Algorithmics

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Algorithmics

• The discipline of algorithms

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Abu Jafar Mohammed ibn Musa al Khowarizmi

• Algorithm comes from the name of a Persian, Abu
  Jafar Mohammed ibn Musa al Khowarizmi (c. 825 A.
  D.).

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al-Khwarizmi

• ???(algorithm)?????al-Khwarizmi,????????780?????
  ??( Iraq )???( Baghdad )????????????????(Abu
  Jafar Muhammad ibn Musa al-Khwarizmi)????????,????
  ?????????????????????(????????????????????),??????
  ??ilm al jabr wal-muqabala???,?????????????????
  ,?????????????12?????????,??Algebra et
  Almucabala, (Algebra?????????al
  jabr),??????(Algebra)??????

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What is an algorithm (1/2)

• Any special method of solving a certain kind of
  problem -- Websters
• A precise method useable by a computer for the
  solution of a problem. -- Viewed by Computer
  Engineers
• In mathematics and computing, an algorithm is a
  procedure (a finite set of well-defined
  instructions) for accomplishing some task which,
  given an initial state, will terminate in a
  defined end-state. -- WiKipedia

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What is an algorithm (2/2)

• An algorithm is composed of a set of steps for
  solving a specific problem. It should satisfy the
  following properties
• Zero or more inputs
• At least one output
• Finiteness (the number of steps should be finite
  and the execution of the steps should terminate)
• Definiteness (to compute 5/0 is not definite)
• Effectiveness (to compute v2 as an infinite
  decimal is not effective)

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A recipe is an algorithm

• Informally, the concept of an algorithm is often
  illustrated by the example of a recipe.
• ??????(4??)???12????6-8???2?????2???1?????
  ??????????
• ??1. ??????????????????????,???????????????????
  ????????,???2. ?????,??,???????,??????????,??????
  3. ?????,?????????,????????????????1??,???????2??
  ???,?????????

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A flowchart is an algorithm
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A program is an algorithm

1. import javax.swing.
2. public class EuclidGCDApplet extends JApplet
3. public void init( )
4. int ??m,??n,?????
5. String ????1,????2,????
6. ????1JOptionPane.showInputDialog("??????m?")
   
7. ????2JOptionPane.showInputDialog("??????n?")
   
8. ??mInteger.parseInt(????1)
9. ??nInteger.parseInt(????2)
10. ?????EuclidGCD(??m, ??n)
11. ????"??"??m"???"??n"???????"?????
12. JOptionPane.showMessageDialog(null,????)
13. //??init() ??????

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Representation of Algorithms

• ??????????(????????)????(flow chart)????(pseudo
  code)???????(high level programming
  language)???????
• ??,???????????(?????)?????????? ?
  ????(Euclid)GCD????
• ????(Euclid)GCD?????????300?????????????,?????????
  ??????(GCD, Greatest Common Divisor)?

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Euclid GCD Algorithm
Euclid GCD????????????m?n,???????????(????????m?n??????)??E1.???? ??m??n???,????r?E2.???0?? ??r0???,??n?GCD?E3.???? ??mn,nr,?????E1?
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Pseudo Code

• ???????????????????????????????,?????????????,????
  ??????????????
• ????Goodrich?Tamassia??Data Structures and
  Algorithms in JAVA?????????????

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Pseudo Code

• ??? Algorithm ????(??1,??2,)?????????????????,A
  lgorithm Euclid-GCD(m, n) ???????????????m?n????E
  uclid-GCD????,?
• ????? ? ??,????????????????????,m?38
  ????38??????m??
• ????? ??,???????????????,m38
  ?????38???????m?????

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Pseudo Code

• ????? if??then??????? else ???????
  ???,????????????????????????????????????,
• if m38 then b?38else c?38
• ?????m???38???,????38??????b???,??????38??????c
  ????

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Pseudo Code

• while??? while??do???? ???,????????????????????
  ???????,???????????????????,
• while m38 do b?38 c?38
• ?????m???38???,??????38??????b??38??????c????

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Pseudo Code

• repeat??? repeat???? until?? ???,???????????????
  ???????????????????????????,???????????,
• repeat b?38 c?38until m38
• ????????38??????b??38??????c???,????m???38?????

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Pseudo Code

• for??? for??????? do ???? ???,?????????????????
  ??????,
• for i?3 to 8 do b?38 c?38
• ?????i???3?8(?i?3?4?5?6?7?8)????,??????38??????b?
  ?38??????c????

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Pseudo Code

• ??????? Ai ????A???(index)?i???,???n??????,??
  ????0,1,,n-1???,A5 ????A????5????
• ????? ??.??(??) ??????,?????????,???????????,M
  ath.random() ?????Math??????random()???
• ????? return ????? ??????,return 38
  ?????38???

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Pseudo Code for Euclid Alg.
Algorithm Euclid_GCD(m, n)Input two integers m and nOutput GCD of m and nr?mnwhile r?0 do m?n n?r r?mnreturn n
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Java Method for Euclid Alg.

• ????????????????????,?????????,???????????????????
  ????????????????,?????????????????????Java????????
  
