概述 - 中山大学信息科学与技术学院

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Algorithms Design and its
Applications
Introduction
Something about me
刘晓铭
• Home Page: http://sist.sysu.edu.cn/~isslxm
• Email: [email protected]
• Office: 信息学院楼526B
Something about this course
• Course homepage:
http://sist.sysu.edu.cn/~isslxm/ada/cs2013/inde
x.html
• BBS:
http://bbs.sysu.edu.cn/bbsdoc?board=ACMICPC
• Office Hours:
每周一下午2:00 – 4:00,信息学院楼526B
其它时间可邮件预约。
Something about this course
• 1. 平时作业:在线编程练习+书面作业
http://soj.me/
• 2. 研究型的小Project :选一个问题,编程
实现解决这一问题的若干种算法,在不同
规模的测试数据上,比较各种算法的性能
• 3. 期末考试:ACM 方式机试, 自动测评
FAQ
• Q: 这门课是为了培养参加ACM / ICPC比赛的
队员而专门开设的吗?
• A: No.
Some important topics not included in ICPC
• Advanced Algorithm Analysis Techniques
• NP-Completeness Proofs
• Randomized and Approximation Algorithms
• Quantum Algorithms
推荐教材
武林秘籍
Why take this course?
You want to be a computer
scientist?
Is your goal to be
a mundane programmer?
Or a great leader and thinker?
Original Thinking
Boss assigns task:
– Given today’s prices of pork, grain, sawdust, …
– Given constraints on what constitutes a hotdog.
– Make the cheapest hotdog.
Everyday industry asks these questions.
Your answer:
• Um? Tell me what to code.
With more sophisticated software engineering systems,
the demand for mundane programmers will diminish.
Your answer:
• I learned this great algorithm that will work.
Soon all known algorithms
will be available in libraries.
Your answer:
• I can develop a new algorithm for you.
Great thinkers
will always be needed.
Course Content
• A list of algorithms.
– Learn their code.
– Trace them until you are convinced that they
work.
class InsertionSortAlgorithm extends
SortAlgorithm {
– Implement them.
void sort(int a[]) throws Exception {
for (int i = 1; i < a.length; i++) {
int j = i;
– Worry about details.
int B = a[i];
while ((j > 0) && (a[j-1] > B)) {
a[j] = a[j-1];
j--; }
a[j] = B;
}
}
Course Content
• A survey of algorithmic design techniques.
• Abstract thinking.
• How to develop new algorithms for any problem
that may arise.
Study:
• Many experienced programmers were asked to
code up binary search.
Study:
• Many experienced programmers were asked to
code up binary search.
80% got it wrong
What did they lack?
What did they lack?
• Formal proof methods?
What did they lack?
• Formal proof methods?
Yes, likely
Industry is starting to
realize that formal
methods are important.
But even without formal
methods …. ?
What did they lack?
• Fundamental understanding of the
algorithmic design techniques.
• Abstract thinking.
Course Content
Notations, analogies, and abstractions
for developing,
thinking about,
and describing algorithms
so correctness is transparent
A survey of fundamental
ideas and algorithmic
design techniques
For example . . .
Some Math
Time Complexity
t(n) = Q(n2)
Logs and Exps
2a × 2b = 2a+b
2log n = n
Classifying Functions
f(i) = nQ(n)
Time
Logic Quantifiers
g "b Loves(b,g)
"b g Loves(b,g)
Input Size
Adding Made Easy
∑i=1 f(i).
Recurrence Relations
T(n) = a T(n/b) + f(n)
Iterative Algorithms
Loop Invariants
<preCond>
codeA
loop
<loop-invariant>
exit when <exit Cond>
codeB
codeC
<postCond>
Code
0
5 km
i-1
i
9 km
One step
at a time
Relay Race
Recursive Algorithms
Recursive Back Tracking
Greedy Algorithms
Dynamic Programing
4
5
3
2
2
8
9
5
8
4
4
6
2
3
5
7
5
6
1
3
2
5
4
8
Graph Search Algorithms
Network Flows
Reduction
=
Rudich www.discretemath.com
Useful Learning Techniques
Read Ahead
You are expected to read the lecture notes before
the lecture.
This will facilitate more productive discussion
during class.
