5 candidates for consideration in the fourth pass. Apriori, on the other hand, generates and counts only one itemset, f1 3 4 5g, because it concludes a priori that the other combinations cannot possibly have minimum support.
Concurrent to our work, the following two-step candidate generation procedure has been proposed in MTV94 : 0 0 0 0 Ck = fX X jX; X 2 Lk ,1 ; jX X j = k , 2g 0 Ck = fX 2 Ck jX contains k members of Lk ,1 g These two steps are similar to our join and prune steps respectively. However, in general, step 1 would produce a superset of the candidates produced by our join step. For example, if L2 were ff1 2g, f2, 3gg, then step 1 of MTV94 will generate the candidate f1 2 3g, whereas our join step will not generate any candidate.
1 Example Let L3 be ff1 2 3g, f1 2 4g, f1 3 4g, f1 3 5g, f2 3 4gg. After the join step, C4 will be ff1 2 3 4g, f1 3 4 5g g. The prune step will delete the itemset f1 3 4 5g because the itemset f1 4 5g is not in L3 . We will then be left with only f1 2 3 4g in C4 . 6 have minimum support. Hence, if we extended each itemset in Lk,1 with all possible items and then deleted all those whose k , 1-subsets were not in Lk,1 , we would be left with a superset of the itemsets in Lk . The join is equivalent to extending Lk,1 with each item in the database and then deleting those itemsets for which the k , 1-itemset obtained by deleting the k , 1th item is not in Lk,1 . The condition p.itemk,1 q .itemk,1 simply ensures that no duplicates are generated. Thus, after the join step, Ck Lk . By similar reasoning, the prune step, where we delete from Ck all itemsets whose k , 1-subsets are not in Lk,1 , also does not delete any itemset that could be in Lk .
0 0 only candidates of size k in the kth pass, we can also count the candidates Ck+1 , where Ck+1 is 0 generated from Ck , etc. Note that Ck+1 Ck+1 since Ck+1 is generated from Lk . This variation can 0 pay o in the later passes when the cost of counting and keeping in memory additional Ck+1 , Ck+1 candidates becomes less than the cost of scanning the database. Correctness We need to show that Ck Lk . Clearly, any subset of a large itemset must also Variation: Counting Candidates of Multiple Sizes in One Pass Rather than counting Membership Test The prune step requires testing that all k ,1-subsets of a newly generated
stored in a hash table. k-candidate-itemset are present in Lk,1 . To make this membership test fast, large itemsets are 2.1.2 Subset Function
Candidate itemsets Ck are stored in a hash-tree. A node of the hash-tree either contains a list of itemsets a leaf node or a hash table an interior node. In an interior node, each bucket of the hash table points to another node. The root of the hash-tree is de ned to be at depth 1. An interior node at depth d points to nodes at depth d + 1. Itemsets are stored in the leaves. When we add an itemset c, we start from the root and go down the tree until we reach a leaf. At an interior node at depth d, we decide which branch to follow by applying a hash function to the dth item of the itemset. All nodes are initially created as leaf nodes. When the number of itemsets in a leaf node exceeds a speci ed threshold, the leaf node is converted to an interior node. Starting from the root node, the subset function nds all the candidates contained in a transaction t as follows. If we are at a leaf, we nd which of the itemsets in the leaf are contained in t and add references to them to the answer set. If we are at an interior node and we have reached it by hashing the item i, we hash on each item that comes after i in t and recursively apply this procedure to the node in the corresponding bucket. For the root node, we hash on every item in t. To see why the subset function returns the desired set of references, consider what happens at the root node. For any itemset c contained in transaction t, the rst item of c must be in t. At 7 the root, by hashing on every item in t, we ensure that we only ignore itemsets that start with an item not in t. Similar arguments apply at lower depths. The only additional factor is that, since the items in any itemset are ordered, if we reach the current node by hashing the item i, we only need to consider the items in t that occur after i. If k is the size of a candidate itemset in the hash-tree, we can nd in Ok time whether the itemset is contained in a transaction by using a temporary bitmap. Each bit of the bitmap corresponds an item. The bitmap is created once for the data structure, and reinitialized for each transaction. This initialization takes Osizetransaction time for each transaction. 2.1.3 Bu er Management
In the candidate generation phase of pass k, we need storage for large itemsets Lk,1 and the candidate itemsets Ck . In the counting phase, we need storage for Ck and at least one page to bu er the database transactions. First, assume that Lk,1 ts in memory but that the set of candidates Ck does not. The apriorigen function is modi ed to generate as many candidates of Ck as will t in the bu er and the database is scanned to count the support of these candidates. Large itemsets resulting from these candidates are written to disk, while those candidates without minimum support are deleted. This procedure is repeated until all of Ck has been counted. If Lk,1 does not t in memory either, we externally sort Lk,1 . We bring into memory a block of Lk,1 in which the rst k , 2 items are the same. We now generate candidates using this block. We keep reading blocks of Lk,1 and generating candidates until the memory lls up, and then make a pass over the data. This procedure is repeated until all of Ck has been counted. Unfortunately, we can no longer prune those candidates whose subsets are not in Lk,1 , as the whole of Lk,1 is not available in memory. 2.2 Algorithm AprioriTid
The AprioriTid algorithm, shown in Figure 2, also uses the apriori-gen function given in Section 2.1.1 to determine the candidate itemsets before the pass begins. The interesting feature of this algorithm is that the database D is not used for counting support after the rst pass. Rather, the set C k is used for this purpose. Each member of the set C k is of the form TID; fXkg , where each Xk is a potentially large k-itemset present in the transaction with identi er TID. For k = 1, C 1 corresponds to the database D, although conceptually each item i is replaced by the itemset fig. For k 1, C k is generated by the algorithm step 10. The member of C k corresponding to transaction t is t:TID, fc 2 Ck jc contained in tg . If a transaction does not contain any candidate k-itemset, then C k will not have an entry for this transaction. Thus, the number of 8 entries in C k may be smaller than the number of transactions in the database, especially for large values of k. In addition, for large values of k, each entry may be smaller than the corresponding transaction because very few candidates may be contained in the transaction. However, for small values for k, each entry may be larger than the corresponding transaction because an entry in Ck includes all candidate k-itemsets contained in the transaction. We further explore this trade-o in Section 4. We establish the correctness of the algorithm in Section 2.2.1. In Section 2.2.2, we give the data structures used to implement the algorithm, and we discuss bu er management in Section 2.2.3.
