Download Orthogonal Range Searching

Orthogonal Range Searching
Advanced Data Structures
Computation and Reasoning Laboratory
Graduate Course - Spring 2007
April 16, 2007
Outline
1
Introduction
2
1-Dimensional Range Searching
3
Kd-Trees
4
Range Trees
5
Fractional Cascading
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
2 / 44
Outline
1
Introduction
2
1-Dimensional Range Searching
3
Kd-Trees
4
Range Trees
5
Fractional Cascading
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
3 / 44
Querying a Database
Do databases have something to do with geometry?
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
4 / 44
Querying a Database
Do databases have something to do with geometry?
Queries in a database can be interpreted geometrically.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
4 / 44
Querying a Database
Do databases have something to do with geometry?
Queries in a database can be interpreted geometrically.
Transform records in a database into points in a
multi-dimensional space.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
4 / 44
Querying a Database
Do databases have something to do with geometry?
Queries in a database can be interpreted geometrically.
Transform records in a database into points in a
multi-dimensional space.
Transform queries about the records into queries on the set of
points.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
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A typical query interpreted geometrically
salary
4000
G. Ometer
born: Aug 19, 1954
salary: $3200
3000
19500000
Advanced Data Structures (Spring 2007)
19559999
Orthogonal Range Searching
date of birth
CoReLab Graduate Course
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A query in 3 dimensions
salary
4000
3000
chlidren
4
2
19500000
Advanced Data Structures (Spring 2007)
19559999
Orthogonal Range Searching
date of birth
CoReLab Graduate Course
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The geometric approach
We are interested in answering queries on d fields of the records
in our database.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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The geometric approach
We are interested in answering queries on d fields of the records
in our database.
Transform the records to points in d-dimensional space.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
7 / 44
The geometric approach
We are interested in answering queries on d fields of the records
in our database.
Transform the records to points in d-dimensional space.
The transformed query asks for all points inside a d-dimensional
axis-parallel box.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
7 / 44
The geometric approach
We are interested in answering queries on d fields of the records
in our database.
Transform the records to points in d-dimensional space.
The transformed query asks for all points inside a d-dimensional
axis-parallel box.
Such a query is called ‘‘rectangular’’ or ‘‘orthogonal’’ range query.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
7 / 44
The geometric approach
We are interested in answering queries on d fields of the records
in our database.
Transform the records to points in d-dimensional space.
The transformed query asks for all points inside a d-dimensional
axis-parallel box.
Such a query is called ‘‘rectangular’’ or ‘‘orthogonal’’ range query.
We are going to present data structures for such queries.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
7 / 44
Outline
1
Introduction
2
1-Dimensional Range Searching
3
Kd-Trees
4
Range Trees
5
Fractional Cascading
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
8 / 44
The problem
Data
A set of points P = {p1 , p2 , . . . , pn } in 1-dimensional space (a set of real
numbers).
Query
Which points lie inside a ‘‘1-dimensional query rectangle’’? (i.e. an
interval [x : x 0 ])
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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Efficient Data Structures
Arrays
Solve the problem, but
do not generalize,
do not allow efficient updates.
Balanced Binary Search Trees (BBST)
The leaves of T store the points of P,
internal nodes store splitting values that guide the search.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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Balanced Binary Search Trees
xv
v
xi ≤ xv
pi
Advanced Data Structures (Spring 2007)
xv < xj
pj
Orthogonal Range Searching
CoReLab Graduate Course
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A search with the interval [18 : 77]
49
23
80
10
37
3
3
19
10
19
30
23
30
62
49
37
59
59
70
62
70
89
80
100
100 105
µ0
µ
Advanced Data Structures (Spring 2007)
89
Orthogonal Range Searching
CoReLab Graduate Course
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A search with the interval [x , x 0 ]
Search for x and x 0 in T . The search ends to leaves µ and µ0 .
Report all the points stored at leaves between µ and µ0 plus,
possibly, the points stored at µ and µ0 .
