CS473-Algorithms I

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CS473-Algorithms I

• Lecture X
• Augmenting Data Structures

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How to Augment a Data Structure

• FOUR STEP PROCEDURE
• 1. Choose an underlying data structure (UDS)
• 2. Determine additional info to be maintained in
  the UDS
• 3. Verify that additional info can be maintained
  for the modifying operations on the UDS
• 4. Develop new operations
• Note Order of steps may vary and be intermixed
• in real design.

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Example

• Design of our order statistic trees
• 1. Choose Red Black (R-B) TREES
• 2. Additional Info Subtree sizes
• 3. INSERT, DELETE gt ROTATIONS
• 4. OS-RANK, OS-SELECT
• Bad design choice for OS-TREES
• 2. Additional Info Store in each node its rank
  in the subtree
• OS-RANK, OS-SELECT would run quickly but
• Inserting a new minimum element would cause a
  change to
• this info in every node of the tree.

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Augmenting R-B Trees Theorem

• Theorem
• Let f be a field that augments a R-B Tree T of n
  nodes
• Suppose that fx for a node x can be computed
  using only
• The info in nodes, x, leftx, rightx
• f leftx and f rightx
• Proof Main idea
• Changing fx gt Update only fpx but
  nothing else
• Updating fpx gt Update only fppx but
  nothing else
• And so on up to the tree until frootx is
  updated
• When froot is updated, no other node depends on
  new value
• So the process terminates
• Changing an f field in a node costs O(lgn) time
  since the height of a R-B tree is O(lgn)

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INTERVALSDEFINITIONS

• DEFINITION A Closed interval
• An ordered pair of real numbers t1,t2 with t1
  t2
• t1,t2 t ? R t1 t t2
• INTERVALS
• Used to represent events that each occupy a
  continuous period of time
• We wish to query a database of time intervals to
  find out what events occurred during a given
  interval
• Represent an interval t1,t2 as an object i,
  with the fields lowi t1 highi t2
• Intervals i i' overlap if i n i' ? Ø that is
  lowi high i' AND lowi' high i

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INTERVALSDEFINITIONS

• Any two intervals satisfy the interval trichotomy
  
• That is exactly one of the following 3 properties
  hold
• a) i and i' overlap
• i i i i
• i' i' i' i'
• b) highi lt lowi'
• i i'
• c) highi' lt lowi
• i' i

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INTERVAL TREES

• Maintain a dynamic set of elements with each
  element x containing an interval intx
• Support the following operations
• INSERT(T,x) Adds an element x whose int field
  contains an interval to the tree
• DELETE(T,x) Removes the element x from the tree
  T
• SEARCH(T,i) Returns a pointer to an element x in
  T such that
• intx overlaps with i'
• NIL if no such element in the set.

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INTERVAL TREES(Cont.)

• S1 Underlying Data Structure
• Choose R-B Tree
• Each node x contains an interval intx
• Key of xlow intx
• Inorder tree walk of the tree lists the intervals
  in sorted order by low endpoints
• S2 Additional information
• Store in each node x the maximum endpoint
  maxx in the subtree rooted at the node

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EXAMPLE
7,10
5,11
17,19
4,8
15,18
21,23
int max
17,19 23
5,11 18
21,23 23
4,8 8
15,18 18
7,10 10
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INTERVAL TREES(cont.)

• S3 Maintaining Additional Info(maxx)
• maxx minimum highintx, maxleftx,
  maxrightx
• Thus, by theorem INSERT DELETE run in O(lgn)
  time

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INTERVAL TREES(cont.)

• INSERT OPERATION
• Fix subtree Maxs on the way down
• As traverse path for INSERTION while comparing
  new low to that of node intervals
• Use new high to update max of nodes as
  appropriate
• Restore balance with rotations updating of max
  fields for rotation
• Z X
• ?
• X Y Right Rotate Y Z
• Thus, fixing max fields for rotation takes O(1)
  time.

No change
11,35 35
6,20 35
6,20 20
30
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11,35 35
14
19
19
14
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INTERVAL TREES(cont.)

• S4 Developing new operations
• INTERVAL-SEARCH(T,i)
• x ? rootT
• while x ?NIL and i nintx Ø do
• if leftx ?NIL and maxleftx lt lowi then
• x ? leftx
• else
• x ? rightx
• return x

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INTERVAL TREES(cont.)

