算法思想 ：

堆排序利用堆數據結構設計的一種排序算法，堆是一種近似完全二叉樹的結構，同時滿足堆積的性質，即對于任意的i均有ki>=k(2i+1),ki>=k(2i+2)

步驟：

將數組轉化為最大堆

將當前堆的根結點和最后一個結點交換，然后將前n-1個元素轉化為最大堆

重復上一過程

為什么要是最大堆呢，因為這樣方便再第二步交換的時候可以將最大的放在最后，得到最后升序的結果

算法實現：

堆排序的主要復雜的地方就是在于生成最大堆的算法，而且這一算法要能被后面調整最大堆的時候用上，因為調整最大堆時候是從根節點開始向下的，所以生成最大堆的算法也設計為 從堆中只有只有一個或兩個子節點的結點開始(很好找n-2//2),一層一層的向上 ，而在每個節點的調整過程中，都是從上到下的，如果和子節點交換了，就再對這個子節點調整。經過對比，這也是生成最大堆的算法中最快的一種。

def heap_sort5(collection):

length=len(collection)

def adjustment(collection,index,heap_length):

#index要加入堆中的一個數，index是collection中的下標

#當隊列已經變成大頂堆的時候，通過讓根節點和最后一個元素交換，然后在對前n-1個元素組成的堆進行調整，重新變為大頂堆

#為了實現上面的功能，就必須在調整大頂堆的算法中加入一個變量，用來標識collection中是堆中的元素的長度，這個變量名為heap_lenght

left_index=index*2+1

right_index=index*2+2

temp=collection[index]

# if right_index>=length-1:#竟然有這樣的錯誤

while right_index<=heap_length-1:

if collection[left_index]>collection[right_index]:

if temp

collection[index]=collection[left_index]

index=left_index

left_index=index*2+1

right_index=index*2+2

else:

collection[index]=temp

break

elif temp

collection[index]=collection[right_index]

index=right_index

right_index=index*2+2

left_index=index*2+1

else:

collection[index]=temp

break

collection[index]=temp

if left_index<=heap_length-1:#只有左子樹說明就到底了

if collection[index]

collection[index],collection[left_index]=collection[left_index],collection[index]

#生成最大堆

for i in range((length-2)//2,-1,-1):

adjustment(collection,i,length)

#堆排序

for i in range(length-1):

collection[0],collection[length-1-i]=collection[length-1-i],collection[0]#注意每次是和第一個交換，不是i是0

adjustment(collection,0,length-i-1)#這個也是從0開始

return collection

效率分析：

對比

1000pic 隨機數據：

與桶排序(平均O(n+K)):比桶排序慢，時間是桶排序的4倍

詳細數據：[0.0030105114, 0.00298571587, 0.00298523903, 0.00296974182, 0.00300073624, 0.00302100182, 0.0019993782, 0.00198626518, 0.00301527977, 0.00298166275, 0.0040140152, 0.00301671028, 0.00198078

156, 0.00299835205, 0.00301647186, 0.00298070908, 0.00299263, 0.00300550461, 0.00298094749, 0.0029938221, 0.00299906731, 0.00301623344, 0.00298047066, 0.00200772285, 0.00198721886, 0.00299811363, 0.00301504135, 0.00399470329, 0.0029861927, 0.00301098824, 0.00198554993, 0.00301480293, 0.00198483467, 0.0029964447, 0.00299835205, 0.0020263195, 0.00401711464, 0.00200629234, 0.00299668312, 0.00302457809, 0.00199437141, 0.00300264359, 0.00302124023, 0.00298285484, 0.00301980972, 0.0029835701, 0.00301551819, 0.00299835205, 0.00199890137, 0.00499534607, 0.00301027298, 0.00401067734, 0.00299453735, 0.00300312042, 0.00299811363, 0.0019774437, 0.0029771328, 0.00199961662, 0.003002882, 0.00298476219, 0.0019993782, 0.00299906731, 0.0020108223, 0.00297951698, 0.00202202797, 0.00199866295, 0.00198745728, 0.00399780273, 0.00299715996, 0.00199818611, 0.00397968292, 0.00300979614, 0.00198554993, 0.00199079514, 0.00398445129, 0.00299572945, 0.00398278236, 0.00300073624, 0.00199818611, 0.00399780273, 0.00301170349, 0.00299835205, 0.00296235085, 0.00202345848, 0.00399947166, 0.00299429893, 0.00398445129, 0.00299811363, 0.00398445129, 0.00300931931, 0.00298428535, 0.00301766396, 0.0029976368, 0.00202298164, 0.00401210785, 0.00299596786, 0.00398468971, 0.00199770927, 0.00298476219, 0.00299715996]