Also please
proof read
assignments
& tests.
Be Creative
•Ask questions.
• Why is it done this way and not that way?
Please feel free
to ask questions!
Help me know what people
are not understanding
Slow down the slides
We do have a lot of material
Please don’t shout out answers
for at least 10sec.
What is algorithm?
Informally, an algorithm is any well – defined
computational procedure that takes some
value, or set of values, as input and produces
some value, or set of values, as output .
-- Introduction to Algorithms
Sorting Problem
Input: A sequence of n keys a1, a2, …, an.
Output: The permutation of the input sequence
such that a1’≤a2’≤…≤an’.
• An instance (实例):
Inpute: <31, 41, 59, 26, 41, 58>
Output: <26, 31, 41, 41, 58, 59>
• 如果对每个输入的实例,算法都能正确的
停机,则称该算法是正确的。
• 不正确的算法对某些输入实例可能根本不
停机,也可能以不正确的输出停机。
• 不正确的算法只要错误率可控有时可能是
有用的。例如,大素数的判定。
Knuth said
Knuth (1968, 1973) has given a list of five properties that are widely
accepted as requirements for an algorithm:
Finiteness: "An algorithm must always terminate after a finite number of
steps ... a very finite number, a reasonable number"
Definiteness: "Each step of an algorithm must be precisely defined; the
actions to be carried out must be rigorously and unambiguously specified
for each case"
Input: "...quantities which are given to it initially before the algorithm
begins. These inputs are taken from specified sets of objects"
Output: "...quantities which have a specified relation to the inputs"
Effectiveness: "... all of the operations to be performed in the algorithm
must be sufficiently basic that they can in principle be done exactly and
in a finite length of time by a man using paper and pencil"
Finally …
The word algorithm does not have a generally
accepted definition.
From wikipedia
Analysis of algorithms
• The theoretical study of computer-program
performance and resource usage.
• What’s more important than performance?
• modularity
• correctness
• maintainability • functionality
• robustness
• user-friendliness
• programmer time • simplicity
• extensibility
• reliability
为什么学习算法
原因之一:广泛的应用
• Human Genome Project
• Internet
• Electronic commerce security
• Other manufacturing and commercial
setting
• Etc.
为什么学习算法
原因之二:强大的力量
排序n个项,插入排序,c1n2 归并排序,c2n lgn
n=1000万
计算机A: 较快,每秒执行百亿条指令,执行插入排序,实
现较好,代码需要2n2条指令。
计算机B: 较慢,每秒执行1000万条指令,执行归并排序,实
现较差,代码需要50nlgn条指令。
计算机A: 2*(107)2条指令/1010条指令每秒=20000秒
计算机B: 50*107lg107条指令/107条指令每秒=1163秒
计算机B比计算机A还快17倍!
为什么学习算法
不懂算法的悲剧
和大姐姐一起来计算!
http://www.bilibili.tv/video/av362069/
http://www.youtube.com/watch?v=Q4gTV4r0zRs
来自日本科学未来馆的趣味教育动画,介绍的是
一个正方形由“起点(S)”到“终点(G)”会有多少种
路线,以及关于大姐姐的催人泪下的故事
分析:http://oeis.org/A007764
为什么学习算法
原因之三:Just for fun!
算法分析基础
Fibonacci number
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …,
Leonardo of Pisa (Fibonacci)
1170-1250
Fibonacci number
• An exponential algorithm
int fib1(n)
{
if ( n==0 ) return 0;
if ( n==1 ) return 1;
return fib1(n-1) + fib1(n-2);
}
How much time does it take?
Let T(n) be the number of computer steps needed to
compute fib1(n).
T(n)<=2 for n<=1
T(n)=T(n-1) + T(n-2) + 3 for n>1
(two recursive invocations of fib1, plus three
computer steps to check on the value of n and a final
addition.)
Therefore, T(n) >= Fn.
The running time of the algorithm grows as fast as the
Fibonacci numbers!
T(n) is exponential in n.
How much time does it take?
How bad exponential time is?
To compute F200, the fib1 algorithm executes
T(200)>=F200>=2138 elementary computer
steps.
On a computer which clocks 40 trillion steps
per second, fib1(200) would take at least 292
seconds.
How much time does it take?