1 L1 = flarge 1-itemsetsg; 2 C 1 = database D; 3 for k = 2; Lk,1 6= ;; k++ do begin 4 Ck = apriori-genLk,1; New candidates see Section 2.1.1 5 C k = ;; 6 forall entries t 2 C k,1 do begin 7 determine candidate itemsets in Ck contained in the transaction with identi er t.TID Ct = fc 2 Ck j c , c k 2 t:set-of-itemsets ^ c , c k , 1 2 t.set-of-itemsetsg; 8 forall candidates c 2 Ct do 9 c:count++; 10 if Ct 6= ; then C k += t:TID; Ct ; 11 end 12 Lk = fc 2 Ck j c:count minsupg 13 end S 14 Answer = k Lk ; Figure 2: Algorithm AprioriTid Calling apriori-gen with L1 at step 4 gives the candidate itemsets C2 . In steps 6 through 10, we count the support of candidates in C2 by iterating over the entries in C 1 and generate C 2. The rst entry in C 1 is f f1g f3g f4g g, corresponding to transaction 100. The Ct at step 7 corresponding to this entry t is f f1 3g g, because f1 3g is a member of C2 and both f1 3g - f1g and f1 3g f3g are members of t.set-of-itemsets. Calling apriori-gen with L2 gives C3 . Making a pass over the data with C 2 and C3 generates C 3. Note that there is no entry in C 3 for the transactions with TIDs 100 and 400, since they do not contain any of the itemsets in C3 . The candidate f2 3 5g in C3 turns out to be large and is the only member of L3 . When we generate C4 using L3 , it turns out to be empty, and we terminate. Example Consider the database in Figure 3 and assume that minimum support is 2 transactions. 2.2.1 Correctness
Rather than using the database transactions, AprioriTid uses the entries in C k to count the support of candidates in Ck . To simplify the proof, we assume that in step 10 of AprioriTid, we always 9 Database TID Items 100 1 3 4 200 2 3 5 300 1 2 3 5 400 2 5 C2 Itemset f1 2g f1 3g f1 5g f2 3g f2 5g f3 5g C3 Itemset f2 3 5g TID 100 200 300 400 TID 100 200 300 400 C1 Set-of-Itemsets f f1g, f3g, f4g g f f2g, f3g, f5g g f f1g, f2g, f3g, f5g g f f2g, f5g g C2 Set-of-Itemsets f f1 3g g f f2 3g, f2 5g, f3 5g g f f1 2g, f1 3g, f1 5g, f2 3g, f2 5g, f3 5g g f f2 5g g L1 Itemset Support f1g 2 f2g 3 f3g 3 f5g 3 L2 Itemset Support f1 3g 2 f2 3g 2 f2 5g 3 f3 5g 2 L3 Itemset Support f2 3 5g 2 C3 TID Set-of-Itemsets 200 f f2 3 5g g 300 f f2 3 5g g Figure 3: Example add t.TID,Ct to C k , rather than adding an entry only when Ct is non-empty. For correctness, we need to establish that the set Ct generated in step 7 in the kth pass is the same as the set of candidate k-itemsets in Ck contained in the transaction with identi er t.TID. We say that the set C k is complete if 8t 2 C k , t.set-of-itemsets includes all large k-itemsets contained in the transaction with identi er t.TID. We say that the set C k is correct if 8t 2 C k , t.set-of-itemsets does not include any k-itemset not contained in the transaction with identi er t.TID. The set Lk is correct if it is the same as the set of all large k-itemsets. We say that the set Ct generated in step 7 in the kth pass is correct if it is the same as the set of candidate k-itemsets in Ck contained in the transaction with identi er t.TID. Lemma 1 8k 1, if C k,1 is correct and complete and Lk,1 is correct, then the set Ct generated
in step 7 in the kth pass is the same as the set of candidate k-itemsets in Ck contained in the transaction with identi er t.TID. By simple rewriting, a candidate itemset c = c 1 c 2 . . . c k is present in transaction t.TID if and only if both c1 = c , c k and c2 = c , c k , 1 are in transaction t.TID. Since Ck was obtained by calling apriori-genLk,1 , all subsets of c 2 Ck must be large. So, c1 and c2 must be large itemsets. Thus, if c 2 Ck is contained in transaction t.TID, c1 and c2 must be members of t.set-of-itemsets since C k,1 is complete Hence c will be a member of Ct . Since C k,1 is correct, if 10 c1 c2 is not present in transaction t.TID then c1 c2 is not contained in t:set-of-itemsets. Hence, if c 2 Ck is not contained in transaction t.TID, c will not be a member of Ct . 2
as the set of candidate k-itemsets in Ck contained in the transaction with identi er t.TID, then the set C k is correct and complete. Lemma 2 8k 1, if Lk,1 is correct and the set Ct generated in step 7 in the kth pass is the same Since the apriori-gen function guarantees that Ck Lk , the set Ct includes all large k-itemsets contained in t.TID. These are added in step 10 to C k and hence C k is complete. Since Ct only includes itemsets contained in the transaction t.TID, and only itemsets in Ct are added to C k , it follows that C k is correct. 2 1, the set Ct generated in step 7 in the kth pass is the same as the set of candidate k-itemsets in Ck contained in the transaction with identi er t.TID. We rst prove by induction on k that the set C k is correct and complete and Lk correct for all k 1. For k = 1, this is trivially true since C 1 corresponds to the database D. By de nition, L1 is also correct. Assume this holds for k = n. From Lemma 1, the set Ct generated in step 7 in the n+1th pass will consist of exactly those itemsets in Cn+1 contained in the transaction with identi er t.TID. Since the apriori-gen function guarantees that Cn+1 Ln+1 and Ct is correct, Ln+1 will be correct. From Lemma 2, the set C n+1 will be correct and complete. Since C k is correct and complete and Lk correct for all k 1, the theorem follows directly from Lemma 1. 2 Theorem 1 8k 2.2.2 Data Structures
