Data-Structure-Notes_ZenDei技術網路在線

Data Structure Notes Chapter 1 Sorting Algorithm Insert Sorting: 對於近乎有序的數組可以降到$ O(n)$的時間複雜度。 Merge Sorting: Tips1 ：Merge Sort Optimize in nearly order ...

Data Structure Notes

Chapter-1 Sorting Algorithm

Selection Sorting:

/*
*   Selection Sort
*/
template<typename T>
void selectionSort(T arr[], int n) {
    for (int i = 0;i < n;i++) {
        int minIndex = i;
        for (int j = i + 1;j < n;j++) {
            if (arr[j] < arr[minIndex])
                minIndex = j;
        }
        swap(arr[i], arr[minIndex]);
    }
}

// From both ends to exchange the elements in original array, it's a better solution optimize the previous Selection Sort.
template<typename T>
void OptimizedselectionSort(T arr[], int n) {

    int left = 0, right = n - 1;
    while (left < right) {
        int minIndex = left;
        int maxIndex = right;

        // In each rounds must assure arr[minIndex] <= arr[maxIndex]
        if (arr[minIndex] > arr[maxIndex])
            swap(arr[minIndex], arr[maxIndex]);

        //Traversing the array to choose the match positon.
        for (int i = left + 1; i < right; i++)
            if (arr[i] < arr[minIndex])
                minIndex = i;
            else if (arr[i] > arr[maxIndex])
                maxIndex = i;

        swap(arr[left], arr[minIndex]);
        swap(arr[right], arr[maxIndex]);

        left++;
        right--;
    }

    return;
}

Bubble Sorting:

/*
*   BubbleSort
*/
template<typename T>
void BubbleSort(T arr[], int n) {

    bool swapped;

    do {
        swapped = false;
        for (int i = 1; i < n; i++)
            if (arr[i - 1] > arr[i]) {
                swap(arr[i - 1], arr[i]);
                swapped = true;

            }

        // 優化, 每一趟Bubble Sort都將最大的元素放在了最後的位置
        // 所以下一次排序, 最後的元素可以不再考慮
        n--;

    } while (swapped);
}


// 我們的第二版bubbleSort,使用newn進行優化
template<typename T>
void OptimizedBubbleSort(T arr[], int n) {

    int newn; // 使用newn進行優化

    do {
        newn = 0;
        for (int i = 1; i < n; i++)
            if (arr[i - 1] > arr[i]) {
                swap(arr[i - 1], arr[i]);

                // 記錄最後一次的交換位置,在此之後的元素在下一輪掃描中均不考慮
                newn = i;
            }
        n = newn;
    } while (newn > 0);
}

Shell Sorting:

template<typename T>
void shellSort(T arr[], int n) {

    // 計算 increment sequence: 1, 4, 13, 40, 121, 364, 1093...
    int h = 1;
    while (h < n / 3)
        h = 3 * h + 1;

    while (h >= 1) {

        // h-sort the array
        for (int i = h; i < n; i++) {

            // 對 arr[i], arr[i-h], arr[i-2*h], arr[i-3*h]... 使用插入排序
            T e = arr[i];
            int j;
            for (j = i; j >= h && e < arr[j - h]; j -= h)
                arr[j] = arr[j - h];
            arr[j] = e;
        }

        h /= 3;
    }
}

Insert Sorting: 對於近乎有序的數組可以降到$ O(n)$的時間複雜度。

template<typename T>
void BinaryInsertionSort(T arr[], int n) {
    int i, j, low, high, mid;
    for (i = 1;i < n;i++) {
        T e = arr[i];
        
        //Binary Searching in the ordered range of array.
        low = 0; high = i - 1;
        while (low<= high)
        {
            mid = (low + high) / 2;
            if (arr[mid] > e) high = mid - 1;
            else low = mid + 1;
        }
        //Moving elements.
        for (j = i - 1;j >= high + 1;--j) {
            arr[j + 1] = arr[j];
        }
        arr[high + 1] = e;
    }
}

template<typename T>
void OptimizedInsertionSort(T arr[], int n) {
    for (int i = 1;i < n;i++) {

        // Find right position without exchange frequently.
        T e = arr[i];
        int j;
        for (j = i;j > 0 && arr[j - 1] > e;j--) {
            arr[j] = arr[j - 1];
        }
        arr[j] = e;
    }
}

Merge Sorting:

Tips1：Merge Sort Optimize in nearly ordered array

void __mergeSort(T arr[], int l, int r) {
    if (l >= r) return;

    int mid = (l + r) / 2;      // variable 'mid' may overflow
    __mergeSort(arr, l, mid);
    __mergeSort(arr, mid+1, r);
    if(arr[mid] > arr[mid+1])   // optimize in nearly ordered array.
        __merge(arr, l, mid, r);
}

Tips2：When the sorting range of array in a short length, using InsertSort replace MergeSort can be more faster.

 template<typename T>
void __mergeSort(T arr[], int l, int r) {
    //if (l >= r) return;
    if (r - l <= 15) {           // The '15' is a constant represent the minmum judge range.
        InsertionSort(arr, l, r);
        return;
    }
    int mid = (l + r) / 2;      // variable 'mid' may overflow
    __mergeSort(arr, l, mid);
    __mergeSort(arr, mid+1, r);
    if(arr[mid] > arr[mid+1])   // optimize in nearly ordered array.
        __merge(arr, l, mid, r);
}

Botton to Up Merge Sorting : The algorithm can be usd in the LinkedList . The original MergeSort may preform better than this algorithm in normal situation.

