最小化代表性整数的错误总和
给定[0,10000]之间的n个整数作为D 1 ，D 2 ...，D n ，其中可能存在重复，n可能很大：
我想在[0,10000]之间找到k个不同的代表性整数（例如k = 5）作为R 1 ，R 2 ，...，R k ，因此所有代表性整数的误差总和最小化。
代表性整数的错误定义如下：
假设我们有k个代表整数按升序排列为{R 1 ，R 2 ...，R k }，错误R i
是：
我希望最小化k个代表整数的错误总和：
如何有效地完成这项工作？
EDIT1： k个代表整数中最小的一个必须是最小的数字 {d <子> 1 ，d <子> 2 ...，d <子>名词}
EDIT2： k代表整数中最大的一个必须是{D 1 中最大的数字，D 2 ......，D n }加1.例如，当{D 1 中的最大数字，D 2 ...，D n }是9787然后R k 是9788。
EDIT3：这里有一个具体的例子：
D = {1,3,3,7,8,14,14,14,30}并且如果k = 5且R被选为{1,6,10,17,31}那么误差总和是：
误差之和=（1-1）+（3-1）* 2 +（7-6）+（8-6）+（14-10）* 3 +（30-17）= 32
这是因为1 <= 1,3,3 <6,6 <= 7,8 <10,10 <= 14,14,14 <17,17 <= 30 <31
8 个答案:
答案 0 :(得分：4)
虽然在社区的帮助下，您已设法陈述您的 在数学上可以理解的形式的问题，你仍然可以 没有提供足够的信息让我（或其他任何人）给予 你一个明确的答案（我会发布这个评论，但是 由于某种原因，我没有看到“添加评论”选项可用 我）。为了对这个问题给出一个好的答案，我们需要知道 n和k相对于彼此的相对大小和10000（或者 D i 的预期最大值（如果不是10000），以及是否 至关重要的是你达到精确最小值（即使这样 需要花费过多的时间进行计算）或者如果a 近似也没关系（如果是这样，你有多接近 需要得到）。另外，为了知道运行什么算法 最短的时间，我们需要了解什么样的 硬件将运行算法（即，我们有CPU 核心并行运行，m相对于k的大小是多少。
另一个重要的信息是这个问题是否存在 只解决了一次，或者它必须多次解决但存在 整数D i 的分布之间的某种联系 从一个问题到下一个问题（例如，整数 D i 都是来自特定的随机样本，不变 概率分布，或者每个连续问题都有 它的输入是一个不断增加的集合，它是前一个集合 问题加上额外的 s 整数）。
没有合理的算法可以及时运行你的问题 由于构建直方图，因此在n中的方式大于线性 n个整数D i 需要O（n）时间，答案为 优化问题本身仅取决于直方图 整数而不是他们的排序。没有算法可以及时运行 小于O（n），因为这是问题输入的大小。
对所有可能性进行蛮力搜索需要（假设 至少有一个D i 是0而另一个是10000），for 小k，比如k＆lt; 10，大约是O（10000 k-2 ）时间，所以如果 log 10 （n）＆gt;＆gt; 4（k-2），这是最优算法（因为在 在这种情况下，强力搜索的时间与之相比微不足道 时间阅读输入）。值得注意的是，如果k是 非常接近10000，然后蛮力搜索只需要 O（10000 10002-k ）（因为我们可以搜索而不是 不用作代表整数的整数。）
如果您使用更多信息更新问题的定义，我 将依次尝试编辑我的答案。
答案 1 :(得分：3)
现在问题澄清了，我们观察到R i 将D x 分成k-1个区间，[R 1 ，R 2 ），[R 2 ，R 3 ），... [R k-1 ，R <子>ķ）。每个D x 恰好属于这些间隔中的一个。令q i 为区间[R i ，R i + 1 ）中D x 的数量，并且让s i 是那些D x 的总和。然后每个错误（R i ）表达式是q i 项的总和，并且计算为s i - q i - [R <子> I
总结在所有i上，我们得到S - sum的总误差（q i R i ），其中S是所有D
任何全局最大值必须是局部最大值;所以我们想象增加或减少一个R i 。如果R i 不原始数据值之一，那么我们可以在不改变任何q i 的情况下增加它并改进我们的目标函数。因此，最优解具有每个R i （除了限制最后一个）作为数据值之一。之后我在数学方面陷入了一些困境，但似乎一个明智的方法是选择初始R i 作为每个（n / k）数据值（简单百分位数），然后迭代地查看是否将R_i移动到上一个或下一个值可以改善分数，从而减少错误。 （q i R i 似乎更容易使用，因为你可以读取数据并计算重复次数并更新q i ，R i 只查看单个数据/计数点。无论数据有多大，您只需要存储10,000个数据计数的数组。
data: 1 3 7 8 14 30
count: 1 2 1 1 3 1 sum(data) = 94
initial R: 1 3 8 14 31
initial Q: 1 3 1 4 sum(QR) = 74 (hence error = 20)
在这个例子中，我们可以尝试将3或8更改为7，例如，如果我们将3增加到7，那么我们看到初始数据中有2个3，所以前两个Q变为1+ 2,3-2 - 事实证明这减少了总和（QR））。我确信有更聪明的模式来检测QR表中的哪些变化是可行的，但这似乎是可行的。
答案 2 :(得分：1)
如果分布接近随机且选择（n）足够大，那么您通常会浪费时间，尝试优化实际成本的时间并计算得比预期平均值减少％。最快的平均解决方案是将低k-1设置在间隔M /（k-1）的低端，其中M是最低上限 - 最大下限（即，M =可能的最大数量 - 0）和M + 1的最后一个k。它将采用顺序k（我们可以用这个问题中提供的信息做得最好）来计算这些值。陈述我刚刚做的事情当然不是证明。
我的观点是这样的。上面的讨论是一个简化，我认为对于一大类集合非常实用。另一方面，可以直接计算所有排列的每个错误，然后选择最小的错误。这种运行时间使得该解决方案在许多情况下难以处理。提出这个问题的人所期望的不仅仅是最直接和最确切（难以处理）的答案，而且还有很多是开放式的。我们可以从边缘修剪到永恒，试图量化沿无限解空间的各种属性，以获得n个数和所有k个值的所有可能的排列（或组合）。
答案 3 :(得分：1)
该程序的完整性部分得到了修改版本的确认，该版本使用here来生成与@mhum独立获得的results匹配的数据。
它找到给定数据集和k值的精确最小误差值和相应的R值。
/************************************************************
This program can be compiled and run (eg, on Linux):
$ gcc -std=c99 minimize-sum-errors.c -o minimize-sum-errors
$ ./minimize-sum-errors
************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
//data: Data set of values. Add extra large number at the end
int data[]={
10,40,90,160,250,360,490,640,810,1000,1210,1440,1690,1960,2250,2560,2890,3240,3610,4000,4410,4840,5290,5760,6250,6760,7290,7840,8410,9000,9610,10240,99999,1,2,3,4,5,6,7,8,9,10,24,24,14,12,41,51,21,41,41,848,21, 547,3,2,888,4,1,66,5,4,2,11,742,95,25,365,52,441,874,51,2,145,254,24,245,54,21,87,98,65,32,25,14,25,36,25,14,47,58,58,69,85,74,71,82,82,93,93,93,12,12,45,78,78,985,412,74,3,62,125,458,658,147,432,124,328,952,635,368,634,637,874,124,35,23,65,896,21,41,745,49,2,7,8,4,8,7,2,6,5,6,9,8,9,6,5,9,5,9,5,9,11,41,5,24,98,78,45,54,65,32,21,12,18,38,48,68,78,75,72,95,92,65,55,5,87,412,158,654,219,943,218,952,357,753,159,951,485,862,1,741,22,444,452,487,478,478,495,456,444,141,238,9,445,421,441,444,436,478,51,24,24,24,24,24,24,247,741,98,99,999,111,444,323,33,5,5,5,85,85,85,85,654,456,5,4,566,445,5664,45,4556,45,5,6,5,4,56,66,5,456,5,45,6,68,7653,434,4,6,7,689,78,8,99,8700,344,65,45,8,899,86,65,3,422,3,4,3,4,7,68,544,454,545,65,4,6,878,423,64,97,8778,5456,5486,5485,545,5455,4548,81,999,8233,5223,8741,7747,7756,54,7884,5477,89,332,5999,9861,12545,9852,11452,5482,9358,9845,577,884,5589,5412,3669,32,6699,396,9629,953,321,45,5484,588,5872,85,872,87,1122,884,2458,471,22685,955,2845,6852,589,5896,2521,35696,5236,32633,56665,6633,326,5486,5487,8541,5495,2155,3,8523,65896,3852,5685,69536,1,1,1,1,1,2,3,4,5,6,
};
//N: size of data set
int N=sizeof(data)/sizeof(int);
//SavedBundle: data type to "hold" memoized values needed (minimized error sums and corresponding "list" of R values for a given round)
typedef struct _SavedBundle {
long e;
int head_index_value;
int tail_offset;
} SavedBundle;
//sb: (pts to) lookup table of all calculated values memoized
SavedBundle *sb; //holds winning values being memoized
//Sort in increasing order.