We assign each candidate itemset a unique number, called its ID. Each set of candidate itemsets Ck is kept in an array indexed by the IDs of the itemsets in Ck . A member of C k is now of the form TID; fIDg . Each C k is stored in a sequential structure. The apriori-gen function generates a candidate k-itemset ck by joining two large k,1-itemsets. We maintain two additional elds for each candidate itemset: i generators and ii extensions. The generators eld of a candidate itemset ck stores the IDs of the two large k ,1-itemsets whose join generated ck . The extensions eld of an itemset ck stores the IDs of all the k +1-candidates that 1 2 are extensions of ck . Thus, when a candidate ck is generated by joining lk,1 and lk,1 , we save the 1 2 IDs of lk,1 and lk,1 in the generators eld for ck . At the same time, the ID of ck is added to the 1 extensions eld of lk,1 . We now describe how Step 7 of Figure 2 is implemented using the above data structures. Recall that the t.set-of-itemsets eld of an entry t in C k,1 gives the IDs of all k,1-candidates contained 11 in transaction t.TID. For each such candidate ck,1 the extensions eld gives Tk , the set of IDs of all the candidate k-itemsets that are extensions of ck,1 . For each ck in Tk , the generators eld gives the IDs of the two itemsets that generated ck . If these itemsets are present in the entry for t.set-of-itemsets, we can conclude that ck is present in transaction t.TID, and add ck to Ct . 2 We actually need to store only lk,1 in the generators eld, since we reached ck starting from the 1 ID of lk,1 in t. We omitted this optimization in the above description to simplify exposition. Given an ID and the data structures above, we can nd the associated candidate itemset in constant time. We can also nd in constant time whether or not an ID is present in the t.set-of-itemsets eld by using a temporary bitmap. Each bit of the bitmap corresponds to an ID in Ck . This bitmap is created once at the beginning of the pass and is reinitialized for each entry t of C k . 2.2.3 Bu er Management
In the kth pass, AprioriTid needs memory for Lk,1 and Ck during candidate generation. During the counting phase, it needs memory for Ck,1 , Ck , and a page each for C k,1 and C k . Note that the entries in C k,1 are needed sequentially and that the entries in C k can be written to disk as they are generated. At the time of candidate generation, when we join Lk,1 with itself, we ll up roughly half the bu er with candidates. This allows us to keep the relevant portions of both Ck and Ck,1 in memory during the counting phase. In addition, we ensure that all candidates with the same rst k , 1 items are generated at the same time. The computation is now e ectively partitioned because none of the candidates in memory that turn out to large at the end of the pass will join with any of the candidates not yet generated to derive potentially large itemsets. Hence we can assume that the candidates in memory are the only candidates in Ck and nd all large itemsets that are extensions of candidates in Ck by running the algorithm to completion. This may cause further partitioning of the computation downstream. Having thus run the algorithm to completion, we return to Lk,1 , generate some more candidates in Ck , count them, and so on. Note that the prune step of the apriori-gen function cannot be applied after partitioning because we do not know all the large k-itemsets. When Lk does not t in memory, we need to externally sort Lk as in the bu er management scheme used for Apriori. 3 Discovering Rules
The association rules that we consider here are somewhat more general than in AIS93b in that we allow a consequent to have more than one item; rules in AIS93b were limited to single item 12 consequents. We rst give a straightforward generalization of the algorithm in AIS93b and then present a faster algorithm. To generate rules, for every large itemset l, we nd all non-empty subsets of l. For every such subset a, we output a rule of the form a = l , a if the ratio of supportl to supporta is at least minconf. We consider all subsets of l to generate rules with multiple consequents. Since the large itemsets are stored in hash tables, the support counts for the subset itemsets can be found e ciently. We can improve the above procedure by generating the subsets of a large itemset in a recursive depth- rst fashion. For example, given an itemset ABCD, we rst consider the subset ABC , then AB , etc. Then if a subset a of a large itemset l does not generate a rule, the subsets of a need not be considered for generating rules using l. For example, if ABC = D does not have enough con dence, we need not check whether AB = CD holds. We do not miss any rules because the support of any subset a of a must be as great as the support of a. Therefore, the con dence of the ~ rule a = l , a cannot be more than the con dence of a = l , a. Hence, if a did not yield a ~ ~ rule involving all the items in l with a as the antecedent, neither will a. The following algorithm ~ embodies these ideas: forall large itemsets lk , k 2 do call genruleslk , lk ;
The genrules generates all valid rules a = lk , ~, for all ~ am ~ a a procedure genruleslk: large k-itemset, am : large m-itemset 1 A = fm,1-itemsets am,1 j am,1 am g; 2 forall am,1 2 A do begin 3 conf = supportlk supportam,1 ; 4 if conf minconf then begin 7 output the rule am,1 = lk , am,1 , with con dence = conf and support = supportlk ; 8 if m , 1 1 then 9 call genruleslk , am,1; to generate rules with subsets of am,1 as the antecedents 10 end 11 end Simple Algorithm 3.1 A Faster Algorithm
We showed earlier that if a = l , a does not hold, neither does ~ = l , ~ for any a a. By a a ~ rewriting, it follows that for a rule l , c = c to hold, all rules of the form l , c = c must ~ ~ also hold, where c is a non-empty subset of c. For example, if the rule AB = CD holds, then the ~ rules ABC = D and ABD = C must also hold. Consider the above property that for a given large itemset, if a rule with consequent c holds then so do rules with consequents that are subsets of c. This is similar to the property that if an 13 itemset is large then so are all its subsets. From a large itemset l, therefore, we rst generate all rules with one item in the consequent. We then use the consequents of these rules and the function apriori-gen in Section 2.1.1 to generate all possible consequents with two items that can appear in a rule generated from l, etc. An algorithm using this idea is given below. The rules having one-item consequents in step 2 of this algorithm can be found by using a modi ed version of the preceding genrules function in which steps 8 and 9 are deleted to avoid the recursive call.
