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February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 1
and now for
Sequences
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 2
Sequences
Sequences represent ordered lists of elements. A sequence is defined as a function from a subset of N to a set S. We use the notation an to denote the image of the integer n. We call an a term of the sequence. Example: subset of N : 1 2 3 4 5
S: 2 4 6 8 10
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 3
Sequences
We use the notation {an} to describe a sequence.
Important: Do not confuse this with the {} used in set notation.
It is convenient to describe a sequence with an equation.
For example, the sequence on the previous slide can be specified as {an}, where an = 2n.
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 4
The Equation Game
1, 3, 5, 7, 9, an = 2n - 1
-1, 1, -1, 1, -1, an = (-1)n
2, 5, 10, 17, 26, an = n^2 + 1
0.25, 0.5, 0.75, 1, 1.25 an = 0.25n
3, 9, 27, 81, 243, an = 3n
What are the equations that describe the following sequences a 1 , a 2 , a 3 , ?
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 5
Strings
Finite sequences are also called strings , denoted by a 1 a 2 a 3 an.
The length of a string S is the number of terms that it consists of.
The empty string contains no terms at all. It has length zero.
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 6
Summations
It represents the sum am + am+1 + am+2 + + an.
The variable j is called the index of summation , running from its lower limit m to its upper limit n. We could as well have used any other letter to denote this index.
What does stand for?
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 7
Summations
It is 1 + 2 + 3 + 4 + 5 + 6 = 21.
We write it as.
What is the value of?
It is so much work to calculate this
What is the value of?
How can we express the sum of the first 1000 terms of the sequence {an} with an=n^2 for n = 1, 2, 3, ?
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 8
Summations
It is said that Carl Friedrich Gauss came up with the following formula:
When you have such a formula, the result of any summation can be calculated much more easily, for example:
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences
9
Double Summations
Corresponding to nested loops in C or Java,
there is also double (or triple etc.) summation:
Example:
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 10
Double Summations
Table 2 in 4 th^ Edition: Section 1. 5 th^ Edition: Section 3. 6 th^ and 7th^ Edition: Section 2. contains some very useful formulas for calculating sums.
In the same Section, Exercises 15 and 17 (7th^ Edition: Exercises 31 and 33) make a nice homework.
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 11
Enough Mathematical Appetizers!
Let us look at something more interesting:
Algorithms
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 12
Algorithms
What is an algorithm?
An algorithm is a finite set of precise instructions for performing a computation or for solving a problem.
This is a rather vague definition. You will get to know a more precise and mathematically useful definition when you attend CS420 or CS620.
But this one is good enough for now
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 19
Algorithm Examples
a c d f g h j l m o p r s u v x z
binary search for the letter ‘j’
center element
search interval
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 20
Algorithm Examples
a c d f g h j l m o p r s u v x z
binary search for the letter ‘j’
center element
search interval
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 21
Algorithm Examples
a c d f g h j l m o p r s u v x z
binary search for the letter ‘j’
center element
search interval
found!
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 22
Algorithm Examples
procedure binary_search(x: integer; a 1 , a 2 , , an: integers) i := 1 {i is left endpoint of search interval} j := n {j is right endpoint of search interval} while (i < j) begin m := (i + j)/2 if x > am then i := m + 1 else j := m end if x = ai then location := i else location := 0 {location is the subscript of the term that equals x, or is zero if x is not found}
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 23
Algorithm Examples
Obviously, on sorted sequences, binary search is more efficient than linear search.
How can we analyze the efficiency of algorithms?
We can measure the
- time (number of elementary computations) and
- space (number of memory cells) that the algorithm requires.
These measures are called computational complexity and space complexity , respectively.
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 24
Complexity
What is the time complexity of the linear search algorithm?
We will determine the worst-case number of comparisons as a function of the number n of terms in the sequence.
The worst case for the linear algorithm occurs when the element to be located is not included in the sequence.
In that case, every item in the sequence is compared to the element to be located.
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 25
Algorithm Examples
Here is the linear search algorithm again: procedure linear_search(x: integer; a 1 , a 2 , , an: integers) i := 1 while (i ≤ n and x ≠ ai) i := i + 1 if i ≤ n then location := i else location := 0 {location is the subscript of the term that equals x, or is zero if x is not found}
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 26
Complexity
For n elements, the loop
while (i ≤ n and x ≠ ai) i := i + 1 is processed n times, requiring 2n comparisons. When it is entered for the (n+1)th time, only the comparison i ≤ n is executed and terminates the loop. Finally, the comparison if i ≤ n then location := i is executed, so all in all we have a worst-case time complexity of 2n + 2.
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 27
Reminder: Binary Search Algorithm
procedure binary_search(x: integer; a 1 , a 2 , , an: integers) i := 1 {i is left endpoint of search interval} j := n {j is right endpoint of search interval} while (i < j) begin m := (i + j)/2 if x > am then i := m + 1 else j := m end if x = ai then location := i else location := 0 {location is the subscript of the term that equals x, or is zero if x is not found} February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 28
Complexity
What is the time complexity of the binary search algorithm?
Again, we will determine the worst-case number of comparisons as a function of the number n of terms in the sequence.
Let us assume there are n = 2k^ elements in the list, which means that k = log n.
If n is not a power of 2, it can be considered part of a larger list, where 2k^ < n < 2k+1.
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 29
Complexity
In the first cycle of the loop
while (i < j) begin m := (i + j)/2 if x > am then i := m + 1 else j := m end
the search interval is restricted to 2k-1^ elements, using two comparisons.
February 7, 2013 Applied Discrete Mathematics Week 2: Functions and Sequences 30
Complexity
In the second cycle, the search interval is restricted to 2 k-2^ elements, again using two comparisons.
This is repeated until there is only one (2^0 ) element left in the search interval.
At this point 2k comparisons have been conducted.
