Thm07 - Augmenting Ds p1

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Augmenting Data Structures

Advanced Algorithms & Data Structures

Lecture Theme 07 Part I

Prof. Dr. Th. Ottmann

Summer Semester 2006


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Augmentation is a process of extending a data structure in order to support additional

functionality. It consists of four steps:

1. Choose an underlying data structure.

2. Determine the additional information to be maintained in the underlying datastructure.

3. Verify that the additional information can be maintained for the basic modifying

operations on the underlying data structure.

4. Develop new operations.

Augmentation Process


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Examples for Augmenting DS

Dynamic order statistics: Augmenting binary search trees by size information

D-dimensional range trees: Recursive construction of (static) d-dim range trees

Min-augmented dynamic range trees: Augmenting 1-dim range trees by min-

information

Interval trees

Priority search trees


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information

Interval trees



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Problem: Given a set S of numbers that changes under insertions and deletions,

construct a data structure to store S that can be updated in O(log n) time and that canreport the k-th order statistic for any k in O(log n) time.

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S

Dynamic Order Statistics


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Binary Search Trees and Order Statistics

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Binary Search Trees and Order Statistics

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Retrieving an element with a given rank:

For a given i, find the i-th smallest key in the

set.

Determining the rank of an element:

For a given (pointer to a) key k, determine the

rank of k in the set of keys.


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Every node v stores two pieces of

information:

Its key

The number of its descendants

(The size of the subtree with root v)

Augmenting the Data Structure

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Find the rank of key x in the tree with root node v:

Rank(v, x)

1 ifx= key(v)

2 then return 1 + size(left(v))

3 ifx< key(v)

4 then return Rank(left(v),x)

5 else return 1 + size(left(v)) + Rank(right(v),x)

How To Determine The Rank of an Element

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How to Find the k-th Order Statistic

Find (a pointer to) the node containing the

k-th smallest key in the subtree rooted

at node v.

Select(v, k)

1 ifk= size(left(v)) + 1

2 then return v

3 ifk size(left(v))

4 then return Select(left(v), k)

5 else return Select(right(v), k 1 size(left(v)))

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Maintaining Subtree Sizes Under Insertions

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11 Insert operationInsert node as into a standardbinary search tree.

Add 1 to the subtree size of every

ancestor of the new node.


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Insert operation

Insert node as into a standardbinary search tree


ancestor of the new node


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Insert operation

Insert node as into a standardbinary search tree


ancestor of the new node


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Maintaining Subtree Sizes Under Deletions

Delete operation

Delete node as from a standardbinary search tree

Subtract 1 from the subtree size of

every ancestor of the deleted node


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Maintaining Subtree Sizes Under Rotations

s1

s2 s3

s4

s5

s1

s3

s5

s4

s5 + s3 + 1


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Theorem: There exists a data structure to represent a dynamically changing set S of

numbers with the following properties:

The data structure can be updated in O(log n) time after every insertion or deletion

into or from S.

The data structure allows us to determine the rank of an element or to find the

element with a given rank in O(log n) time.

The data structure occupies O(n) space.

Dynamic Order StatisticsSummary


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information

Interval trees Priority search trees


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4-Sided Range Queries


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4-Sided Range Queries

Goal: Build a static data structure of size O(nlog n) that can answer 4-sided rangequeries in O(log2 n+ k) time.


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Orthogonal d-dimensional Range Search

Build a static data structure for a set Pofn points in d-space that supports d-dim range

queries:

d-dim range query: Let Rbe a d-dim orthogonal hyperrectangle, given by

d ranges [x1, x1], , [xd, xd]:

Find all pointsp = (p1, ,pd) Psuch that x1 p1 x1,,xd pd xd.

Special cases:

1-dim range query: 2-dim range query:

x1

x1

x1

x1

x2

x2


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1-dim Range Search

Standard binary search trees support also 1-dim range queries:

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1-dim Range Search

Leaf-search-tree:

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1-dim Range Tree

A 1-dim range tree is a leaf-search tree for the x-values (points on the line).

Internal nodes have routers guiding the search to the leaves: We choose the maximal

x-value in left subtree as router.

