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Chapter 1Introduction

- Instructors
- C. Y. Tang and J. S. Roger Jang

All the material are integrated from the textbook

"Fundamentals of Data Structures in C" and some

supplement from the slides of Prof. Hsin-Hsi Chen

(NTU).

How to create programs

- Requirements
- Analysis bottom-up vs. top-down
- Design data objects and operations
- Refinement and Coding
- Verification
- Program Proving
- Testing
- Debugging

Data Structure How data are organized and hence

operated

- Youve been very familiar with arrays in your

programming assignments. They are basic (yet

powerful!) data structures. - They can hold data (objects)e.g., integers.
- They are structuredstructured in a way that the

data held inside can be operated. - Each element in an array has an index. With that,

you can store or retrieve an element.

Learning Data Structures, and Algorithms

- You want your tasks to be performed efficiently.

You need good methods (algorithms). - Data must be structured in some manner to be

operated. - Good structures can be operated efficiently.

Example

- For example, suppose that you are building a

database storing the data of all (past, present,

and future) students of NTHU, which are growing

in size. - Youll need to find anybodys data in the

database (to search/retrieve). - To enter new entries into it (to insert).
- Etc.

Example

- You can use an array to implement the database.
- To insert a new entry, simply add it to the first

empty array cell. - If the current array is full, allocate a new

array whose size is the double of the current.

Then copy all the original entries from the old

array to the new one. - To search for an entry, simply looks at all

entries in the array, one by one.

Example

- Your programming experience, however, tells you

that this is not a good method. - In Chapter 10, youll see sophisticated data

structures that can be operated (searched,

inserted/deleted, etc.) efficiently. - Then, youll just feel how data can be cleverly

structured to serve as the basis of fast

algorithms.

Example 2

- This time you want to work with polynomials,

i.e., functions of the form f(x) an xn an-1

xn-1 a0 - Youll need to store them, as well as to multiply

or add them. - You may allocate an array to store the

coefficients of a polynomial. - E.g., Ai stores ai.

Example 2

- Alternatively, you can use a linked list to store

it. - Each node needs to store the coefficient and the

exponent. E.g., to store 3x2 5, a linked list

like 32 -gt 50 is constructed. - To represent polynomials, both data structures

have their relative advantages and drawbacks in

time and space considerations, etc (see later

chapters).

Searching in Arrays

- You are able to write, in minutes, a program to

search sequentially in an array. - However, when the numbers in an array are sorted,

you can do much faster - using binary search, which you probably are also

familiar with. - You may say that, a sorted array is not the same

structure as a general array.

Binary Search

- Let A0..n - 1 be an array of n integers.
- We want to ask is some integer x stored in A?
- Suppose that A is sorted (say, in ascending

order), i.e., (for simplicity assume that the

numbers are distinct) A0 lt A1 lt lt An -

1. - To be more concrete, let n be 5.

Binary Search

- Observation If x gt A2, then x must fall in

A3..4, (or x is not in A). If x lt A2, then x

must fall in A0..1. - Ex Let x 11.
- A 3 5 7 11 13
- x ( 11) gt A2 ( 7), so we need not consider

A0..1.

Binary Search

- Example Let x 17. Let A contain the following

8 integers. Initially, let left 0 and right 7

(shown in red). - left and right define the range to be searched.
- 2 3 5 7 11 13 17 19
- mid (0 7) / 2 3 (rounded).
- Amid 7 lt x, so let left mid 1 4, and

continue the search.

- 2 3 5 7 11 13 17 19
- mid (4 7) / 2 5, Amid 13, x 17.
- Amid lt x, so let left mid 1 6, and

continue the search. - 2 3 5 7 11 13 17 19
- mid (6 7) / 2 6, Amid 17, x 17.
- Amid x, so return mid 6, the desired

position, and were done!