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
13 / 44
A search with the interval [x , x 0 ]
Search for x and x 0 in T . The search ends to leaves µ and µ0 .
Report all the points stored at leaves between µ and µ0 plus,
possibly, the points stored at µ and µ0 .
Remark
The leaves to be reported are the ones of subtrees that are rooted at
nodes whose parents are on the search paths to µ and µ0 .
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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The selected subtrees
root(T )
vsplit
µ0
µ
Advanced Data Structures (Spring 2007)
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Algorithms
1D-RangeQuery(T , [x : x 0 ])
FindSplitNode(T , x , x 0 )
v ← root (T )
while v is not a leaf and (x 0 ≤ xv or x > xv ) do
if x 0 ≤ xv then
v ← lc (v)
else
v ← rc (v)
return v
vsplit ← FindSplitNode(T , x , x 0 )
if vsplit is a leaf then
check if xvsplit must be reported
else {follow the path to x}
v ← lc (vsplit )
while v is not a leaf do
if x ≤ xv then
ReportSubtree(rc (v)) {subtrees right of path}
v ← lc (v)
else
v ← rc (v)
check if xv must be reported
...
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Correctness and Performance
Any reported point lies in the query range.
Any point in the range is reported.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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Correctness and Performance
Any reported point lies in the query range.
Any point in the range is reported.
O (n ) storage.
O (n log n ) preprocessing.
Θ(n ) worst case case query cost.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
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Correctness and Performance
Any reported point lies in the query range.
Any point in the range is reported.
O (n ) storage.
O (n log n ) preprocessing.
Θ(n ) worst case case query cost.
O (k + log n ) output sensitive query cost: O (k ) to report the points
plus O (log n ) to follow the paths to x, x 0 .
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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Outline
1
Introduction
2
1-Dimensional Range Searching
3
Kd-Trees
4
Range Trees
5
Fractional Cascading
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
17 / 44
The problem
Data
A set of points P = {p1 , p2 , . . . , pn } in the plane.
Query
Which points lie inside a query rectangle [x : x 0 ] × [y : y0 ]?
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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The problem
Data
A set of points P = {p1 , p2 , . . . , pn } in the plane.
Query
Which points lie inside a query rectangle [x : x 0 ] × [y : y0 ]?
Remark
A point p = (px , py ) lies inside this rectangle iff px ∈ [x , x 0 ] and
py ∈ [y, y0 ].
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
18 / 44
The way the plane is subdivided
l5
p1
p2
l1
l7
p8
p3
l8
p6
l9
p7
p5
p9
l3
l2
p4
p10
l4
Advanced Data Structures (Spring 2007)
l6
Orthogonal Range Searching
CoReLab Graduate Course
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The way the plane is subdivided
p1
p2
l1
p8
p3
p6
p7
p5
p9
p4
p10
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
19 / 44
The way the plane is subdivided
p1
p2
l1
p8
p3
p6
p7
p5
p9
l2
p4
p10
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
19 / 44
The way the plane is subdivided
p1
p2
l1
p8
p3
p6
p7
p5
p9
l3
l2
p4
p10
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
19 / 44
The way the plane is subdivided
p1
p2
l1
p8
p3
p6
p7
p5
p9
l3
l2
p4
p10
l4
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
19 / 44
The way the plane is subdivided
l5
p1
p2
l1
p8
p3
p6
p7
p5
p9
l3
l2
p4
p10
l4
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
19 / 44
The way the plane is subdivided
l5
p1
p2
l1
p8
p3
p6
p7
p5
p9
l3
l2
p4
p10
l4
Advanced Data Structures (Spring 2007)
l6
Orthogonal Range Searching
CoReLab Graduate Course
19 / 44
The way the plane is subdivided
