DAT600: Sorting Algorithms and Running Times – Find-Max-Subarray – Engineering Assignment Help

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Internal Code: 3IIA
Engineering Assignment Help
Task: 1
Running Times
For a given problem with input size
n
, algorithms A, B, C are executed. In terms of
running time, one of the algorithms is O(
n
), another is O(
n
log
n
) and the third is O(
n
2
). Some measured running times of these algorithms are given below:

Engineering

Identify which algorithm has which big-Oh running time and explain your answer.

1.2)
Calculate big-Oh running time, in terms of
n
, of the algorithm D shown below:
Algorithm D (n) {
s = 0; for i = 1 to n -1,
for j = n downto 1 in steps j = j / 2,
s = s + i*j; return s; }

Task: 2

Algorithm Design

Let
E
= {
e
1
,
e
2
, …, e
n
} be a sequence of integers. We say that an integer
x
forms a majority in
E
if
x
appears more than floor(
n
/2) times in
E
. For instance, the integer 3 is a majority in sequence
E
= {2, 3, 3, 2, 3, 3}, whereas the sequence
E
= {6, 3, 2, 7, 3, 1} has no majority.
Assume that for each
i
, we have 1 ?
e
i
?
k
, where
k
is an integer. Describe an algorithm, with O(
n + k
) running time, which decides if
E
has a majority.

Task: 3

Master Theorem

Using the Master theorem, find the running time (Big-O) of the following algorithms:
3.1) Merge sort is a classical recursive divide-and-conquer algorithm for sorting an array. The algorithm splits the array in half, recursively sorts the two halves, and then merges the two sorted sub-arrays into the final sorted array. Time taken for the algorithm can be denoted as:

Engineering

3.2) Time taken for the Karatsuba’s multiplication algorithm is given by the following
recurrence equation:

Engineering

Task: 4

Understanding how “Find-Max-Subarray” and “Quicksort” works

4.1) Show step-by-step, how the procedure “find-max-subarray” determines the
maximum subarray of the following row of numbers: A = [10 -2 8 -7 13 23 -18 14 24 8 -10 8]
4.2) Show step-by-step, how quicksort sorts the following row of numbers:
A = [10 2 8 7 13 23 18 14 24 8]
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