前缀、中缀、后缀表达式

它们都是对表达式的记法,因此也被称为前缀记法、中缀记法和后缀记法。它们之间的区别在于运算符相对与操作数的位置不同:前缀表达式的运算符位于与其相关的操作数之前;中缀和后缀同理。

举例:
(3 + 4) × 5 - 6 是中缀表达式
- × + 3 4 5 6 前缀表达式
3 4 + 5 × 6 - 后缀表达式

中缀表达式(中缀记法)


中缀表达式是一种通用的算术或逻辑公式表示方法,操作符以中缀形式处于操作数的中间。中缀表达式是人们常用的算术表示方法。

虽然人的大脑很容易理解与分析中缀表达式,但对计算机来说中缀表达式却是很复杂的(主要是需要考虑运算符的优先级等问题),因此计算表达式的值时,通常需要先将中缀表达式转换为前缀或后缀表达式,然后再进行求值。对计算机来说,计算前缀或后缀表达式的值非常简单。

前缀表达式(前缀记法、波兰式)

前缀表达式的运算符位于操作数之前。

前缀表达式的计算机求值

从右至左扫描表达式,遇到数字时,将数字压入堆栈,遇到运算符时,弹出栈顶的两个数,用运算符对它们做相应的计算(栈顶元素 op 次顶元素),并将结果入栈;重复上述过程直到表达式最左端,最后运算得出的值即为表达式的结果。

例如前缀表达式“- × + 3 4 5 6”:

(1) 从右至左扫描,将6、5、4、3压入堆栈;
(2) 遇到+运算符,因此弹出3和4(3为栈顶元素,4为次顶元素,注意与后缀表达式做比较),计算出3+4的值,得7,再将7入栈;
(3) 接下来是×运算符,因此弹出7和5,计算出7×5=35,将35入栈;
(4) 最后是-运算符,计算出35-6的值,即29,由此得出最终结果。

可以看出,用计算机计算前缀表达式的值是很容易的。

将中缀表达式转换为前缀表达式


遵循以下步骤:

(1) 初始化两个栈:运算符栈S1和储存中间结果的栈S2;
(2) 从右至左扫描中缀表达式;
(3) 遇到操作数时,将其压入S2;
(4) 遇到运算符时,比较其与S1栈顶运算符的优先级:
(4-1) 如果S1为空,或栈顶运算符为右括号“)”,则直接将此运算符入栈;
(4-2) 否则,若优先级比栈顶运算符的较高或相等,也将运算符压入S1;
(4-3) 否则,将S1栈顶的运算符弹出并压入到S2中,再次转到(4-1)与S1中新的栈顶运算符相比较;
(5) 遇到括号时:
(5-1) 如果是右括号“)”,则直接压入S1;
(5-2) 如果是左括号“(”,则依次弹出S1栈顶的运算符,并压入S2,直到遇到右括号为止,此时将这一对括号丢弃;
(6) 重复步骤(2)至(5),直到表达式的最左边;
(7) 将S1中剩余的运算符依次弹出并压入S2;
(8) 依次弹出S2中的元素并输出,结果即为中缀表达式对应的前缀表达式。

例如,将中缀表达式“1+((2+3)×4)-5”转换为前缀表达式的过程如下:

扫描的元素

S2(左为栈底)

S1(左为栈底)

说明

5 5 数字,直接入栈
- 5 - S1为空,运算符直接入栈
) 5 - ) 右括号直接入栈
4 5 4 - ) 数字直接入栈
× 5 4 - ) × S1栈顶是右括号,直接入栈
) 5 4 - ) × ) 右括号直接入栈
3 5 4 3 - ) × ) 数字
+ 5 4 3 - ) × ) + S1栈顶是右括号,直接入栈
2 5 4 3 2 - ) × ) + 数字
( 5 4 3 2 + - ) × 左括号,弹出运算符直至遇到右括号
( 5 4 3 2 + × - 同上
+ 5 4 3 2 + × - + 优先级与-相同,入栈
1 5 4 3 2 + × 1 - + 数字
到达最左端 5 4 3 2 + × 1 + - S1中剩余的运算符

因此结果为“- + 1 × + 2 3 4 5”。


后缀表达式(后缀记法、逆波兰式)

后缀表达式与前缀表达式类似,只是运算符位于操作数之后。

后缀表达式的计算机求值

与前缀表达式类似,只是顺序是从左至右:

从左至右扫描表达式,遇到数字时,将数字压入堆栈,遇到运算符时,弹出栈顶的两个数,用运算符对它们做相应的计算(次顶元素 op 栈顶元素),并将结果入栈;重复上述过程直到表达式最右端,最后运算得出的值即为表达式的结果。

