Download Document

Chapter 1 Introduction
Definition of Algorithm An algorithm is a finite sequence of precise
instructions for performing a computation or for solving a problem.
Example 1 Describe an algorithm for finding the maximum (largest) value in a finite
sequence of integers.
Solution
1.
Set the temporary maximum equal to the first integer in the sequence.
2.
Compare the next integer in the sequence to the temporary maximum, and set
the larger one to be temporary maximum.
3.
Repeat the previous step if there are more integers in the sequence.
4.
Stop when there are no integers left in the sequence. The temporary maximum
at this point is the maximum in the sequence.
Algorithm 1. Finding the Maximum Element in a Finite Sequence
//Input : n integers a1,a2 ,...,an
// Output : max (the maximum of a1,a2 ,...,an )
max : a1 ;
for i:  2 to n
if max  ai then max : ai
return max
Instead of using a
particular computer
language, we use a form
of pseudocode.
Algorithmic Problem Solving
Understand the problem
Design an algorithm and proper data structures
Analyze the algorithm
Code the algorithm
• Ascertaining the Capabilities of a Computational Device
• Choosing between Exact and Approximate Problem Solving
• Deciding on Appropriate Data Structures
• Algorithm Design
• Algorithm Analysis
• Coding
Important Problem Types
• Sorting
• Searching
• String processing (e.g. string matching)
• Graph problems (e.g. graph coloring problem)
• Combinatorial problems (e.g. maximizes a cost)
• Geometric problems (e.g. convex hull problem)
• Numerical problems (e.g. solving equations )
Fundamental Data Structures
1. Array
2. Link List
a[1] a[2] a[3]
Head of list
a1
a[n]
a3
a2
an
10
3. Binary Tree
6
4
1
9
7
17
13
4
9
20
4. Heap
Definition 1 A max-heap (or min-heap) A is an array object that can be reviewed
as a complete binary tree (each node has two children, except the right part of
the last level) , where at each node, there is a key which is larger (or smaller)
or equal to the keys of its children.
A[0]
A[0] A[1] A[2] A[3] A[4] A[5] A[6] A[7] A[8] A[9]
20
19
17
13
4
16
2
Node i’s parent and children
 i  1
Parent:  2 
Left child: 2i +1
10
8
20
3
A[1]
19
A[3]
A[4]
13
A[7]
10
Right child: 2i+2
The height of a max-heap
with n elements is O(log n)
Operations in a max-Heap (or min-heap):
• Extract the item with the maximum (or minimum )
key.
• Insert a new item into the heap
A[2]
A[8]
A[9]
8
3
4
A[5]
17
16
A[6]
2
5. Binary Search Tree
Definition 1 Binary Search Tree is a Binary Tree
satisfying the following condition:
15
(1) Each vertex contains an item called as key
which belongs to a total ordering set and two
links to its left child and right child,
respectively.
(2) In each node, its key is larger than the keys of
all vertices in its left subtree and smaller than
the keys of all the vertices in its right subtree.
6
3
1
18
7
17
13
4
9
Operations a binary search tree support: search, insert, and delete an element
with a given key.
How to select a data structure for a given problem?
20
Fundamental Abstract Data Structures
1. Stack
an
• Operations a stack supports: last-in-first-out (pop, push)
• Implementation: using an array or a list (need to check
underflows and overflows)
a2
a1
2. Queue
• Operations a queue supports: first-in-first-out (enqueue, dequeue)
• Implementation: using an array or a list (need to check underflows
and overflows)
am am1 am  2
front
an
rear
3. Priority Queue (each elements of a priority queue has a key)
• Operations a priority queue supports: inserting an element, returning an element
with maximum/minimum key.
• Implementation: heap
4. Dynamic Dictionary
A Dynamic Dictionary is a data structure of item
with keys that support the following basic operations:
(1) Insert a new item
(2) Remove an item with a given key
(3) Search an item with a given key
Some Advanced Data Structure
Graph
• Operations a graph support: finding neighbors
• Implementation: using list or matrix
G  (V , E )
V  (a, b, c, d , e, f )
E  {( a, c), (b, c), (d , a), (c, e), (e, c), (b, f ), (d , e), (e, f )}
a
c
b
d
e
f
c
a
c
f
b
e
c
a
e
d
f
c
e
f
Adjacent lists
 a b c d e f


