Slide 1Too many frequent itemsetsFrequent itemsets maybe too many to be helpfulSlide 4Borders of frequent itemsetsPositive and negative borderExamples with bordersDescriptive power of the bordersMaximal patternsMaximal Frequent ItemsetMaximal patternsClosed patternsMaximal vs Closed ItemsetsMaximal vs Closed Frequent ItemsetsWhy are closed patterns interesting?Why closed patterns are interesting?Maximal vs Closed setsSlide 18Prototype problems: Covering problemsPrototype covering problemsSet-cover problemTrivial algorithmGreedy algorithm for set coverAs an algorithmHow can this go wrong?How good is the greedy algorithm?How good is the greedy algorithm for set cover?How good is the greedy algorithm for set cover? A tighter boundBest-collection problemGreedy approximation algorithm for the best-collection problemBasic theoremSubmodular functions and the greedy algorithmSlide 33Approximating a collection of frequent patternsSlide 35Reducing the collection of itemsets: alternative representations and combinatorial problemsToo many frequent itemsets•If {a1, …, a100} is a frequent itemset, then there are1.27*1030 frequent sub-patterns!•There should be some more condensed way to describe the data1210010021001100100Frequent itemsets maybe too many to be helpful•If there are many and large frequent itemsets enumerating all of them is costly.•We may be interested in finding the boundary frequent patterns.•Question: Is there a good definition of such boundary?all itemsempty setFrequent itemsetsNon-frequent itemsetsborderBorders of frequent itemsets•Itemset X is more specific than itemset Y if X superset of Y (notation: Y<X). Also, Y is more general than X (notation: X>Y)•The Border: Let S be a collection of frequent itemsets and P the lattice of itemsets. The border Bd(S) of S consists of all itemsets X such that all more general itemsets than X are in S and no pattern more specific than X is in S. then with allfor and , then with allfor )(SWWXPWSYXYPYPXSBdPositive and negative border•Border•Positive border: Itemsets in the border that are also frequent (belong in S)•Negative border: Itemsets in the border that are not frequent (do not belong in S) then with allfor and , then with allfor )(SWWXPWSYXYPYPXSBd then with allfor )( SYYXPYSXSBd then with allfor \)( SYXYPYSPXSBd Examples with borders•Consider a set of items from the alphabet: {A,B,C,D,E} and the collection of frequent sets S = {{A},{B},{C},{E},{A,B},{A,C},{A,E},{C,E},{A,C,E}}•The negative border of collection S isBd-(S) = {{D},{B,C},{B,E}}•The positive border of collection S isBd+(S) = {{A,B},{A,C,E}}Descriptive power of the borders•Claim: A collection of frequent sets S can be fully described using only the positive border (Bd+(S)) or only the negative border (Bd-(S)).Maximal patternsFrequent patterns without proper frequent super patternMaximal Frequent ItemsetBorderInfrequent ItemsetsMaximal ItemsetsAn itemset is maximal frequent if none of its immediate supersets is frequentMaximal patterns•The set of maximal patterns is the same as the positive border•Descriptive power of maximal patterns:–Knowing the set of all maximal patterns allows us to reconstruct the set of all frequent itemsets!!–We can only reconstruct the set not the actual frequenciesClosed patterns•An itemset is closed if none of its immediate supersets has the same support as the itemsetItemset Support{A,B,C} 2{A,B,D} 3{A,C,D} 2{B,C,D} 3{A,B,C,D} 2Maximal vs Closed ItemsetsTID Items1 ABC2 ABCD3 BCE4 ACDE5 DEnullAB AC AD AE BC BD BE CD CE DEA B C D EABC ABD ABE ACD ACE ADE BCD BCE BDE CDEABCD ABCE ABDE ACDE BCDEABCDE124 1231234245 34512 124 24412323243445122244 423 424Transaction IdsNot supported by any transactionsMaximal vs Closed Frequent ItemsetsnullAB AC AD AE BC BD BE CD CE DEA B C D EABC ABD ABE ACD ACE ADE BCD BCE BDE CDEABCD ABCE ABDE ACDE BCDEABCDE124 1231234245 34512 124 24412323243445122244 423 424Minimum support = 2# Closed = 9# Maximal = 4Closed and maximalClosed but not maximalWhy are closed patterns interesting?•s({A,B}) = s(A), i.e., conf({A}{B}) = 1•We can infer that for every itemset X , s(A U {X}) = s({A,B} U X)•No need to count the frequencies of sets X U {A,B} from the database!•If there are lots of rules with confidence 1, then a significant amount of work can be saved–Very useful if there are strong correlations between the items and when the transactions in the database are similarWhy closed patterns are interesting?•Closed patterns and their frequencies alone are sufficient representation for all the frequencies of all frequent patterns•Proof: Assume a frequent itemset X:–X is closed s(X) is known –X is not closed s(X) = max {s(Y) | Y is closed and X subset of Y}Maximal vs Closed sets•Knowing all maximal patterns (and their frequencies) allows us to reconstruct the set of frequent patterns•Knowing all closed patterns and their frequencies allows us to reconstruct the set of all frequent patterns and their frequenciesA more algorithmic approach to reducing the collection of frequent itemsetsPrototype problems: Covering problems•Setting: –Universe of N elements U = {U1,…,UN}–A set of n sets S = {s1,…,sn}–Find a collection C of sets in S (C subset of S) such that UcєCc contains many elements from U•Example:–U: set of documents in a collection–si: set of documents that contain term ti–Find a collection of terms that cover most of the documentsPrototype covering problems•Set cover problem: Find a small collection C of sets from S such that all elements in the universe U are covered by some set in C•Best collection problem: find a collection C of k sets from S such that the collection covers as many elements from the universe U as possible•Both problems are NP-hard•Simple approximation algorithms with provable properties are available and very useful in practiceSet-cover problem•Universe of N elements U = {U1,…,UN}•A set of n sets S = {s1,…,sn} such that Uisi =U•Question: Find the smallest number of sets from S to form collection C (C subset of S) such that UcєCc=U •The set-cover problem is NP-hard (what does this mean?)Trivial algorithm•Try all subcollections of S•Select the smallest
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