• int Euclid_GCD(int m, int n)
• int rmn
• while (r!0)
• mn
• nr
• rmn
• return n

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Java Applet for Euclid Alg.

• ????(??EuclidGCDApplet.java)
• //??EuclidGCDApplet.java
• //????????????,??????GCD??????????(GCD)
• import javax.swing.
• public class EuclidGCDApplet extends JApplet
• public void init( )
• int ??m,??n,?????
• String ????1,????2,????
• ????1JOptionPane.showInputDialog("??????m?")

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Java Applet for Euclid Alg.

• ????2JOptionPane.showInputDialog("??????n?
  ")
• ??mInteger.parseInt(????1)
• ??nInteger.parseInt(????2)
• ?????EuclidGCD(??m, ??n)
• ????"??"??m"???"??n"???????"?????
• JOptionPane.showMessageDialog(null,????)
• //??init() ??????

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Java Applet for Euclid Alg.

• static int EuclidGCD(int m, int n)
• int rmn
• while (r!0)
• mn
• nr
• rmn
• return n
• //??EuclidGCD() ??????
• //??EuclidGCDApplet ??????

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Java Applet for Euclid Alg.

• ????(??EuclidGCDApplet.html)
• lthtmlgt
• lth1gtEuclidGCDApplet???lth1gt
• ltapplet code"EuclidGCDApplet.class" width350
  height100gt
• lt/appletgt
• lt/htmlgt
• ????(????????EuclidGCDApplet.html)

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Correctness and Efficiency

• ???????,???????????(correctness),????????????????,
  ???????????,??????????????(efficiency)?,??????????
  ???????????????,????????????????????,?????????????
  ???

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An Alg. for Testing Primes
Algorithm Prime1(n)Input????2????nOutputtrue?false(??n????????)for i?2 to n-1 do if (ni)0 then return falsereturn true

• ??????,????2??????n,?n???,????Prime1???????????(n
  i)???????((ni)0)??n-2???,?????n??????,?n????,???
  ?Prime1??????????????????1?,?????n??????

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A Theorem for Testing Primes
?? ???????2???n??,???????? ???(1??)?????n,? n??????
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Hint

• ????n??????????,??,16????1?16(11616)?2?8(2816)
  ?4?4(4416)?????????,???????n????,??????????n????
  ,??,??????????n??????????1???????n????????n???????
  

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The Other Alg. for Testing Primes

• ???????????????????Prime2

Algorithm Prime2(n)Input????2????nOutputtrue?false(??n????????)for i?2 to vn do if (ni)0 then return falsereturn true
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Prime1Applet

• ????(??Prime1Applet.java)
• //??Prime1Applet.java
• //????????????2???,???????????(prime number)
• import javax.swing.
• public class Prime1Applet extends JApplet
• public void init()
• int ??n
• String ????,????
• ????JOptionPane.showInputDialog("?????2???n?"
  )

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Prime1Applet

1. ??nInteger.parseInt(????)
2. boolean ???
3. long ????????System.currentTimeMillis()
4. ???Prime1(??n)
5. long ????????System.currentTimeMillis()
6. long ???????????????-????????
7. if(???) ??????n"???"
8. else ??????n"????"
9. ????("\n????????"???????"??(milli-seconds)
   ")

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Prime1Applet

1. JOptionPane.showMessageDialog(null,????)
2. //??init() ??????
3. boolean Prime1(int n)
4. for (int i2iltn-1i)
5. if (ni0) return false
6. return true
8. //??Prime1Applet ??????

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Prime1Applet

• ????(??Prime1Applet.html)
• lthtmlgt
• lth1gtPrime1Applet???lth1gt
• ltapplet code"Prime1Applet.class" width350
  height100gt
• lt/appletgt
• lt/htmlgt
• ????(????????Prime1Applet.html)

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Prime2Applet

• ????(??Prime2Applet.java)
• //??Prime2Applet.java
• //????????????2???,???????????(prime number)
• import javax.swing.
• public class Prime2Applet extends JApplet
• public void init()
• int ??n
• String ????,????
• ????JOptionPane.showInputDialog("?????2???n?"
  )
• ??nInteger.parseInt(????)

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Prime2Applet

• boolean ???
• long ????????System.currentTimeMillis()
• ???Prime2(??n)
• long ????????System.currentTimeMillis()
• long ???????????????-????????
• if(???) ??????n"???"
• else ??????n"????"
• ????("\n????????"???????"??(milli-seconds)
  ")

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Prime2Applet

• JOptionPane.showMessageDialog(null,????)
• //??init() ??????
• boolean Prime2(int n)
• for (int i2iltMath.sqrt(n)i)
• if (ni0) return false
• return true
• //??Prime2Applet ??????

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Prime2Applet

• ????(??Prime2Applet.html)
• lthtmlgt
• lth1gtPrime2Applet???lth1gt
• ltapplet code"Prime2Applet.class" width350
  height100gt
• lt/appletgt
• lt/htmlgt
• ????(????????Prime2Applet.html)

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• QA

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????1-3.???????Prime2??????????

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????1-2.???????Prime1??????????

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Result