l5
p1
p2
l1
l7
p8
p3
p6
p7
p5
p9
l3
l2
p4
p10
l4
Advanced Data Structures (Spring 2007)
l6
Orthogonal Range Searching
CoReLab Graduate Course
19 / 44
The way the plane is subdivided
l5
p1
p2
l1
l7
p8
p3
p6
l8
p7
p5
p9
l3
l2
p4
p10
l4
Advanced Data Structures (Spring 2007)
l6
Orthogonal Range Searching
CoReLab Graduate Course
19 / 44
The way the plane is subdivided
l5
p1
p2
l1
l7
p8
p3
p6
l8
l9
p7
p5
p9
l3
l2
p4
p10
l4
Advanced Data Structures (Spring 2007)
l6
Orthogonal Range Searching
CoReLab Graduate Course
19 / 44
The corresponding binary tree
l1
l2
l3
l4
p5
ll55
p4
pp22
p10
l8
pp33
Advanced Data Structures (Spring 2007)
l6
ll57
p9
pp11
Orthogonal Range Searching
pp27
l9
pp63
pp18
CoReLab Graduate Course
20 / 44
Algorithm
BuildKdTree(P , depth )
if P contains only one point then
return a leaf storing this point
else
if depth is even then
split P with vertical l through median x-coord of points in P
P1 ← the set of points left of l or on l
P2 ← the set of points right of l
else
split P with horizontal l through median y-coord of points in P
P1 ← the set of points below l or on l
P2 ← the set of points above l
vleft ← BuidKdTree(P1 , depth + 1)
vright ← BuidKdTree(P2 , depth + 1)
create a node v storing l
lc (v) → vleft
rc (v) → vright
return v
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
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Algorithm
BuildKdTree(P , depth )
if P contains only one point then
return a leaf storing this point
else
if depth is even then
split P with vertical l through median x-coord of points in P
P1 ← the set of points left of l or on l
P2 ← the set of points right of l
else
split P with horizontal l through median y-coord of points in P
P1 ← the set of points below l or on l
P2 ← the set of points above l
vleft ← BuidKdTree(P1 , depth + 1)
vright ← BuidKdTree(P2 , depth + 1)
create a node v storing l
lc (v) → vleft
rc (v) → vright
return v
Remarks
We should split at the
n
-th
2
smallest coordinate.
Preprocessing involves sorting both on x- and y-coordinate.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
21 / 44
Building time and storage
The building time satisfies the recurrence:
T (n ) =
O (1)
O (n ) + 2T ( n2 )
if n = 1
if n > 1
T (n ) = O (n log n ) which subsums the preprocessing time.
O (n ) storage: each leaf stores a distinct point of P.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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Nodes in a kd-tree and regions in the plane
l1
l1
l2
l2
l3
v
l3
region(v)
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
23 / 44
Regions and the query algorithm
Internal nodes of a Kd-tree correspond to rectangular regions of
the plane which can be unbounded on one or more sides.
Such regions are bounded by splitting lines stored at ancestors of
the internal nodes.
region (root (T )) is the whole plane.
A point is stored in a subtree rooted at a node v iff it lies in
region (v).
We search the subtree rooted at v only if the query rectangle
intersects region (v).
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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A query on a kd-tree
l5
p1
p2
l1
l7
p8
l1
p3
p6
l8
l2
l9
p7
p5
l4
p9
l3
l2
p4
p5
ll55
p4
Advanced Data Structures (Spring 2007)
p2
l6
p10
l8
p3
p10
l4
l3
pp11
ll57
p9
pp27
l9
pp63
pp18
l6
Orthogonal Range Searching
CoReLab Graduate Course
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Algorithm
SearchKdTree(v, R )
if v is a leaf then
report the point stored at v if it lies in R
else
if region (lc (v)) is fully contained in R then
ReportSubtree(lc (v))
else
if region (lc (v)) intersects R then
SearchKdTree(lc (v), R )
if region (rc (v)) is fully contained in R then
ReportSubtree(rc (v))
else
if region (rc (v)) intersects R then
SearchKdTree(rc (v), R )
Advanced Data Structures (Spring 2007)
Works for any query range R (e.g.
triangles).
O (k ), in order to report k points.
