（說明：本博客中的題目、題目詳細說明及參考代碼均摘自 “何海濤《劍指Offer：名企面試官精講典型編程題》2012年”）
題目
實現函數 double Power(double base, int exponent)，求 base 的 exponent 次方。不得使用庫函數，同時不需要考慮大數問題。
算法設計思想
無論是簡單直接方法，還是高效的方法，都需要首先全面考慮 base 和 exponent 的可能的輸入：正數、負數和 0。實現的基本思想是把數值的整數次方，轉化為數值的非負整數次方，然后根據指數的符號，再做相應處理，具體如下：
假設求解 a 的 n 次方，在指數 n 為正整數的前提下，簡單直接方法就是將結果依次，就將結果乘以 base 幾次，此時算法的時間復雜度為 O(n)；
高效算法利用下面的公式，此時算法的時間復雜度為 O(logn)。
若指數 n 為負整數，則可先求 a 的 -n 次方，最后將計算結果取倒數，即可。此時需要注意，分母不能為 0，即 a 的 -n 次方的結果不能為 0，也就是說，當 n 為負整數時，a 不能為 0。
若指數 n 為 0 時，只要 a 不等於 0，則計算結果為 1；若 a 為 0 時，則為 0 的 0 次方，沒有意義。
注：
易錯點，將浮點數（float 或 double）使用 == 符號與 0 直接比較，以判斷此數值是否為 0。因為浮點數在計算機中的表示是有誤差的，所以不是直接使用 == 符號判斷某浮點數是否為 0。在實現時，往往需要判斷浮點數是否在數值 0.0 附近的小范圍之內，若是，則判定此數值為 0。本博文中，取 10 的 -7 次方（1e-7）作為誤差范圍。
C++ 實現
/* * 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 實現
#!/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()
參考代碼
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. //
// 《劍指Offer——名企面試官精講典型編程題》代碼 // 著作權所有者：何海濤
#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; } // ====================測試代碼====================
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[]) { // 底數、指數都為正數
printf("Test1 begins.\n"); Test(2, 3, 8, false); // 底數為負數、指數為正數
printf("Test2 begins.\n"); Test(-2, 3, -8, false); // 指數為負數
printf("Test3 begins.\n"); Test(2, -3, 0.125, false); // 指數為0
printf("Test4 begins.\n"); Test(2, 0, 1, false); // 底數、指數都為0
printf("Test5 begins.\n"); Test(0, 0, 1, false); // 底數為0、指數為正數
printf("Test6 begins.\n"); Test(0, 4, 0, false); // 底數為0、指數為負數
printf("Test7 begins.\n"); Test(0, -4, 0, true); return 0; }
5. 參考代碼下載
項目 11_Power 下載： 百度網盤
何海濤《劍指Offer：名企面試官精講典型編程題》 所有參考代碼下載：百度網盤
參考資料
[1] 何海濤. 劍指 Offer：名企面試官精講典型編程題 [M]. 北京：電子工業出版社，2012. 84-93.