Range search: In order to find all points in a given range [l, r] search for the boundary

values l and r.

This is a forked path; report all leaves of subtrees rooted at nodes v in between the

two search paths whose parents are on the search path.


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The selected subtrees

l r

Split node


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Canonical Subsets

The canonical subset of node v, P(v), is the subset of points of P stored at the leaves

of the subtree rooted at v.

If v is a leaf, P(v) is the point stored at this leaf.

If v is the root, P(v) = P.

Observations:

For each query range [l, r] the set of points with x-coordinates falling into this rangeis the disjoint union of O(log n) canonical subsets of P.

A node v is called an umbrella node for the range [l, r], if the x-coordinates of all

points in its canonical subset P(v) fall into the range, but this does not hold for the

predecessor of v.

All k points stored at the leaves of a tree rooted at node v, i.e. the k points in a

canonical subset P(v), can be reported in time O(k).


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1-dim Range Tree: Summary

Let P be a set of n points in 1-dim space.

P can be stored in a balanced binary leaf-search tree such that the following holds:

Construction time: O(n log n)

Space requirement: O(n)

Insertion of a point: O(log n) time

Deletion of a point: O(log n) time

1-dim-range-query: Reporting all k points falling into a given query range can be

carried out in time O(log n + k).The performance of 1-dim range trees does not depend on the chosen balancing

scheme!


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2-dim Range tree: The Primary Structure

Static binary leaf-search tree overx-

coordinates of points.


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The Primary Structure




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Every leaf represents a vertical slab

of the plane.


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of the plane.

Every internal node represents a slab

that is the union of the slabs of its

children.


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of the plane.



children.


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of the plane.



children.


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of the plane.



children.


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Answering 2-dim Range Queries

Normalize queries to end on slab

boundaries.

Query decomposes into O(log n)

subqueries.

Every subquery is a

1-dimensional range query on y-coordinates of all points in the slab of

the corresponding node.

(x-coordinates do not matter!)


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The selected subtrees

l r

Split node


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Answering Queries


boundaries.


subqueries.

Every subquery is a





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Answering Queries


boundaries.

Query decomposes into O(lg n)

subqueries.

Every subquery is a





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Answering Queries


boundaries.


subqueries.

Every subquery is a





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2-dim Range Tree

v

Tx

Ty(v)

Ix(v)

x

y

2 di R T


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2-dim Range Tree

A 2-dimensional range tree for storing a set P of n points in the x-y-plane is:

A 1-dim-range tree Tx for the x-coordinates of points.

Each node v of Tx has a pointer to a 1-dim-range-tree Ty(v) storing all points which

fall into the interval Ix(v). That is: Ty(v) is a 1-dim-range-tree based on the y-

coordinates of all points p P with p Ix(v).

Leaf-search-tree on x-coordinates of points

Leaf-search-tree on

y-coordinates of poins

v

2 di R T


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2-dim Range Tree

A 2-dim range tree on a set of n points in the plane requires O(n log n) space.

p

p

p

p

A point p is stored in all associated

range trees Ty(v) for all nodes v on the

search path to px in Tx.

Hence, for each depth d, each point p occurs

in only one associated search structure Ty(v)

for a node v of depth d in Tx.

The 2-dim range tree can be constructed in

time O(n log n).

(Presort the points on y-coordinates!)

Th 2 Di i l R T


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The 2-Dimensional Range Tree

Primary structure:

Leaf-search tree onx-coordinates of points

Every node stores a secondary

structure:

Balanced binary search tree on y-

coordinates of points in the nodes

slab.

Every point is stored in secondary

structures ofO(log n) nodes.

Space: O(n log n)

A i Q i


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Answering Queries

Every 2-dimensional range query

decomposes into O(log n) 1-

dimensional range queries

Each such query takes O(log n + k)

time

Total query complexity:

O(log2 n + k)

2 dim Range Query


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2-dim Range Query

Let P be a set of points in the plane stored in a 2-dim range tree and let a 2-dim range

R defined by the two intervals [x, x], [y, y] be given. The all k points of P falling

into the range R can be reported as follows:

1. Determine the O(log n) umbrella nodes for the range [x, x], i.e. determine the

canonical subsets of P that together contain exactly the points with x-coordinates

in the range [x, x]. (This is a 1-dim range query on the x-coordinates.)