運行了100次，平均運行時間差(heap_sort5-bucket_sort3)(正數代表你是個弟弟)是：0.00290829897

前者(heap_sort5)平均運行時間0.00392773628,后者(bucket_sort3)平均運行時間0.00101943731,前者約是后者的3.8528倍

與快排(平均O(nlogn))：比快排慢，時間是快排的2.4倍

詳細數據：[0.00201654434, 0.00297951698, 0.00201129913, 0.00196933746, 0.00198483467, 0.00199818611, 0.0019993782, 0.00100493431, 0.00298690796, 0.00099802017, 0.00199866295, 0.00300002098, 0.002963

78136, 0.00300335884, 0.00201702118, 0.00296258926, 0.00198817253, 0.00298595428, 0.00100684166, 0.00099444389, 0.00298118591, 0.00301027298, 0.0019865036, 0.00302529335, 0.00298523903, 0.00199794769, 0.00299739838, 0.00199866295, 0.00198602676, 0.0019865036, 0.00299787521, 0.00299954414, 0.0029976368, 0.00299859047, 0.00299811363, 0.00199818611, 0.0019993782, 0.00299739838, 0.00299835205, 0.00199794769, 0.00199818611, 0.0009996891, 0.00299787521, 0.00299692154, 0.0029976368, 0.00299930573, 0.0019993782, 0.00199961662, 0.0029976368, 0.0020005703, 0.00199818611, 0.00097441673, 0.00325942039, 0.00203251839, 0.00299954414, 0.0020236969, 0.00298595428, 0.00201702118, 0.00198149681, 0.00298118591, 0.00198149681, 0.00199890137, 0.0020236969, 0.00201678276, 0.00201177597, 0.00199770927, 0.00290203094, 0.00201177597, 0.00197386742, 0.00299882889, 0.00299882889, 0.00199460983, 0.00102114677, 0.00298571587, 0.00301146507, 0.0029861927, 0.00199913979, 0.00199842453, 0.00402283669, 0.0030105114, 0.0029976368, 0.00202226639, 0.00296854973, 0.0020160675, 0.00201106071, 0.00198984146, 0.00101685524, 0.00199794769, 0.00302910805, 0.00299811363, 0.0010368824, 0.00299811363, 0.00199842453, 0.00201964378, 0.00298213959, 0.00199866295, 0.0019993782, 0.00199818611, 0.0019865036, 0.00299906731]

運行了100次，平均運行時間差(heap_sort5-quick_sort2)(正數代表你是個弟弟)是：0.00235058308

前者(heap_sort5)平均運行時間0.00402917624,后者(quick_sort2)平均運行時間0.00167859316,前者約是后者的2.4003倍

與歸并(平均O(nlogn))：與歸并差不多，實際浮動10%

詳細數據：[1.764297e-05, 0.00099945068, 1.740456e-05, -0.00101423264, 0.00097084045, 0.0009701252, -0.00096297264, 1.811981e-05, 0.00096440315, 0.00198078156, -0.00097417831, 0.00101184845, -5.48363

e-06, 0.00099921227, -4.7684e-07, 0.00101065636, 0.0, 0.00098347664, 0.00100016594, -0.00098752975, 0.00098752975, -1.144409e-05, 9.77516e-06, -1.573563e-05, 0.00099873543, -0.00099945068, 0.00101733208, -1.001358e-05, 9.5367e-07, 0.00097441673, 0.00098729134, -0.00097298622, 0.00099992752, 0.0, -0.00097846985, 1.645088e-05, 0.00097513199, 0.00097751617, -0.00098800659, -9.5367e-07, 0.00098729134, 0.0009868145, 0.00099849701, -0.0009624958, 1.764297e-05, -2.14577e-06, 0.00197386742, -0.00200939178, 0.00098705292, -1.072884e-05, 0.00099825859, -0.00097560883, 0.00101137161, 0.00097537041, 0.00096106529, -1.66893e-06, -7.1526e-07, 0.00100588799, -1.716614e-05, 0.00101280212, 0.00096774101, 0.00096273422, 1.40667e-05, -0.00098156929, 0.00098705292, 3.8147e-06, -0.00098156929, 0.0010008812, 9.5367e-07, -2.3842e-07, 3.671646e-05, 3.623962e-05, -4.7684e-07, 0.00097966194, -3.576279e-05, -1.66893e-05, -3.647804e-05, 7.86781e-06, 0.00100302696, 3.433228e-05, 4.05312e-06, 0.00099158287, 3.433228e-05, 4.05312e-06, 0.00098752975, 0.00198554993, 0.00102448463, 0.00101280212, -1.239777e-05, 2.098083e-05, 0.00099825859, -1.358986e-05, 0.00098633766, -0.00098776817, 0.00101685524, -1.311302e-05, -2.598763e-05, 1.144409e-05, 1.192093e-05, 0.00097417831]