例如后缀表达式“3 4 + 5 × 6 –”:

(1) 从左至右扫描,将3和4压入堆栈;
(2) 遇到+运算符,因此弹出4和3(4为栈顶元素,3为次顶元素,注意与前缀表达式做比较),计算出3+4的值,得7,再将7入栈;
(3) 将5入栈;
(4) 接下来是×运算符,因此弹出5和7,计算出7×5=35,将35入栈;
(5) 将6入栈;
(6) 最后是-运算符,计算出35-6的值,即29,由此得出最终结果。

将中缀表达式转换为后缀表达式

与转换为前缀表达式相似,遵循以下步骤:

(1) 初始化两个栈:运算符栈S1和储存中间结果的栈S2;
(2) 从左至右扫描中缀表达式;
(3) 遇到操作数时,将其压入S2;
(4) 遇到运算符时,比较其与S1栈顶运算符的优先级:
(4-1) 如果S1为空,或栈顶运算符为左括号“(”,则直接将此运算符入栈;
(4-2) 否则,若优先级比栈顶运算符的高,也将运算符压入S1(注意转换为前缀表达式时是优先级较高或相同,而这里则不包括相同的情况);
(4-3) 否则,将S1栈顶的运算符弹出并压入到S2中,再次转到(4-1)与S1中新的栈顶运算符相比较;
(5) 遇到括号时:
(5-1) 如果是左括号“(”,则直接压入S1;
(5-2) 如果是右括号“)”,则依次弹出S1栈顶的运算符,并压入S2,直到遇到左括号为止,此时将这一对括号丢弃;
(6) 重复步骤(2)至(5),直到表达式的最右边;
(7) 将S1中剩余的运算符依次弹出并压入S2;
(8) 依次弹出S2中的元素并输出,结果的逆序即为中缀表达式对应的后缀表达式(转换为前缀表达式时不用逆序)。
例如,将中缀表达式“1+((2+3)×4)-5”转换为后缀表达式的过程如下:

扫描的元素 S2(左侧为栈底) S1(左侧为栈底) 说明
1 1 数字,直接入栈
+ 1 + S1为空,运算符直接入栈
( 1 + ( 左括号,直接入栈
( 1 + ( ( 同上
2 1 2 + ( ( 数字
+ 1 2 + ( ( + S1栈顶为左括号,运算符直接入栈
3 1 2 3 + ( ( + 数字
) 1 2 3 + + ( 右括号,弹出运算符直至遇到左括号
× 1 2 3 + + ( × S1栈顶为左括号,运算符直接入栈
4 1 2 3 + 4 + ( × 数字
) 1 2 3 + 4 × + 右括号,弹出运算符直至遇到左括号
- 1 2 3 + 4 × + - -与+优先级相同,因此弹出+,再压入-
5 1 2 3 + 4 × + 5 - 数字
到达最右端 1 2 3 + 4 × + 5 - S1中剩余的运算符

因此结果为“1 2 3 + 4 × + 5 -”(注意需要逆序输出)。

编写Java程序将一个中缀表达式转换为前缀表达式和后缀表达式,并计算表达式的值。其中的toPolishNotation()方法将中缀表达式转换为前缀表达式(波兰式)、toReversePolishNotation()方法则用于将中缀表达式转换为后缀表达式(逆波兰式)。程序很长且注释比较少,但如果将上面的理论内容弄懂之后再将程序编译并运行起来,还是比较容易理解的。有耐心的话可以研究一下。

 

