Minimize the sum of errors of representative integers
Given n integers between [0,10000] as D1,D2...,Dn, where there may be duplicates, and n can be huge:
I want to find k distinct represent
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one first obvious point is that your Rs must be chosen amongst your Ds. (if you choose an R that is any D - 1 (but not another D), you'll improve the answer by incrementing your R by 1)
a second point is that your last R is useless and its error will always be 0
讨论(0) -
The integrity of this program was partly confirmed by a modified version of it used here to produce data that matched results obtained independently by @mhum.
It finds the exact minimal error value(s) and corresponding R values for a given data set and k value(s).
/************************************************************ 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; }Some sample results obtained:
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}The output started slowing down on the PC too much before it reached 100 rows (of the possible 375).
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Now the question is clarified, we observe the Ri divide the Dx into k-1 intervals, [R1,R2), [R2,R3), ... [Rk-1,Rk). Every Dx belongs to exactly one of those intervals. Let qi be the number of Dx in the interval [Ri,Ri+1), and let si be the sum of those Dx. Then each error(Ri) expression is the sum of qi terms and evaluates to si - qiRi.
Summing that over all i, we get a total error of S - sum(qiRi), where S is the sum of all the Dx. So the problem is to choose the Ri to maximize sum(qiRi). Remember each qi is the number of original data at least as large as Ri, but smaller than the next one.
Any global maximum must be a local maximum; so we imagine increasing or decreasing one of the Ri. If Ri is not one of the original data values, then we can increase it without changing any of the qi and improve our target function. So an optimal solution has each Ri (except the limiting last one) as one of the data values. I got a bit bogged down in math after that, but it seems a sensible approach is to pick the initial Ri as every (n/k)th data value (simple percentiles), then iteratively seeing if moving the R_i to the previous or next value improves the score and thus decreases the error. (The qiRi seems easier to work with, since you can read the data and count repetitions and update qi, Ri by only looking at a single data/count point. You only need to store an array of 10,000 data counts, no matter how huge the data).
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)In this example, we could try changing the 3 or the 8 to a 7, For example if we increase the 3 to 7, then we see there are 2 3's in the initial data, so the first two Q's become 1+2, 3-2 - it turns out this decreases sum(QR)). I'm sure there are smarter patterns to detect what changes in the QR table are viable, but this seems workable.
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This algorithm does not produce exact answers, but it allows very large data sets to be handled much faster than the memoized algorithm and with results that tends to be very close to the exact answer. Algorithm still needs improvement/debugging to prevent potential infinite loops in some cases, but that is not a deal-breaker since code can be added to stop that. Meanwhile, the alg excels with data sets too large to be managed by some other generally preferable algorithms (like the memoized alg answer submitted earlier). For example, on nearly 10,000 samples from the range [1-100000], k=500 is calculated in seconds on an old PC where the memo version appeared would take over an hour to do a much smaller k=90 on a much much smaller data set of size 375. For this kind of added performance, not getting the absolute lowest error sum is a very small price to pay. [I have not derived the quality of the results, but all comparisons made on data values where memo could keep up yielded not much over 10% worse, if that.]
/************************************************************ 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 91971,84841,7108,25538,61927,311,13293,49323,82575,42047,42621,4528,33492,40233,8207,19313,17418,20046,97930,91319,21352,75522,80884,92887,3172,3402,30154,53295,45129,64875,76120,95241,75935,50600,14969,24058,64668,10739,74264,82103,95766,81604,31825,48253,98824,53223,50979,74839,22673,6901,6628,40582,11625,16851,74329,34832,99379,67076,64535,32430,87878,39846,87266,94771,68911,30598,78570,11443,96418,82912,14659,57422,88738,73430,37122,14757,65752,64413,55350,47566,40052,5269,63245,91024,62122,96172,73761,32491,9914,22246,56477,72743,31766,539,8060,51233,94746,38936,82773,4027,49755,75621,27878,36503,63731,96923,93088,71466,7829,91854,96506,5351,9372,792,8237,29526,10003,45061,84050,64869,44551,18686,31130,92931,77843,257,81465,33118,41736,19277,23252,95069,38862,84583,1510,78924,52875,83591,88760,51204,55668,31803,28820,72180,85375,31097,61709,65438,76378,50339,69786,24471,37894,62870,61760,37134,19589,41610,54127,65701,23447,47115,77960,13598,42731,76482,51722,53980,83969,68876,28247,64097,74556,89852,32215,28318,66235,62950,5848,45470,40770,50000,20546,47738,5013,56026,69247,9403,14276,78600,52114,49300,57225,70920,41405,25704,72529,85561,95069,24490,22578,66416,10333,42579,7541,34835,89226,88650,29651,87181,47493,73420,73326,86056,96184,881,3074,34043,12385,62809,32617,30558,47161,95675,18317,95487,1691,30156,70901,86281,29738,59373,94311,11038,62245,98438,48944,35946,67426,98144,37638,39288,90091,2419,74368,5501,53487,4721,45268,92114,77645,92420,55346,24469,10418,80834,72980,57352,54643,47955,28398,59555,4432,64450,94353,18022,7363,88904,18304,75731,28145,77099,37077,51892,77769,78618,58440,76279,93078,66569,40061,11341,95239,42097,34627,455,45190,15006,70919,28975,52242,94947,81103,98508,18289,49883,93925,10329,28593,59948,62807,53107,82485,46257,99603,81315,69200,57179,81100,70139,56208,71697,58216,18287,56682,80797