Moore’s law: computer speeds doubles roughly every
18 months. Computers get roughly 1.6 times faster
each year. It helps?
The running time of fib1(n) is proportional to 20.694n
= (1.6)n, so it takes 1.6 times longer to compute
Fn+1 than Fn.
So if we reasonably compute F100 ith this year’s
technology, then next year we will manage F101. And
the year after, F102. And so on.
Can we do better?
• Why fib1 is so slow?
Can we do better?
• An polynomial algorithm
int fib2(n)
{
if ( n==0 ) return 0;
int f[n]; // create an array f[0…n]
f[0] = 0, f[1] = 1;
for (int i=2; i<=n; i++)
f[i] = f(i-1) + f(i-2);
return f[n];
}
How long does fib2 take?
• The inner loop consists of a single computer
step and it executed n-1 times. Therefore the
number of computers steps used by fib2 is
linear in n.
• It is now perfectly reasonable to compute F200
or even F200000.
Can we do even better?
Write the equations F1 = F1 and F2 = F0+F1 in matrix notation:
So, in order to compute Fn, it suffices to raise this 2*2 matrix,
call it X, to the nth power.
Two 2*2 matrices can be multiplied using 4 additions and 8
multiplications. How many matrix multiplications suffice for
computing Xn?
Efficient Exponentiation
long pow( long x, int n)
{
if( n == 0 ) return 1;
if( n == 1 ) return x;
if( n%2==0 )
return pow(x*x,n/2);
else
return pow(x*x,n/2) * x;
}
O(log n) matrix multiplications suffice for computing Xn?
Can we do even better?
Thus the number of arithmetic operations
needed by our matrix-based algorithm, call it
fib3, is just O(logn), as compared to O(n) for
fib2.
Sicily 1863.
Elegant fibonacci numbers again
• Description
Fibonacci numbers are nice simple integers. All of you are
familiar with it, aren’t you?
The Fibonacci sequence <F[n]> are defined by the recurrence:
F[0]=0;
F[1]=1;
F[n]=F[n-1]+F[n-2], for n>1
You know that the value of F[n] increases rapidly when the n
becomes larger. So for each testcase, output the value F[n]
mod m will be ok.
Sicily 1863.
Elegant fibonacci numbers again
• Input
The first line of the input is a positive integer.It is the number of the test
cases followed. Each test case contains two integers n (0<=n<2^32) and m
(0<m<10000). There may be one or several spaces between the two
integers.
• Output
The output of the program should consist of one line of output for each test
case. The output of each test case only contains one integer which equals to
the value F[n] mod m. No any redundant spaces are needed.
Sample Input
2
1 1000
2 100
Sample Output
1
1
Sicily 1863.
Elegant fibonacci numbers again
The RAM Model of Computation
• 用基本指令条数来精确度量一个算法的效
率很困难:
– 从理论上分析及其繁琐,从实际中估计也颇为
困难。
– 不同的计算机采用的指令集、缓存策略等技术
细节不同。
• 寻找一种简便易行的,与具体机器无关的
度量方法,它是以输入规模为自变量的函
数。
The RAM Model of Computation
• Random Access Machine (RAM)
– Each simple operation (+, *, -, =, if, call) takes
exactly one time step.
– Loops and subroutines are not considered simple
operations. Instead, they are the composition of
many single-step operations.
– Each memory access takes exactly one time step.
Further, we have as much memory as we need. The
RAM model takes no notice of whether an item is in
cache or on the disk.
The Big-O Notation
Definition:
f (n),g (n) are positive function about natural
number. If there is a constant c >0,and natural
number n 0,When n>n 0,there is f (n) <c g (n), so
f (n)=O ( g (n))
If there is a constant c >0,and natural number n 0,
When n>n 0,there is f (n) >c g (n), so
f (n)= Ω(g (n))
if f (n)=O( g (n)), f (n)= Ω( g (n)) , then
f (n)= Θ( g (n))
The Big Oh Notation
Working with the Big Oh
• Adding Functions
Working with the Big Oh
• Multiplying Functions
O(cf(n)) = O(f(n))
Ω(cf(n)) = Ω(f(n))
Θ(cf(n)) = Θ(f(n))
O(f(n)) * O(g(n)) = O(f(n) * g(n))
Ω(f(n)) * Ω(g(n)) = Ω(f(n) * g(n))
Θ(f(n)) * Θ(g(n)) = Θ(f(n) * g(n))
Working with the Big Oh
• Transitive Experience
If f(n) = O(g(n)) and g(n) = O(h(n)), then f(n)=O(h(n)).