int sortfunc (const void *a, const void *b) {
return (*(int *)a - *(int *)b);
}
/****************************
Most interesting code in here
****************************/
long full_memh(int l, int n) {
long e;
long e_min=-1;
int ti;
if (sb[l*N+n].e) {
return sb[l*N+n].e; //convenience passing
}
for (int i=l+1; i<N-1; i++) {
e=0;
//sum first part
for (int j=l+1; j<i; j++) {
e+=data[j]-data[l];
}
//sum second part
if (n!=1) //general case, recursively
e+=full_memh(i, n-1);
else //base case, iteratively
for (int j=i+1; j<N-1; j++) {
e+=data[j]-data[i];
}
if (e_min==-1) {
e_min=e;
ti=i;
}
if (e<e_min) {
e_min=e;
ti=i;
}
}
sb[l*N+n].e=e_min;
sb[l*N+n].head_index_value=ti;
sb[l*N+n].tail_offset=ti*N+(n-1);
return e_min;
}
/**************************************************
Call to calculate and print results for the given k
**************************************************/
int full_memoization(int k) {
char *str;
long errorsum; //for convenience
int idx;
//Call recursive workhorse
errorsum=full_memh(0, k-2);
//Now print
str=(char *) malloc(k*20+100);
sprintf (str,"\n%6d %6d {%d,",k,errorsum,data[0]);
idx=0*N+(k-2);
for (int i=0; i<k-2; i++) {
sprintf (str+strlen(str),"%d,",data[sb[idx].head_index_value]);
idx=sb[idx].tail_offset;
}
sprintf (str+strlen(str),"%d}",data[N-1]);
printf ("%s",str);
free(str);
return 0;
}
/******************************************
Initialize and seek result for all k values
******************************************/
int main (int x, char **y) {
int t;
int i2;
qsort(data,N,sizeof(int),sortfunc);
sb = (SavedBundle *)calloc(sizeof(SavedBundle),N*N);
printf("\n Total data size: %d",N);
printf("\n k errSUM R values",N);
for (int i=3; i<=N; i++) {
full_memoization(i);
}
free(sb);
return 0;
}
获得了一些样本结果：
Total data size: 375
k errSUM R values
3 475179 {1,5223,99999}
4 320853 {1,5223,56665,99999}
5 260103 {1,5223,7653,56665,99999}
6 210143 {1,5223,7653,32633,56665,99999}
7 171503 {1,421,5223,7653,32633,56665,99999}
8 142865 {1,412,2458,5223,7653,32633,56665,99999}
9 124403 {1,412,2458,5223,7653,32633,56665,65896,99999}
10 106790 {1,412,2458,5223,7653,9610,32633,56665,65896,99999}
11 93715 {1,412,2458,5223,7653,9610,22685,32633,56665,65896,99999}
12 81507 {1,412,848,2458,5223,7653,9610,22685,32633,56665,65896,99999}
13 71495 {1,412,848,2155,3610,5223,7653,9610,22685,32633,56665,65896,99999}
14 64243 {1,412,848,2155,3610,5223,6633,7747,9610,22685,32633,56665,65896,99999}
15 58355 {1,412,848,2155,3610,5223,6633,7653,8523,9610,22685,32633,56665,65896,99999}
16 53363 {1,65,412,848,2155,3610,5223,6633,7653,8523,9610,22685,32633,56665,65896,99999}
17 48983 {1,65,412,848,2155,3610,4548,5412,6633,7653,8523,9610,22685,32633,56665,65896,99999}
18 45299 {1,65,412,848,2155,3610,4548,5412,6633,7653,8523,9610,11452,22685,32633,56665,65896,99999}
19 41659 {1,65,412,848,2155,3610,4548,5412,6633,7653,8523,9610,11452,22685,32633,56665,65896,69536,99999}
20 38295 {1,65,321,441,848,2155,3610,4548,5412,6633,7653,8523,9610,11452,22685,32633,56665,65896,69536,99999}
21 35232 {1,65,321,441,848,2155,3610,4548,5412,6633,7653,8523,9610,11452,22685,32633,35696,56665,65896,69536,99999}
22 32236 {1,65,321,441,848,2155,3610,4410,5223,5455,6633,7653,8523,9610,11452,22685,32633,35696,56665,65896,69536,99999}
23 29323 {1,65,321,432,634,872,2155,3610,4410,5223,5455,6633,7653,8523,9610,11452,22685,32633,35696,56665,65896,69536,99999}
24 26791 {1,65,321,432,634,862,1690,2458,3610,4410,5223,5455,6633,7653,8523,9610,11452,22685,32633,35696,56665,65896,69536,99999}
25 25123 {1,65,321,432,634,862,1690,2458,3610,4410,5223,5455,5872,6633,7653,8523,9610,11452,22685,32633,35696,56665,65896,69536,99999}
26 23658 {1,65,321,432,634,862,1690,2458,3610,4410,5223,5455,5872,6633,7653,8233,8700,9610,11452,22685,32633,35696,56665,65896,69536,99999}
27 22333 {1,41,78,321,432,634,862,1690,2458,3610,4410,5223,5455,5872,6633,7653,8233,8700,9610,11452,22685,32633,35696,56665,65896,69536,99999}
28 21073 {1,41,78,321,432,634,862,1440,2155,2845,3610,4410,5223,5455,5872,6633,7653,8233,8700,9610,11452,22685,32633,35696,56665,65896,69536,99999}
29 19973 {1,41,78,321,432,634,848,951,1960,2458,2845,3610,4410,5223,5455,5872,6633,7653,8233,8700,9610,11452,22685,32633,35696,56665,65896,69536,99999}
30 18879 {1,41,78,321,432,634,848,951,1960,2458,2845,3610,4410,5223,5455,5872,6633,7653,8233,8700,9358,9845,11452,22685,32633,35696,56665,65896,69536,99999}
31 17786 {1,41,78,321,432,634,848,951,1960,2458,2845,3610,4410,5223,5455,5872,6633,7653,8233,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
32 16801 {1,41,78,321,432,634,848,943,1440,1960,2458,2845,3610,4410,5223,5455,5872,6633,7653,8233,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
33 15821 {1,41,78,218,321,432,634,848,943,1440,1960,2458,2845,3610,4410,5223,5455,5872,6633,7653,8233,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
34 14900 {1,41,78,218,321,421,544,634,848,943,1440,1960,2458,2845,3610,4410,5223,5455,5872,6633,7653,8233,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
35 14185 {1,41,78,218,321,421,544,634,848,943,1440,1960,2458,2845,3610,4410,5223,5455,5872,6633,7290,7747,8410,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
36 13503 {1,41,78,218,321,421,544,634,741,862,951,1440,1960,2458,2845,3610,4410,5223,5455,5872,6633,7290,7747,8410,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
37 12859 {1,21,45,78,218,321,421,544,634,741,862,951,1440,1960,2458,2845,3610,4410,5223,5455,5872,6633,7290,7747,8410,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
38 12232 {1,21,45,78,218,321,421,544,634,741,862,951,1440,1960,2458,2845,3610,4410,5223,5455,5664,5872,6633,7290,7747,8410,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
39 11662 {1,21,45,78,218,321,412,444,544,634,741,862,951,1440,1960,2458,2845,3610,4410,5223,5455,5664,5872,6633,7290,7747,8410,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
