Advanced SearchTechniques for Large Scale Data Analytics Pavel Zezula and Jan Sedmidubsky Masaryk University http://disa.fi.muni.cz Supermarket shelf management – Market-basket model:  Goal: Identify items that are bought together by sufficiently many customers  Approach: Process the sales data collected with barcode scanners to find dependencies among items  A classic rule:  If someone buys diaper and milk, then he/she is likely to buy beer  Don’t be surprised if you find six-packs next to diapers! Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 2  A large set of items  e.g., things sold in a supermarket  A large set of baskets  Each basket is a small subset of items  e.g., the things one customer buys on one day  Want to discover association rules  People who bought {x,y,z} tend to buy {v,w}  Amazon! Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 3 Rules Discovered: {Milk} --> {Coke} {Diaper, Milk} --> {Beer} TID Items 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4 Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk Input: Output:  Items = products; Baskets = sets of products someone bought in one trip to the store  Real market baskets: Chain stores keep TBs of data about what customers buy together  Tells how typical customers navigate stores, lets them position tempting items  Suggests tie-in “tricks”, e.g., run sale on diapers and raise the price of beer  Need the rule to occur frequently, or no $$’s  Amazon’s people who bought X also bought Y Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 4  Baskets = sentences; Items = documents containing those sentences  Items that appear together too often could represent plagiarism  Notice items do not have to be “in” baskets  Baskets = patients; Items = drugs & side-effects  Has been used to detect combinations of drugs that result in particular side-effects  But requires extension: Absence of an item needs to be observed as well as presence Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 5  A general many-to-many mapping (association) between two kinds of things  But we ask about connections among “items”, not “baskets”  For example:  Finding communities in graphs (e.g., Twitter) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 6  Finding communities in graphs (e.g., Twitter)  Baskets = nodes; Items = outgoing neighbors  Searching for complete bipartite subgraphs Ks,t of a big graph Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 7  How?  View each node i as a basket Bi of nodes i it points to  Ks,t = a set Y of size t that occurs in s buckets Bi  Looking for Ks,t  set of support s and look at layer t – all frequent sets of size t … … A dense 2-layer graph snodes tnodes  Simplest question: Find sets of items that appear together “frequently” in baskets Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 8 TID Items 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4 Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk Support of {Beer, Bread} = 2  Simplest question: Find sets of items that appear together “frequently” in baskets  Support for itemset I: Number of baskets containing all items in I  (Often expressed as a fraction of the total number of baskets) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 8 TID Items 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4 Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk Support of {Beer, Bread} = 2  Simplest question: Find sets of items that appear together “frequently” in baskets  Support for itemset I: Number of baskets containing all items in I  (Often expressed as a fraction of the total number of baskets)  Given a support threshold s, then sets of items that appear in at least s baskets are called frequent itemsets Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 8 TID Items 1 Bread, Coke, Milk 2 Beer, Bread 3 Beer, Coke, Diaper, Milk 4 Beer, Bread, Diaper, Milk 5 Coke, Diaper, Milk Support of {Beer, Bread} = 2  Items = {milk, coke, pepsi, beer, juice}  Support threshold = 3 baskets B1 = {m, c, b} B2 = {m, p, j} B3 = {m, b} B4 = {c, j} B5 = {m, p, b} B6 = {m, c, b, j} B7 = {c, b, j} B8 = {b, c}  Frequent itemsets: {m}, {c}, {b}, {j}, Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 9  Items = {milk, coke, pepsi, beer, juice}  Support threshold = 3 baskets B1 = {m, c, b} B2 = {m, p, j} B3 = {m, b} B4 = {c, j} B5 = {m, p, b} B6 = {m, c, b, j} B7 = {c, b, j} B8 = {b, c}  Frequent itemsets: {m}, {c}, {b}, {j}, Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 9 {m,b}  Items = {milk, coke, pepsi, beer, juice}  Support threshold = 3 baskets B1 = {m, c, b} B2 = {m, p, j} B3 = {m, b} B4 = {c, j} B5 = {m, p, b} B6 = {m, c, b, j} B7 = {c, b, j} B8 = {b, c}  Frequent itemsets: {m}, {c}, {b}, {j}, Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 9 , {b,c}{m,b}  Items = {milk, coke, pepsi, beer, juice}  Support threshold = 3 baskets B1 = {m, c, b} B2 = {m, p, j} B3 = {m, b} B4 = {c, j} B5 = {m, p, b} B6 = {m, c, b, j} B7 = {c, b, j} B8 = {b, c}  Frequent itemsets: {m}, {c}, {b}, {j}, Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 9 , {b,c} , {c,j}.