,74856,68581,24932,56111,79553,44985,1078,33601,5052,46698,58454,21591,22216,75724,51901,19814,34312,93688,12404,1472,74226,42104,88751,74560,27770,52677,1257,3921,14543,38065,62154,80166,41952,83753,27875,96367,46870,56989,70061,29349,30417,14600,15638,9381,12672,32427,52193,63465,21644,79884,1788,84165,86538,32588,14481,62895,18922,17814,52043,27770,90651,30220,54177,684,12877,79534,9521,9151,62696,49504,17889,92016,34501,79437,49929,35694,79281,81751,61146,37207,14690,88139,71934,37867,42414,14138,68956,66459,78179,98301,41906,28393,66701,39038,98593,78928,19123,89097,7903,86555,7229,72289,30837,26828,75810,85795,52580,23946,52315,75066,6195,6247,10422,36205,85037,85639,37868,40653,13242,14990,17400,87468,33841,76043,15413,52200,15840,43988,4222,1163,97877,5894,27907,49478,82287,62434,88319,1326,96296,19314,63080,94678,65175,46033,18353,721,50185,87762,48604,70941,57076,15778,83744,24345,72384,93133,60848,51265,34558,58951,16594,45325,19575,41243,4129,36254,47318,68398,85336,10464,81489,49839,17483,40148,36113,1869,68571,90880,26744,26872,80029,40512,50642,85233,39595,61899,73401,33864,32744,45026,35147,62806,66004,75647,32795,25836,22709,46475,18975,89237,63503,37520,62019,72519,66694,78254,11971,26555,38208,51235,82437,73811,30071,60979,42083,59457,17922,53300,69295,14213,79140,14106,93565,39018,98767,12898,19065,99290,55406,96661,81503,17804,17835,45522,36121,15560,90373,29672,82686,95100,85898,30209,39965,18232,96036,83814,67533,31902,91084,43548,81247,34779,24890,45285,10364,29152,94940,1995,28647,63798,74587,51510,61728,52559,95367,41582,56753,92546,45668,73055,76292,80820,71398,87558,10149,25260,95802,56610,94918,65816,83004,32247,89064,94486,43603,91064,9278,44821,43852,46724,55095,8366,4778,36327,75601,71599,3061,64696,56375,58868,1881,13519,7193,3729,55724,98000,686,20422,84697,6823,99729,51581,9345,3230,33531,62041,75483,78380,13008,92322,73680,95761,3407,73779,1497,25348,4410,4715,97954,27151,96981,33027,16691,4754,50716,26714,15603,3877,63828,2177,78364,78663,90410,32799,40001,88635,31521,62240,71126,88550,45596,35836,30578,14734,48055,78423,99670,66613,25034,16271,95578,39832,59491,64164,90110,24612,94666,98316,36945,84526,23957,35914,74261,10148,89869,7362,96525,28747,19389,54348,30954,83866,24346,62858,96355,25336,89159,17438,19877,26213,67260,19395,50133,17429,80909,44168,77546,44149,40791,21306,59121,22933,97532,24283,47625,60143,50324,31150,79093,34412,15694,57816,56400,30645,44351,91535,47481,71120,45186,25358,96844,2731,37108,15691,10876,85188,81006,56378,7416,80928,73845,50342,33962,45379,14001,62637,45,66328,28684,75003,63335,81237,31773,34202,32170,51647,64902,65287,23594,39435,72560,25085,34321,96756,31878,39290,10456,313,58353,87017,45851,46863,88919,38035,94970,67059,21063,13281,91385,94599,5249,34230,96221,35681,18889,64631,49931,51949,23519,37007,59540,76583,70018,97867,98583,94493,13835,55055,56230,57409,48797,81045,97777,38919,4967,75806,36522,29159,64195,58832,51397,5911,47348,72203,31621,61132,32046,47295,81259,92105,61855,46985,8173,15735,16105,14233,36084,27771,77334,39122,50253,25481,17826,63048,64197,80649,172,94257,41669,22848,45634,72586,11604,36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//+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; }讨论(0) -
You may want to look into Ward's method using your particular distance function.
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Although with help from the community you have managed to state your problem in a form which is understandable mathematically, you still do not supply enough information to enable me (or anyone else) to give you a definitive answer (I would have posted this as a comment, but for some reason I don't see the "add comment" option is available to me). In order to give a good answer to this problem, we need to know the relative sizes of n and k versus each other and 10000 (or the expected maximum of the Di if it isn't 10000), and whether it is critical that you attain the exact minimum (even if this requires an exorbitant amount of time for the calculation) or if a close approximation would be OK, also (and if so, how close do you need to get). In addition, in order to know what algorithm runs in minimum time, we need to have an understanding about what kind of hardware is going to run the algorithm (i.e., do we have m CPU cores to run on in parallel and what is the size of m relative to k).
Another important piece of information is whether this problem will be solved only once, or it must be solved many times but there exists some connection between the distributions of the integers Di from one problem to the next (e.g., the integers Di are all random samples from a particular, unchanging probability distribution, or perhaps each successive problem has as its input an ever-increasing set which is the set from the previous problem plus an extra s integers).
No reasonable algorithm for your problem should run in time which depends in a way greater than linear in n, since building a histogram of the n integers Di requires O(n) time, and the answer to the optimization problem itself is only dependent on the histogram of the integers and not on their ordering. No algorithm can run in time less than O(n), since that is the size of the input of the problem.
A brute force search over all of the possibilities requires, (assuming that at least one of the Di is 0 and another is 10000), for small k, say k < 10, approximately O(10000k-2) time, so if log10(n) >> 4(k-2), this is the optimal algorithm (since in this case the time for the brute force search is insignificant compared to the time to read the input). It is also interesting to note that if k is very close to 10000, then a brute force search only requires O(1000010002-k) (because we can search instead over the integers which are not used as representative integers).
If you update the definition of the problem with more information, I will attempt to edit my answer in turn.
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