40 11127 {1,21,45,78,218,321,412,444,544,634,741,862,951,1440,1960,2458,2845,3610,4410,5223,5455,5664,5872,6633,7290,7747,8233,8523,8700,9358,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
41 10623 {1,21,45,78,218,321,412,444,544,634,741,862,951,1440,1960,2458,2845,3610,4410,5223,5455,5664,5872,6633,7290,7747,8233,8523,8700,9358,9610,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
42 10121 {1,21,41,65,85,218,321,412,444,544,634,741,862,951,1440,1960,2458,2845,3610,4410,5223,5455,5664,5872,6633,7290,7747,8233,8523,8700,9358,9610,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
43 9637 {1,21,41,65,85,218,321,412,444,544,634,741,862,951,1440,1960,2458,2845,3610,3852,4410,5223,5455,5664,5872,6633,7290,7747,8233,8523,8700,9358,9610,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
44 9207 {1,21,41,65,85,218,321,412,444,544,634,741,862,951,1440,1960,2458,2845,3610,3852,4410,4840,5223,5455,5664,5872,6633,7290,7747,8233,8523,8700,9358,9610,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
45 8804 {1,21,41,65,85,218,321,412,444,544,634,741,862,943,1122,1690,2155,2458,2845,3610,3852,4410,4840,5223,5455,5664,5872,6633,7290,7747,8233,8523,8700,9358,9610,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
46 8409 {1,21,41,65,85,218,321,412,444,544,634,741,862,943,1122,1690,2155,2458,2845,3240,3610,3852,4410,4840,5223,5455,5664,5872,6633,7290,7747,8233,8523,8700,9358,9610,9845,11452,12545,22685,32633,35696,56665,65896,69536,99999}
47 8014 {1,21,41,65,85,218,321,412,444,544,634,741,862,943,1122,1690,2155,2458,2845,3240,3610,3852,4410,4840,5223,5455,5664,5872,6633,7290,7747,8233,8523,8700,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
48 7636 {1,21,41,65,85,218,321,412,444,544,634,741,862,943,1122,1690,2155,2458,2845,3240,3610,3852,4410,4840,5223,5455,5664,5872,6250,6633,7290,7747,8233,8523,8700,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
49 7273 {1,21,41,65,85,218,321,412,444,544,634,741,862,943,1122,1690,2155,2458,2845,3240,3610,3852,4410,4840,5223,5455,5664,5872,6250,6633,7290,7653,7747,8233,8523,8700,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
50 6922 {1,21,41,65,85,124,218,321,412,444,544,634,741,862,943,1122,1690,2155,2458,2845,3240,3610,3852,4410,4840,5223,5455,5664,5872,6250,6633,7290,7653,7747,8233,8523,8700,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
51 6584 {1,21,41,65,85,124,218,321,412,444,544,634,741,862,943,1122,1440,1960,2155,2458,2845,3240,3610,3852,4410,4840,5223,5455,5664,5872,6250,6633,7290,7653,7747,8233,8523,8700,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
52 6283 {1,21,41,65,85,124,218,321,412,444,544,634,741,862,943,1122,1440,1960,2155,2458,2845,3240,3610,3852,4410,4840,5223,5412,5477,5664,5872,6250,6633,7290,7653,7747,8233,8523,8700,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
53 5983 {1,21,41,65,85,124,218,321,412,444,544,634,741,862,943,1122,1440,1960,2155,2458,2845,3240,3610,3852,4410,4840,5223,5412,5477,5664,5872,6250,6633,7290,7653,7747,8233,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
54 5707 {1,21,41,65,85,124,218,321,412,444,544,634,741,862,943,1122,1440,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,7290,7653,7747,8233,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
55 5450 {1,21,41,65,85,124,218,321,412,441,478,544,634,741,862,943,1122,1440,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,7290,7653,7747,8233,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
56 5196 {1,21,41,65,85,124,218,321,412,441,478,544,634,741,862,943,1122,1440,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,8233,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
57 4946 {1,21,41,65,85,124,218,321,412,441,478,544,634,741,862,943,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,8233,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
58 4722 {1,5,21,41,65,85,124,218,321,412,441,478,544,634,741,862,943,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,8233,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
59 4536 {1,5,21,41,65,85,124,218,321,412,441,478,544,634,741,862,943,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,7840,8233,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
60 4352 {1,5,21,41,65,85,124,218,321,412,441,478,544,634,741,848,872,951,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,7840,8233,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
61 4172 {1,5,21,41,65,85,124,218,321,357,412,441,478,544,634,741,848,872,951,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,7840,8233,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
62 3995 {1,5,21,41,65,85,124,218,321,357,412,441,478,544,634,741,848,872,951,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,7840,8233,8410,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
63 3828 {1,5,21,41,65,85,124,218,321,357,412,441,478,544,634,741,848,872,943,985,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,7840,8233,8410,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
64 3680 {1,5,21,41,65,85,124,218,321,357,412,441,478,544,634,741,848,872,943,985,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4000,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,7840,8233,8410,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
65 3545 {1,5,21,32,45,65,85,124,218,321,357,412,441,478,544,634,741,848,872,943,985,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4000,4410,4548,4840,5223,5412,5477,5664,5872,6250,6633,6760,7290,7653,7747,7840,8233,8410,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
66 3418 {1,5,21,32,45,65,85,124,218,321,357,412,441,478,544,634,741,848,872,943,985,1122,1440,1690,1960,2155,2458,2845,3240,3610,3852,4000,4410,4548,4840,5223,5412,5477,5664,5872,5999,6250,6633,6760,7290,7653,7747,7840,8233,8410,8523,8700,9000,9358,9610,9845,10240,11452,12545,22685,32633,35696,56665,65896,69536,99999}
在PC达到100行（可能是375行）之前，PC上的输出开始减慢太多。
答案 4 :(得分：1)