Binary Search

- Binsearch(A, x, left, right) / finds x in

Aleft..right / - while left lt right do
- mid (left right) / 2
- if x lt Amid then right mid 1
- else if x Amid then return mid
- else left mid 1
- return -1 / not found /

Recursive Functions

- A function that invokes (or is defined in terms

of) itself directly or indirectly is a recursive

function. - Fibonacci sequence Fn Fn-1 Fn-2
- Summation sum(n) sum(n - 1) An
- Where sum(i) is the sum of the first i items in A
- Binomial coefficient (n choose k) C(n, k) C(n

1, k) C(n 1, k 1) - The combinatorial interpretation of this is

profoundtry it out if youve not learned it yet!

Recursive Binary Search

- Binsearch(A, x, left, right)
- if left gt right then return -1
- mid (left right) / 2
- if x lt Amid then return Binsearch(A, x,

left, right 1) - else if x Amid then return mid
- else return Binsearch(A, x, left 1, right)

Data Abstraction

- Before implementation, we need first to know the

specification of the objects to be stored as well

as the operations that should be supported,

before we can implement it. - In our previous polynomial example, first we have

the demand to store polynomials (the objects),

supporting multiplications and other operations.

The specification is independent of how it is

implemented (e.g. using arrays or linked lists).

Abstract Data Type (ADT)

- An abstract data type (ADT) is a data type that

is organized in such a way that the specification

of the objects and the specification of the

operations on the objects is separated from the

representation of the objects and the

implementation of the operations. - No fixed syntax to describe them. The

specifications need only be clear.

ADT for Natural Numbers (an example)

- Structure Natural_Number
- Objects an ordered subrange of the integers

starting at 0 and ending at the maximum integer

(INT_MAX) on the computer. - Functions
- Nat_Num Zero() 0
- Nat_Num Add(x, y) if (xy) lt INT_MAX then

return x y, else return INT_MAX. - And so on. This example is actually too simple so

that you may feel that the implementation of the

functions have been stated. But this is generally

not the case.

Structure 1.1Abstract data type Natural_Number

(p.17)structure Natural_Number is objects

an ordered subrange of the integers starting at

zero and ending at the maximum integer (INT_MAX)

on the computer functions for all x,

y ? Nat_Number TRUE, FALSE ? Boolean and

where , -, lt, and are the usual integer

operations. Nat_No Zero ( )

0 Boolean Is_Zero(x) if (x) return

FALSE

else return TRUE Nat_No Add(x, y)

if ((xy) lt INT_MAX) return xy

else

return INT_MAX Boolean Equal(x,y)

if (x y) return TRUE

else return FALSE

Nat_No Successor(x) if (x INT_MAX) return

x

else return x1 Nat_No Subtract(x,y)

if (xlty) return 0

else return x-y end

Natural_Number

is defined as

Performance Analysis

- Were most interested in the time and space

requirements of an algorithm. - The space complexity of a program is the amount

of memory that it needs to run to completion. The

time complexity is the amount of computer time

that it needs to run to completion.

??? Algorithm ?

- A number of rules, which are to be followed in a

prescribed order, for solving a specific type of

problems. - Computer Algorithm?????
- Finiteness(???steps)
- Definiteness(???step?????)
- Effectiveness(?????????????)
- Input/Output(O.S.??terminate????computational

procedure)

Algorithm is everywhere !

- Operating Systems
- System Programming
- Numerical Applications
- Non-numerical Applications
- ???field???Algorithm????field???????
- Algorithm Implement???
- Software
- Hardware
- Firmware

?????Algorithm ?

- 1. ????,?????????(Time, Space)?Algorithm?????
- Life-time Job
- ????????????Algorithm???
- ?????????????paper,update???algorithms??????????

?????Algorithm ?

- 2. ????,?????NP-Complete?????efficient??
- Life-time Job
- ????????NP-Complete?
- Real Application
- Average Performance??
- ?Approximating??
- TSP
- n 20 771??
- N3log n in average (B B)
- Planar Graph Coloring (Maximum 4 ????)

