Download Range tree solution: Input Range: 3.5

  Problem: Given a set of n points in
Rd, preprocess them such that
reporting or counting the k points
inside a d-dimensional axis-parallel
box will be most efficient. "
Y
  Desired output-sensitive query time
complexity – O(k+f(n)) for reporting
and O(f(n)) for counting, where f(n)=o
(n), e.g. f(n)=logn."
X
  Sample application: Report all cities
within 100 mile radius of Boston."
15.
25.
 Range tree solution:"
  In a one-dimensional world, points are
real numbers and the query is two
numbers (a,b)."
"   Sort points."
"   Construct a binary balanced
tree, storing the points in its
leaves. "
"   Each tree node stores the
largest value of its left sub-tree."
  Simple O(logn) algorithm:"
a
"   Preprocessing: Sort points in O(nlogn)."
"   Query: (Binary) search for a and b in list in O(logn).!
" "
List all values inbetween."
b
-4 -2
01
3 5
1
-2
5
  Cannot be easily generalized to higher
dimensions (why not ?)."
4-
-4
-2
0
0
3
1
3
7
5
35.
-4
  Mark the vertex at which the search
paths diverge as V-split. "
-4
 Build a data structure storing a “small” number of
canonical subsets, such that:"
1
-2
-2
5
0
0
V-split
3
7
5
  Continue to find the two boundaries, reporting values in the
subtrees:"
"When going left (right), report the entire right (left) subtree."
  When reaching a leaf, check it exhaustively."
55.
"   The c.s. may overlap."
"   Every query may be answered as the union of a “small”
number of c.s."
 The geometry of the space enables this."
3
1
11
45.
Input Range: 3.5-8.2
  Required time for finding a leaf: O(log
n)."
  Find the two boundaries of the given
range in the leaves u and v. "
  Report all the leaves in maximal
subtrees between u and v."
7
7
 Two extremes:"
11
"   Singletons – O(k) query time, even for counting."
"   Power set – O(1) query time. O(2n) storage."
65.
7
11
15
7
  Given a set of points in 2D."
  Bound the points by a rectangle."
  Split the points into two (almost) equal size
groups, using a horizontal or vertical line."
  Continue recursively to partition the subsets,
until they are small enough."
  Canonical subsets are subtrees."
V-split
24
canonical subset
{9,12,14,15}
3
1
1
4
{3}
3
12
{4,7}
4
{17,20}
14
9
7
9
12
14
17
15
17
20
27
20
25
22
{22}
22
24
25
29
27
29
31
v
u
2
23
canonical subsets are subtrees
75.
L6
L4
A
  Partitions 2D space into
axis-aligned rectangular
regions."
  Nodes represent partition
lines, and leaves represent
input points."
L2
85.
L1
F
E
B
G
C
L4
L3
H
D
L7
L5
L1
construction complexity:
L2
L3
L4
A
L5
B
C
E
L7
F
G
2
H
"   Counting: Add its
count."
"   Reporting:
Report entire
subtree."
A
B
C
D
L3
D
L6
E
2
F
L7
H
D
L8
B
L7
4
2
L5
C
H L3
D
L6
E
F
2
L7
G
H
105.
  k nodes are reported. How much time is spent on internal nodes?
The nodes visited are those that are stabbed by R but not
contained in R. How many such nodes are there ?"
  Theorem: Every side of R stabs O(√n) cells of the tree."
  Proof: Extend the side to a full line (WLOG - horizontal): "
"   In the first level it stabs two children."
"   In the next level it stabs (only) two of the four grandchildren.
"   Thus, the recursive equation is:"
"
3
  Total query time: O(√n + k)."
L8
G
L3
I
L7
5
LL55
E
G H
L5
2
2
B
L2 C
9
LL11
L6
F
A
R
2
LL44
L1
L4
A
G
L5
F
L3
2
L4
L6
E
B
L2 C
4
L6
D
A
L2
95.
  Given an axis-parallel
range query R, search
for this range in the tree. "
  Traverse only subtrees
which represent regions
intersecting R."
  If a subtree is contained
entirely in R:"
L4
  Each node in the tree defines
an axis-parallel rectangle in the
plane, bounded by the lines
marked by this vertexʼs
ancestors."
  Label each node with the
8
number of points in that
L1
rectangle. "
L1
2
I
115.
125.
 For a d-dimensional space:"
"   Construction time: O(d nlogn)."
"   Space Complexity: O(dn)."
"   Query time complexity: O(dn1-1/d+k)."
Question: Are kd trees useful for non-orthogonal
range queries, e.g. disks, convex polygons ?
Fact: Using interval trees, orthogonal range
queries may be solved in O(logd-1n+k) time and
O(nlogd-1n) space.
135.