Chapter 06 - Changing Education Paradigm

28 CHAPTER 6. MINING ASSOCIATION RULES IN LARGE DATABASES
1-var Constraint Anti-Monotone Succinct
S v, 2f=; ; g yes yes
v 2 S no yes
S V no yes
S V yes yes
S = V partly yes
min(S) v no yes
min(S) v yes yes
min(S) =v partly yes
max(S) v yes yes
max(S) v no yes
max(S) =v partly yes
count(S) v yes weakly
count(S) v no weakly
count(S) =v partly weakly
sum(S) v yes no
sum(S) v no no
sum(S) =v partly no
avg(S) v, 2f=; ; g no no
(frequency constraint) (yes) (no)
Table 6.3: Characterization of 1-variable constraints: anti-monotonicity and succinctness.
either. This property is used at each iteration of the Apriori algorithm to reduce the number of candidate itemsets
examined, thereby reducing the search space for association rules.
Other examples of anti-monotone constraints include \min(J.price) 500" and \S.year = 1998". Any itemset
which violates either of these constraints can be discarded since adding more items to such itemsets can never satisfy
the constraints. A constraint such as\avg(I.price) 100" is not anti-monotone. For a given set that does not satisfy
this constraint, a superset created by adding some (cheap) items may result in satisfying the constraint. Hence,
pushing this constraint inside the mining process will not guarantee completeness of the data mining query response.
A list of 1-variable constraints, characterized on the notion of anti-monotonicity, is given in the second column of
Table 6.3.
\What other kinds of constraints can we use for pruning the search space?" Apriori-like algorithms deal with other
constraints by rst generating candidate sets and then testing them for constraint satisfaction, thereby following a
generate-and-test paradigm. Instead, is there a kind of constraint for which we can somehow enumerate all and only
those sets that are guaranteed to satisfy the constraint? This property of constraints is called succintness. If a rule
constraint is succinct, then we can directly generate precisely those sets that satisfy it, even before support counting
begins. This avoids the substantial overhead of the generate-and-test paradigm. In other words, such constraints are
pre-counting prunable. Let's study an example of how succinct constraints can be used in mining association rules.
Example 6.8 Based on Table 6.3, the constraint \min(J:price) 500" is succinct. This is because we can explicitly
and precisely generate all the sets of items satisfying the constraint. Speci cally, such a set must contain at least
one item whose price is less than $500. It is of the form: S1 [ S2, where S1 6= ; is a subset of the set of all those
items with prices less than $500, and S2, possibly empty, is a subset of the set of all those items with prices > $500.
Because there is a precise \formula" to generate all the sets satisfying a succinct constraint, there is no need to
iteratively check the rule constraint during the mining process.
What about the constraint \min(J:price) 500", which occurs in Example 6.7? This is also succinct, since we
can generate all sets of items satisfying the constraint. In this case, we simply do not include items whose price is
less than $500, since they cannot be in any set that would satisfy the given constraint. 2
Note that a constraint such as\avg(I:price) 100" could not be pushed into the mining process, since it is
neither anti-monotone nor succinct according to Table 6.3.
Although optimizations associated with succinctness (or anti-monotonicity) cannot be applied to constraints like
\avg(I:price) 100", heuristic optimization strategies are applicable and can often lead to signi cant pruning.