How many other nodes are visited?
For these nodes v, the query range
intersects region (v).
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Query time analysis
Any vertical line intersects region (lc (root (T ))) or
region (rc (root (T ))) but not both.
If a vertical line intersects region (lc (root (T ))) it will always
intersect the regions corresponding to both children of lc (root (T )).
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Query time analysis
The number of intersected regions in a kd-tree storing n points,
satisfies the recurrence:
Q (n ) =
O (1)
2 + 2Q ( n4 )
√
if n = 1
if n > 1
√
Q (n ) = O ( n ). The total query time is O ( n + k )
The analysis is rather pessimistic: In many practical situations
the query range is small and will intersect much fewer regions.
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Kd-trees in higher-dimensional spaces
Kd-trees can be also used for point sets in 3- or
higher-dimensional spaces.
Assume the dimension d to be a constant:
O (d · n ) storage.
O (d · n log n ) construction time.
1
O (n 1− d + k ) query time.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
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Outline
1
Introduction
2
1-Dimensional Range Searching
3
Kd-Trees
4
Range Trees
5
Fractional Cascading
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
30 / 44
The Range Tree approach
2D range queries are two 1D range queries one on x − and one on
y−coordinate.
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The Range Tree approach
2D range queries are two 1D range queries one on x − and one on
y−coordinate.
Find first the points whose x −coordinate lies in [x : x 0 ] and worry
about the y−coordinate later.
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CoReLab Graduate Course
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The Range Tree approach
2D range queries are two 1D range queries one on x − and one on
y−coordinate.
Find first the points whose x −coordinate lies in [x : x 0 ] and worry
about the y−coordinate later.
During the 1D range query a logarithmic number of subtrees is
selected.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
31 / 44
The Range Tree approach
2D range queries are two 1D range queries one on x − and one on
y−coordinate.
Find first the points whose x −coordinate lies in [x : x 0 ] and worry
about the y−coordinate later.
During the 1D range query a logarithmic number of subtrees is
selected.
The leaves of these subtrees contain exactly the points whose
x −coordinate lies in [x : x 0 ].
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The Range Tree approach
Canonical Subset of a node v
The subset of points P (v) of P stored in the leaves of the subtree rooted
at v.
P (root (T )) = P
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The Range Tree approach
Canonical Subset of a node v
The subset of points P (v) of P stored in the leaves of the subtree rooted
at v.
P (root (T )) = P
The subset of points whose x −coordinate lies in the query range is
a disjoint union of O (log n ) canonical subsets.
Advanced Data Structures (Spring 2007)
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CoReLab Graduate Course
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The Range Tree approach
Canonical Subset of a node v
The subset of points P (v) of P stored in the leaves of the subtree rooted
at v.
P (root (T )) = P
The subset of points whose x −coordinate lies in the query range is
a disjoint union of O (log n ) canonical subsets.
We are not interested in all the points in such subsets.
Advanced Data Structures (Spring 2007)
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CoReLab Graduate Course
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The Range Tree approach
Canonical Subset of a node v
The subset of points P (v) of P stored in the leaves of the subtree rooted
at v.
P (root (T )) = P
The subset of points whose x −coordinate lies in the query range is
a disjoint union of O (log n ) canonical subsets.
We are not interested in all the points in such subsets.
Report the ones whose y−coordinate lies in [y : y0 ]: This is
another 1D query.
Advanced Data Structures (Spring 2007)
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A 2-dimensional Range Tree
T
binary search tree
on x-coordinates
Tassoc (v)
binary search tree
on y-coordinates
v
P (v)
P (v)
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Algorithm
Build2DRangeTree(P )
Build a BST Tassoc on the set Py
Store the points of P at the leaves of Tassoc
if P contains only one point then
Create a leaf v storing this point
Associate Tassoc with v
else
Split P into Pleft and Pright through xmid
vleft ← Build2DRangeTree(Pleft )
vright ← Build2DRangeTree(Pright )
create a node v storing xmid
lc (v) ← vleft
rc (v) ← vright
Associate Tassoc with v
return v
Advanced Data Structures (Spring 2007)
Preprocessing involves maintaining
two lists of points.
