University of Hawaii
ICS141:
Discrete Mathematics for
Computer Science I
Dept. Information & Computer Sci., University of Hawaii
Jan Stelovsky
based on slides by Dr. Baek and Dr. Still
Originals by Dr. M. P. Frank and Dr. J.L. Gross
Provided by McGraw-Hill
ICS 141: Discrete Mathematics I – Fall 2011
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Lecture 14
Chapter 3. The Fundamentals
3.1 Algorithms
ICS 141: Discrete Mathematics I – Fall 2011
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Algorithms
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Previously…
n  Characteristics of algorithms
n  Pseudocode
n  Examples: Max algorithm
Today…
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Examples: Sum algorithm
Problem of searching an ordered list
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Linear search & binary search algorithms
Sorting problem
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Bubble sort & insertion sort algorithms
ICS 141: Discrete Mathematics I – Fall 2011
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Practice Exercises
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Devise an algorithm that finds the sum of all
the integers in a list.
procedure sum(a1, a2, …, an: integers)
s := 0 {sum of elements so far}
for i := 1 to n {go thru all elements}
s := s + ai {add current item}
{now s is the sum of all items}
return s
ICS 141: Discrete Mathematics I – Fall 2011
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Searching Algorithms
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Problem of searching an ordered list.
n  Given a list L of n elements that are sorted into
a definite order (e.g., numeric, alphabetical),
n  And given a particular element x,
n  Determine whether x appears in the list,
n  And if so, return its index (position) in the list.
Problem occurs often in many contexts.
Let’s find an efficient algorithm!
ICS 141: Discrete Mathematics I – Fall 2011
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Linear Search (Naïve)
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{Given a list of integers and an integer x to look up,
returns the index of x within the list or 0 if x is not in the list}
procedure linear_search
(x: integer, a1, a2, …, an: distinct integers)
i := 1 {start at beginning of list}
while (i ≤ n ∧ x ≠ ai) {not done and not found}
i := i + 1 {go to the next position}
if i ≤ n then index := i {it was found}
else index := 0 {it wasn’t found}
return index {index where found or 0 if not found}
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Alg. #2: Binary Search
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Basic idea: At each step, look at the middle
element of the remaining list to eliminate half of it,
and quickly zero in on the desired element.
<x
ICS 141: Discrete Mathematics I – Fall 2011
<x <x >x
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Search Alg. #2: Binary Search
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procedure binary_search
(x: integer, a1, a2, …, an: increasing integers)
i := 1 {left endpoint of search interval}
j := n {right endpoint of search interval}
while i < j begin {while interval has > 1 item}
m := ⎣(i + j)/2⎦ {midpoint}
if x > am then i := m + 1 else j := m
end
if x = ai then location := i else location := 0
return location {index or 0 if not found}
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Search Example
Search for 19 in the list
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Index: 1
2
3
4
5
6
7
8
9
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10
11
12
13
14
15
16
1, 2, 3, 5, 6, 7, 8, 10, 12, 13, 15, 16, 18, 19, 20, 22
i
i
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using linear search
using binary search
i
j
i
j
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Entering while loop: i = 1, j = 16
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m = ⎣(i+j)/2⎦ = ⎣(1+16)/2⎦ = ⎣8.5⎦ = 8,
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m = ⎣(i+j)/2⎦ = ⎣(9+16)/2⎦ = ⎣12.5⎦ = 12, i = 13, j = 16
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m = ⎣(i+j)/2⎦ = ⎣(13+16)/2⎦ = ⎣14.5⎦ = 14, i = 13, j = 14
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m = ⎣(i+j)/2⎦ = ⎣(13+14)/2⎦ = ⎣13.5⎦ = 13, i = 14, j = 14
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Exit loop
ICS 141: Discrete Mathematics I – Fall 2011
i = 9, j = 16
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Sorting Algorithms
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Sorting is common in many applications.
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E.g. spreadsheets and databases
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We can search quickly when data is odrered!
Sorting is also widely used as a subroutine in
other data-processing algorithms.
Two sorting algorithms shown in textbook:
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Bubble sort
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Insertion sort
However, these are not
very efficient, and you should
not use them on large data sets!
We’ll see some more efficient algorithms later in the course.
ICS 141: Discrete Mathematics I – Fall 2011
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Bubble Sort
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Smaller elements “float” up to the top of the list, like
bubbles in a container of liquid, and the larger
elements “sink” to the bottom.
ICS 141: Discrete Mathematics I – Fall 2011
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Bubble Sort Algorithm
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procedure bubble_sort
(a1, a2, …, an: real numbers, n ≥ 2)
for i := 1 to n – 1
{iterate n – 1 passes}
for j := 1 to n – i
if aj > aj+1 then interchange aj and aj+1
{an-i+1, …, an is sorted and ≤ a1, …, an-i}
{a1, a2, …, an is sorted}
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Example 4: Use the bubble sort to put 3, 2, 4,
1, 5 into increasing order. (See previous slide)
ICS 141: Discrete Mathematics I – Fall 2011
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Insertion Sort
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English description of algorithm:
n  Start with the second element,
for each item in the input list:
n  “Insert” it into the correct place in the sorted
output list generated so far. Like so:
n  Find the location where the new item should
be inserted using linear or binary search.
n  Then, shift the items from that position
onwards up by one position.
n  Put the new item in the remaining hole.
ICS 141: Discrete Mathematics I – Fall 2011
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Insertion Sort Example
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Use the insertion sort to put 3, 2, 4, 1, 5 into increasing order
nd element 2 in the right position:
n  Insert the 2
n  3 > 2 ⇒ put 2 in front of 3. ⇒ 2, 3, 4, 1, 5
rd element 4 in the right position:
n  Insert the 3
n  4 > 2 ⇒ do nothing. Move to the next comparison.
n  4 > 3 ⇒ do nothing. Done. ⇒ 2, 3, 4, 1, 5
th element 1 in the right position:
n  Insert the 4
n  2 > 1 ⇒ put 1 in front of 2. ⇒ 1, 2, 3, 4, 5
th element 5 in the right position:
n  Insert the 5
n  5 > 1 ⇒ do nothing. Move to the next comparison.
n  5 > 2 ⇒ do nothing. Move to the next comparison.
n  5 > 3 ⇒ do nothing. Move to the next comparison.
n  5 > 4 ⇒ do nothing. Done. ⇒ 1, 2, 3, 4, 5
ICS 141: Discrete Mathematics I – Fall 2011
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Insertion Sort Algorithm
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procedure insertion_sort
(a1, a2, …, an: real numbers, n ≥ 2)
for i := 2 to n begin
m := ai {the element to be inserted}
j := 1
while aj < m {look for the index of the hole with j }
j := j + 1
{now a1, …, aj-1 < m ≤ aj, …, ai }
{the hole is at j; j ≤ i, i.e. possibly j = i }
for k := j + 1 to i
ak := ak+1
ai := m
{a1, a2, …, ai are sorted in increasing order}
end {a1, a2, …, an are sorted in increasing order}
ICS 141: Discrete Mathematics I – Fall 2011
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Efficiency of Algorithms
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Intuitively we see that binary search is much
faster than linear search,
Also, we may see that insertion sort is better
than bubblesort in some cases (why? when?),
But how do we analyze the efficiency of
algorithms formally?
Use methods of algorithmic complexity,
which utilize the order-of-growth concepts
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