快速選擇(quick_select) 演算法分析
阿新 • • 發佈:2019-01-06
快速選擇演算法,就是從給定的一個集合S={a1,a2,...an}中選出第K個大小的數,或者給出其所在的下標之類的。
如果使用排序,比如merge_sort,然後返回第K個元素的下標,複雜度是O(NlogN)
如果使用heap_sort,或者優先佇列,則複雜度是O(NlogK)
如果使用quick _sort的一個變種,叫 quick select,則平均複雜度為O(N),最壞複雜度為O(N^2)
如果使用排序,比如merge_sort,然後返回第K個元素的下標,複雜度是O(NlogN)
如果使用heap_sort,或者優先佇列,則複雜度是O(NlogK)
如果使用quick _sort的一個變種,叫 quick select,則平均複雜度為O(N),最壞複雜度為O(N^2)
如果使用一種線性選擇演算法,則可以達到最壞O(N)的複雜度,不過實際應用中,該演算法通常比quick select慢1到2倍,所以並不常用(參考Blum, Floyd, Pratt, Rivest, and Tarjan 1973 Time bounds for selection)
演算法思想:
(1)利用快速排序的分治思想,求得待搜尋陣列按照的主元S[q](pivot)(主元的選定有好幾種方法,這裡不詳細討論,可參考快速排序),以主元為界分成左右兩個區間
(2)通過比較主元的位置,判斷第K個大小的數在主元左區間?在主元又區間?還是就是主元?(還要注意邊界條件的判斷,有可能在邊界)
(3)進入子區間遞迴呼叫
這裡實現了stl風格的quick select,僅僅作為一個mark
#include <algorithm> #include <cassert> namespace algorithm { template<typename _Tp> const _Tp& choose_pivot(const _Tp& x, const _Tp& y, const _Tp& z) { if( (x < y && y < z)||(z < y && y < x) ) return y; else if( (z < x && x < y)||(y < x && x < z) ) return x; else return z; } template<typename _Tp,typename _Compare> const _Tp& choose_pivot(const _Tp& x, const _Tp& y,const _Tp& z, _Compare comp) { if( (comp(x,y) && comp(y,z))||(comp(z,y)&&comp(y,x)) ) return y; else if( (comp(z,x) && comp(x,y))||(comp(y,x)&&comp(x,z))) return x; return z; } template<typename _RandomAccessIterator,typename _Tp> _RandomAccessIterator quick_partition(_RandomAccessIterator first, _RandomAccessIterator last,_Tp pivot) { while( true ){ while( *first < pivot ) ++first; --last; while( pivot < *last ) --last; if( first >= last ) return first; std::swap(*first,*last); ++first; } } template<typename _RandomAccessIterator,typename _Tp, typename _Compare> _RandomAccessIterator quick_partition(_RandomAccessIterator first, _RandomAccessIterator last, _Tp pivot, _Compare comp) { while( true ){ while( comp(*first,pivot) == true ) ++first; --last; while( comp(pivot,*last) == true ) --last; if( first >= last ) return first; std::swap(*first,*last); ++first; } } template<typename _RandomAccessIterator> _RandomAccessIterator quick_select(_RandomAccessIterator first, _RandomAccessIterator last, size_t kth) { typedef typename std::iterator_traits<_RandomAccessIterator>::value_type _ValueType; typedef typename std::iterator_traits<_RandomAccessIterator>::difference_type _DistanceType; if( first == last || last-first <=(_DistanceType)kth )//out of range return last; _ValueType pivot; _RandomAccessIterator mid; while( true ) { if( kth == 0 ) return std::min_element(first,last); else if( first+kth == last - 1 ) return std::max_element(first,last); else{ mid = first+(last-first)/2; pivot = choose_pivot(*first,*mid,*(last-1)); mid = quick_partition(first,last,pivot); if( mid-first > (_DistanceType)kth ) last = mid; else{ kth -= mid-first; first = mid; } } assert( last-first > (_DistanceType)kth); } } template<typename _RandomAccessIterator,typename _Compare> _RandomAccessIterator quick_select(_RandomAccessIterator first, _RandomAccessIterator last, size_t kth,_Compare comp) { typedef typename std::iterator_traits<_RandomAccessIterator>::value_type _ValueType; typedef typename std::iterator_traits<_RandomAccessIterator>::difference_type _DistanceType; if( first == last || last-first <=(_DistanceType)kth )//out of range return last; _ValueType pivot; _RandomAccessIterator mid; while( true ) { if( kth == 0 ) return std::min_element(first,last,comp); else if( first+kth == last - 1 ) return std::max_element(first,last,comp); else{ mid = first+(last-first)/2; pivot = choose_pivot(*first,*mid,*(last-1),comp); mid = quick_partition(first,last,pivot,comp); if( mid-first > (_DistanceType)kth ) last = mid; else{ kth -= mid-first; first = mid; } } assert( last-first > (_DistanceType)kth); } } } //namespace