Faster Algorithm 1 forall large k-itemsets lk , k 2 do begin 2 H1 = f consequents of rules derived from lk with one item in the consequent g; 3 call ap-genruleslk , H1; 4 end procedure ap-genruleslk: large k-itemset, Hm: set of m-item consequents if k m + 1 then begin Hm+1 = apriori-genHm ; forall hm+1 2 Hm+1 do begin conf = supportlk supportlk , hm+1 ; if conf minconf then output the rule lk , hm+1 = hm+1 with con dence = conf and support = supportlk ; else delete hm+1 from Hm+1; end call ap-genruleslk , Hm+1; end
As an example of the advantage of this algorithm, consider a large itemset ABCDE . Assume that ACDE = B and ABCE = D are the only one-item consequent rules derived from this itemset that have the minimum con dence. If we use the simple algorithm, the recursive call genrulesABCDE , ACDE will test if the two-item consequent rules ACD = BE , ADE = BC , CDE = BA, and ACE = BD hold. The rst of these rules cannot hold, because E BE , and ABCD = E does not have minimum con dence. The second and third rules cannot hold for similar reasons. The call genrulesABCDE , ABCE will test if the rules ABC = DE , ABE = DC , BCE = DA and ACE = BD hold, and will nd that the rst three of these rules do not hold. In fact, the only two-item consequent rule that can possibly hold is ACE = BD, where B and D are the consequents in the valid one-item consequent rules. This is the only rule that will be tested by the faster algorithm. 4 Performance
To assess the relative performance of the algorithms for discovering large itemsets, we performed several experiments on an IBM RS 6000 530H workstation with a CPU clock rate of 33 MHz, 64 14 MB of main memory, and running AIX 3.2. The data resided in the AIX le system and was stored on a 2GB SCSI 3.5" drive, with measured sequential throughput of about 2 MB second. We rst give an overview of the AIS AIS93b and SETM HS93 algorithms against which we compare the performance of the Apriori and AprioriTid algorithms. We then describe the synthetic datasets used in the performance evaluation and show the performance results. Next, we show the performance results for three real-life datasets obtained from a retail and a direct mail company. Finally, we describe how the best performance features of Apriori and AprioriTid can be combined into an AprioriHybrid algorithm and demonstrate its scale-up properties. 4.1 The AIS Algorithm
Figure 4 summarizes the essence of the AIS algorithm see AIS93b for further details. Candidate itemsets are generated and counted on-the- y as the database is scanned. After reading a transaction, it is determined which of the itemsets that were found to be large in the previous pass are contained in this transaction step 5. New candidate itemsets are generated by extending these large itemsets with other items in the transaction step 7. A large itemset l is extended with only those items that are large and occur later in the lexicographic ordering of items than any of the items in l. The candidates generated from a transaction are added to the set of candidate itemsets maintained for the pass, or the counts of the corresponding entries are increased if they were created by an earlier transaction step 9.
1 L1 = flarge 1-itemsetsg; 2 for k = 2; Lk,1 6= ;; k++ do begin 3 Ck = ;; 4 forall transactions t 2 D do begin 5 Lt = subsetLk,1 , t; Large itemsets contained in t 6 forall large itemsets lt 2 Lt do begin 7 Ct = 1-extensions of lt contained in t; Candidates contained in t 8 forall candidates c 2 Ct do 9 if c 2 Ck then add 1 to the count of c in the corresponding entry in Ck add c to Ck with a count of 1; 10 end 11 Lk = fc 2 Ck j c:count minsupg 12 end S 13 Answer = k Lk ; else Figure 4: Algorithm AIS Data Structures The data structures required for maintaining large and candidate itemsets were not speci ed in AIS93b . We store the large itemsets in a dynamic multi-level hash table to make the subset operation in step 5 fast, using the algorithm described in Section 2.1.2. Candidate 15 itemsets are kept in a hash table associated with the respective large itemsets from which they originate in order to make the membership test in step 9 fast. Bu er Management When a newly generated candidate itemset causes the bu er to over ow,
we discard from memory the corresponding large itemset and all candidate itemsets generated from it. This reclamation procedure is executed as often as necessary during a pass. The large itemsets discarded in a pass are extended in the next pass. This technique is a simpli ed version of the bu er management scheme presented in AIS93b . 4.2 The SETM Algorithm
The SETM algorithm HS93 was motivated by the desire to use SQL to compute large itemsets. Our description of this algorithm in Figure 5 uses the same notation as used for the other algorithms, but is functionally identical to the SETM algorithm presented in HS93 . C k Lk in Figure 5 represents the set of candidate large itemsets in which the TIDs of the generating transactions have been associated with the itemsets. Each member of these sets is of the form TID; itemset . Like AIS, the SETM algorithm also generates candidates on-the- y based on transactions read from the database. It thus generates and counts every candidate itemset that the AIS algorithm generates. However, to use the standard SQL join operation for candidate generation, SETM separates candidate generation from counting. It saves a copy of the candidate itemset together with the TID of the generating transaction in a sequential structure step 9. At the end of the pass, the support count of candidate itemsets is determined by sorting step 12 and aggregating this sequential structure step 13. SETM remembers the TIDs of the generating transactions with the candidate itemsets. To avoid needing a subset operation, it uses this information to determine the large itemsets contained in the transaction read step 6. Lk C k and is obtained by deleting those candidates that do not have minimum support step 13. Assuming that the database is sorted in TID order, SETM can easily nd the large itemsets contained in a transaction in the next pass by sorting Lk on TID step 15. In fact, it needs to visit every member of Lk only once in the TID order, and the candidate generation in steps 5 through 11 can be performed using the relational merge-join operation HS93 . The disadvantage of this approach is mainly due to the size of candidate sets C k . For each candidate itemset, the candidate set now has as many entries as the number of transactions in which the candidate itemset is present. Moreover, when we are ready to count the support for candidate itemsets at the end of the pass, C k is in the wrong order and needs to be sorted on itemsets step 12. After counting and pruning out small candidate itemsets that do not have minimum support, the resulting set Lk needs another sort on TID step 15 before it can be used 16 for generating candidates in the next pass.
