K-Nearest Neighbors (kNN)

• Given a case base CB, a new problem P, and a similarity metric sim

• Obtain: the k cases in CB that are most similar to P according to sim

• Reminder: we used a priority list with the top k most similar cases obtained so far

Forms of Retrieval

•Sequential Retrieval

•Two-Step Retrieval

•Retrieval with Indexed Cases

Retrieval with Indexed Cases

Sources:–Bergman’s b`ook

–Davenport & Prusack’s book on Advanced Data Structures

–Samet’s book on Data Structures

Range Search

Red light on? YesBeeping? Yes…

Transistor burned!

Space of known problems

K-D Trees

•Idea: Partition of the case base in smaller fragments

•Representation of a k-dimension space in a binary tree

•Similar to a decision tree: comparison with nodes

•During retrieval:

Search for a leaf, but

Unlike decision trees backtracking may occur

Definition: K-D Trees•Given:

K types: T1, …, Tk for the attributes A1, …, Ak

A case base CB containing cases in T1 … Tk A parameter b (size of bucket)

•A K-D tree T(CB) for a case base CB is a binary tree defined as follows:

If |CB| < b then T(CB) is a leaf node (a bucket)Else T(CB) defines a tree such that:

The root is marked with an attribute Ai and a

value v in Ai andThe 2 k-d trees T({c CB: c.i-attribute < v}) and T({c CB: c.i-attribute v}) are the left and right subtrees of the root

Example

(0,0)

(0,100)

(25,35)Omaha

(5,45)Denver

(35,40)Chicago

(50,10)Mobile (90,5)

Miami

Atlanta (85,15)

(80,65)Buffalo

(60,75)Toronto

(100,0)

A1

<35 35

DenverOmaha

A2

<40

40A1

<85 85

Mobile AtlantaMiami A1

<60 60

Chicago TorontoBuffalo

Notes:•Supports Euclidean distance•May require backtracking

Closest city to P(32,45)?•Priority lists are used for computing kNN

P(32,45)

Using Decision Trees as Index

Ai

v1 v2

… vn

Standard Decision Tree

Ai

v1v2

… vn

Variant: InReCA Tree

unknown

Can be combined with numeric attributes

Ai

v1

>v1

v2

… >vn

unknown

Notes:•Supports Hamming distance•May require backtracking

Operates in a similar fashion as kd-trees•Priority lists are used for computing kNN

Variation: Point QuadTree

•Particularly suited for performing range search (i.e, similarity assessment)

•Adequate with fewer numerical and known-important attributes

A node in a (point) quadtree contains:•4 Pointers: quad [‘NW’], quad [‘NE’], quad[‘SW’], and quad[‘SE’] •point, of type DataPoint, which in turn contains:

•name•(x,y) coordinates

Example

(0,0)

(0,100)

(25,35)Omaha

(5,45)Denver

(35,40)Chicago

(50,10)Mobile (90,5)

Miami

Atlanta (85,15)

(80,65)Buffalo

(60,75)Toronto

(100,0)

Insertion order: Chicago, Mobile, Toronto, Buffalo, Denver, Omaha, Atlanta and Miami

Insertion in Quadtree

Chicago

Denver Toronto Omaha Mobile

Buffalo Atlanta Miami

Insertion Procedure

We define a new type: quadrant: ‘NW’, ‘NE’, ‘SW’, ‘SE’

function PT_compare(DataPoint dP, dR): quadrant//quadrant where dP belongs relative to dR

if (dP.x < dR.x) then if (dP.y < dR.y) then return ‘SW’ else return ‘NW’ else if (dP.y < dR.y) then return ‘SE’ else return ‘NE’

Insertion Procedure (Cont.)

procedure PT_insert(Pointer P, R)//inserts P in the tree rooted at RPointer T //points to the current node being examinedPointer F // points to the parent of TQuadrant Q //auxiliary variableT R F null

while not(T == null) && not(equalCoord(P.point,T.point)) do F T Q PT_compare(P.point, T.point) T T.quad[Q]if (T == null) then F.quad[Q] P

Search

Typical query: “find all cities within 50 miles of Washington,DC”

In the initial example: “find all cities within 8 data units from (83,13)”

Solution:•Discard NW, SW and NE of Chicago (that is, only examine SE)•There is no need to search the NW and SW of Mobile

Search (II)

Ar

1 2 3

4 5

6 7 8

9 10

11 12

Let R be the root of the quadtree, what regions need to be inspected if R is in the quadrant:

1: SE

2: SW, SE

8: NW

11: NW, NE, SE

Priority Queues•Typical example: printing in a Unix/Linux environment. Printing jobs have different priorities.

