Download Algorithms Lecture 5 Name:___________________________

Algorithms
Lecture 5
Name:___________________________
In "divide-and-conquer" algorithms (e.g., recursive fibonacci, binary search, merge sort, ...):
Large Problem
Smaller Problem
. . .
1) divide large problem into
one or more smaller problems
Smaller Problem
2) "conquer"/solve
smaller problem(s)
(usually recursively)
3) combine solution(s) of
smaller problem(s) to get
solution to larger problem
1. I want you to think about when a divide-and-conquer algorithm might be “fast enough.”
a.) Why was recursive fibonacci so slow?
b) Why is binary search so fast?
c) Simple sorts (e.g., bubble, selection, insertion) are all Θ(n2), we can employ divide-and-conquer to develop
Θ(nlog n) “advanced” sorting algorithms: Merge sort and Quick Sort. Consider why a divide-and-conquer sort
might be more efficient. Assume that I had a simple Θ(n2) sorting algorithm with n = 100, then there is roughly
1002 / 2 or 5,000 amount of work. Suppose I split the problem down into two smaller problems of size 50.
If I run the n2 algorithm on both smaller problems of size 50, then what would be the approximate amount of
work?
If I further solve the problems of size 50 by splitting each of them into two problems of size 25, then what
would be the approximate amount of work?
What other work do we need to consider in this analysis?
Lecture 5 Page 1
Algorithms
Lecture 5
Name:___________________________
2. Quick sort is another advanced sort that often is quicker than merge sort (hence its name). The general idea
is as follows. Assume “n” items to sort.
Select a “random” item in the unsorted part as the pivot
Rearrange (called partitioning) the unsorted items such that:
Pivot Index
All items < to Pivot
Pivot
Item
All items >= to Pivot
Quick sort the unsorted part to the left of the pivot
Quick sort the unsorted part to the right of the pivot
a) What base case(s) would we have?
b) What would be the theta notation (Θ( )) of the partition function which returns the index of the pivot
after this rearrangement?
c) What would the recurrence relation be for the following cases?
Worst-case recurrence
Best-case recurrence
Lecture 5 Page 2
Algorithms
Lecture 5
Name:___________________________
3. Section 2.6: Arithmetic with Large Integers: (too big for built-in integer representation)
Representation: one decimal digit per array element with sign (0 means positive and 1 means negative)
u = 432,581
6 5 4 3 2 1
4 3 2 5 8 1
0
v = 6,741,312
7 6 5 4 3 2 1
6 7 4 1 3 1 2
0
Let n be the number of digits in larger integers. What are the Θ( ) for the large integer operations:
addition/subtraction?
u ∗ 10m ?
u divide 10m ?
u rem 10m ?
multiplication of two large integers using grammar school “long multiplication” algorithm?
u = 432,581
v = 6,741,312
4
x 6 7
8
4 3
3
4
6
2
...
2
1
5
5
4
5
3
1
8
3
8 1
1 2
6 2
1
..
+
...
2 7 2
4. To apply divide-and-conquer to multiplication, we can divide the n-digit value roughly in two where m =
n/2. For example, u = 432,581 = 432 x 103 + 581 and v = 6741 x 103 + 312. In general,
m
m
m
v = w * 10 + z


n
n/2
digits digits
m
uv = (x * 10 + y) (w * 10 + z)
u = x * 10 + y
Complete:
 n/2  = m
digits
uv =
2m
m
* 10 + (
) * 10 +
5. Write the recurrence for multiplying uv assuming n-digit values. T(n) =
6. Use the “Master Theorem” to determine the Θ( ) for the large-integer-multiplication recurrence.
Suppose a complexity function T(n) is eventually nondecreasing and satisfies the recurrence
T(n) = aT(n/b) + cn k for n > 1, n a power of b
T(1) = d,
where constant a > 0, b is an integer ≥ 2, c > 0, d ≥ 0, and k is an integer ≥ 0. Then
T(n) ∈
{
θ ( nk )
θ ( nk log n)
θ ( nlogba )
if a < bk
if a = bk
if a > bk
a=
b=
k=
Lecture 5 Page 3
Algorithms
Lecture 5
Name:___________________________
7. The “trick” to improving performance:
Let
r = (x + y) (w + z)
r = xw + (xz + wy) + yz
(xz + wy) = r - xw - yz, substitute for r
(xz + wy) = (x + y) (w + z) - xw - yz
a) Rewrite the uv equation by replacing (xz + wy). Recall, u = x * 10m + y, v = w * 10m + z, and m = n/2.
uv = xw * 102m + (xz + wy)*10m + yz
uv = xw * 102m + (
)*10m + yz
b) Write the recurrence for this version of multiplying uv assuming n-digit values. T(n) =
c) Use the “Master Theorem” to determine the Θ( ) for this large-integer-multiplication recurrence.
Lecture 5 Page 4