1 L1 = flarge 1-itemsetsg; 2 L1 = fLarge 1-itemsets together with the TIDs in which they appear, sorted on TIDg; 3 for k = 2; Lk,1 6= ;; k++ do begin 4 C k = ;; 5 forall transactions t 2 D do begin 6 Lt = fl 2 Lk,1 j l:TID = t:TIDg; Large k , 1-itemsets contained in t 7 forall large itemsets lt 2 Lt do begin 8 Ct = 1-extensions of lt contained in t; Candidates in t 9 C k += f t:TID; c j c 2 Ct g; 10 end 11 end 12 sort C k on itemsets; 13 delete all itemsets c 2 C k for which c.count minsup giving Lk ; 14 Lk = f l.itemset, count of l in Lk j l 2 Lk g; Combined with step 13 15 sort Lk on TID; 16 end S 17 Answer = k Lk ; . Figure 5: Algorithm SETM of the set C k relative to the size of memory. If C k ts in memory, the two sorting steps can be performed using an in-memory sort. In HS93 , C k was assumed to t in main memory and bu er management was not discussed. If C k is too large to t in memory, we write the entries in C k to disk in FIFO order when the bu er allocated to the candidate itemsets lls up, as these entries are not required until the end of the pass. However, C k now requires two external sorts. Bu er Management The performance of the SETM algorithm critically depends on the size 4.3 Generation of Synthetic Data
We generated synthetic transactions to evaluate the performance of the algorithms over a large range of data characteristics. These transactions mimic the transactions in the retailing environment. Our model of the real" world is that people tend to buy sets of items together. Each such set is potentially a maximal large itemset. An example of such a set might be sheets, pillow case, comforter, and ru es. However, some people may buy only some of the items from such a set. For instance, some people might buy only sheets and pillow case, and some only sheets. A transaction may contain more than one large itemset. For example, a customer might place an order for a dress and jacket when ordering sheets and pillow cases, where the dress and jacket together form another large itemset. Transaction sizes are typically clustered around a mean and a few transactions have many items. Typical sizes of large itemsets are also clustered around a mean, with a few large itemsets having a large number of items. 17 To create a dataset, our synthetic data generation program takes the parameters shown in Table 2. Table 2: Parameters jDj jT j jI j jLj
N Number of transactions Average size of the Transactions Average size of the maximal potentially large Itemsets Number of maximal potentially large itemsets Number of items We rst determine the size of the next transaction. The size is picked from a Poisson distribution with mean equal to jT j. Note that if each item is chosen with the same probability p, and there are N items, the expected number of items in a transaction is given by a binomial distribution with parameters N and p, and is approximated by a Poisson distribution with mean Np. We then assign items to the transaction. Each transaction is assigned a series of potentially large itemsets. If the large itemset on hand does not t in the transaction, the itemset is put in the transaction anyway in half the cases, and the itemset is moved to the next transaction the rest of the cases. Large itemsets are chosen from a set T of such itemsets. The number of itemsets in T is set to jLj. There is an inverse relationship between jLj and the average support for potentially large itemsets. An itemset in T is generated by rst picking the size of the itemset from a Poisson distribution with mean equal to jI j. Items in the rst itemset are chosen randomly. To model the phenomenon that large itemsets often have common items, some fraction of items in subsequent itemsets are chosen from the previous itemset generated. We use an exponentially distributed random variable with mean equal to the correlation level to decide this fraction for each itemset. The remaining items are picked at random. In the datasets used in the experiments, the correlation level was set to 0.5. We ran some experiments with the correlation level set to 0.25 and 0.75 but did not nd much di erence in the nature of our performance results. Each itemset in T has a weight associated with it, which corresponds to the probability that this itemset will be picked. This weight is picked from an exponential distribution with unit mean, and is then normalized so that the sum of the weights for all the itemsets in T is 1. The next itemset to be put in the transaction is chosen from T by tossing an jLj-sided weighted coin, where the weight for a side is the probability of picking the associated itemset. To model the phenomenon that all the items in a large itemset are not always bought together, we assign each itemset in T a corruption level c. When adding an itemset to a transaction, we keep dropping an item from the itemset as long as a uniformly distributed random number between 0 and 1 is less than c. Thus for an itemset of size l, we will add l items to the transaction 1 , c of 18 the time, l , 1 items c1 , c of the time, l , 2 items c21 , c of the time, etc. The corruption level for an itemset is xed and is obtained from a normal distribution with mean 0.5 and variance 0.1. We generated datasets by setting N = 1000 and jLj = 2000. We chose 3 values for jT j: 5, 10, and 20. We also chose 3 values for jI j: 2, 4, and 6. The number of transactions was to set to 100,000 because, as we will see in Section 4.4, SETM could not be run for larger values. However, for our scale-up experiments, we generated datasets with up to 10 million transactions 838MB for jT j = 20. Table 3 summarizes the dataset parameter settings. For the same jT j and jDj values, the size of datasets in megabytes were roughly equal for the di erent values of jI j. Table 3: Parameter settings Synthetic datasets
Name T5.I2.D100K T10.I2.D100K T10.I4.D100K T20.I2.D100K T20.I4.D100K T20.I6.D100K jT j jI j
5 10 10 20 20 20 2 2 4 2 4 6 jDj Size in Megabytes 100K 2.4 100K 4.4 100K 100K 8.4 100K 100K 4.4 Experiments with Synthetic Data
Figure 6 shows the execution times for the six synthetic datasets given in Table 3 for decreasing values of minimum support. As the minimum support decreases, the execution times of all the algorithms increase because of increases in the total number of candidate and large itemsets. For SETM, we have only plotted the execution times for the dataset T5.I2.D100K in Figure 6. The execution times for SETM for the two datasets with an average transaction size of 10 are given in Table 4. We did not plot the execution times in Table 4 on the corresponding graphs because they are too large compared to the execution times of the other algorithms. For the three datasets with transaction sizes of 20, SETM took too long to execute and we aborted those runs as the trends were clear. Clearly, Apriori beats SETM by more than an order of magnitude for large datasets. Table 4: Execution times in seconds for SETM