此算法不会产生精确的答案，但它允许处理非常大的数据集的速度比memoized algorithm快得多，并且结果往往非常接近确切的答案。在某些情况下，算法仍然需要改进/调试以防止潜在的无限循环，但这不是一个交易破坏者，因为可以添加代码来阻止它。同时，alg的数据集太大而无法通过其他一些通常更好的算法（如前面提交的memoized alg答案）进行管理。例如，在[1-100000]范围内的近10,000个样本中，在旧PC上以秒为单位计算k = 500，其中出现备忘录版本需要花费一个多小时才能在小得多的情况下进行小得多的k = 90大小为375的数据集。对于这种增加的性能，没有获得绝对最低的误差总和是一个非常小的代价。 [我没有得出结果的质量，但所有比较数据值的地方备忘录可以保持不会超过10％更糟，如果那样。]
/************************************************************
This program can be compiled and run (eg, on Linux):
$ gcc -std=c99 fast-inexact.c -o fast-inexact
$ .fast-inexact
************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <stdint.h>
//a: Data set of values. Add extra large number at the end
int a[]={
10,40,90,160,250,360,490,640,810,1000,1210,1440,1690,1960,2250,2560,2890,3240,3610,4000,4410,
4840,5290,5760,6250,6760,7290,7840,8410,9000,9610,10240,99999,
1,2,3,4,5,6,7,8,9,10,
24,24,14,12,41,51,21,41,41,848,21, 547,3,2,888,4,1,66,5,4,2,11,742,
95,25,365,52,441,874,51,2,145,254,24,245,54,21,87,98,65,32,25,14,
25,36,25,14,47,58,58,69,85,74,71,82,82,93,93,93,12,12,45,78,78,985,
412,74,3,62,125,458,658,147,432,124,328,952,635,368,634,637,874,
124,35,23,65,896,21,41,745,49,2,7,8,4,8,7,2,6,5,6,9,8,9,6,5,9,5,9,
5,9,11,41,5,24,98,78,45,54,65,32,21,12,18,38,48,68,78,75,72,95,92,65,
55,5,87,412,158,654,219,943,218,952,357,753,159,951,485,862,1,741,22,
444,452,487,478,478,495,456,444,141,238,9,445,421,441,444,436,478,51,
24,24,24,24,24,24,247,741,98,99,999,111,444,323,33,5,5,5,85,85,85,85,
654,456,5,4,566,445,5664,45,4556,45,5,6,5,4,56,66,5,456,5,45,6,68,7653,
434,4,6,7,689,78,8,99,8700,344,65,45,8,899,86,65,3,422,3,4,3,4,7,68,
544,454,545,65,4,6,878,423,64,97,8778,5456,5486,5485,545,5455,4548,81,
999,8233,5223,8741,7747,7756,54,7884,5477,89,332,5999,9861,12545,9852,
11452,5482,9358,9845,577,884,5589,5412,3669,32,6699,396,9629,953,321,
45,5484,588,5872,85,872,87,1122,884,2458,471,22685,955,2845,6852,589,
5896,2521,35696,5236,32633,56665,6633,326,5486,5487,8541,5495,2155,3,
8523,65896,3852,5685,69536,
1,1,1,1,1,2,3,4,5,6,
//375
54,5451,545,54,885,855,8621,5,23,7,54,89,3,8545,196,35338,6412,5338,35512,8545,55483,3548,34878,37846,1545,2489,24534,84234,56465,8643,454,8,548,78,454,85,44,54564,87,85,45,48,54,564,67564,8945,864,54564,864,5453,554,7894,65456,45,5489,8424,84248,543,5454,82,54548,44,54654,8454,54,684,54,34,8,454,87,84,4,548,45456,48454,86465,4,454,45,4445,4564,484,4564,64654,56456,54,45,121,2851,15,248,24853,845,8485,384,3484,3484,3853,183,4835,83545,82,1851,6851,854,83,48434,87,34,854,943,849,468,4654,97,35,494,6549,878,65,2184,4845,4564,64,8,44,84,5,4454,4845,484,8513,897,47,8789,764,54,454,54894,454,842,181,54,81348,4518,548,51,813,1851,1841,5484,51,8431,8484,5487,79,4,31,31,84,87,74111,1,7272,7814,18,781,1,7,823,27872,8,8178,4156,485,184,84,45,18,75,18,715,48,78174,6541,8,54,8,41,8,4564,187,154,841,9,4194,53,4194,15,48,941,48,941,5,489,7415,41,49,41,54,54545,494,15,98,4189,5641,841,5145,41,416,48,414,4,841,5414,61,41,9891,61,169,19,1989,173,48154,56116,187,191,61,61418,8719,8187,51842,815,4815,4984,5,484,15,4897,18,4151,81,8941,549,1,5498,15,89,12,4,97,97,1,591,519,1,51,9,15,1655,65,2,3214,2365,8,77899,6565,6589,586,5,66,5,23669,5,9,59,9,8,569,3,3,6,96,99,955,5,96,9595,95,629,8971,81,5715,45,141,4819,84,518,81,87,2,41,5,98,41,54,9415,49,841,54,591,54,918,781,794,1221,2891,5,19878,154,9,4154,94,1518,41,49,415,49,15,4541,954,78,219,45,4515,49,9187,1549,15,985,14984,1597,91978,1541,41,5491,54197,815,914,91,78195,4179,1984,971,54,91,5198,71914,97,194,914,59419,49,4194,941,94191,41,9419,1941,914,9149,4191,1,19149,4949,454,141,1,9,489,415,4941,9841,24,8941,54,5915,198,419,24949,194,8545,4591,5498,714,54,984,5491,54,978,154,978,154,91495,41945,49,41954,8,154,94,149,4594,54,98,154,594,984,815,45,9148,4191,19,84,15,1,948,7897,184,5419,71,4194,8419,41984,954,54,1941,81798,789,459,45,4198,787,184,941,921,987,181,541,48,971,894,9145,594,19,78,48,4984,184,945,4,194,19849,454,978,4154,944,84154,9871,8489,4154,841,8945,198,710,45,4,51,541,984,982,1954,81,491,2465,498,5419,7481,5497,8515,498,4154,979,871,41,148,11971,184,94145,498,15,48,154,9418,41894,815,494,145,419,8,151,25,18940,5415,64348,74851,541,9481,24,9841,2,498,124,91,594,34614,64,8491,456,4164,81,3496,494,324,16498,15,4917,9841,546,4841,546,484,54541,654,81,246,518,4841,65,486,4165,4654,8415,4646,846,41654,864,165,45,4188,165,481,31,354,9415,491,549,484,87131,828,284,842,2,434,8434,64835,4313,143,48,35,498,7,154,8,7897,7154,654,987,564,3546,8789,715,4684,864,234,864,615,467,89,135,4198,7,654,64,189,7817,56,4,654,98,465,46,48,4,354,96,8413,54,8,768,45,165,46,81,654,3,48,7,41,54,6,71654,5618,745,4687,56461,8415,46841,654,18415,4641,684,8641,654,6848,
//1030 data points where 0 sum was reached around k=700
91971,84841,7108,25538,61927,311,13293,49323,82575,42047,42621,4528,33492,40233,8207,19313,17418,20046,97930,91319,21352,75522,80884,92887,3172,3402,30154,53295,45129,64875,76120,95241,75935,50600,14969,24058,64668,10739,74264,82103,95766,81604,31825,48253,98824,53223,50979,74839,22673,6901,6628,40582,11625,16851,74329,34832,99379,67076,64535,32430,87878,39846,87266,94771,68911,30598,78570,11443,96418,82912,14659,57422,88738,73430,37122,14757,65752,64413,55350,47566,40052,5269,63245,91024,62122,96172,73761,32491,9914,22246,56477,72743,31766,539,8060,51233,94746,38936,82773,4027,49755,75621,27878,36503,63731,96923,93088,71466,7829,91854,96506,5351,9372,792,8237,29526,10003,45061,84050,64869,44551,18686,31130,92931,77843,257,81465,33118,41736,19277,23252,95069,38862,84583,1510,78924,52875,83591,88760,51204,55668,31803,28820,72180,85375,31097,61709,65438,76378,50339,69786,24471,37894,62870,61760,37134,19589,41610,54127,65701,23447,47115,77960,13598,42731,76482,51722,53980,83969,68876,28247,64097,74556,89852,32215,28318,66235,62950,5848,45470,40770,50000,20546,47738,5013,56026,69247,9403,14276,78600,52114,49300,57225,70920,41405,25704,72529,85561,95