(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.
原文地址:https://www.cnblogs.com/klchang/p/8283908.html