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Augmented Data Structures

Associating further information with the nodes of a data structure.

Dynamic Order-Statistics Trees

Abstract data type dynamic order-statistics set:

INSERT SEARCH DELETE RANK SELECT

RANK(S,x) computes the rank ofx in S, i.e., |{s S: s x}|.

SELECT(S,k) computes the k-th smallest element in S.

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Implementation: Balanced binary search tree with additional infor-

mation stored at each node v: The number size(v) is the number of

internal nodes in the subtree rooted at v.

size(v) = size(lc(v)) + size(rc(v)) + 1

Here and later on, lc(v) and rc(v) denote the left and right child of a node v, respectively.

size(v) can be updated during insertion and deletion in O(log n) time.

46 42

89 89

59

4217

28

1728

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SELECT is implemented using SELECT(v,k), which returns the k-thsmallest key in the subtree rooted at v.

SELECTNODE(v,k)

1 r size(lc(v)) + 12 if r= k3 then return v

4 else ifr> k

5 then return SELECTNOD E(lc(v),k)6 else return SELECTNOD E(rc(v),k r)

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NODE RANK(T,v) computes the rank of a node v in an inorder traver-sal of tree T. In order to compute the rank of key x, search for the

node w containing x and call NODERANK(T,w).

NODE RAN K(T,v)

1 r size(lc(v)) + 12 w v

3while

(w = root(T))4 do ifw = rc(parent(w))5 then r r+ size(lc(parent(w)))+ 16 w parent(w)7 return r

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Theorem: Operations SEARCH, INSERT, DELETE, SELECT, and

RANK in an order-statistics tree storing n elements take time O(log n)each.

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Interval Overlap

Given a set of closed intervals

I= {[x1,x1], [x2,x

2], . . . , [xn,x

n]}

check whether a closed query interval

[q,q]

overlaps any of the intervals in I.

All cases:

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Application:

Intersection detection of axis-parallel rectangles using plane-sweep

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

0 0 0

0 0 0

0 0 0

0 0 0

0 0 00 0 00 0 00 0 00 0 0

1 1 1

1 1 1

1 1 1

1 1 1

1 1 11 1 11 1 11 1 11 1 1

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1

0 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 11 1 11 1 11 1 11 1 11 1 1

0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 11 1 11 1 11 1 11 1 11 1 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

0 0 0 0 0 0

1 1 1 1 1 11 1 1 1 1 11 1 1 1 1 1

1 1 1 1 1 1

Sweep a horizontal line from top to bottom.Maintain the intersection intervals of the rectangles and the sweep

line in a data structure appropriate for interval overlap queries.

Check for overlap at the upper sides of the rectangles.

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Sweep line reaches an upper side:

00000000

00000000

00000000

00000000

00000000

11111111

11111111

11111111

11111111

11111111

?

OVERLAP(I) INSERT(I)

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Sweep line reaches a lower side:

00000000

00000000

00000000

00000000

00000000

11111111

11111111

11111111

11111111

11111111

DELETE(I)

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Implementation:

Balanced binary search tree for the intervals ordered according to left

endpoints xi.

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maxright(v

Additional information stored at each node v:

maxright(v) is the largest right endpoint of theintervals stored in the subtree rooted at v.

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FIN DOVERLAPPINGINTERVAL(T, [q,q])

1 v root(T)2 while (v is not a leaf and [q,q] does not overlap interval(v))3 do if(lc(v) exists and maxright(lc(v)) q)4 then v lc(v)5 else v rc(v)

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FIN DOVERLAPPINGINTERVAL(T, [q,q])

1 v root(T)2 while (v is not a leaf and [q,q] does not overlap interval(v))3 do if(lc(v) exists and maxright(lc(v)) q)4 then v lc(v)5 else v rc(v)

Lemma:

1. If line 5 is executed, then vs left subtree contains no interval

that overlaps [q,q].

2. If line 4 is executed, then vs left subtree contains an intervalthat overlaps [q,q] or no interval in vs right subtree overlaps[q,q].

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Theorem: Let I be a set ofn intervals. There is a data structure for I

of size O(n), such that the operations I NSERT, DELETE, and OVER-LAP each take time O(log n).

Theorem: Checking whether n axis-parallel rectangles in the plane

are pairwise disjoint can be done in O(n log n) time.

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Secondary Data Structures

Let v be a node in a search-tree with leaf-oriented storage. The

canonical subsetof v is the set of items stored in the subtree rooted

at v.

In a number of data structures the canonical subsets of the nodes are

stored in a secondary data structures associated with a node.

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A secondary data structure might be an augmented data structure as

well:

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