数值的整数次方（C++ 和 Python 实现）
（说明：本博客中的题目、题目详细说明及参考代码均摘自 “何海涛《剑指Offer：名企面试官精讲典型编程题》2012年”）
题目
实现函数 double Power(double base, int exponent)，求 base 的 exponent 次方。不得使用库函数，同时不需要考虑大数问题。
算法设计思想
无论是简单直接方法，还是高效的方法，都需要首先全面考虑 base 和 exponent 的可能的输入：正数、负数和 0。实现的基本思想是把数值的整数次方，转化为数值的非负整数次方，然后根据指数的符号，再做相应处理，具体如下：
假设求解 a 的 n 次方，在指数 n 为正整数的前提下，简单直接方法就是将结果依次，就将结果乘以 base 几次，此时算法的时间复杂度为 O(n)；
高效算法利用下面的公式，此时算法的时间复杂度为 O(logn)。
若指数 n 为负整数，则可先求 a 的 -n 次方，最后将计算结果取倒数，即可。此时需要注意，分母不能为 0，即 a 的 -n 次方的结果不能为 0，也就是说，当 n 为负整数时，a 不能为 0。
若指数 n 为 0 时，只要 a 不等于 0，则计算结果为 1；若 a 为 0 时，则为 0 的 0 次方，没有意义。
注：
易错点，将浮点数（float 或 double）使用 == 符号与 0 直接比较，以判断此数值是否为 0。因为浮点数在计算机中的表示是有误差的，所以不是直接使用 == 符号判断某浮点数是否为 0。在实现时，往往需要判断浮点数是否在数值 0.0 附近的小范围之内，若是，则判定此数值为 0。本博文中，取 10 的 -7 次方（1e-7）作为误差范围。
C++ 实现
/*
* Author: klchang
* Date: 2018.1.14
* Description: Compute the integer power of a numeric value.
*/
#include <iostream>
#include <exception>
// Exception class for invalid input: base = 0 when exponent is negative.
class InvalidInputException: public std::exception {
// virtual function does not throw any exception
virtual const char* what() const throw()
{
return "Invalid input exception happened.";
}
} invalid_input;
// power function with non-negative exponent in the common method
// parameters:
// base - <0, =0, >0; exponent - =0 or >0
double power_common(double base, unsigned int exponent)
{
double result = 1;
for (int i = 0; i < exponent; ++ i) {
result *= base;
}
return result;
}
// power function with non-negative exponent in the common method
// parameters:
// base - <0, =0, >0; exponent - =0 or >0.
double power_fast(double base, unsigned int exponent)
{
double result = 1;
if (0 == exponent)
return 1;
else if (1 == exponent)
return base;
else {
// odd number
result = power_fast(base, exponent >> 1);
if (exponent & 1) {
// odd number
return result * result * base;
} else {
// even number
return result * result;
}
}
}
// Check if a double value is zero
bool is_zero(double value)
{
double zero_limit = 1e-7;
return (value >= -1 * zero_limit) && (value <= zero_limit);
}
// generic interface for power function with integer exponent including positives, zero and negatives
// parameters:
// method: 1 -- fast method; others -- common method
double Power(double base, int exponent, int method=0)
{
int sign = 1; // default: positive exponent
double result;
if (exponent <= 0) {
if (is_zero(base)) { // fallibility: use 0 == base(double type)
// illegal input: 0^0 no meaning; 0^negative_integer error
throw invalid_input;
}
sign = -1;
exponent = - exponent;
}
if (1 == method) // fast method
result = power_fast(base, (unsigned int)exponent);
else // common method
result = power_common(base, (unsigned int)exponent);
if (sign < 0) {
result = 1.0 / result;
}
return result;
}
void unitest()
{
try {
std::cout << "---------------- Power function in Fast Method Test ----------------" << std::endl
<< "The result of -2^-3 is " << Power(-2, -3, 1) << std::endl
<< "The result of -2^3 is " << Power(-2, 3, 1) << std::endl
<< "The result of 2^-3 is " << Power(2, -3, 1) << std::endl
<< "The result of 2^3 is " << Power(2, 3, 1) << std::endl;
std::cout << "---------------- Power function in Common Method Test ----------------" << std::endl
<< "The result of -2^-3 is " << Power(-2, -3) << std::endl
<< "The result of -2^3 is " << Power(-2, 3) << std::endl
<< "The result of 2^-3 is " << Power(0, -3) << std::endl
<< "The result of 2^3 is " << Power(2, 3) << std::endl;
}
catch(std::exception& e) {
std::cerr << e.what() << '\n';
}
}
int main()
{
unitest();
return 0;
}
Python 实现
#!/usr/bin/python
# -*- coding: utf8 -*-
"""