運行了100次，平均運行時間差(heap_sort5-merge_sort3)(正數代表你是個弟弟)是：0.00031897068

前者(heap_sort5)平均運行時間0.00396824121,后者(merge_sort3)平均運行時間0.00364927053,前者約是后者的1.0874倍

與希爾排序()：比希爾慢，時間是希爾的1.2倍

詳細數據：[1.335144e-05, 0.00201821327, -5.24521e-06, 0.00099253654, -1.311302e-05, 0.0020070076, 0.00099611282, 0.00096917152, 2.43187e-05, -5.24521e-06, -1.43051e-06, 1.28746e-05, 0.00096964836, 0

.00100708008, 0.00100922585, 0.00099921227, -0.00100970268, 5.24521e-06, 0.00099039078, 0.00099992752, -3.57628e-06, 0.0009829998, 0.00099992752, 0.00198245049, -1.478195e-05, 0.00100398064, 0.00098252296, -4.7684e-07, 0.00198626518, 0.00099468231, 0.0009970665, 0.00097560883, 0.00099992752, 0.00099849701, 0.00199985504, 0.00099897385, 0.00203323364, 0.00099825859, -7.1526e-07, -7.1526e-07, 0.00099921227, -2.3842e-07, 0.00099921227, 0.00099921227, -1.43051e-06, 0.00100111961, 2.3842e-07, 0.00100064278, -7.1526e-07, -2.3842e-07, 2.38419e-06, 0.0, 0.00099992752, -0.00100040436, 0.00099897385, 0.00199961662, 0.0010099411, 0.00102591515, 0.00099945068, 1.66893e-06, 0.00101184845, 0.00199985504, 0.00017309189, -2.31266e-05, 0.00097489357, 0.00198602676, 0.00102424622, -2.3842e-07, 0.00098109245, 0.00099468231, 1.788139e-05, -2.3842e-07, 0.00098013878, 1.43051e-06, 0.00096201897, 0.00100421906, -0.00098609924, 1.263618e-05, 0.00099945068, 0.00099945068, 0.00099897385, 0.00096940994, 1.788139e-05, -0.0010035038, 0.0009906292, 0.0019724369, -5.96046e-06, 0.00100493431, 6.19888e-06, -1.239777e-05, 9.77516e-06, 0.00098276138, -0.00100016594, 1.192093e-05, 0.00200486183, 7.86781e-06, 4.7684e-07, -3.09944e-06, 8.34465e-06, 0.00099611282]

運行了100次，平均運行時間差(heap_sort5-shell_sort3)(正數代表你是個弟弟)是：0.00062998772

前者(heap_sort5)平均運行時間0.00394880772,后者(shell_sort3)平均運行時間0.00331882000,前者約是后者的1.1898倍

與選擇排序(O(n^2)):比選擇快，是選擇時間的九分之一

詳細數據：[-0.03098297119, -0.03199076653, -0.03198218346, -0.03198122978, -0.03098106384, -0.03198075294, -0.0309817791, -0.03097701073, -0.03196048737, -0.03194522858, -0.03298187256, -0.030964136

12, -0.03097605705, -0.03295016289, -0.02898311615, -0.03000020981, -0.0329887867, -0.02998185158, -0.03198099136, -0.03098106384, -0.03298187256, -0.03098273277, -0.03100085258, -0.03200554848, -0.03200006485, -0.03198170662, -0.03196406364, -0.02997612953, -0.03097915649, -0.03096842766, -0.03198003769, -0.03099584579, -0.03197383881, -0.03098273277, -0.03098297119, -0.0319545269, -0.03097319603, -0.0320045948, -0.02999281883, -0.03101181984, -0.03199768066, -0.03097224236, -0.03094625473, -0.03100132942, -0.03198194504, -0.03099894524, -0.03197240829, -0.03098273277, -0.03099250793, -0.03298354149, -0.03197526932, -0.03198218346, -0.03200125694, -0.03199028969, -0.02902030945, -0.02996492386, -0.03098249435, -0.0299885273, -0.03198456764, -0.03000354767, -0.03098773956, -0.02900099754, -0.03199529648, -0.03196907043, -0.03195810318, -0.03095602989, -0.03194713593, -0.03198242188, -0.03098273277, -0.03198242188, -0.03098154068, -0.02998065948, -0.03198289871, -0.03298139572, -0.0309817791, -0.03099584579, -0.03101348877, -0.03197312355, -0.03200078011, -0.02999806404, -0.03199219704, -0.03098058701, -0.03095769882, -0.02999162674, -0.03095960617, -0.03098225594, -0.0319507122, -0.03199577332, -0.03299593925, -0.03000020981, -0.03098058701, -0.03100943565, -0.0309586525, -0.03098368645, -0.03195118904, -0.03098011017, -0.03100037575, -0.03000617027, -0.03198218346, -0.03198266029]

