Cell Probe Complexity

Cell Probe Complexity

Balanced Trees 1 Motivation Example Reporting (2-D) Given a set of points S on the plane, preprocess them to build structure that allows efficient queries of the from: Given an rectangle R=[x1,x2][y1,y2] find all points in S that are in the rectangle. P3 P8 P5 P2 P6 P4 P1

P7 2 Summary Add Delete Find Arrays Linked List Bal. Trees Simple, fast Inflexible O(1) O(n) inc sort O(n) Simple

Flexible O(1) O(n) - inc sort O(1) - any O(n) - specific O(n) (no binary search) Not-So-simple Flexible O(logn) O(n) O(logn) binary search O(log n) O(log n) 3 Order statistics Trees

rank and select 4 The dictionary ADT Insert(x,D) Delete(x,D) Find(x,D): Returns a pointer to x if x D, and a pointer to the successor or predecessor of x if x is not in D 5 Suppose we want to add to the dictionary ADT Select(i,D): Returns the ith element in the dictionary: An element x such that i-1 elements are smaller than x 6

Select(5,D) 89 19 20 26 90 21 34 4 67 73 70

77 7 Can we still use trees ? 4 19 20 21 26 34 67 70

73 77 89 90 8 For each node v store # of leaves in the subtree of v 12 4 8 2 4 19

2 20 4 4 2 2 21 26 34 67 70 2

2 73 77 89 90 9 Select(7,T) 12 4 8 2 4 19

2 20 4 4 2 2 21 26 34 67 70 2

2 73 77 89 90 10 Select(7,T) 12 Select(3, ) 4 8 2 4

19 2 20 4 4 2 2 21 26 34 67 70

2 2 73 77 89 90 11 Select(7,T) 12 4 Select(3, ) 2 4

19 8 2 20 4 4 2 2 21 26 34 67

70 2 2 73 77 89 90 12 Select(7,T) 12 4 8

2 4 19 2 20 4 4 Select(1,) 2 2 21 26 34 67

70 2 2 73 77 89 O(logn) worst case time, in case of balanced tree 90 13 Rank(x,T) Return the index of x in T 14

Rank(x,T) Need to return 9 x 15 12 4 8 2 4 19 2 20 4

4 2 2 21 26 34 67 70 2 2 73 77

x Sum up the sizes of the subtrees to the left of the path 89 90 16 Insert (non-balanced tree) 12 8 4 2 17 Insert (cont) 13 9

5 3 2 update only O(h) sizes 18 Summary Insertion and deletion and other dictionary operations still take O(log n) time for balanced trees 19 Orthogonal range searching Range trees 20 Reporting (1-D)

Given a set of points S on the line, preprocess them to build structure that allows efficient queries of the from: Given an interval I=[x1,x2] find all points in S that are in the interval. 6 2 4 5 17 7 8 12 15 19 21 Reporting (1-D)

Build a balanced tree over the points Concatenate them in a list 6 2 4 5 17 7 8 12 15 19 query: O(log n+k) space: O(n) 7

4 12 2 2 5 4 5 8 7 8 15

12 15 22 19 Counting (1-D) Given a set of points S on the line, preprocess them to build structure that allows efficient queries of the from: Given an interval I=[x1,x2] return the # of points in the interval 6 2 4 5 17 7 8 12

15 19 23 Counting (1-D) Build a balanced tree over the points, with subtree sizes. Return: Rank(x2,T)- Rank(x1,T)+1 6 2 4 5 17 7 8 12 15

19 query: O(log n) space: O(n) 7 4 12 2 5 8 2 1 2

4 5 7 15 1 8 12 15 24 19 Reporting (2-D) Given a set of points S on the plane, preprocess them to build structure that allows efficient

queries of the from: Given an rectangle R=[x1,x2][y1,y2] find all points in S that are in the rectangle. P3 P8 P5 P2 P6 P4 P1 P7 25 Range Trees (2-D) Maintain the points in a balanced search tree ordered by x-coor. In each internal node maintain the points in its subtree in a balanced search tree ordered by y