# Author: klchang
# Date: 2018.1.14
# Description: Compute the integer power of a numeric value.
"""
# Invalid input exception class
class InvalidInput(Exception):
def __init__(self, value):
self.value = value
def __str__(self):
return repr(self.value)
# power function with non-negative exponent in the common method
def power_common(base, exponent):
result = 1
for i in range(exponent):
result *= base;
return result
# power function with non-negative exponent in the fast method
def power_fast(base, exponent):
if 0 == exponent:
return 1
elif 1 == exponent:
return base
else:
result = power_fast(base, exponent >> 1)
if exponent & 1:
# odd integer
return result * result * base
else:
# even integer
return result * result
# Check if value (int/float) is zero
# parameters:
# value - int type or float type
def is_zero(value):
# Check the type that value belongs to
if isinstance(value, float):
# float type
zero_limit = 1e-7
return (value >= -zero_limit) and (value <= zero_limit)
else:
# int type
return value == 0
# Generic interface for power function with integer exponent including positives, zero and negatives
# parameters:
# method: 1 -- fast method; others -- common method
def power(base, exponent, method=0):
# sign flag: positive(default)
is_positive_exponent = True
if exponent <= 0:
if is_zero(base):
raise InvalidInput(base)
exponent = - exponent
is_positive_exponent = False
# computation result
result = 0
if 1 == method:
result = power_fast(base, exponent)
else:
result = power_common(base, exponent)
# check the sign of the exponent
if not is_positive_exponent:
result = 1.0 / result
return result
def unitest():
try:
print("---------------- Power function in Fast Method Test ----------------")
print("The result of -2^-3 is %f." % power(-2, -3, 1))
print("The result of -2^3 is %f." % power(-2, 3, 1))
print("The result of 2^-3 is %f." % power(2, -3, 1))
print("The result of 2^3 is %f."% power(2, 3, 1))
print("---------------- Power function in Common Method Test ----------------")
print("The result of -2^-3 is %f." % power(-2, -3))
print("The result of -2^3 is %f." % power(-2, 3))
print("The result of 2^-3 is " % power(0, -3))
print("The result of 2^3 is " % power(2, 3))
except Exception as e:
print("Invalid input exception happened: input %s with negative exponent" % e)
if __name__ == '__main__':
unitest()
参考代码
1. targetver.h
#pragma once
// The following macros define the minimum required platform. The minimum required platform
// is the earliest version of Windows, Internet Explorer etc. that has the necessary features to run
// your application. The macros work by enabling all features available on platform versions up to and
// including the version specified.
// Modify the following defines if you have to target a platform prior to the ones specified below.
// Refer to MSDN for the latest info on corresponding values for different platforms.
#ifndef _WIN32_WINNT // Specifies that the minimum required platform is Windows Vista.
#define _WIN32_WINNT 0x0600 // Change this to the appropriate value to target other versions of Windows.
#endif
2. stdafx.h
// stdafx.h : include file for standard system include files,
// or project specific include files that are used frequently, but
// are changed infrequently
//
#pragma once
#include "targetver.h"
#include <stdio.h>
#include <tchar.h>
// TODO: reference additional headers your program requires here
3. stdafx.cpp
// stdafx.cpp : source file that includes just the standard includes
// Power.pch will be the pre-compiled header
// stdafx.obj will contain the pre-compiled type information
#include "stdafx.h"
// TODO: reference any additional headers you need in STDAFX.H
// and not in this file
4. Power.cpp
// Power.cpp : Defines the entry point for the console application.
//
// 《剑指Offer——名企面试官精讲典型编程题》代码
// 著作权所有者：何海涛
#include "stdafx.h"
#include <math.h>
bool g_InvalidInput = false;
bool equal(double num1, double num2);
double PowerWithUnsignedExponent(double base, unsigned int exponent);
double Power(double base, int exponent)
{
g_InvalidInput = false;
if(equal(base, 0.0) && exponent < 0)
{
g_InvalidInput = true;
return 0.0;
}
unsigned int absExponent = (unsigned int)(exponent);
if(exponent < 0)
absExponent = (unsigned int)(-exponent);
double result = PowerWithUnsignedExponent(base, absExponent);
if(exponent < 0)
result = 1.0 / result;
return result;
}
/*
double PowerWithUnsignedExponent(double base, unsigned int exponent)
{
double result = 1.0;
/
for(int i = 1; i <= exponent; ++i)
result *= base;
return result;
}
*/
double PowerWithUnsignedExponent(double base, unsigned int exponent)
{
if(exponent == 0)
return 1;
if(exponent == 1)
return base;
double result = PowerWithUnsignedExponent(base, exponent >> 1);
result *= result;
if((exponent & 0x1) == 1)
result *= base;
return result;
}
bool equal(double num1, double num2)
{
if((num1 - num2 > -0.0000001)
&& (num1 - num2 < 0.0000001))
return true;
else
return false;
}
// ====================测试代码====================
void Test(double base, int exponent, double expectedResult, bool expectedFlag)
{
double result = Power(base, exponent);
if(abs(result - expectedResult) < 0.00000001
&& g_InvalidInput == expectedFlag)
printf("Test passed.\n");
else
printf("Test failed.\n");
}
int _tmain(int argc, _TCHAR* argv[])
{
// 底数、指数都为正数
printf("Test1 begins.\n");
Test(2, 3, 8, false);
// 底数为负数、指数为正数
printf("Test2 begins.\n");
Test(-2, 3, -8, false);
// 指数为负数
printf("Test3 begins.\n");
Test(2, -3, 0.125, false);
// 指数为0
printf("Test4 begins.\n");
Test(2, 0, 1, false);
// 底数、指数都为0
printf("Test5 begins.\n");
Test(0, 0, 1, false);
// 底数为0、指数为正数
printf("Test6 begins.\n");
Test(0, 4, 0, false);
// 底数为0、指数为负数
printf("Test7 begins.\n");
Test(0, -4, 0, true);
return 0;
}
5. 参考代码下载
项目 11_Power 下载： 百度网盘
何海涛《剑指Offer：名企面试官精讲典型编程题》 所有参考代码下载：百度网盘
参考资料
[1] 何海涛. 剑指 Offer：名企面试官精讲典型编程题 [M]. 北京：电子工业出版社，2012. 84-93.