069,24490,22578,66416,10333,42579,7541,34835,89226,88650,29651,87181,47493,73420,73326,86056,96184,881,3074,34043,12385,62809,32617,30558,47161,95675,18317,95487,1691,30156,70901,86281,29738,59373,94311,11038,62245,98438,48944,35946,67426,98144,37638,39288,90091,2419,74368,5501,53487,4721,45268,92114,77645,92420,55346,24469,10418,80834,72980,57352,54643,47955,28398,59555,4432,64450,94353,18022,7363,88904,18304,75731,28145,77099,37077,51892,77769,78618,58440,76279,93078,66569,40061,11341,95239,42097,34627,455,45190,15006,70919,28975,52242,94947,81103,98508,18289,49883,93925,10329,28593,59948,62807,53107,82485,46257,99603,81315,69200,57179,81100,70139,56208,71697,58216,18287,56682,80797,74856,68581,24932,56111,79553,44985,1078,33601,5052,46698,58454,21591,22216,75724,51901,19814,34312,93688,12404,1472,74226,42104,88751,74560,27770,52677,1257,3921,14543,38065,62154,80166,41952,83753,27875,96367,46870,56989,70061,29349,30417,14600,15638,9381,12672,32427,52193,63465,21644,79884,1788,84165,86538,32588,14481,62895,18922,17814,52043,27770,90651,30220,54177,684,12877,79534,9521,9151,62696,49504,17889,92016,34501,79437,49929,35694,79281,81751,61146,37207,14690,88139,71934,37867,42414,14138,68956,66459,78179,98301,41906,28393,66701,39038,98593,78928,19123,89097,7903,86555,7229,72289,30837,26828,75810,85795,52580,23946,52315,75066,6195,6247,10422,36205,85037,85639,37868,40653,13242,14990,17400,87468,33841,76043,15413,52200,15840,43988,4222,1163,97877,5894,27907,49478,82287,62434,88319,1326,96296,19314,63080,94678,65175,46033,18353,721,50185,87762,48604,70941,57076,15778,83744,24345,72384,93133,60848,51265,34558,58951,16594,45325,19575,41243,4129,36254,47318,68398,85336,10464,81489,49839,17483,40148,36113,1869,68571,90880,26744,26872,80029,40512,50642,85233,39595,61899,73401,33864,32744,45026,35147,62806,66004,75647,32795,25836,22709,46475,18975,89237,63503,37520,62019,72519,66694,78254,11971,26555,38208,51235,82437,73811,30071,60979,42083,59457,17922,53300,69295,14213,79140,14106,93565,39018,98767,12898,19065,99290,55406,96661,81503,17804,17835,45522,36121,15560,90373,29672,82686,95100,85898,30209,39965,18232,96036,83814,67533,31902,91084,43548,81247,34779,24890,45285,10364,29152,94940,1995,28647,63798,74587,51510,61728,52559,95367,41582,56753,92546,45668,73055,76292,80820,71398,87558,10149,25260,95802,56610,94918,65816,83004,32247,89064,94486,43603,91064,9278,44821,43852,46724,55095,8366,4778,36327,75601,71599,3061,64696,56375,58868,1881,13519,7193,3729,55724,98000,686,20422,84697,6823,99729,51581,9345,3230,33531,62041,75483,78380,13008,92322,73680,95761,3407,73779,1497,25348,4410,4715,97954,27151,96981,33027,16691,4754,50716,26714,15603,3877,63828,2177,78364,78663,90410,32799,40001,88635,31521,62240,71126,88550,45596,35836,30578,14734,48055,78423,99670,66613,25034,16271,95578,39832,59491,64164,90110,24612,94666,98316,36945,84526,23957,35914,74261,10148,89869,7362,96525,28747,19389,54348,30954,83866,24346,62858,96355,25336,89159,17438,19877,26213,67260,19395,50133,17429,80909,44168,77546,44149,40791,21306,59121,22933,97532,24283,47625,60143,50324,31150,79093,34412,15694,57816,56400,30645,44351,91535,47481,71120,45186,25358,96844,2731,37108,15691,10876,85188,81006,56378,7416,80928,73845,50342,33962,45379,14001,62637,45,66328,28684,75003,63335,81237,31773,34202,32170,51647,64902,65287,23594,39435,72560,25085,34321,96756,31878,39290,10456,313,58353,87017,45851,46863,88919,38035,94970,67059,21063,13281,91385,94599,5249,34230,96221,35681,18889,64631,49931,51949,23519,37007,59540,76583,70018,97867,98583,94493,13835,55055,56230,57409,48797,81045,97777,38919,4967,75806,36522,29159,64195,58832,51397,5911,47348,72203,31621,61132,32046,47295,81259,92105,61855,46985,8173,15735,16105,14233,36084,27771,77334,39122,50253,25481,17826,63048,64197,80649,172,94257,41669,22848,45634,72586,11604,36415,75842,95214,81968,86722,8491,5522,778,68350,83144,72919,27675,98142,63391,76649,1091,61181,77909,10498,4311,1144,73887,86234,49497,2192,89204,27685,19088,12111,74087,63381,72931,39497,22860,73816,96460,82602,26617,90907,742,77501,54128,6263,58682,81642,54077,13337,55144,73541,30715,98031,19841,26379,51787,48035,81621,81003,63135,71207,857,53082,49846,33006,69020,32600,28809,93781,27697,28789,84895,40154,42393,46255,83968,38531,59098,23078,2388,31081,47343,91678,12450,54226,9212,68542,55477,6778,75148,15625,15970,58963,76847,7532,43793,56065,6579,15151,54887,15814,80796,62039,38595,82848,30052,22450,42599,2606,11555,17245,8693,90166,98322,3856,43958,78906,24069,91181,74155,11157,26701,75147,79735,83698,59368,99053,27406,35721,38162,72535,90580,98451,36614,74207,57638,43118,31493,54616,3525,14593,70458,65804,2371,29952,95822,16967,46585,85324,44495,40046,40188,28571,49601,87926,46314,89084,54871,51785,30464,40750,88002,46775,99857,41941,70369,49355,82416,67822,88126,72305,68090,42573,6664,50620,8171,54154,64323,71018,70255,49214,19102,13961,38126,13767,75255,89885,24285,6784,45907,76710,69512,96761,36343,9178,43610,26232,20416,35417,79808,48812,1442,12738,14060,27780,73339,72251,22224,984,99484,3129,95242,5406,45172,93152,17698,79263,91020,2372,96955,93000,16632,24974,80075,22770,78679,52026,87169,97389,60924,95753,22470,73104,14341,89258,27802,15165,44009,16116,65558,26768,84349,42048,38158,69626,54520,6232,89607,70649,89678,64649,61427,73712,23429,60767,97914,19092,55872,67273,72611,17408,58426,45902,1158,3151,12460,6843,12175,39110,13795,48488,598,88102,62734,30051,85108,83685,4614,16221,89546,22251,33607,22389,28056,97714,97847,69668,14514,25876,47436,98820,80096,38333,73919,10210,53350,38424,71994,95426,16011,63218,46060,88059,54803,17782,4764,89636,75816,20450,71524,51424,66346,38996,51636,65503,35668,16180,35424,16688,71067,19510,20900,81505,44392,88822,66810,54956,22721,4020,55164,7768,53816,29369,97493,22693,50851,53883,40911,91519,8328,3488,89357,265,68837,37347,68925,53993,39617,21956,81340