a 0 0 1 0 0 0 
b 0 0 1 0 0 1 


c 0 0 0 0 1 0 
d1 0 0 0 1 0 


e 0 0 1 0 0 1 


f
0
0
0
0
0
0


Adjacent matrix
Assignment 1
(1) Read the sections of the text book about array, linked list, heap, binary search tree, stack,
queue, priority queue, and dynamic dictionary.
(2)
(i) Implement stack S and queue Q using both of array and linked list.
(ii) Write a main function to do the following operations for S: pop, push 10 times, pop, pop,
repeat push and pop 7 times. Do the following operations for Q: dequeue, enqueue 10 times,
dequeue, depueue, repeat enqueue and dequeue 7 times. When you use array, declare the
size of the array to be 10. Printout all the elements in S and in Q.
Select one of (3) and (4)
(3) Implement a min-heap of integer. The program must contain the following operations:
• Heapify the min-heap
• Build the min-heap from a given array.
• Extract an item with the minimum key.
• Insert a new item into the min-heap.
• Print the heap.
Also, write a Main method to test your program as follows:
•
Create an array A of integers.
•
Create an empty min-heap.
• Assign 10 integers one by one into the array A in the following order: 12, 34, 76, 21, 4, 3, 33,
88, 90, 15.
• Build the min-heap from array A, and print the heap.
• Extract the smallest integer, and print the heap.
• Insert a new integer into the heap, and print the heap.
(4) Write the program for BST of student records (student’s id is used as key). The program must
contain the following 5 operations:
• Search a node (given a student id), and print the record when it is found.
• Insert a node (given a new student record)
• Delete a node (given a student id)
• Print out of all student records in the BST in in-order.
Also, write a Main method to test your program as follows:
• Create an empty BST.
• Insert 10 student records one by one into the BST in the following order:
record1: 1884, name1, score1
record2: -174, name2, score2
record3: 1464, name3, score3
record4: 527, name4, score4
record5: -1158, name5, score5
record6: 1860, name6, score6
record7: -1783, name7, score7
record8: -2014, name8, score8
record9: 1979, name9, score9
record10: 1892, name10, score10.
and print the BST in in-order to confirm 10 records are inserted correctly.
• Search a student record (giving a student id), and print the student record if the id is found.
• Delete a student record (giving a student id), and print the BST in in-order to confirm that the
record is deleted.
Chapter 2 Fundamentals of the Analysis of Algorithm
Implementation and Empirical Analysis
Challenge in empirical analysis:
•
Develop a correct and complete implementation.
•
Determine the nature of the input data and other factors
influencing on the experiment. Typically, There are three
choices: actual data, random data, or perverse data.
•
Compare implementations independent to programmers,
machines, compilers, or other related systems.
•
Consider performance characteristics of algorithms, especially
for those whose running time is big.
Can we analyze algorithms that haven’t run yet?
Example 1 Describe an algorithm for finding an element x in a
list of distinct elements a1 , a2 ,..., an .
Algothm 1 The linear Search Algorithm.
Procedure linear sea rch ( x : integer, a1,a2 ,..., an : distinct integers)
i:  1;
while (i  n and x  ai )
i:  i  1;
if i  n then location:  i
else locatiton:  0;
{location is the subscript of
term that equals x, or is 0 if x is not found}
Algorithm 2 The Binary Search Algorithm.
Procedure binary sea rch ( x : integer, a1,a2 ,..., an : increasing integers)
i:  1;
j:  n;
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
term that equals x, or is 0 if x is not found}
Linear Search and Binary Search written in C++
1.
2.
Linear search
int lsearch(int a[], int v, int l, int r)
{
for (int i=l; i<=n; i++)
if (v==a[i]) return i;
return -1;
}
Binary search
int bsearch(int a[], int v, int l, int n)
{
while (r>=l)
{int m=(l+r)/2;
if (v==a[m]) return m;
if (v<a[m]) r=m-1; else l=m+1;
}
return -1;
}
Complexity of Algorithms
Assume that both algorithms A and B solve the problem P. Which
one is better?
• Time complexity: the time required to solve a problem of
a specified size.
• Space complexity: the computer memory required to
solve a problem of a specified size.
The time complexity is expressed in terms of the
number of operations used by the algorithm.
• Worst case analysis: the largest number of operations
needed to solve the given problem using this algorithm.
• Average case analysis: the average number of
operations used to solve the problem over all inputs.
Example 2 Analyze the time complexities of linear search
algorithm and binary search algorithm
Linear Search Algorithm.
Procedure linear sea rch ( x : integer, a1,a2 ,..., an : distinct integers)
i:  1;
while (i  n and x  ai )
i:  i  1;
if i  n then location:  i
else locatiton:  0;
{location is the subscript of
term that equals x, or is 0 if x is not found}
1
2(n+1)
n
2
3n+5
Binary Search Algorithm.
Procedure binary sea rch ( x : integer, a1,a2 ,..., an : increasing integers)
1
i:  1;
1
j:  n;
while (i  j )
1
begin
1
m:  (i  j)/ 2;
if x  am then i:  m  1
else j:  m;
end;
if x  ai then location:  i
2
Let n  2k.
It repeats at most k (k  log 2 n) times.
2
else location:  0;
{location is the subscript of
term that equals x, or is 0 if x is not found}
Number of operations = 4 log n+4
2
Orders of Growth
Running time for a problem with size n  106
Running
Time
Operation
Per second
10
6
1012
necessary
lg n
n
operations
n
2
2n
instant
1 second
11.5 days
Never end
259350 days
instant
Instant
1 second
Never end
259340 days
Using silicon computer, no matter how fast CPU will be you can
never solve the problem whose running time is exponential !!!
Asymptotic Notations: O-notation
Definition 2.1 A function t(n) is said to be O(g(n)) if there exist
some constant c0  0 and n0  0 such that t (n) c 0 g (n) for all n  n0 .
c0 g (n)
t (n)
n0
If lim n
n
t ( n)
 c (c  0 is a constant ) , then t(n)  O(g(n)).
g ( n)
Example 3 Prove 2n+1=O(n)
Example 4 Prove 10n 2  12n  5  O(n 2 )
Example 5
List the following function in O-notation in increasing order:
lg n, n, n 2 , n lg n, n3 , n!,2n.
Example 6 What is the big - oh of the following functions?
5n 5  100n 2 1000,
n lg n 2  100n lg n  1000n,
0.0001 2 n  n
Asymptotic Analysis of algorithms (using O-notation)
Example 7 Analyze the time complexities of linear search
algorithm and binary search algorithm asymptotically.
Linear Search Algorithm.
Procedure linear sea rch ( x : integer, a1,a2 ,..., an : distinct integers)
i:  1;
while (i  n and x  ai )
i:  i  1;
if i  n then location:  i
It repeats n times.
else locatiton:  0;
{location is the subscript of
term that equals x, or is 0 if x is not found} Totally(addition): O(n)
Binary Search Algorithm.
Procedure binary sea rch ( x : integer, a1,a2 ,..., an : increasing integers)
i:  1;
j:  n;
while (i  j )
begin
m:  (i  j)/ 2;
if x  am then i:  m  1
else j:  m;
Let n  2k.
It repeats at most k (k  log 2 n) times.
end;
if x  ai then location:  i
else location:  0;
{location is the subscript of
term that equals x, or is 0 if x is not found} Totally(comparison): O(log n)
Example 8 Analyze the time complexities of following algorithm
asymptotically.
Matrix addition algorithm
Procedure MatricAddition(A[0..n-1,0..n-1],B[0..n-1,0..n-1])
for i=0 to n-1 do
for j=0 to n-1 do
C[i,j] = A[i,j] + B[i,j];
return C;
Repeat n times
Repeat n times
2
Totally(addition): O(n )
Recursive Algorithms
Example 9 Computing the factorial function F(n)=n!.
F(n) can be defined recursively as follows:
 F (0)  1