{m,b} 10  Association Rules: If-then rules about the contents of baskets  {i1, i2,…,ik} → j means: “if a basket contains all of i1,…,ik then it is likely to contain j”  In practice there are many rules, want to find significant/interesting ones!  Confidence of this association rule is the probability of j given I = {i1,…,ik} )support( )support( )conf( I jI jI   Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Not all high-confidence rules are interesting  The rule X → milk may have high confidence for many itemsets X, because milk is just purchased very often (independent of X) and the confidence will be high  Interest of an association rule I → j: difference between its confidence and the fraction of baskets that contain j  Interesting rules are those with high positive or negative interest values (usually above 0.5) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 11 ]Pr[)conf()Interest( jjIjI  B1 = {m, c, b} B2 = {m, p, j} B3 = {m, b} B4= {c, j} B5 = {m, p, b} B6 = {m, c, b, j} B7 = {c, b, j} B8 = {b, c}  Association rule: {m, b} →c  Confidence = 2/4 = 0.5  Interest = |0.5 – 5/8| = 1/8  Item c appears in 5/8 of the baskets  Rule is not very interesting! Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 12  Problem: Find all association rules with support ≥s and confidence ≥c  Note: Support of an association rule is the support of the set of items on the left side Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 13 )support( )support( )conf( I jI jI    Problem: Find all association rules with support ≥s and confidence ≥c  Note: Support of an association rule is the support of the set of items on the left side  Hard part: Finding the frequent itemsets!  If {i1, i2,…, ik} → j has high support and confidence, then both {i1, i2,…, ik} and {i1, i2,…,ik, j} will be “frequent” Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 13 )support( )support( )conf( I jI jI    Step 1: Find all frequent itemsets I  (we will explain this next)  Step 2: Rule generation  For every subset A of I, generate a rule A → I \ A  Since I is frequent, A is also frequent  Variant 1: Single pass to compute the rule confidence  confidence(A,B→C,D) = support(A,B,C,D) / support(A,B)  Variant 2:  Observation: If A,B,C→D is below confidence, so is A,B→C,D  Can generate “bigger” rules from smaller ones!  Output the rules above the confidence threshold Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 14 B1 = {m, c, b} B2 = {m, p, j} B3 = {m, c, b, n} B4= {c, j} B5 = {m, p, b} B6 = {m, c, b, j} B7 = {c, b, j} B8 = {b, c}  Support threshold s = 3, confidence c = 0.75  1) Frequent itemsets:  {b,m} {b,c} {c,m} {c,j} {m,c,b} Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 15 B1 = {m, c, b} B2 = {m, p, j} B3 = {m, c, b, n} B4= {c, j} B5 = {m, p, b} B6 = {m, c, b, j} B7 = {c, b, j} B8 = {b, c}  Support threshold s = 3, confidence c = 0.75  1) Frequent itemsets:  {b,m} {b,c} {c,m} {c,j} {m,c,b}  2) Generate rules:  b→m: c=4/6 b→c: c=5/6 b,c→m: c=3/5  m→b: c=4/5 … b,m→c: c=3/4  b→c,m: c=3/6 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 15 B1 = {m, c, b} B2 = {m, p, j} B3 = {m, c, b, n} B4= {c, j} B5 = {m, p, b} B6 = {m, c, b, j} B7 = {c, b, j} B8 = {b, c}  Support threshold s = 3, confidence c = 0.75  1) Frequent itemsets:  {b,m} {b,c} {c,m} {c,j} {m,c,b}  2) Generate rules:  b→m: c=4/6 b→c: c=5/6 b,c→m: c=3/5  m→b: c=4/5 … b,m→c: c=3/4  b→c,m: c=3/6 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 15  To reduce the number of rules we can post-process them and only output:  Maximal frequent itemsets: No immediate superset is frequent  Gives more pruning or  Closed itemsets: No immediate superset has the same count (> 0)  Stores not only frequent information, but exact counts Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 16 Support Maximal(s=3) Closed A 4 No No B 5 No Yes C 3 No No AB 4 Yes Yes AC 2 No No BC 3 Yes Yes ABC 2 No Yes Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 17 Support Maximal(s=3) Closed A 4 No No B 5 No Yes C 3 No No AB 4 Yes Yes AC 2 No No BC 3 Yes Yes ABC 2 No Yes Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 17 Frequent, but superset BC also frequent. Support Maximal(s=3) Closed A 4 No No B 5 No Yes C 3 No No AB 4 Yes Yes AC 2 No No BC 3 Yes Yes ABC 2 No Yes Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 17 Frequent, but superset BC also frequent. Frequent, and its only superset, ABC, not freq. Support Maximal(s=3) Closed A 4 No No B 5 No Yes C 3 No No AB 4 Yes Yes AC 2 No No BC 3 Yes Yes ABC 2 No Yes Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 17 Frequent, but superset BC also frequent. Frequent, and its only superset, ABC, not freq. Superset BC has same count. Support Maximal(s=3) Closed A 4 No No B 5 No Yes C 3 No No AB 4 Yes Yes AC 2 No No BC 3 Yes Yes ABC 2 No Yes Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 17 Frequent, but superset BC also frequent. Frequent, and its only superset, ABC, not freq. Superset BC has same count. Its only superset, ABC, has smaller count.  