CS 473
Homework 1 (due February 2, 2009)
Spring 2010
1.
(a) How many cells are there, as a function of
n
? Prove your answer is correct.
Solution:
A kdtree is a binary tree, where each internal node corresponds to a cut through
one of the points, and each leaf corresponds to a cell. A kdtree for
n
points is a binary tree
with
n
internal nodes and therefore has
n
+
1
leaves.
We can also prove this fact by induction. A kdtree for 0 points has a single cell: the entire
plane. For any
n
>
0, a kdtree for
n
points consists of a root node and two subkdtrees.
Suppose the left subtree contains
k
points; then the right subtree contains
n

k

1 points.
The induction hypothesis implies that the left subtree has
k
+
1 cells, and the right subtree
has
n

k
cells. So the total number of cells is
(
k
+
1
)+(
n

k
) =
n
+
1, as claimed.
In both proofs, the fact that the tree is balanced is irrelevant!
±
Rubric:
1 point max: ½ for answer (
n
+
1
) + ½ for proof. Only one proof is necessary;
these are not the only correct proofs!
(b) In the worst case,
exactly
how many cells can a horizontal line cross, as a function of
n
? Prove your
answer is correct. Assume that
n
=
2
k

1
for some integer
k
.
Solution:
Let
H
(
n
)
be the number of cells crossed by a horizontal line in a kdtree for
n
points whose toplevel cut is horizontal, and let
V
(
n
)
be the number of cells crossed if
the toplevel cut is vertical. We have a pair of mutual recurrences
H
(
n
) =
V
(
b
n
/
2
c
)
and
V
(
n
) =
2
H
(
b
n
/
2
c
)
, with base cases
H
(
0
) =
V
(
0
) =
1. we can combine these into a single
recurrence
V
(
n
) =
2
V
(
b
n
/
4
c
)
, with base cases
V
(
0
) =
1 and
V
(
1
) =
2. The related function
v
(
k
) =
lg
V
(
2
k

1
)
satisﬁes the even simpler recurrence
v
(
k
) =
v
(
k

2
)+
1 with base cases
v
(
0
) =
0 and
v
(
1
) =
1. The solution
v
(
k
) =
d
k
/
2
e
now follows by a oneline induction proof,
which implies that
V
(
2
k

1
) =
2
d
k
/
2
e
. Rewriting in terms of
n
, we ﬁnd that
V
(
n
) =
p
n
+
1
if k is even, and V
(
n
) =
p
2
(
n
+
1
)
if k is odd.
±
Rubric:
2 points max: 1 point for answer and 1 point for proof.

½ if the answer is
not given as a function of
n
, or is not simpliﬁed. This is not the only correct proof!
(c) Suppose we have
n
points stored in a kdtree. Describe and analyze an algorithm that counts the
number of points above a horizontal line (such as the dashed line in the ﬁgure) as quickly as possible.
[Hint: Use part (b).]
Solution:
We prepose the following recursive algorithm. The main idea is to recursively
calculate the total number of points above the line in both subtrees, but we don’t recurse if
all points in a subtree lie above the line, or all points in a subtree lie below the line.
Each node
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 Spring '08
 Chekuri,C
 Mathematical Induction, Recursion

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