2. For each umbrella node v obtained in 1, use the associated 1-dim range tree Ty(v)

in order to select the subset P(v) of points with y-coordinates in the range [y, y].

(This is a 1-dim range query for each of the O(log n) canonical subsets obtained

in 1.)

Time to report all k points in the 2-dim range R: O(log2 n + k).

Query time can be reduced to O(log n +k) by a technique known as fractional

cascading.

The 3 Dimensional Range Tree


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The 3-Dimensional Range Tree

Primary structure:Search tree on

x-coordinates of points

Every node stores a secondary

structure:

2-dimensional range tree on points in

the nodes slab.

Every point is stored in secondary

structures ofO(log n) nodes.

Space: O(nlog2 n)

Answering Queries


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Answering Queries

Every 3-dimensional range query

decomposes into O(log n) 2-

dimensional range queries

Each such query takes O(log2 n + k)

time

Total query complexity:

O(log3 n + k)

d Dimensional Range Queries


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d-Dimensional Range Queries

Primary structure:

Search tree on x-coordinates

Secondary structures:

(d 1)-dimensional range trees

Space requirement:

O(n logd 1 n)

Query time:O(n logd 1 n)

Updates are difficult!


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Updates are difficult!

Insertion ordeletion of a point p in a 2-dim range tree requires:

1. Insertion or deletion of p into the primary range tree Tx

according to the x-

coordinate of p

2. For each node v on the search path to the leaf storing p in Tx, insertion or deletion

of p in the associated secondary range tree Ty(v).

Maintaining the primary range tree balanced is difficult, except for the case d = 1!

Rotations in the primary tree may require to completely rebuild the associated range

trees along the search path!

Range TreesSummary


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Range TreesSummary

Theorem: There exists a data structure to represent a static set S of n points in d

dimensions with the following properties:

The data structure allows us to answer range queries in

O(logdn + k) time. The data structure occupies O(n logd 1 n) space.

Note: The query complexity can be reduced to O(logd 1 n + k), ford 2, using a very

beautiful technique called fractional cascading.



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information

Interval trees


minXinRectangle Queries


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Problem: Given a set P of points that changes under insertions and deletions,

construct a data structure to store P that can be updated in O(log n) time and that can

find the point with minimal x-coordinate in a given range below a given

threshold in O(log n) time.

l r

y0minXinRectangle(l, r, y0)

Assumption: All points have pairwise different x-coordinates



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lr

y0

minXinRectangle(l, r, y0)

Assumption: All points have pairwise different x-coordinates

Min-augmented Range Tree


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Min augmented Range Tree

(2, 12) (3, 4) (4, 11)

(5, 3) (8, 5)

(11, 21)

(14, 7)

(21, 8)(15, 2) (17, 30)

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Two data structures in one:

Leaf-search tree on

x-coordinates of points

Min-tournament tree

on y-coordinates of points

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g ( , , y0)

l r

Split node

Search for the boundary values l, r.

Find the leftmost umbrella node with

a min-field y0.



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g ( , , y0)

l r

Split node

Search for the boundary values l, r.

Find the leftmost umbrella node with

a min-field y0.

Proceed to the left son of the current

node, if its min-field is y0, and to

the right son, otherwise.

Return the point at the leaf.

minXinRectangle(l, r, y0) can be found in time O(height of tree).

Updates


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p

Insert operation

Insert node as into a standard binary leaf search tree.

Adjust min-fields of every ancestor of the new node by playing a min tournament for

each node and its sibling along the search path.

Delete operation: Similar

Maintaining min-fields under Rotations


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g

s1

s2

s3

s4

s5

s1

s3s5

s4

min{s5, s3}

Min-augmented Range TreesSummary


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Theorem: There exists a data structure to represent a dynamic set S of n points in

the plane with the following properties:

The data structure allows updates and to answerminXinRectangle(l, r, y0) queries inO(log n) time. The data structure occupies O(n) space.

Note: The data structure can be based on an arbitrary scheme of balanced binary leafsearch trees.

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Thm07 - Augmenting Ds p1