Dataset Algorithm T10.I2.D100K SETM Apriori T10.I4.D100K SETM Apriori Minimum Support 2.0 1.5 1.0 0.75 0.5 74 161 838 1262 1878 4.4 5.3 11.0 14.5 15.3 41 91 659 929 1639 3.8 4.8 11.2 17.4 19.3 19 T5.I2.D100K
80 70 60
Time (sec) T10.I2.D100K
160 SETM AIS AprioriTid Apriori 140 120
Time (sec) AIS AprioriTid Apriori 50 40 30 20 10 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 100 80 60 40 20 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 T10.I4.D100K
350 300 250
Time (sec) T20.I2.D100K
1000 AIS AprioriTid Apriori 900 800 700
Time (sec) AIS AprioriTid Apriori 200 150 100 50 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 600 500 400 300 200 100 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 T20.I4.D100K
1800 1600 1400 2500 1200
Time (sec) Time (sec) T20.I6.D100K
3500 AIS AprioriTid Apriori 3000 AIS AprioriTid Apriori 1000 800 600 2000 1500 1000 400 200 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 500 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 Figure 6: Execution times: Synthetic Data 20 Apriori beats AIS for all problem sizes, by factors ranging from 2 for high minimum support to more than an order of magnitude for low levels of support. AIS always did considerably better than SETM. For small problems, AprioriTid did about as well as Apriori, but performance degraded to about twice as slow for large problems. 4.5 Explanation of the Relative Performance
To explain these performance trends, we show in Figure 7 the sizes of the large and candidate sets in di erent passes for the T10.I4.D100K dataset for the minimum support of 0.75. Note that the Y-axis in this graph has a log scale.
Number of Itemsets 100000 10000 1000 100 10 1 1 2 3 C-bar-k (SETM) C-bar-k (AprioriTid) C-k (AIS, SETM) C-k (Apriori, AprioriTid) L-k Figure 7: Sizes of the large and candidate sets T10.I4.D100K, minsup = 0.75 The fundamental problem with the SETM algorithm is the size of its C k sets. Recall that P the size of the set C k is given by candidate itemsets c support-countc. Thus, the sets C k are roughly S times bigger than the corresponding Ck sets, where S is the average support count of the candidate itemsets. Unless the problem size is very small, the C k sets have to be written to disk, and externally sorted twice, causing the SETM algorithm to perform poorly.2 This explains the jump in time for SETM in Table 4 when going from 1.5 support to 1.0 support for datasets with transaction size 10. The largest dataset in the scale-up experiments for SETM in HS93 was still small enough that C k could t in memory; hence they did not encounter this jump in execution time. Note that for the same minimum support, the support count for candidate itemsets increases linearly with the number of transactions. Thus, as we increase the number of transactions for the same values of jT j and jI j, though the size of Ck does not change, the size of C k goes up linearly. Thus, for datasets with more transactions, the performance gap between SETM and the other
The cost of external sorting in SETM can be reduced somewhat as follows. Before writing out entries in C k to disk, we can sort them on itemsets using an internal sorting procedure, and write them as sorted runs. These sorted runs can then be merged to obtain support counts. However, given the poor performance of SETM, we do not expect this optimization to a ect the algorithm choice.
2 4 5 Pass Number 6 7 21 algorithms will become even larger. The problem with AIS is that it generates too many candidates that later turn out to be small, causing it to waste too much e ort. Apriori also counts too many small sets in the second pass recall that C2 is really a cross-product of L1 with L1. However, this wastage decreases dramatically from the third pass onward. Note that for the example in Figure 7, after pass 3, almost every candidate itemset counted by Apriori turns out to be a large set. AprioriTid also has the problem of SETM that C k tends to be large. However, the apriori candidate generation used by AprioriTid generates signi cantly fewer candidates than the transactionbased candidate generation used by SETM. As a result, the C k of AprioriTid has fewer entries than that of SETM. AprioriTid is also able to use a single word ID to store a candidate rather than requiring as many words as the number of items in the candidate.3 In addition, unlike SETM, AprioriTid does not have to sort C k . Thus, AprioriTid does not su er as much as SETM from maintaining C k . AprioriTid has the nice feature that it replaces a pass over the original dataset by a pass over the set C k . Hence, AprioriTid is very e ective in later passes when the size of C k becomes small compared to the size of the database. Thus, we nd that AprioriTid beats Apriori when its C k sets can t in memory and the distribution of the large itemsets has a long tail. When C k doesn't t in memory, there is a jump in the execution time for AprioriTid, such as when going from 0.75 to 0.5 for datasets with transaction size 10 in Figure 6. In this region, Apriori starts beating AprioriTid. 4.6 Reality Check
To con rm the relative performance trends we observed using synthetic data, we experimented with three real-life datasets: a sales transactions dataset obtained from a retail chain and two customerorder datasets obtained from a mail order company. We present the results of these experiments below. store over a short period of time. A transaction contains the names of the departments from which a customer bought a product in a visit to the store. There are a total of 63 items, representing departments. There are 46,873 transactions with an average size of 2.47. The size of the dataset is
3 Retail Sales Data The data from the retail chain consists of the sales transactions from one For SETM to use IDs, it would have to maintain two additional in-memory data structures: a hash table to nd out whether a candidate has been generated previously, and a mapping from the IDs to candidates. However, this would destroy the set-oriented nature of the algorithm. Also, once we have the hash table which gives us the IDs of candidates, we might as well count them at the same time and avoid the two external sorts. We experimented with this variant of SETM and found that, while it did better than SETM, it still performed much worse than Apriori or AprioriTid. 22 very small, only 0.65MB. Some performance results for this dataset were reported in HS93 . Figure 8 shows the execution times of the four algorithms.4 The C k sets for both SETM and AprioriTid t in memory for this dataset. Apriori and AprioriTid are roughly three times as fast as AIS and four times faster than SETM.