Standard

template<typename T>
void mergeSortBottonToUp(T arr[], int n) {
    for(int size = 1; size <= n; size += size)
        // In order to assure exist two sperate array, setting (i+size < n) not (i < n)
        for (int i = 0; i + size < n ; i += size + size) {      
            // merge arr[i ... i+size-1] and arr[i+size ... i+2*size-1]
            // In order to assure latter array isn't overflow so use min(i + size + size - 1, n-1) to choosing a right part.
            __merge(arr, i, i + size - 1, min(i + size + size - 1, n-1));
        }
}

Optimization

template <typename T>
void mergeSortBU2(T arr[], int n){

    // 對於小規模數組, 使用插入排序
    for( int i = 0 ; i < n ; i += 16 )
        insertionSort(arr,i,min(i+15,n-1));

    // 一次性申請aux空間, 並將這個輔助空間以參數形式傳遞給完成歸併排序的各個子函數
    T* aux = new T[n];
    for( int sz = 16; sz <= n ; sz += sz )
        for( int i = 0 ; i < n - sz ; i += sz+sz )
            // 對於arr[mid] <= arr[mid+1]的情況,不進行merge
            // 對於近乎有序的數組非常有效,但是對於一般情況,有一定的性能損失
            if( arr[i+sz-1] > arr[i+sz] )
                __merge2(arr, aux, i, i+sz-1, min(i+sz+sz-1,n-1) );
    delete[] aux; // 使用C++, new出來的空間不要忘記釋放掉:)
}

QuickSort (Divide-and-Conquer Algorithm)

Partition
Insert Sort Optimization

    // sort the range of [l ... r]
template <typename T>
void __quickSort(T arr[], int l, int r) {
    //if (l >= r) return;
    if (r - l <= 15) {
        OptimizedInsertionSort(arr, l, r);
        return;
    }
    int p = __partition(arr, l, r);
    __quickSort(arr, l, p - 1);
    __quickSort(arr, p + 1, r);
}

Optimization in the face of nearly ordered array
Compare to MergeSort, the Sorting Tree generate by Quick Sort is more unbalanced.The worst situation the effience of quick sort can be deteriorate to $O(n^2)$
Tradinational Method using the left element to be demarcating element. In order to solving the problem, we select the demarcating element randomly.

```cpp

template
int __partition(T arr[], int l, int r) {

  swap(arr[l], arr[rand() % (r - l + 1) + l]);  // Add this process to randomly choose demarcating element.
  T v = arr[l];

  //arr[l+i ... j] < v;arr[j+1 ... i] > v
  int j = l;
  for (int i = l + 1;i <= r;i++) {
      if (arr[i] < v) {
          swap(arr[j + 1], arr[i]);
          j++;
      }
  }

  swap(arr[l], arr[j]);
  return j;

}

template
void quickSort(T arr[], int n) {
srand(time(NULL)); // The partial of randomly select.
__quickSort(arr, 0, n - 1);
}

```

Optimization in the face of many repeating Numbers. (Dual Qucik Sort)
When face many repeating numbers, the speration of array may unbalanced. In this situation, Quick Sort can be degraded to $O(n^2)$.

Solution :


template <typename T>
int __partition2(T arr[], int l, int r) {
    swap(arr[l], arr[rand() % (r - l + 1) + l]);  // Add this process to randomly choose demarcating element.
    T v = arr[l];

    //arr[l+i ... j] < v; arr[j+1 ... i] > v
    int i = l + 1, j = r;
    while (true) {
        //From front to behind to find a even bigger number.
        //From behind to front to find a even smaller number.
        while (i <= r&& arr[i] < v) i++;
        while (j >= l + 1 && arr[j] > v) j--;
        if (i > j) break;
        swap(arr[i], arr[j]);
        i++;
        j--;
    }

    swap(arr[l], arr[j]);

    return j;
}

Optimization in the face of many repeating Numbers. (Qucik Sort 3 Ways)

template <typename T>
void __quickSort3(T arr[], int l, int r) {
    //if (l >= r) return;
    if (r - l <= 15) {
        OptimizedInsertionSort(arr, l, r);
        return;
    }

    // partition
    swap(arr[l], arr[rand() % (r - l + 1) + l]);
    T v = arr[l];

    int lt = l;     //arr[l+1 ... lt] < v
    int gt = r + 1; //arr[gt ... r] > v
    int i = l + 1;  //arr[lt+1 ... i] == v
    while (i < gt) {
        if (arr[i] < v) {
            swap(arr[i], arr[lt + 1]);
            lt++;
            i++;
        }
        else if(arr[i] > v) {
            swap(arr[i], arr[gt - 1]);
            gt--;
        }
        else {// arr[i] == v
            i++;
        }
    }

    swap(arr[l], arr[lt]);

    __quickSort3(arr, l, lt - 1);
    __quickSort3(arr, gt, r);
}

template <typename T>
void quickSort(T arr[], int n) {
    srand(time(NULL));      // The partial of randomly select.
    __quickSort3(arr, 0, n - 1);
}