Grow Rates
Grow Rates
Popular:
O(1) < O (logn) <O(n) < O(nlogn) <O(n2)<O(n3)
O(2n) < O(n!) < O(nn)
Some rules:
O( f )+O( g )=O(max( f , g ))
O( f )+O( g )=O( f+g )
O( f )O( g )=O( f *g )
O( c f (n)=O( f (n))
•Popular: O(1), O(log2 n), O(n), O(nlog2 n) , O(n2), O(2n), O(n!)
2n
T(n)
5n2
2000
100n
20
200log2n
10
20
n
Best, Worst and Average-Case
Complexity
空间复杂度
• 计算算法运行过程所占用的存储量作为问题规模
的函数 S(n) = O(f(n)).
• 通常以简单变量的存储空间为单位
• 算法的存储空间包括:
1)程序本身、常量和变量等所需空间
2)程序运行中动态申请的空间和递归调用所
需空间
• 通常考虑除输入和程序以外的空间. 如果相对于
输入是常数, 则称为原地工作. 如sum 和
selectSort.
Running Time Calculations
• A Simple Example
• General Rules
• Solutions for the maximum Subsequence Sum Problem
A Simple Example
A simple program to calculate 13+23+..+n3.
int sum(int n)
{
int partialSum;
1 partialSum = 0;
2 for (int i=0; i<=n; i++)
3
partialSum += i*i*i;
4 return partialSum;
}
General Rules
Rule 1-For loops.
The running time of a for loop is at most
the running time of the statements inside
the for loop times the number of
iterations.
Rule 2-Nested loops.
Analyze these inside out. The total
running time of a statement inside a group
of nested loops is the running time of the
statement multiplied by the product of the
size of all the loops.
General Rules
Rule 3-Consecutive Statements.
These just add.
for (i=0; i<n; i++)
a[i] = 0;
for (i=0; i<n; i++)
for (j=0; j<n; j++)
a[i] += a[j] + i + j;
General Rules
Rule 4-If/Else.
For the fragment
if (condition)
S1
else
S2
the running time of an if/else statement
is never more than the running time of the
test plus the larger of the running times
of S1 and S2.
The Maximum Subsequence Sum
Problem
• 给一串整数a[1…n],求出它和最大的子序
列,即找出1<=i<=j<=n,使
a[i]+a[i+1]+…+a[j]最大
• 介绍四个算法并分析时间复杂度
–
–
–
–
枚举:O(n3)
优化的枚举:O(n2)
分治:O(nlogn)
贪心:O(n)
算法一
max := a[1];
for i:=1 to n do
for j:=i to n do begin
sum := 0;
for k:=i to j do
sum := sum + a[k];
if sum > max then max := sum;
end;
• 思想:枚举所有的i和j,计算a[i]+…+a[j],选择最大的
• 时间复杂度如何分析?
– 三层循环
– 内层操作次数取决于i, j
– 中层操作次数取决于i
算法一分析
•
•
•
•
当i和j一定的时候,内层循环执行j-i+1次
总次数为   ( j  i  1)
应该如何计算?
方法一:直接计算
1
– 先计算里层求和号,得  ( n  i  1)( n  i  2)
2
n
n
i 1
ji
n
i 1
– 再加起来?好复杂…
– 直接算法需要利用公式12+…+n2=O(n3)
– 一般地,有1k+…+nk=O(nk+1). 证明: 数学归纳法
算法一分析(续)
n
n
i 1
ji
• 总次数为:  ( j  i  1)
– 完全的计算太麻烦
– 能不能不动笔就知道渐进时间复杂度呢?
• 何必非要计算出详细的公式再化简呢?
– 里层求和计算出的结果是O((n-i)2)
– 12+22+…+n2=O(n3)
– 每步都化简!但是要保留外层需要的变量
• 结论:算法一时间复杂度为O(n3)
• 经验:一般只能支持 n<=200
算法二
• 思路
–
–
–
–
枚举i和j后,能否快速算出a[i]+…+a[j]呢?
