有趣算法-堆排序学习整理

来源：互联网发布：淘宝借贷额度怎么提高编辑：程序博客网时间：2024/05/29 10:21

    最近,对数据结构比较迷,利用业余的时间看了些文档,发现中文文档比较难理解,参考了些外文文档,总结一下,希望大家能看的明白,堆排序的简单实现.

解析:

1. 最大堆:根结点的键值是所有堆结点键值中最大者的堆.2. 最小堆:根结点的键值是所有堆结点键值中最小者的堆.3. 上述两种堆插入操作: 只需要将节点插在二叉树的最后一个叶子结点位置，然后比较它对它父亲节点的大小，如果大则停止；如果小则交换位置，然后对父亲节点递归该过程直至根节点。复杂度为O(log(n))4. 存储二叉堆:一般用数组来表示。如果根节点在数组中的位置是1，第n个位置的子节点分别在2n和 2n+1。因此，第1个位置的子节点在2和3，第2个位置的子节点在4和5。以此类推。这种基于1的数组存储方式便于寻找父节点和子节点。

概述:

1. 将原始需排序的数据定义为origins[]2. 定义两个下标ori、tri,ori为origins[]数组中元素下标,tri为将要构建的完全二叉树trs(本例子按最大堆建树)的下标.(ps:下文中的堆和树是同一样东西)3. 从ori=0开始,向trs输入元素        建立节点Node时,检查该输入元素origins[ori]是否存在父亲节点,如果存在,则进行判断是否满足最大堆条件,不满足进行递归判断,满足就接着输入origins[ori+1]4. 生成最大堆trs后,进行排序.    4-i)将生成的trs存储(见:本文解析中的二叉堆存储)ns[]中    4-ii)将数组ns[]的第一个元素即(当前树的根元素)与数组ns[]最后的一个元素交换,并且输出根元素(当前最大元素),且从树中删除该节点,ns[]也删除改元素.    4-iii)处理(4-ii)置换后的当前根元素,按最大堆原则,将该元素放到适当位置    4-iv)重复执行(4-ii)直到树和数组ns[]只剩下一个元素.

例子:

这里写图片描述

注:此图出于https://en.wikipedia.org/wiki/Heapsort

    说实话,没有这幅gif,感觉我应该对堆排序还是迷迷糊糊,在这感谢维基百科,希望对大家有帮助.

代码实现:

#coding:utf8def get_parent_id(id):    '获取父亲节点'    if id < 1:        return -1    return (id - 1) // 2def swap(nums, pid,cid):    '交换'    temp = nums[pid]    nums[pid] = nums[cid]    nums[cid] = tempdef shift_up(nums, pid, cid):    '从下至上判断是否符合最大堆'    if nums[pid] < nums[cid]:        print 'pid:%d, pdata:%d, 与 cid:%d, cdata:%d 交换' % (pid,nums[pid],cid,nums[cid])        #交换        swap(nums, pid, cid)        #递归判断        gpid = get_parent_id(pid)        if gpid == -1:            return True        else:            return shift_up(nums, gpid, pid)    else:        return Truedef shift_down(nums,id):    '从上到下判断是否符合最大堆'    length = len(nums)    if id == (length - 1):        return True    lid = id * 2 + 1    rid = id * 2 + 2    #具有两个节点的情况    if lid <= (length - 2):        #取出最大的节点换上去        if nums[rid] > nums[lid]:            temp_id = rid            if nums[temp_id] > nums[id]:                swap(nums, id, temp_id)                #递归检查                return shift_down(nums, temp_id)            else:                return True        else:            temp_id = lid            if nums[temp_id] > nums[id]:                    swap(nums, id, temp_id)                #递归检查                return shift_down(nums, temp_id)            else:                return True    #只有一个节点的情况    if lid == (length -1):        if nums[lid] > nums[id]:            temp_id = lid            swap(nums, id, temp_id)            #递归检查            return shift_down(nums, temp_id)        else:            return True def build_max_heap(nums):    '构建最大堆'    max_heap = []    index = 0    #构建最大堆    for n in nums:        max_heap.append(nums[index])        #获取父亲节点        pid = get_parent_id(index)        print '第%d次,pid: %d,cid %d'%(index, pid, index)        if pid != -1:            shift_up(max_heap, pid, index)        index += 1    return max_heapdef heap_sort(nums):    '堆排序'    result = []    #第一步:建立最大堆    max_heap = build_max_heap(nums)    #第二步:首尾互换    while len(max_heap) > 1:        tail_id = len(max_heap) - 1        swap(max_heap, 0, tail_id)        result.append(max_heap[tail_id])        del max_heap[tail_id]        shift_down(max_heap, 0)    result += max_heap     print max_heap    return result