One sorted on x −coordinate and one
sorted on y−coordinate.
The time spend at a node in the
main tree is linear in the size of its
canonical subset.
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Range Tree storage
p
p
p
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
p
CoReLab Graduate Course
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Storage and costruction time
Each point is stored only once at a given depth.
The total depth is O (log n ): the amount of storage is O (n log n ).
Since the time spend at a node in the main tree is linear in the
size of its canonical subset the total construction time is the same
as the amount of storage.
Presorting is O (n log n ).
Total construction time is O (n log n ).
Advanced Data Structures (Spring 2007)
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Algorithm
2DRangeQuery(T , [x : x 0 ] × [y : y0 ])
vsplit ← FindSplitNode(T , x , x 0 )
if vsplit is a leaf then
check if xvsplit must be reported
else {follow the path to x}
v ← lc (vsplit )
while v is not a leaf do
if x ≤ xv then
1DRangeQuery(Tassoc (rc (v)), [y : y0 ])
v ← lc (v)
else
v ← rc (v)
check if xv must be reported
...
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Query time analysis
The time spend to report the points whose y−coordinate lie in the
range [y : y0 ] is O (log n + kv ) where kv is the number of points
reported in this call.
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CoReLab Graduate Course
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Query time analysis
The time spend to report the points whose y−coordinate lie in the
range [y : y0 ] is O (log n + kv ) where kv is the number of points
reported in this call.
Q (n ) =
visited.
P
v
O ((log n ) + kv ) where the summation is over all nodes
Advanced Data Structures (Spring 2007)
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CoReLab Graduate Course
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Query time analysis
The time spend to report the points whose y−coordinate lie in the
range [y : y0 ] is O (log n + kv ) where kv is the number of points
reported in this call.
Q (n ) =
visited.
P
v
O ((log n ) + kv ) where the summation is over all nodes
P
points. The search paths
v kv = k, the total number of reported
P
of x and x 0 have length O (log n ):
O
(
log
n ) = O (log2 n ).
v
Advanced Data Structures (Spring 2007)
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CoReLab Graduate Course
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Query time analysis
The time spend to report the points whose y−coordinate lie in the
range [y : y0 ] is O (log n + kv ) where kv is the number of points
reported in this call.
Q (n ) =
visited.
P
v
O ((log n ) + kv ) where the summation is over all nodes
P
points. The search paths
v kv = k, the total number of reported
P
of x and x 0 have length O (log n ):
O
(
log
n ) = O (log2 n ).
v
Q (n ) = O (log2 n + k ).
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
38 / 44
Higher-Dimensional Range Trees
Advanced Data Structures (Spring 2007)
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CoReLab Graduate Course
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Higher-Dimensional Range Trees
P is a set on n points in d −dimensional space (d ≥ 2):
O (n logd −1 n ) storage,
O (n logd −1 n ) construction time,
O (logd n + k ) query time.
Advanced Data Structures (Spring 2007)
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Outline
1
Introduction
2
1-Dimensional Range Searching
3
Kd-Trees
4
Range Trees
5
Fractional Cascading
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
41 / 44
The idea of Fractional Cascading
S1 , S2 are two set of objects with real number keys.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
42 / 44
The idea of Fractional Cascading
S1 , S2 are two set of objects with real number keys.
The problem is to report all objects in S1 and S2 whose keys lie in
[y : y0 ].
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
42 / 44
The idea of Fractional Cascading
S1 , S2 are two set of objects with real number keys.
The problem is to report all objects in S1 and S2 whose keys lie in
[y : y0 ].
The keys are in sorted order in arrays A1 and A2 .
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
42 / 44
The idea of Fractional Cascading
S1 , S2 are two set of objects with real number keys.
The problem is to report all objects in S1 and S2 whose keys lie in
[y : y0 ].
