前缀、中缀、后缀表达式

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在函数式编程语言中,为了表示方便,出现了一些新的语法格式。所谓前缀、中缀、后缀表达式,它们之间的区别在于运算符相对与操作数的位置不同,为了说明它们的概念,首先来看一下中缀表达式。

所谓中缀表达式,就是将函数名放到两个操作数中间的表达式,其中,左侧的操作数代表函数对象或值,右侧的操作数代表函数的参数值。例如:

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

前缀表达式

前缀表达式又称为前缀记法、波兰式,主要用于表示运算符位于操作数之前的表达式。

中缀表达式

中缀表达式又称为中缀记法,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,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。

后缀表达式

后缀表达式又称为后缀记法、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。

前缀表达式求值

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”计算的步骤如下:
(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。

后缀表达式求值

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 -”的计算步骤如下:
(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。

前缀、中缀、后缀表达式相互转换

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。

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)。

示例

package com.xzh;

import java.util.Scanner;
import java.util.Stack;

public class Demo {
    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";

    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
        }
    }

    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=='/');
    }
}

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