代码测试:

'''    测试部分'''def test(nums):    for i in range(len(nums)-1):        if(nums[i] < nums[i + 1]):            print '检验不合格'            print '第%d个:%d < 第%d个:%d' % (i, nums[i], i + 1, nums[i + 1])            return    print '检验通过'nums = 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414,3083,3370,350,6328,5947,8020,3409,8831,4419,1580,7634,1886,7319,175,2597,7500,9773,9773,3393,2773,9524,2382,35,5583,8665,4811,3026,3178,2969,722,9953,1799,6647,3055,8729,7133,3213,335,1733,4664,618,3289,7333,5787,5505,7166,6581,4694,2802,5016,4261,8386,8173,9284,6760,1387,8420,9675,2757,7785,7954,1855,1922,333,4922,8226,8995,6215,7691,7620,5968,2211,1637,881,2536,8358,5157,9798,8242,2983,4966,9354,1757,6241,485,1674,8432,3665,5795,6542,1239,6766,7297,1701,3070,8142,5284,1659,8980,1838,4563,2284,9972,2300,8653,2728,9871,6509,7209,1852,5980,5864,2532,9272,4548,5614,1443,9045,4638,2162,4076,2655,7654,8361,2269,2339,6069,8338,9908,1455,1845,4373,7631,8091,9967,6398,1102,8783,5912,8500,805,5450,1203,138,6911,6758,1979,2444,836,8901,738,8776,4256,5262,6467,4072,3021,5373,1199,5114,8622,1925,1380,9555,9864,9578,9800,2618,4304,2579,8484,6850,2874,6004,9161,6224,8892,6835,7984,5498,7060,8464,6331,492,2235,9936,7781,5024,5163,7118,1172,8536,3711,2291,1092,2122,5895,471,3965,1388,8307,7922,4130,661,6373,3555,9939,8100,1636,3899,4722,4655,3001,7841,5559,6097,5020,5964,2641,9046,2868,6674,6810,8917,2902,2565,4635,5978,9882,1590,1613,9516,4569,2012,5831,5027,1307,52,4848,2416,3999,4765,7115,227,9769,8273,2499,5933,3554,8163,5577,7625,4802,2067,390,6810,876,3936,70,4543,3127,327,7848,4375,9133,5451,6775,7305,4961,4608,1852,7270,6544,7192,1018,3144,1474,6034,6278,1035,4560,5308,8969,6872,7786,3371,4826,4451,4830,2166,7840,4337,248,146,9162,2445,1360,3116,6855,6310,8514,1220,9261,4536,7977,4389,49,9779,8261,7629,7572,1744,9603,2381,832,9844,2018,5846,981,4596,5835,5579,4966,4442,7903,1179,280,5685,7799,8581,9185,6696,124,2500,7321,8574,4760,6988,8432,8365,714,8888,3714,4809,5437,7059,5206,4166,6668,7465,1062,5289,1888,1614,2798,6890,8295,1018,7618,2065,1818,3922,8573,528,5283,2977,3682,6042,9118,9345,3352,9917,5930,1352,7392,6079,2616,80,4865,6254,7351,1890,6689,7754,7847,3866,8563,2053,3733,3654,338,2636,1689,9315,4582,2513,8787,4343,1120,7223,1984,8479,7291,5705,6450,7330,4883,3144,5118,8071,7979,917,3698,7732,1810,6386,5442,8251,3826,7320,1029,5163,8616,8911,2524,5002,4040,5641,9918,1287,8342,6044,457,9600,1281,7515,8676,2356,3512,9804,7773,352,5246,2544,9668,1091,7735,4963,7092,9473,5520,5717,2225,5214,6109,6367,8391,2443,6982,5964,2897,1282,1977,2823,9656,9835,5292,6177,2762,1521,7424,3621,4221,6330,7572,6984,6546,9277,6777,3127,2195,715,4585,3954,9288,8815,7069,9426,9923,351,4656,8982,3167,7267,1611,2164,4858,8213,6990,8042,3283,8190,1340,3041,7236,4982,1617,7885,810,9581,9970,5882,3096,4905,9652,912,8862,4859,8092,6718,1943,355,5261,757