運行了100次，平均運行時間差(heap_sort5-select_sort2)(正數代表你是個弟弟)是：-0.03132260799

前者(heap_sort5)平均運行時間0.00397776127,后者(select_sort2)平均運行時間0.03530036926,前者約是后者的0.1127倍

與插入排序(O(n^2)):比插入快，是插入的九分之一

詳細數據：[-0.02899861336, -0.02996730804, -0.03100061417, -0.02997756004, -0.03021883965, -0.02996373177, -0.03097391129, -0.03092384338, -0.03199481964, -0.02900743484, -0.0319890976, -0.028993606

57, -0.02900671959, -0.02995681763, -0.02899456024, -0.0300078392, -0.02999472618, -0.03012704849, -0.03000092506, -0.03001236916, -0.02998304367, -0.0309548378, -0.03105306625, -0.03097820282, -0.02950978279, -0.03198122978, -0.02900028229, -0.02998232841, -0.03000259399, -0.02996039391, -0.02998280525, -0.03099131584, -0.02996897697, -0.03098130226, -0.02997732162, -0.03198027611, -0.0309650898, -0.03096795082, -0.03196811676, -0.03000283241, -0.02998304367, -0.03099298477, -0.03100728989, -0.02998328209, -0.02994346619, -0.03098249435, -0.0319852829, -0.02908825874, -0.03198170662, -0.03098082542, -0.02995967865, -0.03098034859, -0.02999520302, -0.03000998497, -0.02799153328, -0.02997493744, -0.03000164032, -0.02906346321, -0.03098011017, -0.02997016907, -0.03105521202, -0.02998423576, -0.03099560738, -0.02998161316, -0.03099656105, -0.02998256683, -0.02899646759, -0.02998304367, -0.03199863434, -0.02996826172, -0.03199887276, -0.03097033501, -0.02926707268, -0.03097510338, -0.03095555305, -0.03206801414, -0.02901434898, -0.03094601631, -0.02995657921, -0.0289683342, -0.02895879745, -0.02997756004, -0.02998709679, -0.02998375893, -0.03096938133, -0.0302259922, -0.02998423576, -0.03197407722, -0.02994465828, -0.02999520302, -0.02899694443, -0.03098726273, -0.02901959419, -0.03000187874, -0.02998280525, -0.02896547318, -0.02996993065, -0.02997756004, -0.02914905548, -0.03098154068]

運行了100次，平均運行時間差(heap_sort5-insertion_sort4)(正數代表你是個弟弟)是：-0.03028253078

前者(heap_sort5)平均運行時間0.00394899845,后者(insertion_sort4)平均運行時間0.03423152924,前者約是后者的0.1154倍

與雙向冒泡(O(n^2)):比雙向冒泡快，是其0.06倍

詳細數據：[-0.06394600868, -0.06296277046, -0.06496214867, -0.06496310234, -0.06196260452, -0.06313633919, -0.06294035912, -0.06199359894, -0.06495571136, -0.06296420097, -0.06696176529, -0.06396317

