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Augmenting Data Structures
Advanced Algorithms & Data Structures
Lecture Theme 07 – Part I
Prof. Dr. Th. Ottmann
Summer Semester 2006
Augmentation Process
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 data
structure.
3. Verify that the additional information can be maintained for the basic modifying
operations on the underlying data structure.
4. Develop new operations.
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Examples for Augmenting DS
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Dynamic order statistics: Augmenting binary search trees by size information
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D-dimensional range trees: Recursive construction of (static) d-dim range trees
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Min-augmented dynamic range trees: Augmenting 1-dim range trees by mininformation
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Interval trees
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Priority search trees
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Examples for Augmenting DS
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Dynamic order statistics: Augmenting binary search trees by size information
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D-dimensional range trees: Recursive construction of (static) d-dim range trees
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Min-augmented dynamic range trees: Augmenting 1-dim range trees by mininformation
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Interval trees
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Priority search trees
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Dynamic Order Statistics
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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
can report the k-th order statistic for any k in O(log n) time.
S
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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.
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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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Augmenting the Data Structure
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Every node v stores two pieces
of information:
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Its key
The number of its
descendants (The size of
the subtree with root v)
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How To Determine The Rank of an Element
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Find the rank of key x in the tree with root node v:
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Rank(v, x)
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1
if x = key(v)
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2
then return 1 + size(left(v))
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if x < key(v)
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then return Rank(left(v), x)
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else return 1 + size(left(v)) + Rank(right(v), x)
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How to Find the k-th Order Statistic
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Find (a pointer to) the node containing the
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k-th smallest key in the subtree rooted
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at node v.
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Select(v, k)
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if k = size(left(v)) + 1
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then return v
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if k ≤ size(left(v))
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then return Select(left(v), k)
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else return Select(right(v), k – 1 – size(left(v)))
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Maintaining Subtree Sizes Under Insertions
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Insert operation
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Insert node as into a standard
binary search tree.
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Add 1 to the subtree size of
every ancestor of the new
node.
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Maintaining Subtree Sizes Under Insertions
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Insert operation
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Insert node as into a standard
binary search tree
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Add 1 to the subtree size of
every ancestor of the new node
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Maintaining Subtree Sizes Under Insertions
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Insert operation
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Insert node as into a standard
binary search tree
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Add 1 to the subtree size of
every ancestor of the new node
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Maintaining Subtree Sizes Under Deletions
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Delete operation
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Delete node as from a standard
binary search tree
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Subtract 1 from the subtree size
of every ancestor of the deleted
node
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Maintaining Subtree Sizes Under Rotations
s1
s1
s3
s2
s4
s5
s5 + s3 + 1
s4
s5
s3
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Dynamic Order Statistics—Summary
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Theorem: There exists a data structure to represent a dynamically changing set S
of numbers with the following properties:
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The data structure can be updated in O(log n) time after every insertion or deletion
into or from S.
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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.
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The data structure occupies O(n) space.
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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
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Min-augmented dynamic range trees: Augmenting 1-dim range trees by mininformation
•
Interval trees
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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(n log n) that can answer 4-sided range
queries in O(log2 n + k) time.
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Orthogonal d-dimensional Range Search
Build a static data structure for a set P of n points in d-space that supports d-dim range
queries:
d-dim range query: Let R be a d-dim orthogonal hyperrectangle, given by
d ranges [x1, x1‘], …, [xd, xd‘]:
Find all points p = (p1, …, pd)  P such that x1 ≤ p1≤ x1‘,…,xd ≤ pd ≤ xd.
Special cases:
1-dim range query:
2-dim range query:
x2‘
x1
x1‘
x2
x1
x1‘
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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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∞
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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
Split node
l
r
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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 range
is 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
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Static binary leaf-search tree over
x-coordinates of points.
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The Primary Structure
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Static binary leaf-search tree over
x-coordinates of points.
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The Primary Structure
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Static binary leaf-search tree over
x-coordinates of points.
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The Primary Structure
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Static binary leaf-search tree over
x-coordinates of points.
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Every leaf represents a vertical
slab of the plane.
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The Primary Structure
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Static binary leaf-search tree over
x-coordinates of points.
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Every leaf represents a vertical
slab of the plane.
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Every internal node represents a
slab that is the union of the slabs
of its children.
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The Primary Structure
•
Static binary leaf-search tree over
x-coordinates of points.
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Every leaf represents a vertical
slab of the plane.
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Every internal node represents a
slab that is the union of the slabs
of its children.
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The Primary Structure
•
Static binary leaf-search tree over
x-coordinates of points.
•
Every leaf represents a vertical
slab of the plane.