•These priorities may override the FIFO policy of the queues (i.e., jobs with the highest priorities will get printed first).

Operations supported in a priority queue:

•Insert a new element•Extract/Delete of the element with the lowest priority•In search trees, the priority is based on the distance

•Insertion, deletion can be done in O(Log N) and look-head in O(1)

Nearest-Neighbor Search

Problem:Given a point quadtree T and a point P find the node in T that is the closest to PIdea: traverse the quadtree maintaining a priority list, candidates, based on the distance from P to the quadrants containing the candidate nodes

(25,35)Omaha

(5,45)Denver

(35,40)Chicago

(50,10)Mobile (90,5)

Miami

(85,15)Atlanta

(80,65)Buffalo

(60,75)Toronto

P(95,15)

Distance from P to a Quadrant

1

2 3

PP1

P2 P3

distance(P,SW) = f-1(sim(P,(P.y,0))(x,y)

distance(P,NW) = f-1(sim(P,(x,y))

distance(P,NE) = f-1(sim(P,(P.x,0))

4

P4

distance(P,SE) = 0

Let f-1 be the inverse of the distance-similarity compatible function

Idea of the Algorithm

(25,35)Omaha

(5,45)Denver

(35,40)Chicago

(50,10)Mobile

(60,75)Toronto

P = (95,15)

Candidates = [Chicago (4225)] Buffer: null ()

Candidates = [Mobile(0),Toronto (25), Omaha (60), Denver(4225)]Buffer: Chicago (4225)

List of Candidates

(50,10)Mobile (90,5)

Miami

(85,15)Atlanta P(95,15)

•Examine the quadrant of the top of candidates (Mobile) and make it the new buffer:

Buffer: Mobile (1625)

distance(P,NE) = 0distance(P,SE) = 5

•Termination test: Buffer.distance < distance(candidates.top,P) if “yes” then return Buffer if “no” then continue

•In this particular example, is “no” since Mobile is closer to P than Chicago

Finally the Nearest Neighbor is Found

Candidates = [Atlanta(0), Miami(5), Toronto (25), Omaha (60), Denver(4225)]

Buffer: Atlanta(100)

Candidates = [Miami(5), Toronto (25), Omaha (60), Denver(4225)]

A new iteration:

The algorithm terminates since the distance from Atlanta to P is less than the distance from Miami to P

Complexity

•Experiments show that random insertion of N nodes is roughly O(N log4N)

•Thus, insertion of a single node is O(log4N)

•But worst case (actual complexity) can be much worse

•Range search can be performed in O(2 N ½)

Delete

First idea:

•Find the node N that you want to delete•Delete N and all of its descendants ND•For each node N’ in ND, add N’ back into the tree

Terrible idea; it is too inefficient!.

Idealized Deletion in Quadtrees

If a point A is to be deleted find a point B such that the region between A and B is empty and replaced A with B

A

B

“Hatched Region”

Why? Because all the remaining points will be in the same quadrants relative to B as they are relative to A. For example, Omaha could replace Chicago as the root.

Problem with Idealized Situation

First Problem: A lot of effort is required to find such a B.

C

A

D

E

F

In the following example which point (C, F, D or A) has a hatched region with A?

Answer: none!. Second problem: No such a B may exit!

Problem with Defining a New Root

Several points will have to be re-positioned

Old root

New root

SW NE

NW NE

SW NWSE NE

SW SE

Deletion Process

Delete P:1. If P is a leaf then just delete it!.

2. If P has a single child C, then replace P with C

3. For all other cases: 3.1 Compute 4 candidate nodes, one for each quadrant under P 3.2 Select one of the candidate node, N according to certain criteria 3.3 Delete several nodes under P and collect them in a list, ADD. Also delete N. 3.4 Make N.point the new root: P.point N.point 3.5 Re-insert all nodes in ADD

A Word of Warning About Deletion

•In databases frequently deletion is not done immediately because it is so time-consuming.

•Sometimes they don’t even do insertions immediately!

•Instead they keep a log with all deletions (and additions), and periodically (i.e., every night, weekend), the log is traversed to update the database. The technique is called Differential Databases.

•Deleting cases is part of the general problem of case base maintenance.

Properties of Retrieval with Indexed Cases

•Advantage:

•Disadvantages:

Efficient retrievalIncremental: don’t need to rebuild index again every time a new case is entered-error does not occur

Cost of construction is highOnly work for monotonic similarity relations