[算法] 排序

排序算法有很多种类，可以从时空复杂度、稳定性等角度来分析。

排序算法	平均时间复杂度	最坏时间复杂度	最好时间复杂度	空间复杂度	稳定性
快速排序	$O(nlogn)$	$O(n^2)$	$O(nlogn)$	$O(logn)$	不稳定
冒泡排序	$O(n^2)$	$O(n^2)$	$O(n)$	$O(1)$	稳定
归并排序	$O(nlogn)$	$O(nlogn)$	$O(nlogn)$	$O(n)$	稳定
选择排序	$O(n^2)$	$O(n^2)$	$O(n^2)$	$O(1)$	不稳定
堆排序	$O(nlogn)$	$O(nlogn)$	$O(nlogn)$	$O(1)$	不稳定
插入排序	$O(n^2)$	$O(n^2)$	$O(n)$	$O(1)$	稳定
希尔排序	$O(n^{1.3})$	$O(n^2)$	$O(n)$	$O(1)$	不稳定
计数排序	$O(n+k)$	$O(n+k)$	$O(n+k)$	$O(k)$	稳定
桶排序	$O(n)$	$O(n^2)$	$O(n)$	$O(n+b)$	稳定
基数排序	$O(nd)$	$O(nd)$	$O(nd)$	$O(n)$	稳定

1. 快速排序

class Solution {
private:
    int partition(vector<int> &nums, int lo, int hi) {
        swap(nums[lo], nums[lo + rand() % (hi - lo + 1)]);
        int pivot = nums[lo];
        while (lo < hi) {
            while (lo < hi && pivot <= nums[hi])
                --hi;
            nums[lo] = nums[hi];
            while (lo < hi && nums[lo] <= pivot)
                ++lo;
            nums[hi] = nums[lo];
        }
        nums[lo] = pivot;
        return lo;
    }

    void quickSort(vector<int> &nums, int lo, int hi) {
        if (lo >= hi)
            return;
        int mi = partition(nums, lo, hi);
        quickSort(nums, lo, mi - 1);
        quickSort(nums, mi + 1, hi);
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        int n = nums.size();
        quickSort(nums, 0, n - 1);
        return nums;
    }
};

2. 冒泡排序

class Solution {
private:
    void bubbleSort(vector<int> &nums) {
        int n = nums.size();
        for (int i = n; i >= 2; --i) {
            bool noSwap = true;
            for (int j = 0; j < i - 1; ++j) {
                if (nums[j] > nums[j + 1]) {
                    swap(nums[j], nums[j + 1]);
                    noSwap = false;
                }
            }
            if (noSwap)
                break;
        }
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        bubbleSort(nums);
        return nums;
    }
};

3. 归并排序

class Solution {
private:
    void merge(vector<int> &nums, int lo, int mi, int hi) {
        vector<int> temp(hi - lo);
        int i = lo, j = mi, k = 0;
        while (i < mi && j < hi) {
            if (nums[i] <= nums[j])
                temp[k++] = nums[i++];
            else
                temp[k++] = nums[j++];
        }
        while (i < mi)
            temp[k++] = nums[i++];
        while (j < hi)
            temp[k++] = nums[j++];
        for (i = lo, k = 0; i < hi; ++i, ++k)
            nums[i] = temp[k];
    }

    void mergeSort(vector<int> &nums, int lo, int hi) {
        if (lo + 1 >= hi)
            return;
        int mi = (lo + hi) / 2;
        mergeSort(nums, lo, mi);
        mergeSort(nums, mi, hi);
        merge(nums, lo, mi, hi);
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        int n = nums.size();
        mergeSort(nums, 0, n);
        return nums;
    }
};

4. 选择排序

class Solution {
private:
    void selectionSort(vector<int> &nums) {
        int n = nums.size();
        for (int i = n; i >= 2; --i) {
            int maxIdx = 0;
            for (int j = 0; j < i; ++j)
                if (nums[j] > nums[maxIdx])
                    maxIdx = j;
            swap(nums[maxIdx], nums[i-1]);
        }
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        selectionSort(nums);
        return nums;
    }
};