预处理!
记s[i] = a[1] + a[2] + … + a[i], 规定s[0] = 0
则可以在O(n)的时间内计算出s[1], …, s[n]
s[0] := 0;
for i:=1 to n do
s[i] := s[i–1] + a[i];
算法二(续)
• 有了s[i],怎么快速求a[i]+…+a[j]呢?
a[i]+…+a[j]
= (a[1] + … + a[j]) – (a[1] + … + a[i-1])
=s[j] – s[i-1]
而s[i]经过预处理以后可以直接读出!
max := a[1];
for i:=1 to n do
for j:=i to n do
if s[j] – s[i-1] > max then
max := s[j] – s[i-1];
时间复杂度:预处理+主程序=O(n+n2)=O(n2). n<=5000
算法三
• 用一种完全不同的思路
• 最大子序列的位置有三种可能
– 完全处于序列的左半,即j<=n/2
– 完全处于序列的右半,即i>=n/2
– 跨越序列中间,即i<n/2<j
• 用递归的思路解决!
– 设max(i, j)为序列a[i…j]的最大子序列, 那么…
算法三(续)
• 用递归的思路解决!
– 设max(i, j)为序列a[i…j]的最大子序列
– 设mid = (i + j)/2,即区间a[i…j]的中点
• 最大的第一种序列为max(i, mid)
• 最大的第二种序列为max(mid+1, j)
• 问题:最大的第三种序列为???
– 既然跨越中点,把序列a[i…j]划分为两部分
• a[i…mid]:最大序列用扫描法在n/2时间内找到
– 一共只有mid-1=n/2种可能的序列,一一比较即可
• a[mid+1…j]:同理
• 一共花费时间为j-i+1
算法三(续)
//返回数组在区间[x,y)中的最大连续和
int maxsum(int *A, int x, int y)
{
int i, m, v, L, R, max;
if (y-x == 1) return A[x]; //只有一个元素,直接返回
m = x + (y – x) / 2; // 分治第一步:划分成[x,m)和[m,y)
//分治第二步,递归求解
max = maxsum(A, x, m) >? maxsum(A, m, y);
//分治第三步:合并(1),求分界点开始向左的最大连续和;
v = 0; L = A[m-1];
for(i=m-1;i>=x;i--) L >?= v += A[i];
//分治第三步:合并[2],求分界点开始向右的最大连续和;
v = 0; R = A[m];
for(i=m;i<y;i++) R >?= v += A[i];
return max >? (L+R);
}
算法三分析
• 如何分析时间复杂度呢?
– 我们没有直接得到具体的T(n)的式子
– 但是容易得到递推关系T(n) = 2T(n/2) + n
• 设时间复杂度的精确式子是T(n)
• 第一、二种序列的计算时间是T(n/2),因为序列长度
缩小了一半
• 第三种序列的计算时间是n
– 由递归式四, 得T(n)=O(nlogn)
算法四
• 算法二的实质是求出i<=j,让s[j]-s[i-1]最大
– 对于给定的j,能否直接找到在j之前的最小s值呢?
– 从小到大扫描j
• j=1时,只有一个合法的i,即i=1, s[1-1]=0
• 如果s[j]变大,则最小的s值和上次一样
• 如果s[j]再创新低,应该让s[j]作为今后的最优s值
min_s := 0;
For j :=1 to n do begin
if s[j] < min_s then min_s := s[j];
if s[j] – min_s > max then
max := s[j] – min_s;
end;
时间复杂度很明显:O(n). n <= 1,000,000
总结
算法
时间复杂度
规模
分析方法
枚举
O(n3)
约200
分层求和
优化枚举
O(n2)
约3000
明显
分治
O(nlogn)
约105
递归树
扫描
O(n)
约106
明显
Why study algorithms and
performance?
• Algorithms help us to understand scalability.
• Performance often draws the line between
what is feasible and what is impossible.
• Algorithmic mathematics provides a language
for talking about program behavior.
• Performance is the currency of computing.
• The lessons of program performance
generalize to other computing resources.
• Speed is fun! Need for speed!
Three questions on a algorithm
1. Is it correct?
2. How much time does it take, as a function of
n?
3. And can we do better?

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