    package qmk.simple_test;
    import java.util.Scanner;
    import java.util.Stack;
    /**
     * Example of converting an infix-expression to
     * Polish Notation (PN) or Reverse Polish Notation (RPN).
     * Written in 2011-8-25
     * @author QiaoMingkui
     */
    public class Calculator {
          public static final String USAGE = "== usage ==\n"
                + "input the expressions, and then the program "
                + "will calculate them and show the result.\n"
                + "input 'bye' to exit.\n";
          /**
           * @param args
           */
          public static void main(String[] args) {
                System.out.println(USAGE);
                Scanner scanner = new Scanner(System.in);
                String input = "";
                final String CLOSE_MARK = "bye";
                System.out.println("input an expression:");
                input = scanner.nextLine();
                while (input.length() != 0
                      && !CLOSE_MARK.equals((input))) {
                      System.out.print("Polish Notation (PN):");
                      try {
                            toPolishNotation(input);
                      } catch (NumberFormatException e) {
                            System.out.println("\ninput error, not a number.");
                      } catch (IllegalArgumentException e) {
                            System.out.println("\ninput error:" + e.getMessage());
                      } catch (Exception e) {
                            System.out.println("\ninput error, invalid expression.");
                      }
                      System.out.print("Reverse Polish Notation (RPN):");
                      try {
                            toReversePolishNotation(input);
                      } catch (NumberFormatException e) {
                            System.out.println("\ninput error, not a number.");
                      } catch (IllegalArgumentException e) {
                            System.out.println("\ninput error:" + e.getMessage());
                      } catch (Exception e) {
                            System.out.println("\ninput error, invalid expression.");
                      }
                      System.out.println("input a new expression:");
                      input = scanner.nextLine();
                }
                System.out.println("program exits");
          }
          /**
           * parse the expression , and calculate it.
           * @param input
           * @throws IllegalArgumentException
           * @throws NumberFormatException
           */
          private static void toPolishNotation(String input)
                      throws IllegalArgumentException, NumberFormatException {
                int len = input.length();
                char c, tempChar;
                Stack<Character> s1 = new Stack<Character>();
                Stack<Double> s2 = new Stack<Double>();
                Stack<Object> expression = new Stack<Object>();
                double number;
                int lastIndex = -1;
                for (int i=len-1; i>=0; --i) {
                      c = input.charAt(i);
                      if (Character.isDigit(c)) {
                            lastIndex = readDoubleReverse(input, i);
                            number = Double.parseDouble(input.substring(lastIndex, i+1));
                            s2.push(number);
                            i = lastIndex;
                            if ((int) number == number)
                                  expression.push((int) number);
                            else
                                  expression.push(number);
                      } else if (isOperator(c)) {
                            while (!s1.isEmpty()
                                        && s1.peek() != ')'
                                        && priorityCompare(c, s1.peek()) < 0) {
                                  expression.push(s1.peek());
                                  s2.push(calc(s2.pop(), s2.pop(), s1.pop()));
                            }
                            s1.push(c);
                      } else if (c == ')') {
                            s1.push(c);
                      } else if (c == '(') {
                            while ((tempChar=s1.pop()) != ')') {
                                  expression.push(tempChar);
                                  s2.push(calc(s2.pop(), s2.pop(), tempChar));
                                  if (s1.isEmpty()) {
                                        throw new IllegalArgumentException(
                                              "bracket dosen't match, missing right bracket ')'.");
                                  }
                            }
                      } else if (c == ' ') {
                            // ignore
                      } else {
                            throw new IllegalArgumentException(
                                        "wrong character '" + c + "'");
                      }
                }
                while (!s1.isEmpty()) {
                      tempChar = s1.pop();
                      expression.push(tempChar);
                      s2.push(calc(s2.pop(), s2.pop(), tempChar));
                }
                while (!expression.isEmpty()) {
                      System.out.print(expression.pop() + " ");
                }
                double result = s2.pop();
                if (!s2.isEmpty())
                      throw new IllegalArgumentException("input is a wrong expression.");
                System.out.println();
                if ((int) result == result)
                      System.out.println("the result is " + (int) result);
                else
                      System.out.println("the result is " + result);
          }
          /**
           * parse the expression, and calculate it.
           * @param input
           * @throws IllegalArgumentException
           * @throws NumberFormatException
           */
          private static void toReversePolishNotation(String input)
                      throws IllegalArgumentException, NumberFormatException {
                int len = input.length();
                char c, tempChar;
                Stack<Character> s1 = new Stack<Character>();
                Stack<Double> s2 = new Stack<Double>();
                double number;
                int lastIndex = -1;
                for (int i=0; i<len; ++i) {
                      c = input.charAt(i);
                      if (Character.isDigit(c) || c == '.') {
                            lastIndex = readDouble(input, i);
                            number = Double.parseDouble(input.substring(i, lastIndex));
                            s2.push(number);
                            i = lastIndex - 1;
                            if ((int) number == number)
                                  System.out.print((int) number + " ");
                            else
                                  System.out.print(number + " ");
                      } else if (isOperator(c)) {
                            while (!s1.isEmpty()
                                        && s1.peek() != '('
                                        && priorityCompare(c, s1.peek()) <= 0) {
                                  System.out.print(s1.peek() + " ");
                                  double num1 = s2.pop();
                                  double num2 = s2.pop();
                                  s2.push(calc(num2, num1, s1.pop()));
                            }
                            s1.push(c);
                      } else if (c == '(') {
                            s1.push(c);
                      } else if (c == ')') {
                            while ((tempChar=s1.pop()) != '(') {
                                  System.out.print(tempChar + " ");
                                  double num1 = s2.pop();
                                  double num2 = s2.pop();
                                  s2.push(calc(num2, num1, tempChar));
                                  if (s1.isEmpty()) {
                                        throw new IllegalArgumentException(
                                              "bracket dosen't match, missing left bracket '('.");
                                  }
                            }
                      } else if (c == ' ') {
                            // ignore
                      } else {
                            throw new IllegalArgumentException(
                                        "wrong character '" + c + "'");
                      }
                }
                while (!s1.isEmpty()) {
                      tempChar = s1.pop();
                      System.out.print(tempChar + " ");
                      double num1 = s2.pop();
                      double num2 = s2.pop();
                      s2.push(calc(num2, num1, tempChar));
                }
                double result = s2.pop();
                if (!s2.isEmpty())
                      throw new IllegalArgumentException("input is a wrong expression.");
                System.out.println();
                if ((int) result == result)
                      System.out.println("the result is " + (int) result);
                else
                      System.out.println("the result is " + result);
          }
          /**
           * calculate the two number with the operation.
           * @param num1
           * @param num2
           * @param op
           * @return
           * @throws IllegalArgumentException
           */
          private static double calc(double num1, double num2, char op)
                      throws IllegalArgumentException {
                switch (op) {
                case '+':
                      return num1 + num2;
                case '-':
                      return num1 - num2;
                case '*':
                      return num1 * num2;
                case '/':
                      if (num2 == 0) throw new IllegalArgumentException("divisor can't be 0.");
                      return num1 / num2;
                default:
                      return 0; // will never catch up here
                }
          }
          /**
           * compare the two operations' priority.
           * @param c
           * @param peek
           * @return
           */
          private static int priorityCompare(char op1, char op2) {
                switch (op1) {
                case '+': case '-':
                      return (op2 == '*' || op2 == '/' ? -1 : 0);
                case '*': case '/':
                      return (op2 == '+' || op2 == '-' ? 1 : 0);
                }
                return 1;
          }
          /**
           * read the next number (reverse)
           * @param input
           * @param start
           * @return
           * @throws IllegalArgumentException
           */
          private static int readDoubleReverse(String input, int start)
                      throws IllegalArgumentException {
                int dotIndex = -1;
                char c;
                for (int i=start; i>=0; --i) {
                      c = input.charAt(i);
                      if (c == '.') {
                            if (dotIndex != -1)
                                  throw new IllegalArgumentException(
                                        "there have more than 1 dots in the number.");
                            else
                                  dotIndex = i;
                      } else if (!Character.isDigit(c)) {
                            return i + 1;
                      } else if (i == 0) {
                            return 0;
                      }
                }
                throw new IllegalArgumentException("not a number.");
          }