Measurements

- Criteria
- Is it correct?
- Is it readable?
- Performance Analysis (machine independent)
- space complexity storage requirement
- time complexity computing time
- Performance Measurement (machine dependent)

Space Complexity

- The space needed includes two parts
- (1) Fixed space requirement Not dependent on the

number and size of the programs inputs and

outputs. - Instruction space, simple variables (e.g., int),

fixed-size structure variables (such as struct),

and constants.

Space Complexity

- (2) Variable space requirement S(I) Depends on

the instance I involved. - In recursive calls, many copies of simple

variables (e.g. many ints) may exist. Such space

requirement is included in S(I). - S(I) may depend on some characteristics of I. The

characteristic well most often encounter is n,

the size of the instance. - In this case we denote S(I) as S(n).

Space Complexity

- float abc(float a, float b, float c)
- return abbc (ab-c) / (ab) 4
- Sabc(I) 0.
- Only has fixed space requirement.

Space Complexity

- float sum(float list, int n) / adds up list

/ - float tempsum 0
- int i
- for (i0 i lt n i) tempsum listi
- return tempsum
- In C, the array is passed using the address. So

Ssum(n) 0. - In Pascal, the array may be passed by copying

values. If this is the case, then Ssum(n) n.

Space Complexity

- float rsum(float list, int n)
- if (n gt 0) return rsum(list, n-1) listn-1
- return 0
- For each recursive call, the OS must save

parameters, local variables, and the return

address. - In this example, two parameters (list and n)

and the return address (internally) are saved for

each recursive call.

- To add a list of n numbers, there are n recursive

calls in total. - So Srsum(n) (c1 c2 c3) n, where c1, c2

and c3 are the number of bytes (or other unit of

interest) needed for each of these types (list,

n, and return address).

Time Complexity

- Were interested in the number of steps taken by

a program. But what is a step? - A program step is a syntactically or semantically

meaningful program segment whose execution time

is independent of the instance characteristics. - For a program, everybody can have his/her own

steps defined. The important thing is the

independency of the instance size, etc.

Statement s/e Freq. Tot.

float rsum(float list, int n) if (n gt 0) return rsum(list, n-1) listn-1 return 0 1 n1 n1 1 n n 1 1 1

Total 2n2

s/e steps per execution.

Time Complexity

- For some programs, for a fixed n, the time taken

by different instances may still be different. - For example, the binary search algorithm depends

on the position of x in array A. - Let A lt3, 5, 7, 11gt.
- If x 5, then we need less steps than what if x

7. - But in both cases, n 4.

Time Complexity

- So we may consider the worst case, average case,

or best case time complexity of an algorithm. - Worst case the maximum number of steps needed

for any possible instance of size n. - Most commonly used. The concept of guarantee.
- Average case under some assumption of instance

distribution, the expected number of steps

needed. - Useful. But usually most complicated to analyze.
- Best case the opposite of worst case.
- Rarely seen.

Asymptotic Notations

- To make exact step counts is often not necessary.

- The concept of step is even inexact itself.
- It is more impressive to obtain a functional

improvement in the time complexity than an

improvement by a constant multiple. - It is good to improve 2n2 to n2.
- It is even better to improve 2n2 to 1000n.
- For n large enough, you know that n2 is much

larger than 1000n.

Figure 1.7Function values (p.38)

(No Transcript)

Figure 1.9Times on a 1 billion instruction per

second computer(p.40)

Asymptotic Notations

- Therefore, in many cases we do not worry about

the coefficients in the time complexity. - Not in all cases since, when we cannot have

functional improvement, well still want

improvements in the coefficients. - We regard 2n2 as equivalent to n2. They belong to

the same class in this sense. - Similarly, 2n2 100n and n2 belong to the same

class, since the quadratic term is dominant. - n2 and n belong to different classes.