•
Every internal node represents a
slab that is the union of the slabs
of its children.
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The Primary Structure
•
Static binary leaf-search tree over
x-coordinates of points.
•
Every leaf represents a vertical
slab of the plane.
•
Every internal node represents a
slab that is the union of the slabs
of its children.
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Answering 2-dim Range Queries
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Normalize queries to end on slab
boundaries.
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Query decomposes into O(log n)
subqueries.
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Every subquery is a
1-dimensional range query on ycoordinates of all points in the slab
of the corresponding node.
(x-coordinates do not matter!)
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The selected subtrees
Split node
l
r
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Answering Queries
•
Normalize queries to end on slab
boundaries.
•
Query decomposes into O(log n)
subqueries.
•
Every subquery is a
1-dimensional range query on ycoordinates of all points in the slab
of the corresponding node.
(x-coordinates do not matter!)
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Answering Queries
•
Normalize queries to end on slab
boundaries.
•
Query decomposes into O(lg n)
subqueries.
•
Every subquery is a
1-dimensional range query on ycoordinates of all points in the slab
of the corresponding node.
(x-coordinates do not matter!)
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Answering Queries
•
Normalize queries to end on slab
boundaries.
•
Query decomposes into O(log n)
subqueries.
•
Every subquery is a
1-dimensional range query on ycoordinates of all points in the slab
of the corresponding node.
(x-coordinates do not matter!)
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2-dim Range Tree
y
Ty(v)
x
Ix(v)
v
Tx
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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:
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A 1-dim-range tree Tx for the x-coordinates of points.
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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 ycoordinates of all points p  P with p  Ix(v).
Leaf-search-tree on
y-coordinates of poins
v
Leaf-search-tree on x-coordinates of points
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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.
A point p is stored in all associated
range trees Ty(v) for all nodes v on the
search path to px in Tx.
p
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!)
p
p
p
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The 2-Dimensional Range Tree
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Primary structure:
Leaf-search tree on
x-coordinates of points
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Every node stores a secondary
structure:
Balanced binary search tree on ycoordinates of points in the
node’s slab.
Every point is stored in secondary
structures of O(log n) nodes.
Space: O(n log n)
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Answering Queries
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Every 2-dimensional range query
decomposes into O(log n) 1dimensional range queries
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Each such query takes O(log n +
k′) time
• Total query complexity:
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O(log2 n + k)
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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.
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The 3-Dimensional Range Tree
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Primary structure:
Search tree on
x-coordinates of points
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Every node stores a secondary
structure:
2-dimensional range tree on
points in the node’s slab.
Every point is stored in secondary
structures of O(log n) nodes.
Space: O(n log2 n)
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Answering Queries
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Every 3-dimensional range query
decomposes into O(log n) 2dimensional range queries
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Each such query takes O(log2 n +
k′) time
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Total query complexity:
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O(log3 n + k)
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d-Dimensional Range Queries
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•
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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)
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Updates are difficult!
Insertion or deletion 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 xcoordinate 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!
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Range Trees–Summary
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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(logd n + 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), for d ≥ 2, using a
very beautiful technique called fractional cascading.
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Examples for Augmenting DS
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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 mininformation
•
Interval trees
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Priority search trees
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minXinRectangle Queries
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.
y0
minXinRectangle(l, r, y0)
l
r
Assumption: All points have pairwise different x-coordinates
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minXinRectangle Queries
minXinRectangle(l, r, y0)
y0
l
r
Assumption: All points have pairwise different x-coordinates
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Min-augmented Range Tree
Two data structures in one:
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(2, 12)
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(3, 4)
(4, 11)
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Leaf-search tree on
x-coordinates of points
Min-tournament tree
on y-coordinates of points
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(14, 7)
2
8
15
21
3
2
8
(11, 21)
(15, 2)
(17, 30)
(21, 8)
3
(5, 3)
(8, 5)
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minXinRectangle(l, r, y0)
Search for the boundary values l, r.
Find the leftmost umbrella node with
a min-field ≤ y0.
l
Split node
r
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minXinRectangle(l, r, y0)
Search for the boundary values l, r.
Find the leftmost umbrella node with
a min-field ≤ y0.
Split node
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.
l
minXinRectangle(l, r, y0) can be found in time O(height of tree).
r
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Updates
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
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Maintaining min-fields under Rotations
s1
s1
s3
s2
s4
s5
min{s5, s3}
s4
s5
s3
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Min-augmented Range Trees–Summary
•
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 answer minXinRectangle(l, r, y0) queries
in
O(log n) time. The data structure occupies O(n) space.
•
Note: The data structure can be based on an arbitrary scheme of balanced binary
leaf search trees.
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