 F (n)  F (n  1)  n
Factorial Algorithm
Procedure factorial(n)
Algorithm factorial calls itself
in its body!
if n = 0 return 1
else return factorial(n-1) * n;
Time complexity(multiplication): T(0) = 0
T(n) = T(n-1) +1 when n>0
recurrence
Basic Recurrences
Example 10
Solving the following recurrence
T(0) = 1
T(n) = T(n-1) + 1
T(n) = T(n-1) + 1
Example 11
Solve the following recurrence
Tn  2Tn / 2  n
n 1
T1  c'
n>0
T (n)  2T (n / 2)  cn
 2(2T (n / 2 2 )  c(n / 2))  cn  2 2 T (n / 2 2 )  cn  cn
2
3
2
3
3
= T(n-2) + 1 + 1 = T(n-2) + 2  2 (2T (n / 2 )  c(n / 2 ))  cn  cn  2 T (n / 2 )  3cn
= T(n-3) + 1 + 2 = T(n-3) + 3 ......
…
 2i T (n / 2i )  icn
......
= T(n-i) + i
 2 k T (n / 2 k )  kcn (k  log n)
…
 nT1  cn log n  c'cn log n
= T(n-n) + n
 O(n log n)
=n
Example 12 Solve the recurrence Example 13 Solve the recurrence
T (n)  T (n  1)  n n  1
T (n)  T (n / 2)  1
n 1
T1  1
T (1)  1
(Assume that n is a power of 2.
T (n)  T (n  1)  n
 T (n  2)  (n  1)  n
That is, n  2 k , where k  log n).
 T (n  3)  (n  2)  (n  1)  n
T ( n )  T ( n / 2)  1
......
 T (n / 2 2 )  1  1
T 12  3  ...  (n  2)  (n  1)  n
 T (n / 23 )  1  1  1
......
 1  2  3  ...  n
 n(n  1) / 2
 T (n / 2i )  i
......
 T (n / 2 k )  k (k  log n)
 1  log n