5. 堆排序

class Solution {
private:
    void siftDown(vector<int> &nums, int start, int end) {
        int parent = start;
        int child = parent * 2 + 1;
        while (child <= end) {
            if (child + 1 <= end && nums[child] < nums[child + 1])
                ++child;
            if (nums[parent] >= nums[child])
                return;
            swap(nums[parent], nums[child]);
            parent = child;
            child = parent * 2 + 1;
        }
    }

    void heapSort(vector<int> &nums) {
        int n = nums.size();
        for (int i = (n - 2) / 2; i >= 0; --i)
            siftDown(nums, i, n - 1);
        for (int i = n - 1; i >= 0; --i) {
            swap(nums[0], nums[i]);
            siftDown(nums, 0, i - 1);
        }
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        heapSort(nums);
        return nums;
    }
};

6. 插入排序

class Solution {
private:
    void insertionSort(vector<int> &nums) {
        int n = nums.size();
        for (int i = 1; i < n; ++i) {
            int target = nums[i], j = i - 1;
            while (j >= 0 && target < nums[j]) {
                nums[j + 1] = nums[j];
                --j;
            }
            nums[j + 1] = target;
        }
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        insertionSort(nums);
        return nums;
    }
};

7. 希尔排序

class Solution {
private:
    void shellSort(vector<int> &nums) {
        int n = nums.size();
        for (int gap = n / 2; gap >= 1; gap /= 2) {
            for (int i = 0; i < gap; ++i) {
                for (int j = i + gap; j < n; j += gap) {
                    int target = nums[j], k = j - gap;
                    while (k >= 0 && target < nums[k]) {
                        nums[k + gap] = nums[k];
                        k -= gap;
                    }
                    nums[k + gap] = target;
                }
            }
        }
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        shellSort(nums);
        return nums;
    }
};

8. 计数排序

class Solution {
private:
    void countingSort(vector<int> &nums, int minVal, int maxVal) {
        vector<short> count(maxVal - minVal + 1, 0);
        for (int &num : nums)
            count[num - minVal] += 1;
        int idx = 0;
        for (int i = minVal; i <= maxVal; ++i) {
            while (count[i - minVal] > 0) {
                nums[idx++] = i;
                count[i - minVal] -= 1; 
            }
        }
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        countingSort(nums, -50000, 50000);
        return nums;
    }
};

9. 桶排序

class Solution {
private:
    void bucketSort(vector<int> &nums, int minVal, int maxVal) {
        int n = nums.size();
        int bucketNum = (maxVal - minVal) / n + 1;
        vector<vector<int> > buckets(bucketNum, vector<int>());
        for (int &num : nums)
            buckets[(num - minVal) / n].push_back(num);
        int idx = 0;
        for (auto &bucket : buckets) {
            sort(bucket.begin(), bucket.end());
            for (int &num : bucket)
                nums[idx++] = num;
        }   
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        bucketSort(nums, -50000, 50000);
        return nums;
    }
};

10. 基数排序

class Solution {
private:
    void radixSort(vector<int> &nums, int minVal, int maxVal) {
        int newMaxVal = maxVal - minVal;
        int maxDigit = 0;
        while (newMaxVal) {
            maxDigit += 1;
            newMaxVal /= 10;
        }
        vector<vector<int> > buckets(10, vector<int>());
        int div = 1;
        for (int i = 0; i < maxDigit; ++i) {
            for (auto &bucket : buckets)
                bucket.clear();
            for (int &num : nums)
                buckets[(num - minVal) / div % 10].push_back(num);
            int idx = 0;
            for (auto &bucket : buckets)
                for (int &num : bucket)
                    nums[idx++] = num;
            div *= 10;
        }
    }
public:
    vector<int> sortArray(vector<int>& nums) {
        radixSort(nums, -50000, 50000);
        return nums;
    }
};