P3 P3 P2 P2 P4 P4 P1 P1 P1 P2 P3 26 P4

Range Trees (Contd.) P3 P8 P5 P2 P6 P4 P7 P1 P3 P2 P4 P1 P1 P2

P3 P4 P5 P6 P7 P8 27 Range Trees (Contd.) P3 P8 P5 P2

P6 P4 P7 P1 P3 P2 P4 P1 P1 P2 P3 P4 P5 P6

P7 P8 28 Query processing Search by the first dimension gives us O(logn) trees which together contain the output. We search each of these trees to get the answer 29 Analysis (2-D) Space O(nlog n) Query O(log2 n+k) 30 Further facts

Generalizes to d-dimensions 31 1 , pi (.)xi,yi insert(x,y), ) delete(x,y), find(x,y ) ,O(log n ),CountPointsAtDistance(d1,d2 : d1,d2 d1 d2 ( ( )x,y ). )CountPointsAtDistance(d1,d2 . 32 1 .d ( )x,y

. ORDER STAT . 33 .d2- .a : .d1- .b : .a-b+1 " .O(log n) W.C . 34 2 insert(x),

) , ,delete(x), find(x ,x )(multiple_of_5 x, y ,y=5x ( .)false WC . 35 2 . .5 . , . ( , ). )(Multiple_of_5 . 36 3

S n , W.C. O(n), find , : W.C find ) ,O(n n/log n , find ). O(log n , insert .delete 37 3 n/log n . ) .O(n . find . ) .O(log n ).O(n 38 4

39 ( )n,m : n,,1,2,3, .1 m . , . ' , . , '. ( 4) : ( )7,3 ( 4,7,3,1,6,2,5 !) ,n (.)n,n/2 :4 )O(nlogn

40 4 - 41 1 n ." ( )1+ .size .1 ,I ( mod k )I+n/2 OS_Select k ( size ). , . 1 2 - .

Recently Viewed Presentations

  • media.luxoworks.com

    media.luxoworks.com

    ZUMTOBEL HELISSA D450 1/55W T16R EVG WH 60811749 Description Wall-/ceiling-mounted luminaire; lamp(s): 1/55W T16-R; control gear: with high frequency ballast ; uniformly illuminated polycarbonate diffuser; age-resistant diffuser for good light output ratio; lighting corona around luminaire for indirect accent lighting;...
  • Critical Transitions in Faculty Learning Please complete the

    Critical Transitions in Faculty Learning Please complete the

    Until recently the Instructional or Teaching-Centered Paradigm has been the traditional paradigm of higher education. The basic assumptions underlying this paradigm are that subject matter content is the primary focus of instruction, and the role of the instructor is to...
  • ORGANIC CHEMISTRY AN INTRODUCTION TO A guide for

    ORGANIC CHEMISTRY AN INTRODUCTION TO A guide for

    AN INTRODUCTION TO ORGANIC CHEMISTRY A guide for A level students KNOCKHARDY PUBLISHING CONTENTS WHICH COMPOUND IS IT? Elucidation of the structures of organic compounds - a brief summary Organic chemistry is so vast that the identification of a compound...
  • CDs (Cont.) CSIT 301 (Blum) 1 Fixing Mistakes

    CDs (Cont.) CSIT 301 (Blum) 1 Fixing Mistakes

    UDF stands for Universal Disk Format. "UDF, defined by the Optical Technology Storage Association (OTSA), is a subset of ISO 13346, an interchange standard for non-sequential recording of data." CSIT 301 (Blum) * UDF (Cont.) It is a file system...
  • Delegation of Adjudicatory Power to Agencies

    Delegation of Adjudicatory Power to Agencies

    This mirrors some of the issues raised by the delegation of rulemaking powers ... they can set how they are awarded Congress can set up compensation that is not related to real facts Remember this later when we see the...
  • ed Endings PowerPoint

    ed Endings PowerPoint

    Uncle Ed is very old. He likes to remember things he did in the past. With some action words or verbs we add Uncle Ed's name to change the verb to past tense.
  • Presidential Succession and the 25th Amendment

    Presidential Succession and the 25th Amendment

    Presidential Succession and Disability Bill Clinton (1st Attempt) September 12, 1994: Frank Eugene Corder flew a single-engine Cessna into the White House lawn, allegedly trying to hit the White House. The President and First Family were not home at the...
  • Norms for Moral Living - Bishop Allen Academy Religion Department

    Norms for Moral Living - Bishop Allen Academy Religion Department

    Norms for Moral Living pp. 147-155 Norms and Obligations Norms are guides for actions that come in the form of laws, rules, principles, commandments and maxims. They also come with varying degrees of obligation. They are declared by an authority...