旋转数组的最小数字（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

　　把一个数组最开始的若干个元素搬到数组的末尾，我们称之为数组的旋转。输入一个递增排序的数组的一个旋转，输出旋转数组的最小元素。例如数组 {3, 4, 5, 1, 2} 为 {1, 2, 3, 4, 5} 的一个旋转，该数组的最小值为 1 。

算法设计思想

1. 暴力查找（Bruteforce Search）：把旋转数组从前到后遍历一遍，其时间复杂度为 O(n)。很明显，这种思想非常直接粗暴，没有用到旋转数组的性质。

2. 二分查找（Binary Search）：每次查找都把旋转数组平均分成两部分，通过比较当前旋转数组两端点和中间点的值，判断最小值在数组的哪一部分，从而达到缩小搜索范围的目的。其中，两端点为当前的旋转数组索引最小和索引最大的元素，中间点为这两部分子数组的连结元素，也可以看做为轴枢点（pivot point），这里取旋转数组的最小索引和最大索引的算术平均值（向下取整）作为索引，其对应的元素作为中间点。具体过程，如图 2.10 所示。

需要注意，当旋转数组的两端点的值都与中间点的值相等时，因为无法判断最小值在哪一部分，因此需要采用顺序查找方法，即暴力查找。其示例如图 2.11 所示。

C++ 实现

#include <stdio.h>
#include <exception>
#include <iostream>

// Declare a parameter exception
class ParamException: public std::exception {
    virtual const char* what() const throw()
    {
        return "Error: input parameters exception!";
    }
};

// find minimum in data[staIndex..endIndex]
int findMinInRotatedArray(int data[], int staIndex, int endIndex)
{
    if (staIndex > endIndex || data == NULL)
    {
        throw ParamException();
    }

    int minimum = data[staIndex];

    if (data[staIndex] >= data[endIndex] && staIndex < endIndex - )  // 易错点
    {
        // Use Binary Search
        int midIndex = (staIndex + endIndex) / ;

        if (data[midIndex] > data[staIndex])
            minimum = findMinInRotatedArray(data, midIndex, endIndex);
        else if (data[midIndex] < data[endIndex])
            minimum = findMinInRotatedArray(data, staIndex, midIndex);
        else
        {
            // Find the minimum sequentially
            for (int i = staIndex+; i <= endIndex; ++i)
                if (minimum > data[i])
                    minimum = data[i];
        }
    }
    else if (staIndex == (endIndex - ))
    {
        minimum = data[staIndex] > data[endIndex]? data[endIndex]:data[staIndex];
    }

    return minimum;
}

void unitest()
{
    int arr1[] = {, , , , };
    int arr2[] = {, , , , };
    int arr3[] = {, , , , };
    int arr4[] = {, , , , };

    printf("The minimum of the rotated array {3, 4, 5, 1, 2} is %d.\n", findMinInRotatedArray(arr1, , ));
    printf("The minimum of the rotated array {1, 0, 1, 1, 1} is %d.\n", findMinInRotatedArray(arr2, , ));
    printf("The minimum of the rotated array {1, 1, 1, 0, 1} is %d.\n", findMinInRotatedArray(arr3, , ));
    printf("The minimum of the rotated array {1, 2, 3, 4, 5} is %d.\n", findMinInRotatedArray(arr4, , ));

    // Test index parameter exception
    try {
        findMinInRotatedArray(arr3, , );
    }
    catch(std::exception& e) {
        std::cout << e.what() << std::endl;
    };
}

int main(void)
{
    unitest();

    return ;
}

Python 实现

#!/usr/bin/python
# -*- coding: utf8 -*-

# Define ParamError Exception
class ParamError(Exception):
    def __init__(self, value):
        self.value = value
    def __str__(self):
        return repr(self.value)