Back to finding frequent itemsets  Typically, data is kept in flat files rather than in a database system:  Stored on disk  Stored basket-by-basket  Baskets are small but we have many baskets and many items  Expand baskets into pairs, triples, etc. as you read baskets  Use k nested loops to generate all sets of size k Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 19 Item Item Item Item Item Item Item Item Item Item Item Item Etc. Items are positive integers, and boundaries between baskets are –1. Note: We want to find frequent itemsets. To find them, we have to count them. To count them, we have to generate them. 20  The true cost of mining disk-resident data is usually the number of disk I/Os  In practice, association-rule algorithms read the data in passes – all baskets read in turn  We measure the cost by the number of passes an algorithm makes over the data Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 21  For many frequent-itemset algorithms, main-memory is the critical resource  As we read baskets, we need to count something, e.g., occurrences of pairs of items  The number of different things we can count is limited by main memory  Swapping counts in/out is a disaster (why?) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  The hardest problem often turns out to be finding the frequent pairs of items {i1, i2}  Why? Freq. pairs are common, freq. triples are rare  Why? Probability of being frequent drops exponentially with size; number of sets grows more slowly with size Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 22  The hardest problem often turns out to be finding the frequent pairs of items {i1, i2}  Why? Freq. pairs are common, freq. triples are rare  Why? Probability of being frequent drops exponentially with size; number of sets grows more slowly with size  Let’s first concentrate on pairs, then extend to larger sets  The approach:  We always need to generate all the itemsets  But we would only like to count (keep track) of those itemsets that in the end turn out to be frequent Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 22  Naïve approach to finding frequent pairs  Read file once, counting in main memory the occurrences of each pair:  From each basket of n items, generate its n(n-1)/2 pairs by two nested loops Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 23  Naïve approach to finding frequent pairs  Read file once, counting in main memory the occurrences of each pair:  From each basket of n items, generate its n(n-1)/2 pairs by two nested loops  Fails if (#items)2 exceeds main memory  Remember: #items can be 100K (Wal-Mart) or 10B (Web pages)  Suppose 105 items, counts are 4-byte integers  Number of pairs of items: 105(105-1)/2 = 5*109  Therefore, 2*1010 (20 gigabytes) of memory needed Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 23 Two approaches:  Approach 1: Count all pairs using a matrix  Approach 2: Keep a table of triples [i, j, c] = “the count of the pair of items {i, j} is c.”  If integers and item ids are 4 bytes, we need approximately 12 bytes for pairs with count > 0  Plus some additional overhead for the hashtable Note:  Approach 1 only requires 4 bytes per pair  Approach 2 uses 12 bytes per pair (but only for pairs with count > 0) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 24 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 25 4 bytes per pair Triangular Matrix Triples 12 per occurring pair  Approach 1: Triangular Matrix  n = total number items  Count pair of items {i, j} only if i 0)  Beats Approach 1 if less than 1/3 of possible pairs actually occur Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 26  Approach 1: Triangular Matrix  n = total number items  Count pair of items {i, j} only if i 0)  Beats Approach 1 if less than 1/3 of possible pairs actually occur Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 27 Problem is if we have too many items so the pairs do not fit into memory. Can we do better?  A two-pass approach called A-Priori limits the need for main memory  Key idea: monotonicity  If a set of items I appears at least s times, so does every subset J of I  Contrapositive for pairs: If item i does not appear in s baskets, then no pair including i can appear in s baskets  So, how does A-Priori find freq. pairs? Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 29  Pass 1: Read baskets and count in main memory the occurrences of each individual item  Requires only memory proportional to #items  Items that appear ≥ 𝒔 times are the frequent items  Pass 2: Read baskets again and count in main memory only those pairs where both elements are frequent (from Pass 1)  Requires memory proportional to square of frequent items only (for counts)  Plus a list of the frequent items (so you know what must be counted) Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 30 31 Item counts Pass 1 Pass 2 Frequent items Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Mainmemory Counts of pairs of frequent items (candidate pairs)  You can use the triangular matrix method with n = number of frequent items  May save space compared with storing triples  Trick: re-number frequent items 1,2,… and keep a table relating new numbers to original item numbers Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 32 Item counts Pass 1 Pass 2 Counts of pairs of frequent items Frequent items Old item #s Mainmemory Counts of pairs of frequent items 33  For each k, we construct two sets of k-tuples (sets of size k):  Ck = candidate k-tuples = those that might be frequent sets (support > s) based on information from the pass for k–1  Lk = the set of truly frequent k-tuples Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) C1 L1 C2 L2 C3 Filter Filter ConstructConstruct 33  For each k, we construct two sets of k-tuples (sets of size k):  Ck = candidate k-tuples = those that might be frequent sets (support > s) based on information from the pass for k–1  Lk = the set of truly frequent k-tuples Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) C1 L1 C2 L2 C3 Filter Filter ConstructConstruct All items 33  For each k, we construct two sets of k-tuples (sets of size k):  Ck = candidate k-tuples = those that might be frequent sets (support > s) based on information from the pass for k–1  Lk = the set of truly frequent k-tuples Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) C1 L1 C2 L2 C3 Filter Filter ConstructConstruct All items Count the items 33  For each k, we construct two sets of k-tuples (sets of size k):  Ck = candidate k-tuples = those that might be frequent sets (support > s) based on information from the pass for k–1  Lk = the set of truly frequent k-tuples Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) C1 L1 C2 L2 C3 Filter Filter ConstructConstruct All items All pairs of items from L1 Count the items 33  For each k, we construct two sets of k-tuples (sets of size k):  Ck = candidate k-tuples = those that might be frequent sets (support > s) based on information from the pass for k–1  Lk = the set of truly frequent k-tuples Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) C1 L1 C2 L2 C3 Filter Filter ConstructConstruct All items All pairs of items from L1 Count the pairs Count the items 33  For each k, we construct two sets of k-tuples (sets of size k):  Ck = candidate k-tuples = those that might be frequent sets (support > s) based on information from the pass for k–1  Lk = the set of truly frequent k-tuples Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) C1 L1 C2 L2 C3 Filter Filter ConstructConstruct All items All pairs of items from L1 Count the pairs To be explained Count the items  Hypothetical steps of the A-Priori algorithm  C1 = { {b} {c} {j} {m} {n} {p} }  Count the support of itemsets in C1  Prune non-frequent: L1 = { b, c, j, m }  Generate C2 = { {b,c} {b,j} {b,m} {c,j} {c,m} {j,m} }  Count the support of itemsets in C2  Prune non-frequent: L2 = { {b,m} {b,c} {c,m} {c,j} }  Generate C3 = { {b,c,m} {b,c,j} {b,m,j} {c,m,j} }  Count the support of itemsets in C3  Prune non-frequent: L3 = { {b,c,m} } Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 34 ** Note here we generate new candidates by generating Ck from Lk-1 and L1. But that one can be more careful with candidate generation. For example, in C3 we know {b,m,j} cannot be frequent since {m,j} is not frequent **  One pass for each k (itemset size)  Needs room in main memory to count each candidate k–tuple  For typical market-basket data and reasonable support (e.g., 1%), k = 2 requires the most memory  Many possible extensions:  Association rules with intervals:  For example: Men over 65 have 2 cars  Association rules when items are in a taxonomy  Bread, Butter → FruitJam  BakedGoods, MilkProduct → PreservedGoods  Lower the support s as itemset gets bigger Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 35  Observation: In pass 1 of A-Priori, most memory is idle  We store only individual item counts  Can we use the idle memory to reduce memory required in pass 2? Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 37  Observation: In pass 1 of A-Priori, most memory is idle  We store only individual item counts  Can we use the idle memory to reduce memory required in pass 2?  Pass 1 of PCY: In addition to item counts, maintain a hash table with as many buckets as fit in memory  Keep a count for each bucket into which pairs of items are hashed  For each bucket just keep the count, not the actual pairs that hash to the bucket! Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 37 FOR (each basket) : FOR (each item in the basket) : add 1 to item’s count; FOR (each pair of items) : hash the pair to a bucket; add 1 to the count for that bucket;  Few things to note:  Pairs of items need to be generated from the input file; they are not present in the file  We are not just interested in the presence of a pair, but we need to see whether it is present at least s (support) times Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 38 New in PCY  Observation: If a bucket contains a frequent pair, then the bucket is surely frequent  However, even without any frequent pair, a bucket can still be frequent   So, we cannot use the hash to eliminate any member (pair) of a “frequent” bucket  But, for a bucket with total count less than s, none of its pairs can be frequent   Pairs that hash to this bucket can be eliminated as candidates (even if the pair consists of 2 frequent items)  Pass 2: Only count pairs that hash to frequent buckets Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 39  Replace the buckets by a bit-vector:  1 means the bucket count exceeded the support s (call it a frequent bucket); 0 means it did not  4-byte integer counts are replaced by bits, so the bit-vector requires 1/32 of memory  Also, decide which items are frequent and list them for the second pass Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 40 41  Count all pairs {i, j} that meet the conditions for being a candidate pair: 1. Both i and j are frequent items 2. The pair {i, j} hashes to a bucket whose bit in the bit vector is 1 (i.e., a frequent bucket)  Both conditions are necessary for the pair to have a chance of being frequent Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 42 Hash table Item counts Bitmap Pass 1 Pass 2 Frequent items Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Hash table for pairs Mainmemory Counts of candidate pairs 43  Buckets require a few bytes each:  Note: we do not have to count past s  #buckets is O(main-memory size)  On second pass, a table of (item, item, count) triples is essential (we cannot use triangular matrix approach, why?)  Thus, hash table must eliminate approx. 2/3 of the candidate pairs for PCY to beat A-Priori Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  Limit the number of candidates to be counted  Remember: Memory is the bottleneck  Still need to generate all the itemsets but we only want to count/keep track of the ones that are frequent  Key idea: After Pass 1 of PCY, rehash only those pairs that qualify for Pass 2 of PCY  i and j are frequent, and  {i, j} hashes to a frequent bucket from Pass 1  On middle pass, fewer pairs contribute to buckets, so fewer false positives  Requires 3 passes over the data Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 44 45 First hash table Item counts Bitmap 1 Bitmap 1 Bitmap 2 Freq. items Freq. items Counts of candidate pairs Pass 1 Pass 2 Pass 3 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) Count items Hash pairs {i,j} Hash pairs {i,j} into Hash2 iff: i,j are frequent, {i,j} hashes to freq. bucket in B1 Count pairs {i,j} iff: i,j are frequent, {i,j} hashes to freq. bucket in B1 {i,j} hashes to freq. bucket in B2 First hash table Second hash table Counts of candidate pairs Mainmemory  Count only those pairs {i, j} that satisfy these candidate pair conditions: 1. Both i and j are frequent items 2. Using the first hash function, the pair hashes to a bucket whose bit in the first bit-vector is 1 3. Using the second hash function, the pair hashes to a bucket whose bit in the second bit-vector is 1 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 46 1. The two hash functions have to be independent 2. We need to check both hashes on the third pass  If not, we would end up counting pairs of frequent items that hashed first to an infrequent bucket but happened to hash second to a frequent bucket Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 47  Key idea: Use several independent hash tables on the first pass  Risk: Halving the number of buckets doubles the average count  We have to be sure most buckets will still not reach count s  If so, we can get a benefit like multistage, but in only 2 passes Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 48 Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212) 49 First hash table Second hash table Item counts Bitmap 1 Bitmap 2 Freq. items Counts of candidate pairs Pass 1 Pass 2 First hash table Second hash table Counts of candidate pairs Mainmemory  Either multistage or multihash can use more than two hash functions  In multistage, there is a point of diminishing returns, since the bit-vectors eventually consume all of main memory  For multihash, the bit-vectors occupy exactly what one PCY bitmap does, but too many hash functions makes all counts > s 50Pavel Zezula, Jan Sedmidubsky. Advanced Search Techniques for Large Scale Data Analytics (PA212)  A-Priori, PCY, etc., take k passes to find frequent itemsets of size k  Can we use fewer passes?  Use 2 or fewer passes for all sizes, but may miss some frequent itemsets  Random sampling  SON (Savasere, Omiecinski, and Navathe)  Toivonen (see textbook) Pavel Zezula, Jan Sedmidubsky. 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