,8831,8277,3750,8323,474,6239,6231,4288,3427,3319,7302,6633,6854,4679,5208,5326,8258,2299,4680,4871,6290,636,6361,6646,9781,8509,2132,791,2328,6575,5975,1865,9250,4260,5477,5740,3117,753,6745,9082,582,9447,8977,9423,8206,1122,4879,6561,3118,3319,5792,7253,1338,6191,1176,5802,3075,9104,3542,7547,1769,2373,7134,7747,5687,6601,1402,4994,2820,1673,4962,5291,3094,9618,5417,3626,1561,5743,63,6317,9833,2258,6227,1211,1536,9245,9264,1297,8459,8605,3357,7770,5047,9927,790,7888,6957,5472,7266,6798,7589,1982,6843,7426,984,1015,227,9065,3626,2785,4178,9530,577,6191,2629,2197,6387,4369,7345,1312,995,8053,5648,8657,1179,8356,9706,3099,329,845,7441,8701,1522,3788,2729,9828,8756,9412,1582,7409,4701,5694,6541,2209,689,6440,953,830,5152,7208,7410,8884,1797,106,3258,7129,5468,3498,57,9442,6091,7085,3082,5704,6607,1253,3806,5032,9632,3106,2057,9650,8840,2013,8863,7299,3510,5189,2532,9222,8888,7912,6592,8761,3423,7932,8477,6293,8896,3595,772,5821,3235,4176,7553,8535,753,7788,2718,698,800,9054,4920,2159,1097,2255,4127,3259,4541,9747,9102,9495,7504,9757,5406,2755,9889,4076,7952,7042,2868,6878,2268,9334,425,8325,6328,3684,9850,6434,3490,8444,3286,6969,349,3086,805,8013,3745,5520,8625,5238,7556,3821,6405,4763,5495,5123,7822,4003,60,8130,9326,7981,4191,6815,1989,6286,5662,6970,6958,829,7367,344,1420,6929,4355,3543,8147,4240,3239,5722,5613,831,224,1784,5453,5483,2537,4398,5891,7222,5816,2194,7365,6181,4728,432,166,6430,9422,3525,363,5034,4806,2933,9759,1697,3379,3026,5311,2734,5656,403,3364,2529,5170,1253,7863,4802,9590,6365,8370,5795,8023,5452,4627,5667,647,8610,8140,9474,9169,2586,6156,9020,6247,2709,7353,857,3332,7487,534,1198,9754,6603,8182,828,6877,5733,3301,5556,1955,4209,4729,9406,7655,933,2068,444,4326,8067,233,9965,9933,2086,6311,1137,903,6051,9357,6085,1562,7753,9577,6437,2284,6768,5697,9872,4996,2738,6620,4543,5671,9494,4761,9703,8009,5134,3437,8546,183,7072,4661,1064,9950,9206,8483,2111,5318,3697,4992,3636,6762,9988,3555,1676,7532,8967,5058,5319,4191,6919,5114,9243,4825,6954,6424,415,2271,4628,9665,6938,3593,7100,3647,2140,4442,36,8547,4465,2501,950,6430,3906,5797,1666,224,6801,441,9208,7494,8771,2770,805,8715,653,5637,8047,9343,6273,6311,3688,1230,4724,2970,6620,4473,975,8868,8510,6489,1988,8063,6258,119,6765,8626,5765,4284,7499,1496,166,8023,2761,7783,3013,4310,1013,6348,8250,1592,3689,8539,243,3363,9259,1942,5228,2596,4105,8271,1633,2471,3435,2205,625,7108,8709,951,6184,5416,4001,289,1038,5800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11,2927,2924,1712,1854,5049,1343,3533,493,1936,2375,8039,7814,2948,360,8932,2531,2296,2252,8661,6852,5180,1774,497,2036,2711,4156,6764,7285,4626,55,3320,8789,1705,5240,9429,9710,1996,5706,1814,14,7523,3524,7813,128,8096,9348,3501,5257,5277,1926,5877,2811,8125,4123,7849,3466,3585,2210,5950,9995,45,7153,7166,8682,4697,4596,2342,9427,1457,7159,5176,7713,3422,4378,8485,7519,2803,6634,4529,5