,90625,17603,82990,55479,96397,54300,90079,19013,58286,80248,93752,48825,87804,38548,82925,79145,68161,26215,27595,28166,84134,53883,72828,14699,57729,9756,21219,71888,58735,27888,77657,9862,29308,5713,10369,5132,16637,36379,74924,73424,7622,67815,44654,61976,37575,67544,41394,765,60364,48627,28929,35016,65876,16879,35727,58510,44848,68747,63314,45271,38285,19974,31022,46601,79594,28293,41943,93783,73472,25540,42352,12406,76008,60580,97316,35941,23328,63611,42353,32625,86073,50162,36848,48968,26200,44694,79594,18595,96664,3781,66827,18775,76745,23087,49444,9680,44804,1139,95993,35979,73285,78351,51555,55892,72987,919,6576,58724,74645,55748,15929,5263,9385,31276,81207,26297,13715,36839,23698,80161,78030,22099,23672,18946,52224,63113,52239,62193,69534,6218,9437,72951,26121,99384,89559,22585,41905,50820,17497,62181,95609,11040,48450,4672,19090,74922,25774,27276,59533,82413,90879,54002,1927,80107,48775,3426,13476,8,56421,57833,11826,84730,13248,44960,22347,77712,38664,51496,10446,26342,92613,2537,50518,15298,50077,95138,59074,38391,25995,51757,68041,45079,73503,44795,73578,73714,86823,85676,86652,24529,65036,8034,84053,27255,51289,41191,54733,32906,43556,99492,36724,54294,58745,21879,39742,9760,84910,89882,22040,74779,87799,6733,85094,51541,2561,42018,50703,77647,48273,51943,25317,76893,43156,11204,23775,31223,90188,61128,28268,90392,44615,8268,29091,49155,85684,99848,45164,39604,29565,7404,45835,49245,63540,86092,99420,29373,37648,26577,49013,58660,15547,88827,60902,28092,21057,57412,65051,95942,67104,62021,81178,64070,25825,74358,939,257,2738,85713,59958,34270,45916,48325,86031,68099,13435,96,26509,74600,67943,48462,59415,62719,63542,46788,33966,56201,74797,49645,34663,70653,13993,75123,64971,56299,24290,55613,17019,16469,10228,41530,65684,20980,5657,34485,74200,65094,45106,82980,23313,89679,78589,76463,44692,92419,28224,67769,79223,10358,12046,64508,5664,84873,98898,65128,22810,40006,78824,85890,28783,24195,88541,17804,77888,98005,3163,40585,37812,83540,65091,2368,10153,10123,67755,4906,26772,91241,61110,81443,96255,55859,82616,50686,39787,2824,67188,75051,72755,88297,70263,10886,52465,90967,67740,38048,41855,57897,36010,98311,39800,34970,61919,53605,35960,34556,75179,62847,99582,53397,64903,97681,21363,89165,18184,63832,93788,41877,73907,40862,72219,55696,34322,70520,86524,58044,96077,51867,68264,68773,42545,23291,87772,43135,35583,34601,78101,33345,33146,17877,55345,47317,93056,55095,65630,83641,32938,20800,94894,86836,18949,33463,96351,90210,15136,93306,57484,4970,44591,98902,99319,64390,71429,9453,10288,25967,27543,35135,10277,22490,95493,40130,69518,34173,68057,99926,85945,64916,85557,25386,77470,46018,32982,8214,45880,16878,64891,93415,13796,54925,65390,91184,73091,97527,20617,56281,20972,61849,6063,22834,23121,50132,23168,65229,36501,7732,61662,97901,80938,5449,4324,17177,98702,90374,47638,78220,8025,631,14868,7109,99196,81197,61445,13447,88088,61439,81253,11602,79082,91878,78818,23302,98516,17627,35548,90154,93396,36910,78363,42642,78977,42405,77288,36169,14039,62197,8267,34957,14627,80164,49675,44941,46282,38844,80471,92596,57433,129,85146,35443,63604,8925,2513,16232,75329,65547,12865,65564,40061,37036,2476,18330,67599,77393,27915,21717,39812,67331,39831,91457,82078,86552,84524,33187,78388,93499,31834,51610,48374,9599,62442,58001,79427,25427,10503,79516,447,67385,67312,58505,44860,57318,34878,5784,24556,75323,42995,80230,66273,70899,98707,35982,28881,77311,13652,56025,39197,99183,60848,78778,12611,55865,32717,30740,98863,41178,61939,7501,15980,32740,35422,11145,47642,95924,26318,2168,1409,43150,76210,83168,74739,80236,49366,56557,97408,79222,60346,18130,37973,34753,29330,6319,90128,81966,20999,93645,6116,58983,64143,64283,28777,72347,88576,31292,32646,60348,32715,27780,57558,71558,62441,13266,17590,44902,70808,47002,35206,76340,61551,21960,23786,90629,38566,37365,31258,7344,89896,68418,32815,88560,1462,20635,98402,15486,87818,87832,36665,62422,19706,36102,59077,73157,88408,62456,94333,70225,5552,24127,22076,11571,53084,13717,63016,69793,18229,59206,5782,57379,89482,91636,32108,36221,34750,75596,45483,42593,34406,5029,18772,57724,84624,39233,86922,77772,79550,46276,51367,81047,3458,57390,26930,10328,59690,9573,24463,84201,38847,23755,11125,21505,48379,61368,21367,57139,84731,655,79392,8614,574,39349,68808,91270,98549,33131,12817,91148,95310,90468,67711,69769,23201,54384,94404,70804,1585,8258,46100,64826,22817,47627,40503,7975,62228,2710,71282,16290,73345,9938,11950,57669,2767,47660,95717,4262,87231,23613,3132,5303,35138,15020,78273,55237,44423,43500,60718,8549,35593,34035,87978,94337,43352,66386,24438,4444,99250,15905,7122,12769,9057,38677,31656,91946,36395,6718,17485,60118,28635,16146,62946,37135,31640,81708,28255,76004,97645,67914,42872,64750,74295,54037,32473,77988,34434,87058,97884,71812,99386,59090,54572,94425,88778,26076,81966,74456,58589,75413,95109,35565,47090,62943,21610,15980,51616,83195,68938,77421,95579,43972,33475,80601,75068,71451,46172,53235,10594,229,14538,15779,3535,18792,74958,52376,39650,78112,30435,94447,23664,53290,14297,36564,57415,42577,62973,60178,54379,23471,86489,62865,63912,73881,27530,83805,88588,50955,79897,13092,25428,21032,55713,59695,44564,91593,38587,88118,40408,3458,52854,31271,82336,65550,75963,6379,94470,95815,57597,93874,49015,2623,78929,3542,96688,85728,93123,63370,93923,76560,81218,49577,40321,83277,75622,40868,20647,46612,53977,72510,27750,35045,37285,27490,11203,31005,67918,16305,66145,40471,68625,23301,84038,20189,36517,98936,39855,69536,39650,69495,4564,22889,54425,52056,70852,2908,50233,74545,75399,24963,43468,48828,24219,51897,9941,2651,36619,17298,43353,88712,82010,31618,63076,36919,84544,79994,83762,88616,25184,25358,90200,24379,1593,4588,99452,57834,24314,49316,55537,35259,88574,38989,461,34754,4295,90437,72079,88103,58951,94197,6143,66948,69639,73061,73794,18524,36274,36020,97571,22422,94371,55337,70290,23719