# Find the minimum in rotated array
def min_in_rotated_array(data, length):
    if data is None or length <= 0:
        raise ParamError("Error: input parameters exception!")
    # Index initialization
    sta, mid, end = 0, 0, length-1
    # Ensure this requisite before binary search
    while data[sta] >= data[end]:
        if end - sta == 1:
            mid = end
            break
        # Get the middle index
        mid = (sta + end) / 2
        # Find the minimum in order
        if (data[sta] == data[mid]) and (data[mid] == data[end]):
            minimum = data[sta]
            for i in range(sta+1, end+1):
                if minimum > data[i]:
                    minimum = data[i]
            return minimum

        if data[sta] <= data[mid]:
            sta = mid
        elif data[end] >= data[mid]:
            end = mid

    return data[mid]

def unitest():
    arr1 = [3, 4, 5, 1, 2]
    arr2 = [1, 0, 1, 1, 1]
    arr3 = [1, 1, 1, 0, 1]
    arr4 = [1, 2, 3, 4, 5]

    print("The minimum of the rotated array [3, 4, 5, 1, 2] is %d." % min_in_rotated_array(arr1, 5));
    print("The minimum of the rotated array [1, 0, 1, 1, 1] is %d." % min_in_rotated_array(arr2, 5));
    print("The minimum of the rotated array [1, 1, 1, 0, 1] is %d." % min_in_rotated_array(arr3, 5));
    print("The minimum of the rotated array [1, 2, 3, 4, 5] is %d." % min_in_rotated_array(arr4, 5));

    try:
        min_in_rotated_array(arr1, -2)
    except Exception, e:
        print "\nFunction call: min_in_rotated_array(arr1, -2)"
        print 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
// MinNumberInRotatedArray.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. MinNumberInRotatedArray.cpp

// MinNumberInRotatedArray.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<exception>

int MinInOrder(int* numbers, int index1, int index2);

int Min(int* numbers, int length)
{
    if(numbers == NULL || length <= )
        throw new std::exception("Invalid parameters");

    int index1 = ;
    int index2 = length - ;
    int indexMid = index1;
    while(numbers[index1] >= numbers[index2])
    {
        // 如果index1和index2指向相邻的两个数，
        // 则index1指向第一个递增子数组的最后一个数字，
        // index2指向第二个子数组的第一个数字，也就是数组中的最小数字
        if(index2 - index1 == )
        {
            indexMid = index2;
            break;
        }

        // 如果下标为index1、index2和indexMid指向的三个数字相等，
        // 则只能顺序查找
        indexMid = (index1 + index2) / ;
        if(numbers[index1] == numbers[index2] && numbers[indexMid] == numbers[index1])
            return MinInOrder(numbers, index1, index2);

        // 缩小查找范围
        if(numbers[indexMid] >= numbers[index1])
            index1 = indexMid;
        else if(numbers[indexMid] <= numbers[index2])
            index2 = indexMid;
    }

    return numbers[indexMid];
}

int MinInOrder(int* numbers, int index1, int index2)
{
    int result = numbers[index1];
    for(int i = index1 + ; i <= index2; ++i)
    {
        if(result > numbers[i])
            result = numbers[i];
    }

    return result;
}

// ====================测试代码====================
void Test(int* numbers, int length, int expected)
{
    int result = ;
    try
    {
        result = Min(numbers, length);

        for(int i = ; i < length; ++i)
            printf("%d ", numbers[i]);

        if(result == expected)
            printf("\tpassed\n");
        else
            printf("\tfailed\n");
    }
    catch (...)
    {
        if(numbers == NULL)
            printf("Test passed.\n");
        else
            printf("Test failed.\n");
    }
}

int _tmain(int argc, _TCHAR* argv[])
{
    // 典型输入，单调升序的数组的一个旋转
    int array1[] = {, , , , };
    Test(array1, sizeof(array1) / sizeof(int), );

    // 有重复数字，并且重复的数字刚好的最小的数字
    int array2[] = {, , , , , };
    Test(array2, sizeof(array2) / sizeof(int), );

    // 有重复数字，但重复的数字不是第一个数字和最后一个数字
    int array3[] = {, , , , , };
    Test(array3, sizeof(array3) / sizeof(int), );

    // 有重复的数字，并且重复的数字刚好是第一个数字和最后一个数字
    int array4[] = {, , , , };
    Test(array4, sizeof(array4) / sizeof(int), );

    // 单调升序数组，旋转0个元素，也就是单调升序数组本身
    int array5[] = {, , , , };
    Test(array5, sizeof(array5) / sizeof(int), );

    // 数组中只有一个数字
    int array6[] = {};
    Test(array6, sizeof(array6) / sizeof(int), );

    // 输入NULL
    Test(NULL, , );

    return ;
}

7. 参考代码下载

项目 08_MinNumberInRotatedArray 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. 63-71.