071,765,3711,6813,866,4374,2503,175,3403,6756,6232,6496,607,549,4629,7771,4391,2673,919,4700,5985,2064,3822,8736,4744,6809,2763,1628,3883,5135,263,4743,5448,5313,5830,9701,5980,8066,454,9080,8840,6952,5721,4415,796,6075,4341,75,7644,6075,1232,8036,8551,3033,9691,9385,4142,9350,5685,3449,8068,5828,2117,7468,4814,5324,9626,5732,4547,2496,477,3861,6098,1607,3140,8909,2282,787,1866,4515,818,5405,5872,9874,3877,5930,2199,5468,8151,8949,6269,9973,708,3449,3663,5244,4963,9441,5812,4613,7607,2509,2829,426,7337,9825,7061,817,5847,5723,1852,2221,8885,5468,6026,878,2645,2690,9106,6805,4881,8467,7562,7055,8209,2050,8953,7912,5471,5629,5941,569,8756,2611,2265,2305,5639,4598,255,8389,4165,3580,3223,1090,529,6097,5769,4297,3068,351,2917,6781,3161,7522,3647,2257,3462,5080,6130,5552,5779,8201,9715,2353,4105,6594,4980,9128,3577,1468,1213,3426,6507,9485,5917,7906,7451,8569,8920,2879,5031,126,891,8530,4841,3716,124,8249,7715,4229,7947,274,6500,1033,1136,6682,8211,1930,9553,3410,8247,6695,2955,8247,3181,4347,8652,6119,6489,1959,1836,7904,7290,1097,3044,1937,1569,900,6583,245,6570,4366,6613,9464,1531,1845,4839,4851,785,3532,4297,604,3340,2495,4447,9340,2251,7427,1487,3346,8107,851,7656,6152,9356,6704,9582,7889,3584,427,7548,4382,1538,3398,874,8693,6916,1305,1989,582,8302,8191,8077,7458,3757,3170,7360,3345,4667,2204,9805,5925,3858,9175,7924,2466,4900,4298,2607,3468,8855,7235,7441,2745,6523,6203,9608,1664,3050,8027,235,717,4992,3531,8708,2740,582,6205,551,4854,6856,1881,8784,7510,7381,1892,1113,3543,8949,5420,8589,3902,4837,2695,2841,5790,823,7993,6314,8584,376,5057,7894,307,746,5741,8780,199,7936,9986,3338,6165,8333,2134,6868,3220,5997,3449,9537,2184,8341,1012,5774,2773,9828,9089,2237,4937,9521,5795,6460,7791,8110,8278,9174,1266,5970,2186,2817,4713,498,5298,4401,6777,1966,9658,3787,4308,5219,7874,9539,5344,8508,2126,6046,1672,4493,8827,4135,2959,6425,1102,9501,9161,3222,2949,4191,8252,2310,9767,5479,9555,9310,8031,5744,85,5286,6729,8786,6240,337,4676,5369,5271,5887,2858,2252,2656,162,9223,1522,6956,6422,4888,8171,5918,3274,3787,7224,2719,2026,9022,1568,4909,825,7020,2874,2749,5201,5384,4870,9431,3497,6879,4098,7523,6185,9239,6808,1621,2787,1020,6245,3464,6119,6836,7115,2166,990,1080,1691,8519,2773,7286,8473,479,2808,791,1679,1211,9270,5614,328,445,7667,4514,2541,7692,2955,8634,5533,2695,6598,9421,797,6093,9787,9681,7139,9388,8464,1419,1928,3643,9920,2588,89,6241,3357,5033,6154,1944,5030,1910,2134,2359,2803,9024,403,184,8653,3604,1620,3072,4659,1323,1122,2465,2573,2215,2818,6889,6335,3346,1974,7792,8073,7858,9936,250,641,6673,9114,191,1560,3369,2230,3657,4768,2247,6787,6791,412,5381,9082,4978,5461,2188,1691,1691,9400,9805,1924,8606,1603,2780,1009,8170,5547,3106,7576,9260,5879,6652,7675,9933,5799,6569,2187,1006,5671,9082,805,2721,6920,5120,1982,9074,7471,5649,5802,2403,4778,7862,3324,3168,8586,2779,7923,6028,1504,4959,7079,7666,4098,2681,9437