,53295,26714,29618,44899,26361,6649,77299,54330,43848,2630,5462,43877,47405,17621,71980,48556,69593,10270,46270,96883,82522,97178,9692,38037,86735,12843,18547,90318,86142,36102,54725,67779,33960,70407,88337,68586,7416,32155,69809,76500,70306,61447,80616,15737,87763,27999,73308,44107,53341,12657,44004,61361,90129,11435,7085,83481,28657,86953,13639,35151,52055,98933,42279,8555,15128,98397,55654,21379,32574,97085,84159,36599,94225,9609,46510,85077,48658,2029,91337,35690,89749,37788,70610,10988,40687,76143,75614,96737,58393,53231,6038,72579,98172,68388,22202,75978,76105,62423,49984,89596,86908,52089,45258,42748,81535,12974,16453,27924,36973,55034,86639,71309,45107,17565,24195,39749,79635,82267,38262,24281,66895,22641,63707,26083,77311,11325,69871,66911,58230,47237,86189,86797,39368,87093,27872,48833,92246,72298,38752,73197,52211,18861,26297,48056,20065,65745,77341,96440,93208,79154,17581,91398,79828,59172,11980,31520,93006,90766,66947,25505,25662,51023,65611,68719,63162,26980,62111,33503,1486,79944,60541,97670,31407,32663,2175,36753,80740,98049,11417,70007,45707,20439,23306,81475,18876,79912,99032,38020,70540,25705,47679,54785,65712,60704,17881,63622,70029,11128,12249,58989,86318,73313,81666,73660,35613,13506,18101,38049,72504,24305,72834,59070,86071,40740,89168,78606,20938,68498,4058,52276,17717,47543,69021,7429,70137,18260,99292,15612,36939,26462,98679,74311,89545,57406,22151,89872,75116,77927,708,45972,38956,82904,59329,66204,82768,20511,72448,62524,87094,72666,36882,35918,2861,47804,24993,4741,22743,21102,23629,94258,42562,64222,95958,20392,86930,90823,49038,70593,4626,44967,17031,30479,24644,445,39185,42883,78137,45683,17756,54827,56304,89580,68790,85188,79520,52051,50933,47318,99246,22670,38591,17499,28045,35897,10805,97972,59912,69796,47591,87583,71044,9546,60276,47273,54530,64316,78565,55598,85509,68537,15822,53298,6913,78820,34806,23461,284,67945,80858,85813,9375,38770,83815,85870,58022,85829,52337,86324,79505,83821,11453,90476,36126,3399,87985,3568,71839,500,4500,88954,76271,29306,7336,56562,80863,6939,11635,25426,90038,43766,11098,1442,14002,76769,11714,51146,37615,88888,44783,94305,70233,19140,88140,54117,69619,93691,63860,64113,38961,85274,83126,16814,59257,35115,47597,69520,86248,82673,58660,20880,41290,12433,47215,34085,89592,19558,40045,76312,14199,74436,86458,70215,91487,71433,842,52231,56386,50702,87931,25493,45389,77371,50364,58718,16481,47618,59975,34313,10748,46240,47191,67951,40516,61168,65880,38343,23913,43060,82008,62035,75518,40273,83224,93027,21877,77435,88307,71632,1290,632,50128,9736,88768,37080,45712,4640,24781,93644,30882,33452,46119,42674,37702,89906,3306,6481,4953,12024,59540,87372,48001,2808,86196,55303,4471,18788,16408,59537,96648,92719,33499,6722,48955,33862,37879,13396,51670,43929,3939,48079,7524,30503,53480,5113,51939,94524,21433,17288,99724,16560,84499,11638,87050,60679,30010,88386,23716,28451,49719,22809,4109,25706,42582,55513,37422,47324,48847,53170,43576,84234,70617,40713,83624,15968,10641,63638,32104,65516,91641,5415,11173,5659,26470,28816,5998,33061,37595,42178,89808,43363,5269,23750,61805,51709,85293,88466,97116,4958,53628,55383,7265,38854,41885,40104,76385,44247,11543,30538,2550,62201,6462,67803,58608,73861,24575,92339,66125,11831,69331,66295,92341,93284,77231,44467,36331,47688,61093,39930,11186,17004,50702,88419,87123,4120,75947,73915,56934,53118,9829,22476,31866,85915,78365,50359,87479,80483,43982,24880,11468,69964,23984,20071,95228,63841,65270,25149,25133,58416,90652,74823,57118,96185,80077,2593,71310,39144,50045,38367,99790,79818,24103,43742,15546,60521,37955,28865,40358,85080,82134,23407,26816,91597,55285,34872,40824,81081,96452,96514,19764,63241,8287,7009,94878,93697,19786,50498,812,16033,1477,5958,72682,24719,2862,24640,75466,72096,62615,59825,51657,88987,31905,7150,1124,89274,93786,93492,12603,77640,93879,50029,38429,46416,14540,60096,99854,53701,21337,14776,23149,88425,87612,14118,73275,95677,39736,3861,38363,43532,86976,15608,57283,51214,31728,86864,3893,27587,60312,59186,75516,88981,54955,51563,28305,65703,39850,31114,26250,3584,46705,77618,3806,65896,68972,86255,44699,33004,84586,3957,78670,42706,9693,82971,71501,40885,48798,242,43439,36694,26933,51184,3285,42256,95213,33537,49816,36308,90626,951,40183,83542,11529,28352,57885,13323,48035,12880,7877,91954,93393,52484,52896,23149,50824,21749,49257,22391,26697,86114,57421,78006,81506,93889,24402,6114,92508,14331,62855,31552,80177,74401,2284,59442,18156,14048,16802,43234,64081,10157,79349,27857,18695,43181,37814,81532,19176,12963,37409,62303,51145,82240,82460,24582,65136,51998,26422,3929,13113,49884,43757,5622,55423,97518,40118,86731,92631,27205,90926,96293,52068,23709,65996,26947,96154,90337,7128,61897,4087,1991,23993,72488,86899,79374,11324,65148,88418,54336,12508,45647,16415,24710,13391,49148,95397,52338,72425,10023,68818,69355,49133,6885,23866,96638,15108,4489,91949,27222,5337,23033,81313,53666,77066,51356,802,88481,73212,23401,9683,58821,34532,78650,75983,37081,45008,45518,28358,73269,62559,78852,33061,22244,40088,55471,93262,69180,69656,21966,99573,56305,78275,32777,50891,53474,49154,20607,2148,90109,39213,57031,23313,61937,36049,86539,44626,86424,72189,79772,35617,2010,10973,83124,61716,63266,76741,70735,42465,12545,50465,55141,56235,27373,11852,54391,62949,35903,11676,42076,94570,98170,21346,17823,34684,94320,5241,17876,22221,86826,60898,6228,79471,82826,96582,25270,22077,78881,45064,40919,62442,72087,63729,34985,31142,15176,70720,52435,18755,39650,40171,90443,81261,36559,81823,67630,78532,56197,16870,77809,5654,18834,96386,3089,20120,95531,2744,78053,81987,16283,43645,86248,80595,11559,18234,59452,81379,53882,15148,5439,15665,98296,52359,40524,34081,
//+8000 rand ... cut to about 3000 to fit stackoverflow posting limits
};
//numofa: size of data set
int numofa=sizeof(a)/sizeof(int);
//Sort in increasing order. Used by slow algo to be not nearly as slow.
