Introduction to Algorithms

1
Introduction to Algorithms

• Unit 14. Augmenting Data Structures
• Chih-Hung Wang

• Dynamic Order Statistics
• How to Augment a Data Structure
• Interval Trees

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Dynamic Order Statistics

• We shall also see the rank of an element?its
  position in the linear order of the set?can
  likewise be determined in O(lg n) time.

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Dynamic Order Statistics

• Beside the usual red-black tree fields keyx,
  colorx, px, leftx, and rightx in a node
  x, we have another field sizex. This field
  contains the number of (internal) nodes in the
  subtree rooted at x (including x itself), that
  is, the size of the subtree.
• If we define the sentinels size to be 0, that
  is, we set sizenilT to be 0, then we have the
  identity
• sizex sizeleftx sizerightx 1

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An order-statistic Tree
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Retrieving an element with a given rank
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Analysis

• Analysis Each recursive call goes down one
  level. Since R-B tree has O(lg n) levels, have
  O(lg n) calls? O(lg n) time.

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Determining the rank of an element
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Example

• Find the rank of the node with key 38
• Iteration keyy r
• 38 2
• 30 4
• 41 4
• 26 17
• Return 17

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Analysis

• Analysis y goes up one level in each iteration ?
  O(lg n) time.

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Maintaining subtree size

• Referring to the code for LEFT-ROTATE(T, x) in
  section 13.2 we add the following lines
• 12 sizey ? sizex
• 13 sizex ? sizeleftx sizerightx 1

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Updating subtree sizes during rotations
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How to augment a data structure

• 1.Choosing an underlying data structure,
• 2.Determining additional information to be
  maintained in the underlying data structure,
• 3.Verifying that the additional information can
  be maintained for the basic modifying operations
  on the underlying data structure, and
• 4.Developing new operations.

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Augmenting red-black trees

• Theorem 14.1
• Augment a R-B tree with field f , where f x
  depends only on information in x, leftx, and
  rightx (including f leftx and f
  rightx). Then can maintain values of f in all
  nodes during insert and delete without affecting
  O(lg n) performance.

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Augmenting red-black trees

• Proof
• Since f x depends only on x and its children,
  when we alter information in x, changes propagate
  only upward (to px, ppx, . . . , root).
• Height O(lg n)? O(lg n) updates, at O(1) each.

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Interval Trees

• We can represent an intervalt1,t2 as an object
  i, with fields lowi t1 (the low endpoint) and
  highi t2 (the high endpoint). We say that
  intervals i and i overlap if i ? i ? Ø, that
  is, if lowi highi and lowi highi.
  Any two intervals i and i satisfy the interval
  trichotomy that exactly one of the following
  three properties holds
• a. i and i overlap,
• b. i is to the left of i (i.e., highi lt
  lowi),
• c. i is to the right of i (i.e., highi
  lt lowi)

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Interval Trees
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Data Structure of Interval Trees

• An interval tree is a red-black tree that
  maintains a dynamic set of elements, with each
  element x containing an interval intx.

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Operations

• INTERVAL-INSERT(T,x)
• Add the element x in T
• INTERVAL-DELETE(T,x)
• Remove the element x in T
• INTERVAL-SEARCH(T,i)
• Returns a pointer to an element x in T such that
  intx overlaps interval i, or the sentinel
  nilT if no such element is in the set.

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Data Structure(1)
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Data Structure (2)
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Maintaining the Information

• Maintaining the information
• We determine maxx given interval intx and the
  max values of node xs children
• maxx max(highintx, maxleftx,
  maxrightx)
• Thus, by Theorem 14.1, insertion and deletion run
  in O(lg n) time.

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New OperationInterval Search
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Example

• Successful Search
• i22, 25
• 8, 9 -gt 15, 23 (answer)
• Unsuccessful Search
• i 11, 14
• 8, 9 -gt 15, 23 -gt nilT

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Theorem

• Theorem 14.2
• Any execution of INTERVAL-SEARCH(T,i) either
  returns a node whose interval overlaps i, or it
  returns nilT and the tree T contain no node
  whose interval overlaps i.