,146,5167,9995,3537,756,9817,1333,2495,2713,8307,7224,6506,2801,3381,8123,2899,6562,2817,370,3450,5837,5548,703,6906,6353,7514,8109,4774,198,4422,2652,2529,2465,1524,1949,7366,9801,5407,5826,2951,1207,3641,9715,6129,7811,2258,3610,1007,2627,6745,4513,2974,424,6090,4926,9052,7867,812,4524,8644,6496,3637,3788,723,3727,3589,6806,5683,8694,9119,3997,3896,2621,1921,3491,2888,3405,1731,2959,279,3629,7818,1279,7554,3488,7261,5083,9200,9125,9355,5398,1540,4169,1340,6933,8826,9891,2740,2914,2604,4064,9508,1558,5892,3954,8225,5593,3813,1808,8325,1104,8707,9861,770,5911,1183,7390,2703,336,2996,9796,4413,7313,5009,3994,7305,7776,560,983,5479,1883,5797,4352,8900,3385,1140,1161,269,5737,2660,4120,8729,6851,9524,9160,2879,5538,3116,4319,20,7753,2839,1571,8922,2675,9590,5031,3846,8867,702,5191,751,3644,3620,2290,8386,9566,8792,4844,6167,285,6114,2897,7589,9613,7184,368,3178,427,4438,641,3762,7376,6406,5533,8748,5379,9490,8314,5322,7417,1634,1427,3280,3478,4259,2980,3097,1054,7657,4364,7687,8175,3033,9508,7120,7716,3022,9030,3057,3903,5544,1110,4156,9754,1536,7662,2947,4376,7889,6879,506,8076,861,5383,883,7171,2398,2311,3453,9083,6845,1554,9692,2627,2045,4201,3329,4890,5601,716,9672,2234,8711,3489,8043,5444,7958,8883,2700,2682,5194,9448,4872,9165,4045,5625,1883,546,7188,3367,9103,1172,969,9953,7501,9441,3308,5546,8039,1979,7953,4418,8040,1702,8926,8212,7609,9366,1638,9368,2531,9838,2290,9261,3977,4766,5365,5146,7930,4758,5141,9973,1070,2303,7283,2071,7004,1642,3627,5325,3022,5149,2605,2853,1737,1321,110,9197,483,1510,8947,6595,3595,1912,4847,9557,2345,8224,549,6618,4762,6693,8256,1490,6013,7406,5460,1936,5335,3206,653,7873,1539,8491,1604,9047,7525,2516,8935,7638,7826,7262,5392,8445,4856,660,8414,6450,6571,2295,4149,5174,7965,6488,668,3154,6022,7553,1173,5843,2939,6453,517,3612,8274,5853,2047,6388,7599,7253,6623,78,1724,6944,7295,1874,7054,3548,6673,3628,3761,287,429,7545,1539,5782,2405,5160,6269,1808,6314,3121,8121,9224,40,621,6756,1271,7096,9880,1974,6986,4136,986,2314,2506,3916,4089,2508,3846,7333,7651,8667,2150,6103,720,3968,1389,1507,3527,5247,1518,8620,8184,7181,5722,3906,2035,2594,8919,3733,4586,112,9683,1667,7019,4258,6681,3511,9943,6441,6156,2087,1549,747,9069,5496,1972,9565,8351,9323,4431,4779,6668,9026,6224,2884,7941,4215,1457,3217,1764,1800,4737,2106,8760,9173,261,8427,9001,8263,3973,3404,7116,5066,6122,2381,8468,9502,220,3907,9683,1783,7428,8874,1624,9292,4809,9075,7564,7458,1845,7062,1125,812,4110,527,8183,668,2716,8230,2666,2954,2012,2605,503,1829,6676,937,4486,8415,6329,1229,5612,1194,6557,2882,1581,6619,1141,1800,1939,7317,2863,1680,3440,53,7424,4280,123,6286,5197,8485,7504,9320,2131,6540,4333,4606,8479,432,8745,1813,353,4172,4765,3175,8710,5891,4193,6591,4907,4384,3125,5701,6890,8833,5483,2690,5557,7296,6768,1966,9098,6512,7723,4292,2633,3192,7604,9016,7183,7641,7307,4072,4248,3529,310,3652,4660,3290,6489,9648,7664