9 8 7 6
Time (sec) SETM AIS AprioriTid Apriori 5 4 3 2 1 0 2 1.5 1 0.75 0.5 0.25 Minimum Support 0.1 Figure 8: Execution times: Retail sales data items ordered by a customer in a single mail order. There are a total of 15836 items. The average size of a transaction is 2.62 items and there are a total of 2.9 million transactions. The size of this dataset is 42 MB. A transaction in the second dataset consists of all the items ordered by a customer from the company in all orders together. Again, there are a total of 15836 items, but the average size of a transaction is now 31 items and there are a total of 213,972 transactions. The size of this dataset is 27 MB. We will refer to these datasets as M.order and M.cust respectively. The execution times for these two datasets are shown in Figures 9 and 10 respectively. For both datasets, AprioriTid is initially comparable to Apriori but becomes up to twice as slow for lower supports. For M.order, Apriori outperforms AIS by a factor of 2 to 6 and beats SETM by a factor of about 15. For M.cust, Apriori beats AIS by a factor of 3 to 30. SETM had to be aborted after taking 20 times the time Apriori took to complete because, even for 2 support, the set C 2 became larger than the disk capacity.
4 The execution times for SETM in this gure are a little higher compared to those reported in HS93 . The timings in HS93 were obtained on a RS 6000 350 processor, whereas our experiments have been run on a slower RS 6000 530H processor. The execution time for 1 support for AIS is lower than that reported in AIS93b because of improvements in the data structures for storing large and candidate itemsets. Mail Order data A transaction in the rst dataset from the mail order company consists of 23 5000 4500 4000 3500
Time (sec) 18000 SETM AIS AprioriTid Apriori
Time (sec) 16000 14000 12000 10000 8000 6000 4000 2000 0 2 1.5 1 0.75 0.5 Minimum Support AIS AprioriTid Apriori 3000 2500 2000 1500 1000 500 0 0.1 0.05 0.025 Minimum Support 0.01 0.25 Figure 9: Execution times: M.order Figure 10: Execution times: M.cust 4.7 Algorithm AprioriHybrid
It is not necessary to use the same algorithm in all the passes over data. Figure 11 shows the execution times for Apriori and AprioriTid for di erent passes over the dataset T10.I4.D100K. In the earlier passes, Apriori does better than AprioriTid. However, AprioriTid beats Apriori in later passes. We observed similar relative behavior for the other datasets, the reason for which is as follows. Apriori and AprioriTid use the same candidate generation procedure and therefore count the same itemsets. In the later passes, the number of candidate itemsets reduces see the size of Ck for Apriori and AprioriTid in Figure 7. However, Apriori still examines every transaction in the database. On the other hand, rather than scanning the database, AprioriTid scans C k for obtaining support counts, and the size of C k has become smaller than the size of the database. When the C k sets can t in memory, we do not even incur the cost of writing them to disk.
14 12 10
Time (sec) Apriori AprioriTid 8 6 4 2 0 1 2 3 Figure 11: Per pass execution times of Apriori and AprioriTid T10.I4.D100K, minsup = 0.75 Based on these observations, we can design a hybrid algorithm, which we call AprioriHybrid, 24 4 Pass # 5 6 7 that uses Apriori in the initial passes and switches to AprioriTid when it expects that the set C k at the end of the pass will t in memory. We use the following heuristic to estimate if C k would t in memory in the next pass. At the end of the current pass, we have the counts of the candidates in Ck . From this, we estimate what the size of C k would have been if it had been generated. This P size, in words, is candidates c 2 Ck supportc + number of transactions. If C k in this pass was small enough to t in memory, and there were fewer large candidates in the current pass than the previous pass, we switch to AprioriTid. The latter condition is added to avoid switching when C k in the current pass ts in memory but C k in the next pass may not.5 Switching from Apriori to AprioriTid does involve a cost. Assume that we decide to switch from Apriori to AprioriTid at the end of the kth pass. In the k +1th pass, after nding the candidate itemsets contained in a transaction, we will also have to add their IDs to C k+1 see the description of AprioriTid in Section 2.2. Thus there is an extra cost incurred in this pass relative to just running Apriori. It is only in the k +2th pass that we actually start running AprioriTid. Thus, if there are no large k +1-itemsets, or no k +2-candidates, we will incur the cost of switching without getting any of the savings of using AprioriTid. Figure 12 shows the performance of AprioriHybrid relative to Apriori and AprioriTid for large datasets. AprioriHybrid performs better than Apriori in almost all cases. For T10.I2.D100K with 1.5 support, AprioriHybrid does a little worse than Apriori since the pass in which the switch occurred was the last pass; AprioriHybrid thus incurred the cost of switching without realizing the bene ts. In general, the advantage of AprioriHybrid over Apriori depends on how the size of the C k set decline in the later passes. If C k remains large until nearly the end and then has an abrupt drop, we will not gain much by using AprioriHybrid since we can use AprioriTid only for a short period of time after the switch. This is what happened with the M.cust and T20.I6.D100K datasets. On the other hand, if there is a gradual decline in the size of C k , AprioriTid can be used for a while after the switch, and a signi cant improvement can be obtained in the execution time. 4.8 Scale-up Experiment
Figure 13 shows how AprioriHybrid scales up as the number of transactions is increased from 100,000 to 10 million transactions. We used the combinations T5.I2, T10.I4, and T20.I6 for the average sizes of transactions and itemsets respectively. All other parameters were the same as for the data in Table 3. The sizes of these datasets for 10 million transactions were 239MB, 439MB and 838MB respectively. The minimum support level was set to 0.75. The execution times are normalized with respect to the times for the 100,000 transaction datasets in the rst graph and
Other heuristics are also possible. For example, in a system with multiple disks, it may be faster to switch to AprioriTid as soon as the size of C k is less than the size of the database.