Asymptotic Notation (O)

- Definitionf(n) O(g(n)) iff there exist

positive constants c and n0 such that f(n) ?

cg(n) for all n, n ? n0. - Examples
- 3n2O(n) / 3n2?4n for n?2 /
- 3n3O(n) / 3n3?4n for n?3 /
- 100n6O(n) / 100n6?101n for n?10 /
- 10n24n2O(n2) / 10n24n2?11n2 for n?5 /
- 62nn2O(2n) / 62nn2 ?72n for n?4 /

Example

- Complexity of c1n2c2n and c3n
- for sufficiently large of value, c3n is faster

than c1n2c2n - for small values of n, either could be faster
- c11, c22, c3100 --gt c1n2c2n ? c3n for n ? 98
- c11, c22, c31000 --gt c1n2c2n ? c3n for n ?

998 - break even point
- no matter what the values of c1, c2, and c3, the

n beyond which c3n is always faster than c1n2c2n

- O(1) constant
- O(n) linear
- O(n2) quadratic
- O(n3) cubic
- O(2n) exponential
- O(logn)
- O(nlogn)

Asymptotic Notations

- To make these concepts more precise, asymptotic

notations are introduced. - f(n) O(g(n)) if there exist positive constants

c and n0 such that f(n) ? c g(n) for all n gt n0. - 1000n O(2n2). This is to say that 1000n is no

larger than (?) 2n2 in this sense. - You can choose c 500 and n0 1.
- Then 1000n ? 500 2n2 1000n2 for all n gt 1.

Asymptotic Notations

- 1000n O(n) since you can choose c 1000 and n0

1. - 1000 O(1).
- For constant functions, the choice of n0 is

arbitrary. - n2 ?? O(n) since for any c gt 0 and any n0 gt 0,

there always exists some n gt n0 such that n2 gt

cn.

Asymptotic Notations

- 2n2 100n O(n2), since 2n2 100n lt 200n2

O(n2) - This last equation may be validated by choosing

c 200 and n0 1. - So you can feel that in asymptotic notations we

only care about the most dominant term. Simply

throw out the minor terms. Throw out the

constants, too. - log2 n O(n). May choose c 1 and n0 2.
- n log2 n O(n2).

Asymptotic Notations

- n100 O(2n). You may choose c 1 and n0 1000.

- O(1) constant, O(n) linear, O(n2) quadratic,

O(n3) cubic, O(log n) logarithmic, O(n log n),

O(2n) exponential. - These are the most commonly encountered time

complexities. - The base of the log is not relevant

asymptotically, since logba (1/log2b) log2a,

different only by a constant multiple 1/log2b.

Asymptotic Notations

- f(n) O(g(n)) is just a notation. f(n) and

O(g(n)) are not the same thing. - So you cant write O(g(n)) f(n).
- It is also common to view O(g(n)) as a set of

functions, and f(n) O(g(n)) actually means f(n)

? O(g(n)). - O(g(n)) f(n) for some c gt 0 and n0 gt 0 such

that f(n) lt c g(n) for all n gt n0

Asymptotic Notations

- n O(n), n O(n2), n O(n3),
- Choose the function g(n) closer to f(n) is more

informative. - You may ask, why not use f(n) itself? f(n)

O(f(n)) is the best choice. The difficulty is

when we analyze an algorithm, we may even not

know the exact f(n). We can only obtain an upper

bound in some cases.

Asymptotic Notations

- Thm 1.2 If f(n) am nm a1 n a0, then

f(n) O(nm). - Pf Let a maxam, , a0 1. Then
- f(n) lt a nm a n a lt (m1)a nm

for n gt 1. - So choosing c (m1)a and n0 1 just works.
- So, again, drop the constants and minor terms.

Asymptotic Notations

- We also have a notation for lower bounds.
- f(n) ?(g(n)) if for some c gt 0 and n0 gt 0, f(n)

? c g(n) for all n gt n0. - n2 ?(2n) choose c 1/2 and n0 1.
- 2n ?(n100) choose c 1 and n 1000.
- You can prove that If f(n) O(g(n)), then g(n)

?(f(n)).