int sortfunc (const void *a, const void *b) {
return (*(int *)a - *(int *)b);
}
// Given 3 adjacent k values. Re-calculates the middle k value where (changing only this middle k value) the sum of the error on its left and error on its right is minimized (ie, ke[left] and ke[middle] are minimized).
int minimize_error_3(int *k, int *kai, int64_t *ke, const int left, const int middle, const int right) {
int64_t minerr=-1;
int64_t tmperr;
int64_t l_e=0, e=0; //not necessary to save errors by parts in general but
long minidx=kai[left]+1;
//printf ("%d %d %d %d ",k[middle], left, middle, right);
for (int i=kai[left]; i<kai[right]; i++) {
tmperr=0;
for (int j=kai[left]+1; j<i; j++) { //int j=kai[left]
tmperr+=a[j]-a[kai[left]];
}
e=tmperr;
for (int j=i+1; j<kai[right]; j++) {
tmperr+=a[j]-a[i];
}
if (minerr==-1)
minerr=tmperr;
if (tmperr<minerr) {
minerr=tmperr;
minidx=i;
l_e=e;
}
}
ke[left]=l_e;
ke[middle]=minerr>-1?minerr-l_e:0;
kai[middle]=minidx;
k[middle]=a[minidx];
//printf ("%d %d %d.%d ",ke[left], ke[middle], k[middle], minidx);
return 0;
}
int evenstartitercycles (int numofk) {
char *str, *str2;
int i, idx, err;
int done, moved;
int *k, *kai, *k_old;
int64_t *ke;
qsort(a,numofa,sizeof(int),sortfunc);
k=(int *) calloc(numofk,sizeof(int));
kai=(int *) malloc(numofk*sizeof(int));
k_old=(int *) malloc(numofk*sizeof(int));
ke=(int64_t *) calloc(numofk,sizeof(int64_t));
k[0]=a[0];
kai[0]=0;
k_old[0]=k[0];
for (int i=1; i<numofk-1; i++) {
k[i]=a[(numofa*i)/(numofk-1)];
kai[i]=(numofa*i)/(numofk-1);
k_old[i]=k[i];
}
k[numofk-1]=a[numofa-1];
kai[numofk-1]=numofa-1;
k_old[numofk-1]=k[numofk-1];
ke[numofk-1]=0; //already 0
i=0;
moved=1;
int at_end=0;
int min_x=k[2]-k[1]; //1 doing infin loop 0 ok but violates rule
int min_xi=1; //1 doing infin loop 0 ok but violates rule
int max_e=-1;
int max_ei=0;
while (!at_end || moved) {
if (i==0) {
moved=0;
at_end=0;
min_x=k[2]-k[1]; //?
min_xi=1;
max_e=-1; //?
max_ei=0;
}
minimize_error_3(k, kai, ke, i, i+1, i+2);
if (i>0) {
if (k[i+1]-k[i]<min_x) {
min_x=k[i+1]-k[i];
min_xi=i;
}
if (ke[i]>max_e && i>min_xi+1) {
max_e=ke[i];
max_ei=i;
}
//later do going to left version
}
if (k[i+1]!=k_old[i+1]) {
moved=1;
k_old[i+1]=k[i+1];
}
if (i<numofk-3) {
i++;
} else {
if (ke[i+1]>max_e && i+1>min_xi) {
max_e=ke[i+1];
max_ei=i+1;
}
//here see if can gain from shifting around some
if (max_ei>min_xi+3 && .1*ke[min_xi]*(k[min_xi]-k[min_xi-1])<max_e) { //fix the +3 to make it unnec??? .3??
//printf("1:%d %d %d %d ",min_x,min_xi,max_e,max_ei);
moved=1;
for (int i=min_xi; i<max_ei; i++) {
k[i]=a[kai[i+1]];
kai[i]=kai[i+1];
}
k[max_ei]=a[++kai[max_ei]];
}
i=0;
at_end=1;
}
}
err=0;
for (int i=0; i<numofk; i++)
err+=ke[i];
str=(char *) calloc(numofk,20);
for (int i=0; i<numofk; i++)
sprintf (str+strlen(str),"%d,",k[i]);
str2=(char *) calloc(numofk,20);
for (int i=0; i<numofk; i++)
sprintf (str2+strlen(str2),"%d,",(int)ke[i]);
printf ("\nevenstartitercycles(%d): The mininum error was %d, found at, k={%s} with error parts={%s} ",numofk,err,str,str2);
free(str);
free(str2);
return 0;
}
int main (int x, char **y) {
int t; //to track unique num of data if want this feature
int kmax;
qsort(a,numofa,sizeof(int),sortfunc);
t=1;
for (int i=1; i<numofa; i++)
if (a[i]!=a[i-1]) {
t++;
}
kmax=t; //t is value where we can reach 0 err sum for first time
kmax=numofa; //this will give many cases of 0 sum error for data sets that have many repeated data points.
for (int i=3; i<=kmax; i++) {
evenstartitercycles(i);
}
return 0;
}
答案 5 :(得分：0)
此 类似于一维k-medians clustering。
我之前建议的DP不起作用;我认为我们需要一个从（n'，k'，i）到D 1 ≤...≤D n'的最优解的表，其中k'代表最棒的是我。考虑到D的界限，运行时间大约为n 2 k且具有非常大的常数，因此您应该调整人们用于 k -means。
答案 6 :(得分：0)
您可能希望使用特定的距离函数来查看Ward's method。
答案 7 :(得分：0)
相关问题
最新问题
- 我写了这段代码，但我无法理解我的错误
- 我无法从一个代码实例的列表中删除 None 值，但我可以在另一个实例中。为什么它适用于一个细分市场而不适用于另一个细分市场？
- 是否有可能使 loadstring 不可能等于打印？卢阿
- java中的random.expovariate()
- Appscript 通过会议在 Google 日历中发送电子邮件和创建活动
- 为什么我的 Onclick 箭头功能在 React 中不起作用？
- 在此代码中是否有使用“this”的替代方法？
- 在 SQL Server 和 PostgreSQL 上查询，我如何从第一个表获得第二个表的可视化
- 每千个数字得到
- 更新了城市边界 KML 文件的来源？