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<j
 Keep pair counts in lexicographic order:
 {1,2}, {1,3},…, {1,n}, {2,3}, {2,4},…,{2,n}, {3,4},…
 Pair {i, j} is at position (i –1)(n– i/2) + j –1
 Total number of pairs n(n –1)/2; total bytes= 2n2
 Triangular Matrix requires 4 bytes per pair
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<j
 Keep pair counts in lexicographic order:
 {1,2}, {1,3},…, {1,n}, {2,3}, {2,4},…,{2,n}, {3,4},…
 Pair {i, j} is at position (i –1)(n– i/2) + j –1
 Total number of pairs n(n –1)/2; total bytes= 2n2
 Triangular Matrix requires 4 bytes per pair
 Approach 2 uses 12 bytes per occurring pair
(but only for pairs with count > 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<j
 Keep pair counts in lexicographic order:
 {1,2}, {1,3},…, {1,n}, {2,3}, {2,4},…,{2,n}, {3,4},…
 Pair {i, j} is at position (i –1)(n– i/2) + j –1
 Total number of pairs n(n –1)/2; total bytes= 2n2
 Triangular Matrix requires 4 bytes per pair
 Approach 2 uses 12 bytes per pair
(but only for pairs with count > 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. Advanced Search Techniques for Large Scale Data Analytics (PA212) 51