5 25 M.order
700 600 500
Time (sec) M.cust
1100 AprioriTid Apriori AprioriHybrid 1000 900 800
Time (sec) AprioriTid Apriori AprioriHybrid 700 600 500 400 300 200 100 400 300 200 100 0 0.1 0 0.05 0.025 Minimum Support 0.01 2 1.5 1 0.75 0.5 Minimum Support 0.25 T10.I2.D100K
40 35 30
Time (sec) Time (sec) T10.I4.D100K
55 AprioriTid Apriori AprioriHybrid 50 45 40 35 30 25 20 15 10 AprioriTid Apriori AprioriHybrid 25 20 15 10 5 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 5 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 T20.I4.D100K
700 200 AprioriTid Apriori AprioriHybrid 600 500
Time (sec) T20.I6.D100K
AprioriTid Apriori AprioriHybrid 150
Time (sec) 400 300 200 100 50 100 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 0 2 1.5 1 0.75 0.5 Minimum Support 0.33 0.25 Figure 12: Execution times: AprioriHybrid Algorithm 26 with respect to the 1 million transaction dataset in the second. As shown, the execution times scale quite linearly.
12 10 8 6 4 2 0 100 T20.I6 T10.I4 T5.I2 14 12 10
Relative Time Relative Time T20.I6 T10.I4 T5.I2 8 6 4 2 0 250 500 750 Number of Transactions (in '000s) 1000 1 Figure 13: Number of transactions scale-up
30 T20.I6 T10.I4 T5.I2 25 20
Time (sec) 2.5 5 7.5 Number of Transactions (in Millions) 10 45 40 35 30
Time (sec) 500 750 1000 25 20 15 10 15 10 5 5 0 1000 0 2500 5000 7500 Number of Items 10000 5 10 20 30 Transaction Size 40 50 Figure 14: Number of items scale-up Figure 15: Transaction size scale-up Next, we examined how AprioriHybrid scaled up with the number of items. We increased the number of items from 1000 to 10,000 for the three parameter settings T5.I2.D100K, T10.I4.D100K and T20.I6.D100K. All other parameters were the same as for the data in Table 3. We ran experiments for a minimum support at 0.75, and obtained the results shown in Figure 14. The execution times decreased a little since the average support for an item decreased as we increased the number of items. This resulted in fewer large itemsets and, hence, faster execution times. Finally, we investigated the scale-up as we increased the average transaction size. The aim of this experiment was to see how our data structures scaled with the transaction size, independent of other factors like the physical database size and the number of large itemsets. We kept the 27 physical size of the database roughly constant by keeping the product of the average transaction size and the number of transactions constant. The number of transactions ranged from 200,000 for the database with an average transaction size of 5 to 20,000 for the database with an average transaction size 50. Fixing the minimum support as a percentage would have led to large increases in the number of large itemsets as the transaction size increased, since the probability of a itemset being present in a transaction is roughly proportional to the transaction size. We therefore xed the minimum support level in terms of the number of transactions. The results are shown in Figure 15. The numbers in the key e.g. 500 refer to this minimum support. As shown, the execution times increase with the transaction size, but only gradually. The main reason for the increase was that in spite of setting the minimum support in terms of the number of transactions, the number of large itemsets increased with increasing transaction length. A secondary reason was that nding the candidates present in a transaction took a little more time. 5 Conclusions and Future Work
We presented two new algorithms, Apriori and AprioriTid, for discovering all signi cant association rules between items in a large database of transactions. We compared these algorithms to the previously known algorithms, the AIS AIS93b and SETM HS93 algorithms. We presented experimental results, using both synthetic and real-life data, showing that the proposed algorithms always outperform AIS and SETM. The performance gap increased with the problem size, and ranged from a factor of three for small problems to more than an order of magnitude for large problems. We showed how the best features of the two proposed algorithms can be combined into a hybrid algorithm, called AprioriHybrid, which then becomes the algorithm of choice for this problem. Scale-up experiments showed that AprioriHybrid scales linearly with the number of transactions. In addition, the execution time decreases a little as the number of items in the database increases. As the average transaction size increases while keeping the database size constant, the execution time increases only gradually. These experiments demonstrate the feasibility of using AprioriHybrid in real applications involving very large databases. The algorithms presented in this paper have been implemented on several data repositories, including the AIX le system, DB2 MVS, and DB2 6000. In the future, we plan to extend this work along the following dimensions: Multiple taxonomies is-a hierarchies over items are often available. An example of such a hierarchy is that a dish washer is a kitchen appliance is a heavy electric appliance, etc. We would like to be able to nd association rules that use such hierarchies. 28 We did not consider the quantities of the items bought in a transaction, which are useful for some applications. Finding such rules needs further work. The work reported in this paper has been done in the context of the Quest project at the IBM Almaden Research Center. In Quest, we are exploring the various aspects of the database mining problem. Besides the problem of discovering association rules, some other problems that we have looked into include the enhancement of the database capability with classi cation queries AGI+ 92 and similarity queries over time sequences AFS93 . We believe that database mining is an important new application area for databases, combining commercial interest with intriguing research questions. Acknowledgment We wish to thank Mike Carey for his insightful comments and suggestions. References
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