• Time O(lgn)
• Starts with x at the root and proceeds downward
• On a single path, until
• EITHER an overlapping interval is found
• OR x becomes NIL
• Each iteration takes O(1) time
• Height of the tree O(lgn)

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Correctness of the Search Procedure

• Key Idea Need to check only 1 of the nodes 2
  children
• Theorem
• Case 1 If search goes right then
• Either overlap in the right subtree or
  no overlap
• Case 2 If search goes left then
• Either overlap in the left subtree or
  no overlap

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Correctness of the Search Procedure

• Case 1 Go Right
• If overlap in right, then done
• Otherwise (if no overlap in RIGHT)
• Either leftx NIL ? No overlap in LEFT
• OR leftx ? NIL and maxleftx lt lowi
• For each interval i in LEFT
• highi lt maxleftx
• lt lowi
• Therefore, No overlap in LEFT

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Correctness of the Search Procedure

• Case 2 GO LEFT
• If overlap in left, then done
• Otherwise (if no overlap in LEFT)
• lowi lt maxleftx highi for some i in
  LEFT
• Since i i dont overlap and lowi lt highi
• We have highi lt low i (Interval Trichotomy)
• Since tree is sorted by lows we have
• highi lt lowiltAny lows in RIGHT
• Therefore, no overlap in RIGHT

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Pictorial View of Case 1 Case 2

• i i
• i
• i
• maxleftx maxleftx
• Case 1 t Case 2
• i any interval in left i any interval
  in right
• i in left such that highimaxleftx

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Interval Trees

• How to enumarate all intervals overlapping a
  given interval
• Can do in O(klgn) time,
• where k of overlapping intervals
• Find and Delete overlapping intervals one by one
• When done reinsert them
• Theoritical Best is O(klgn)

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How to maintain a dynamic set of numbers that
support min-gap operations

• MIN-GAP(Q) retuns the magnitude of the
  difference of the two closest numbers in Q
• Example Q1,5,9,15,18,22? MIN-GAP(Q) 18-15
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• Underlying Data Structure
• A R-B Tree containing the numbers keyed on the
  numbers
• Additional Info at each Node
• min-gapx minimum gap value in the subtree TX
  rooted at x
• minx minimum value (key) in TX
• maxx maximum value (key) in TX
• These values are 8 if x is a leaf node

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3. Maintaining the Additional Info

• minleftx if leftx NIL
• minx keyx otherwise
• minleftx if leftx NIL
• minx keyx otherwise
• min-gapleftx
• min-gapx Min min-gaprightx
• keyx maxleftx
• minrightx keyx
• Each field can be computed from info in the node
  its children
• Hence, by theorem, they would be maintained
  during insert delete operation without
  affecting the O(lgn) running time

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How to maintain a dynamic set of numbers that
support min-gap operations(cont.)

• The reason for defining the min max fields is
  to make it possible to compute min-gap from
  the info
• at the node its children
• Develop the new operation MIN-GAP(Q)
• MIN-GAP(Q) simply returns the min-gap value of
  the root
• It is an O(1) time operation
• It is also possible to find the two closest
  numbers in O(lgn) time

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How to maintain a dynamic set of numbers that
support min-gap operations(cont.)

• CLOSEST-NUMBERS(Q)
• x ? rootQ
• gapmin ? min-gapx
• while x ? NIL do
• if gapmin min-gapleftx then
• x ? leftx
• elseif gapmin min-gaprightx
• x ? rightx
• elseif gapmin keyx - maxleftx
• return keyx, maxleftx
• else
• return minrightx, keyx

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How to find the overlap of rectilinearly
rectangles

• Given a set R of n rectilinearly oriented
  rectangles
• i.e sides of all rectangles are paralled to the x
  y axis
• Each rectangle r R is represented with 4
  values
• xminr, xmaxr, yminr, ymaxr
• Give an O(nlgn)-time algorithm
• To decide whether R contains two rectangle that
  overlap

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• OVERLAP(R)
• TY ? Ø
• SORT xmin xmax values of rectangles in R
• for each extremum x value in the sorted order do
• r ? rectangle x
• yint ? yminr, ymaxr
• if x xminr then
• v ? INTERVAL-SEARCH(TY, yint)
• if v ?NIL then
• return TRUE
• else
• z ? MAKE-NEW-NODE()
• leftz ? rightz ? pz ? NIL
• intz ? yint
• INSERT(TY,z)
• else / xxmaxr /
• DELETE(TY,yint)
• return FALSE