          /**
           * read the next number
           * @param input
           * @param start
           * @return
           * @throws IllegalArgumentException
           */
          private static int readDouble(String input, int start)
          throws IllegalArgumentException {
                int len = input.length();
                int dotIndex = -1;
                char c;
                for (int i=start; i<len; ++i) {
                      c = input.charAt(i);
                      if (c == '.') {
                            if (dotIndex != -1)
                                  throw new IllegalArgumentException(
                                  "there have more than 1 dots in the number.");
                            else if (i == len - 1)
                                  throw new IllegalArgumentException(
                                  "not a number, dot can't be the last part of a number.");
                            else
                                  dotIndex = i;
                      } else if (!Character.isDigit(c)) {
                            if (dotIndex == -1 || i - dotIndex > 1)
                                  return i;
                            else
                                  throw new IllegalArgumentException(
                                  "not a number, dot can't be the last part of a number.");
                      } else if (i == len - 1) {
                            return len;
                      }
                }

                throw new IllegalArgumentException("not a number.");
          }
          /**
           * return true if the character is an operator.
           * @param c
           * @return
           */
          private static boolean isOperator(char c) {
                return (c=='+' || c=='-' || c=='*' || c=='/');
          }
    }
View Code

 

下面是程序运行结果(绿色为用户输入):

== usage ==
input the expressions, and then the program will calculate them and show the result.
input 'bye' to exit.
 
input an expression:
3.8+5.3
Polish Notation (PN):+ 3.8 5.3
the result is 9.1
Reverse Polish Notation (RPN):3.8 5.3 +
the result is 9.1
input a new expression:
5*(9.1+3.2)/(1-5+4.88)
Polish Notation (PN):/ * 5 + 9.1 3.2 + - 1 5 4.88
the result is 69.88636363636364
Reverse Polish Notation (RPN):5 9.1 3.2 + * 1 5 - 4.88 + /
the result is 69.88636363636364
input a new expression:
1+((2+3)*4)-5
Polish Notation (PN):- + 1 * + 2 3 4 5
the result is 16
Reverse Polish Notation (RPN):1 2 3 + 4 * + 5 -
the result is 16
input a new expression:
bye
program exits
 
 
posted @ 2013-09-04 19:31  Antineutrino  阅读(849)  评论(0编辑  收藏  举报