The keys are in sorted order in arrays A1 and A2 .
Solution: two binary searches in A1 and A2 .
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
42 / 44
The idea of Fractional Cascading
S1 , S2 are two set of objects with real number keys.
The problem is to report all objects in S1 and S2 whose keys lie in
[y : y0 ].
The keys are in sorted order in arrays A1 and A2 .
Solution: two binary searches in A1 and A2 .
If S2 ⊆ S1 we can avoid the binary search in A2 .
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
42 / 44
The idea of Fractional Cascading
S1 , S2 are two set of objects with real number keys.
The problem is to report all objects in S1 and S2 whose keys lie in
[y : y0 ].
The keys are in sorted order in arrays A1 and A2 .
Solution: two binary searches in A1 and A2 .
If S2 ⊆ S1 we can avoid the binary search in A2 .
A1
3 10 19 23 30 37 59 62 70 80 100 105
A2 10 19 30 62 70 80 100
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
42 / 44
A Layered Range Tree
17
8
52
5
15
2
7
2
5
12
7
8
12
33
17
15
58
21
21
41
33
41
58
67
52
67
93
3 10 19 23 30 37 49 59 62 70 80 89 95 99
3 10 19 37 62 80 99
3 62 99
10 19 37 80
19 80
19
80
Advanced Data Structures (Spring 2007)
10 37
10
23 30 49 59 70 89 95
37
3 99
3
62
99
23 30 49 95
30 49
49
30
59 70 89
23 95
95
Orthogonal Range Searching
23
59
70 89
89
70
CoReLab Graduate Course
43 / 44
A query in a Layered Range Tree
The query range is [x : x 0 ] × [y : y0 ].
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
44 / 44
A query in a Layered Range Tree
The query range is [x : x 0 ] × [y : y0 ].
At vsplit find the entry in A(vsplit ) whose y−coordinate is larger
than or equal to y in O (log n ) time.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
44 / 44
A query in a Layered Range Tree
The query range is [x : x 0 ] × [y : y0 ].
At vsplit find the entry in A(vsplit ) whose y−coordinate is larger
than or equal to y in O (log n ) time.
For all O (log n ) nodes on the paths to x and x 0 maintain pointers
to the entries in A whose y−coordinate is larger than or equal to y
in O (1) time.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
44 / 44
A query in a Layered Range Tree
The query range is [x : x 0 ] × [y : y0 ].
At vsplit find the entry in A(vsplit ) whose y−coordinate is larger
than or equal to y in O (log n ) time.
For all O (log n ) nodes on the paths to x and x 0 maintain pointers
to the entries in A whose y−coordinate is larger than or equal to y
in O (1) time.
Report the points of A(v) in O (1 + kv ) time, kv is the number of
reported points at node v.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
44 / 44
A query in a Layered Range Tree
The query range is [x : x 0 ] × [y : y0 ].
At vsplit find the entry in A(vsplit ) whose y−coordinate is larger
than or equal to y in O (log n ) time.
For all O (log n ) nodes on the paths to x and x 0 maintain pointers
to the entries in A whose y−coordinate is larger than or equal to y
in O (1) time.
Report the points of A(v) in O (1 + kv ) time, kv is the number of
reported points at node v.
Total query time becomes O (log n + k ).
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
44 / 44
A query in a Layered Range Tree
The query range is [x : x 0 ] × [y : y0 ].
At vsplit find the entry in A(vsplit ) whose y−coordinate is larger
than or equal to y in O (log n ) time.
For all O (log n ) nodes on the paths to x and x 0 maintain pointers
to the entries in A whose y−coordinate is larger than or equal to y
in O (1) time.
Report the points of A(v) in O (1 + kv ) time, kv is the number of
reported points at node v.
Total query time becomes O (log n + k ).
Fractional cascading also imporves the query time of
higher-dimensional range trees by a logarithmic factor.
Advanced Data Structures (Spring 2007)
Orthogonal Range Searching
CoReLab Graduate Course
44 / 44