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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

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Forms of Retrieval Sequential Retrieval Two-Step Retrieval Retrieval with Indexed Cases

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Sources: –Bergman’s b`ook –Davenport & Prusack’s book on Advanced Data Structures –Samet’s book on Data Structures

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Range Search Red light on? Yes Beeping? Yes … Transistor burned! Space of known problems

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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

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Definition: K-D Trees Given: K types: T 1, …, T k for the attributes A 1, …, A k A case base CB containing cases in T 1 … T k 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 A i and a value v in A i and The 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

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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) A1A1 <35 35 Denver Omaha A2A2 <40 40 A1A1 <85 85 Mobile Atlanta Miami A1A1 <60 60 Chicago Toronto Buffalo Notes: Supports Euclidean distance May require backtracking Closest city to P(32,45)? Priority lists are used for computing kNN P(32,45)

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Using Decision Trees as Index AiAi v1v1 v2v2 … vnvn Standard Decision Tree AiAi v1v1 v2v2 … vnvn Variant: InReCA Tree unknown Can be combined with numeric attributes AiAi v1v1 >v1v2>v1v2 … >v n unknown Notes: Supports Hamming distance May require backtracking Operates in a similar fashion as kd-trees Priority lists are used for computing kNN

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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

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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

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Insertion in Quadtree Chicago Denver Toronto OmahaMobile Buffalo Atlanta Miami

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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’

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Insertion Procedure (Cont.) procedure PT_insert(Pointer P, R) //inserts P in the tree rooted at R Pointer T //points to the current node being examined Pointer F // points to the parent of T Quadrant Q //auxiliary variable T 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

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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

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Search (II) A r 123 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

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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)

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Nearest-Neighbor Search Problem: Given a point quadtree T and a point P find the node in T that is the closest to P Idea: 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)

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Distance from P to a Quadrant 1 2 3 P P1 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

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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)

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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) = 0 distance(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

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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

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Complexity Experiments show that random insertion of N nodes is roughly O(N log 4 N) Thus, insertion of a single node is O(log 4 N) But worst case (actual complexity) can be much worse Range search can be performed in O(2 N ½ )

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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!.

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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.

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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!

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Problem with Defining a New Root Several points will have to be re-positioned Old root New root SW NE NW NE SW NW SE NE SW SE

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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

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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.

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Properties of Retrieval with Indexed Cases Advantage: Disadvantages: Efficient retrieval Incremental: don’t need to rebuild index again every time a new case is entered -error does not occur Cost of construction is high Only work for monotonic similarity relations

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