482, -0.06496310234, -0.06396389008, -0.06396532059, -0.06496357918, -0.06495523453, -0.06297588348, -0.06297802925, -0.06296396255, -0.06296420097, -0.06310391426, -0.06396174431, -0.06396341324, -0.0639629364, -0.06496334076, -0.06396412849, -0.06493639946, -0.06296396255, -0.06394505501, -0.06496310234, -0.06296372414, -0.06297707558, -0.06298160553, -0.06497168541, -0.06299304962, -0.06345295906, -0.06507635117, -0.06296348572, -0.06296467781, -0.0630466938, -0.06496119499, -0.06198263168, -0.06491446495, -0.06296181679, -0.06296348572, -0.0639629364, -0.06296348572, -0.06397724152, -0.06397509575, -0.06496334076, -0.06397604942, -0.06198978424, -0.06342983246, -0.06496357918, -0.06196522713, -0.06296300888, -0.06216788292, -0.06396389008, -0.06412982941, -0.06496143341, -0.0642516613, -0.06396579742, -0.06517672539, -0.06194090843, -0.06196451187, -0.06305217743, -0.06396389008, -0.06396245956, -0.06389951706, -0.06196451187, -0.06396150589, -0.06296372414, -0.06428790092, -0.06198167801, -0.06294608116, -0.06409454346, -0.06496357918, -0.0639629364, -0.06296539307, -0.06296277046, -0.06398153305, -0.06296467781, -0.06396436691, -0.06497693062, -0.06397914886, -0.06296396255, -0.06497597694, -0.06421637535, -0.06296396255, -0.06402182579, -0.06401872635, -0.06594610214, -0.06696248055, -0.06395411491, -0.0639629364, -0.06396508217, -0.06296205521, -0.0639629364, -0.06596231461]

運行了100次，平均運行時間差(heap_sort5-bidirectional_bubble_sort3)(正數代表你是個弟弟)是：-0.06377598286

前者(heap_sort5)平均運行時間0.00407264948,后者(bidirectional_bubble_sort3)平均運行時間0.06784863234,前者約是后者的0.0600倍

其他的版本：

主要是生成堆積的算法

版本1：

依舊是采用尋找結點中的數值，不同的是從根節點向下。要找根結點，就必須和左右子樹對比，所以就必須找左右結點的。。。這樣的話，每一個結點都必須要和他所有的子節點比較；

而且一旦最上面的點發生了變化，即把根節點取出來的時候，也不能只對index=0的結點進行調整，因為?adjustment(0)#只會對任何結點下面的子節點的做一次最大堆調整，而遞歸結束后加入發生了交換，是不會對下面頂點再進行調整的。即不能利用之前已經有序的堆積。采用這樣的辦法排序所需時間是最快的4000多倍

是最快的生成堆積的8倍(不過我很奇怪，他到底比最快的慢在哪里，最快的不也是從上到下嗎)

def adjustment(root):

length=len(collection)

left_index=root*2+1

right_index=root*2+2

#分別尋找左右子樹上的最小值

#左右子樹都存在

if right_index

adjustment(left_index)

adjustment(right_index)

#其實從底層找，并交換能避免一些額外的交換

if collection[left_index]>collection[right_index]:

if collection[root]>collection[right_index]:

collection[root],collection[right_index]=collection[right_index],collection[root]

#這里要不要return呢

elif collection[root]>collection[left_index]:

collection[root],collection[left_index]=collection[left_index],collection[root]

# collection[left_index],collection[right_index]=collection[right_index],collection[left_index]#左右不必交換

# return min(collection[root],adjustment(right_index),adjustment(left_index))

#只有左子樹存在

elif left_index

adjustment(left_index)

if collection[left_index]

collection[left_index],collection[root]=collection[root],collection[left_index]

# return min(collection[root],adjustment(left_index))

return collection[root]

版本2：

通過不斷想堆中的最末尾插入元素，通過和上面的結點比較進行調整

是最快生成堆算法的1.4倍

def adjustment(collection,add_index):

'''collection是整個集合，add_index是向小堆中添加元素的collection中的下標'''

#找父節點

temp=collection[add_index]

while (add_index-1)//2>=0 and temp

collection[add_index]=collection[(add_index-1)//2]

add_index=(add_index-1)//2

collection[add_index]=temp

版本3

并不是覺得上面的版本2效率不高才改進的，說實話我也不知道這個為什么比那個效率高，不過因為版本2不能用在后面調整最大堆的里面，所以想到希爾排序里里面的一種思想

def adjustment(collection,index,heap_length):

left_index=index*2+1

right_index=index*2+2

# if right_index>=length-1:#竟然有這樣的錯誤

while right_index<=heap_length-1:

if collection[left_index]>collection[right_index]:

if collection[index]

collection[index],collection[left_index]=collection[left_index],collection[index]

index=left_index

left_index=index*2+1

right_index=index*2+2

else:

break

elif collection[index]

collection[index],collection[right_index]=collection[right_index],collection[index]

index=right_index

right_index=index*2+2

left_index=index*2+1

else:

break

if left_index<=heap_length-1:#只有左子樹說明就到底了

if collection[index]

collection[index],collection[left_index]=collection[left_index],collection[index]

總結

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