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## FP-Tree structure

The frequent-pattern tree (FP-tree) is a compact structure that stores quantitative information about frequent patterns in a database .

Han defines the FP-tree as the tree structure defined below:

1. One root labeled as “null” with a set of item-prefix subtrees as children, and a frequent-item-header table (presented in the left side of Figure(1);

2. Each node in the item-prefix subtree consists of three fields:

i)Item-name: registers which item is represented by the node;

II)Count: the number of transactions represented by the portion of the path reaching the node;

iii)Node-link: links to the next node in the FP-tree carrying the same item-name, or null if there is none.

3 .Each entry in the frequent-item-header table consists of two fields

• Item-name : as the same to the node.

• Head of node-link: a pointer to the first node in the FP-tree carrying the item-name.

Additionally the frequent-item-header table can have the count support for an item. The Figure 1 below show an example of a FP-tree. The original algorithm to construct the FP-Tree defined by Han in  is presented below in Algorithm 1.

## Algorithm 1: FP-tree construction

Input: A transaction database DB and a minimum support threshold ?.

Output: FP-tree, the frequent-pattern tree of DB.

Method: The FP-tree is constructed as follows.

1. Scan the transaction database DB once. Collect F, the set of frequent items, and the support of each frequent item. Sort F in support-descending order as FList, the list of frequent items.

2. Create the root of an FP-tree, T, and label it as “null”. For each transaction Trans in DB do the following:

⦁ Select the frequent items in Trans and sort them according to the order of FList. Let the sorted frequent-item list in Trans be [ p | P], where p is the first element and P is the remaining list. Call insert tree([ p | P], T ).

⦁ The function insert tree([ p | P], T ) is performed as follows. If T has a child N such that N.item-name = p.item-name, then increment N ’s count by 1; else create a new node N , with its count initialized to 1, its parent link linked to T , and its node-link linked to the nodes with the same item-name via the node-link structure. If P is nonempty, call insert tree(P, N ) recursively.

By using this algorithm, the FP-tree is constructed in two scans of the database. The first scan collects and sort the set of frequent items, and the second constructs the FP-Tree.

## FP-Growth Algorithm:

After constructing the FP-Tree it’s possible to mine it to find the complete set of frequent patterns. To accomplish this job, Han in  presents a group of lemmas and properties, and thereafter describes the FP-Growth Algorithm as presented below in Algorithm 2.

Algorithm 2: FP-Growth:

Input: A database DB, represented by FP-tree constructed according to Algorithm 1, and a minimum support threshold ?.

Output: The complete set of frequent patterns.

Method: call FP-growth(FP-tree, null). Procedure FP-growth(Tree, a)

{

(01) if Tree contains a single prefix path then { // Mining single prefix-path FP-tree

(02) let P be the single prefix-path part of Tree;

(03) let Q be the multipath part with the top branching node replaced by a null root;

(04) for each combination (denoted as ß) of the nodes in the path P do

(05) generate pattern ß ∪ a with support = minimum support of nodes in ß;

(06) let freq pattern set(P) be the set of patterns so generated;

}

(07) else let Q be Tree;

(08) for each item ai in Q do { // Mining multipath FP-tree

(09) generate pattern ß = ai ∪ a with support = ai .support;

(10) construct ß’s conditional pattern-base and then ß’s conditional FP-tree Tree ß;

(11) if Tree ß ≠ Ø then

(12) call FP-growth(Tree ß , ß);

(13) let freq pattern set(Q) be the set of patterns so generated;

}

(14) return(freq pattern set(P) ∪ freq pattern set(Q) ∪ (freq pattern set(P) × freq pattern set(Q)))

}

When the FP-tree contains a single prefix-path, the complete set of frequent patterns can be generated in three parts: the single prefix-path P, the multipath Q, and their combinations (lines 01 to 03 and 14). The resulting patterns for a single prefix path are the enumerations of its subpaths that have the minimum support (lines 04 to 06). Thereafter, the multipath Q is defined (line 03 or 07) and the resulting patterns from it are processed (lines 08 to 13). Finally, in line 14 the combined results are returned as the frequent patterns found.