Asymptotic Notations

- Thm 1.3 If f(n) am nm a1 n a0 and am

gt 0, then f(n) ?(nm). - Pf Exercise.
- Note that, if am lt 0, then f(n) ? ?(nm). For

example, -n2 1000n ? ?(n2) since for any c gt 0

and n0 gt 0, we have n2 1000n lt n2 for n

large.

Asymptotic Notations

- We also have a notation for equivalence.
- f(n) ?(g(n)) if there exist c1 gt 0, c2 gt 0, and

n0 gt 0 such that c1 g(n) ? f(n) ? c2 g(n) for

all n gt n0. - 2n2 100n ?(n2).
- n2 ? ?(n).
- You can prove that, f(n) ?(g(n)) if and only if

f(n) O(g(n)) and f(n) ?(g(n)).

Asymptotic Notations

- Thm 1.4 If f(n) am nm a1n a0 and am gt

0, then f(n) ?(nm). - Pf Immediate from Thm 1.2 and 1.3.

Time Complexity of Binary Search

- At each iteration, the search range for binary

search is reduced by about a half. - So, in any case, the number of iterations needed

cannot exceed log2n. The time complexity is

O(log n) in any case (each iteration takes

O(1)). - The worst case time complexity is ?(log n), which

occurs when, e.g., the number to search is not in

the array. - Therefore, the worst case time complexity is

?(log n). - In the best case, one iteration suffices. The

best case time complexity is ?(1).

Time Complexity of Binary Search

- Note that the time complexity for a sequential

search is ?(n), which occurs when, e.g., the

number to search is not in the array. - So binary search is faster than sequential

search, but it requires the array to be sorted.

Time Complexity of Binary Search

- The worst case time complexity of the recursive

binary search can be stated elegantly as - T(n) T(n/2) ?(1) for n gt 1 T(1) ?(1).
- The ?(1) term means some anonymous function f(n)

s.t. f(n) ?(1), which is for the time needed in

addition to the time taken by the recursive call.

Time Complexity of Binary Search

- Note that any function f(n) ?(1) is bounded

above by a constant, and is bounded below by a

constant. - By the definition of ?, there exists c gt 0 and n0

gt 0 such that f(n) lt c for n gt n0. So f(n) lt

supf(n) n ? n0 c. Similarly f(n) is

bounded below by a constant. - So T(n) lt T(n/2) c.

Time Complexity of Binary Search

- For simplicity let n 2m, so m log n. Then
- T(n) T(2m) lt T(2m-1) c lt T(2m-2) 2c

lt lt T(1) mc O(m) - So T(n) O(log n).
- This may be seen as an initial guess. To confirm

the answer we may use induction.

Time Complexity of Binary Search

- Ind. Hyp. T(m) lt d log m for m lt n.
- We are free in choosing the constant d, if it

satisfies the base case. So we can choose d to be

larger than c. - Induction T(n) lt T(n/2) c lt d log(n/2) c d

log n d c lt d log n. - Base case T(2) lt T(1) c lt c lt d log 2.
- Need to choose d to satisfy d gt c.
- So T(n) lt d log n, implying T(n) O(log n).

Selection Sort

- Given several numbers, how to sort them?
- You can find the smallest number, set it aside,

find the next smallest number, and so on,

continue until all numbers are done.

Selection Sort

- sort(A, n) / Assume that A is indexed by 1..n

/ - for i 1 to n - 1 do find the index of the

min. elem. in Ai..n swap the min. elem. with

Ai - Observe that, at the end of the i th iteration,

Aj holds the j th smallest element of A, for

all i ? i.

Time Complexity of Selection Sort

- Find the minimum in Ai..n takes ?(n - i) time.

- Total time

Comparison of Two Strategies

- Suppose that you want to do searches in A.
- If the search will be performed only once, then a

sequential search is good. - If the search is to be done very frequently (much

more than n times), then it is worth paying n2

time to sort the array first (preprocessing),

being able to do binary search subsequently.