,3423,5813,1484,7676,520,9816,2463,5991,4551,2955,5471,5308,5953,5850,5040,6692,5923,7226,5847,1214,3234,8930,1116,2117,5534,3169,8208,8178,4144,1476,8081,6844,7453,1162,405,4571,78,3822,7197,7720,2432,487,2621,2228,1557,4014,4003,8177,111,938,8896,2980,7707,7661,5748,1168,4571,7364,2376,9189,5622,1963,9478,17,8825,2859,3635,273,6943,5621,8664,2897,2037,198,2615,7852,9630,275,8904,2496,6762,7246,4972,5668,9479,442,1810,3873,3533,530,3221,4033,8449,9818,7426,7751,6970,474,1923,6767,9319,6593,6850,2805,3447,3698,3531,8067,1740,7443,2664,8494,1843,8533,7399,3580,7241,2334,670,5718,4989,7072,6354,9107,7321,8814,4847,358,8994,599,8413,8047,7842,6524,5489,4265,5619,4503,8334,5371,386,593,5512,5407,4533,9522,803,6778,7462,7924,7629,3063,2495,1869,1735,4993,954,320,6692,7914,7463,8007,3117,4088,4969,844,5627,5544,8197,5967,53,4558,564,6255,6009,4414,2151,4610,1815,635,5994,1720,9273,3805,8460,2797,9833,4045,2627,2647,9060,7448,6116,2218,5947,8091,3835,9632,1083,7279,9571,8703,4385,9958,3239,7169,738,290,8837,3837,7260,6932,2226,4984,6658,4133,2300,900,70,6893,7830,4213,4365,7946,9249,1200,2965,4745,331,2428,5267,1223,8287,9095,8801,2436,440,9832,581,105,2106,7051,8895,6473,8950,634,8024,1474,8422,7032,372,196,8777,9732,7516,2662,3241,9477,5711,4119,4180,7950,7821,3076,304,3749,480,4165,2441,5525,7209,2704,9571,9531,8738,3173,8868,5682,4456,3234,3871,1782,9112,6392,1837,1476,8999,8649,2550,1359,157,5996,9452,2162,3229,6542,7970,6972,1301,5446,6109,9718,1998,1910,6074,5987,992,9276,336,7857,5029,8828,62,5470,3288,4916,628,6189,62,8791,4170,3519,9293,2131,8451,5581,1514,510,102,3003,2220,2287,4008,5671,6659,5980,904,5362,2234,8868,3647,3835,2519,8698,1092,1382,2432,3027,2523,8368,8621,1723,9436,3437,138,4980,655,316,8641,4993,5681,8571,2726,5588,7644,62,3346,2389,2781,2058,9490,8483,1992,7483,3124,5873,4153,1337,552,7960,9854,8629,5820,8296,7323,3151,422,254,6956,4115,3236,6495,7237,214,7966,5092,1531,4683,5833,920,120,1224,2759,2256,5851,5524,2926,6348,9034,8785,2966,2827,3804,2188,945,6507,6801,2894,6681,3758,1268,8884,7872,1058,4122,2213,8674,2757,2498,1709,9205,9188,6799,7744,7418,7863,5566,5903,1514,4705,828,3701,567,2992,9433,2392,7854,2390,348,2897,2927,769,116,5395,4512,1272,630,4600,5626,6010,470,7795,7493,187,2816,4506,7933,4630,7330,4868,4419,6329,980,3645,3726,9808,1477,9210,8387,7849,3259,7989,6173,4560,7500,5091,1517,2468,7063,8318,4906,921,2625,3160,8803,4858,2604,3496,2993,9300,3028,8782,6538,6826,9168,776,4110,508,8265,973,7814,3181,8686,1465,7673,1352,5031,8914,6594,9483,4916,2078,5762,3751,5200,9557,5192,9659,3334,4766,9273,4872,7728,24,2980,5448,8425,6571,2110,2862,9558,3695,6567,1080,8000,743,2505,7227,9228,9078,4486,6614,4052,4066,8162,1359,1559,6778,7942,6794,7369,5003,2160,2350,7684,7138,3391,9946,8677,1048,3310,7269]nums = heap_sort(nums)test(nums)

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