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Augmenting Data Structures
Advanced Algorithms & Data Structures
Lecture Theme 07 – Part I
Prof. Dr. Th. Ottmann
Summer Semester 2006
2
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.
Augmentation
Process
3
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
4
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
5
Problem: Given a set S of numbers that changes under insertions and deletions,
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Binary
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Binary
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How to
Find the k-th
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Statistic
Find (a pointer to) the node containing the
k-th smallest key in the subtree rooted
at node v.
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Maintaining
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Maintaining
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Maintaining
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16
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
Statistics—Summary
17
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
18
4-Sided
Range
Queries
19
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.
20
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:
x1
x1
x1‘
x1‘
x2‘
x2
21
1-dim Range Search
Standard binary search trees support also 1-dim range queries:
37
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12 23
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22
1-dim Range Search
Leaf-search-tree:
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∞
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23
23
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.
24
The selected subtrees
l r
Split node
25
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).
26
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!
27
2-di
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Range tree:
The
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31
The
Pri
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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.
32
The
Pri
mary
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.
33
The
Pri
mary
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.
34
The
Pri
mary
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.
35
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36
The selected subtrees
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37
Answering
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38
Answering
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39
Answering
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40
2-dim Range Tree
v
Tx
Ty(v)
Ix(v)
x
y
41
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 ony-coordinates of poins
v
42
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
pp
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!)
43
The 2-Di
mensional
Range
Tree
• P
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Every point is stored in secondary structures of O(log n) nodes.
Space: O(n log n)
44
Answering
Queries
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45
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.
46
The 3-Di
mensional
Range
Tree
• Primary structure:
S
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secondary structure:
2
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Every point is stored in secondary structures of O(log n) nodes.
Space: O(n log2 n)
47
Answering
Queries
• E
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3
-
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c
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p
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x
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:
O(log3
n + k)
48
d-Di
mensional
Range
Queries
• P
r
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s
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:
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49
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 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!
50
Range
Trees–Summary
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.
51
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
52
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.
l r
y0
minXinRectangle(l, r, y0)
Assumption: All points have pairwise different x-coordinates
53
minXinRectangle Queries
lr
y0
minXinRectangle(l, r, y0)
Assumption: All points have pairwise different x-coordinates
54
Min-augmented
Range
Tree
(2, 12) (3, 4) (4, 11)
(5, 3) (8, 5)
(11, 21)
(14, 7)
(21, 8)(15, 2) (17, 30)
2 4
3
11
8
5
14
17
15 21
2
2
2
3
4 3
3
3
Two data structures in one:
Leaf-search tree on x-coordinates of points
Min-tournament tree on y-coordinates of points
82
55
minXinRectangle(l, r, y0)
l r
Split node
Search for the boundary values l, r.Find the leftmost umbrella node witha min-field ≤ y0.
56
minXinRectangle(l, r, y0)
l r
Split node
Search for the boundary values l, r.Find the leftmost umbrella node witha min-field ≤ y0.
Proceed to the left son of the currentnode, if its min-field is ≤ y0, and tothe right son, otherwise. Return the point at the leaf.
minXinRectangle(l, r, y0) can be found in time O(height of tree).
57
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
58
Maintaining
min-fields under
Rotations
s1
s2s3
s4 s5
s1
s3s5
s4
min{s5, s3}
59
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, y 0) 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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