Minimierung der Fehlersumme der repräsentativen ganzen Zahlen

Question

Bei n ganzen Zahlen zwischen [0,10000] als D ₁ ,D ₂ ...,D _n , wobei es Duplikate geben kann und n sehr groß sein kann:

Ich möchte k verschiedene repräsentative ganze Zahlen (zum Beispiel k=5) zwischen [0,10000] als R ₁ ,R ₂ ,...,R _k , so dass die Summe der Fehler aller repräsentativen ganzen Zahlen minimiert wird.

Der Fehler einer repräsentativen ganzen Zahl wird im Folgenden definiert:

Angenommen, wir haben k repräsentative ganze Zahlen in aufsteigender Reihenfolge als {R ₁ ,R ₂ ...,R _k }, der Fehler von R _i ist:

und ich möchte die Summe der Fehler der k repräsentativen ganzen Zahlen minimieren:

Wie kann dies effizient durchgeführt werden?

EDIT1: Die kleinste der k repräsentativen ganzen Zahlen muss die kleinste Zahl in {D ₁ ,D ₂ ...,D _n }

EDIT2: Die größte der k repräsentativen ganzen Zahlen muss die größte Zahl in {D ₁ ,D ₂ ...,D _n } plus 1. Wenn zum Beispiel die größte Zahl in {D ₁ ,D ₂ ...,D _n } 9787 ist, dann ist R _k ist 9788.

EDIT3: Ein konkretes Beispiel finden Sie hier:

D={1,3,3,7,8,14,14,14,30} und wenn k=5 und R als {1,6,10,17,31} gewählt wird, dann ist die Summe der Fehler :

Summe der Fehler=(1-1)+(3-1)*2+(7-6)+(8-6)+(14-10)*3+(30-17)=32

Dies liegt daran, dass 1<=1,3,3<6 , 6<=7,8<10, 10<=14,14,14<17, 17<=30<31

Answer 1

Obwohl es Ihnen mit Hilfe der Gemeinschaft gelungen ist, Ihre Problem in einer mathematisch verständlichen Form zu formulieren, liefern Sie dennoch nicht genügend Informationen, um mir (oder jemand anderem) eine eine endgültige Antwort zu geben (ich hätte dies als Kommentar gepostet, aber aber aus irgendeinem Grund sehe ich nicht, dass die Option "Kommentar hinzufügen" für mich verfügbar ist. zur Verfügung steht). Um eine gute Antwort auf dieses Problem geben zu können, müssen wir wissen die relativen Größen von n und k im Verhältnis zueinander und 10000 (oder das erwartete Maximum der D _i wenn es nicht 10000 ist), und ob es entscheidend ist, dass Sie die genau Minimum (auch wenn diese einen exorbitanten Zeitaufwand für die Berechnung erfordert) oder ob eine auch eine Annäherung in Ordnung wäre (und wenn ja, wie nahe müssen Sie man erreichen muss). Außerdem müssen wir, um zu wissen, welcher Algorithmus in Zeit läuft, müssen wir wissen, welche Art von Hardware, auf der der Algorithmus ausgeführt werden soll (d. h. haben wir m CPU Kerne, auf denen er parallel laufen kann, und wie groß ist m im Verhältnis zu k).

Eine weitere wichtige Information ist, ob dieses Problem nur einmal gelöst wird, oder ob es viele Male gelöst werden muss, aber ein Zusammenhang besteht eine Verbindung zwischen den Verteilungen der ganzen Zahlen D _i von einem Problem zum nächsten (z. B. die ganzen Zahlen D _i sind alle Zufallsstichproben aus einer bestimmten, unveränderlichen Wahrscheinlichkeitsverteilung, oder vielleicht hat jedes nachfolgende Problem als eine immer größer werdende Menge, die aus der Menge des vorherigen Problem plus eine zusätzliche s ganze Zahlen).

Kein vernünftiger Algorithmus für Ihr Problem sollte in einer Zeit laufen, die mehr als linear von n abhängt, da die Erstellung eines Histogramms der n ganzen Zahlen D _i benötigt O(n) Zeit, und die Antwort auf des Optimierungsproblems selbst hängt nur vom Histogramm der der ganzen Zahlen und nicht von ihrer Reihenfolge ab. Kein Algorithmus kann in einer Zeit weniger als O(n) laufen, da dies die Größe der Eingabe des Problems ist.

Eine Brute-Force-Suche über alle Möglichkeiten erfordert (unter der Annahme, dass dass mindestens eine der D _i 0 und eine andere 10000 ist), für kleines k, sagen wir k < 10, ungefähr O(10000 k-2 ) Zeit, wenn also log ₁₀ (n) >> 4(k-2), ist dies der optimale Algorithmus (da in diesem Fall die Zeit für die Brute-Force-Suche unbedeutend ist im Vergleich zu der Zeit zum Lesen der Eingabe). Interessant ist auch die Feststellung, dass, wenn k sehr nahe an 10000 liegt, dann benötigt eine Brute-Force-Suche nur O(10000 10002-k ) (denn wir können stattdessen über die Ganzzahlen, die pas als repräsentative Ganzzahlen verwendet).

Wenn Sie die Definition des Problems mit weiteren Informationen aktualisieren, werde ich werde ich versuchen, meine Antwort im Gegenzug zu ändern.

Answer 2

Jetzt ist die Frage geklärt, wir beobachten die R _i teilen Sie die D _x in k-1 Intervalle, [R ₁ ,R ₂ ), [R ₂ ,R ₃ ), ... [R _k-1 ,R _k ). Jedes D _x genau zu einem dieser Intervalle gehört. Sei q _i ist die Anzahl der D _x in dem Intervall [R _i ,R _i+1 ), und lassen Sie s _i ist die Summe derjenigen D _x . Dann wird jeder Fehler(R _i ) ist die Summe von q _i Begriffe und wertet zu s _i - q _i R _i .

Summiert man dies über alle i, erhält man einen Gesamtfehler von S - sum(q _i R _i ), wobei S die Summe aller D _x . Das Problem besteht also darin, die R _i à maximieren Summe(q _i R _i ). Denken Sie daran, dass jedes q _i ist die Anzahl der Originaldaten, die mindestens so groß wie R _i , aber kleiner als die nächste.

Jedes globale Maximum muss ein lokales Maximum sein; wir stellen uns also vor, dass wir einen der R _i . Wenn R _i est pas einen der ursprünglichen Datenwerte, dann können wir ihn erhöhen, ohne dass sich einer der q _i und unsere Zielfunktion zu verbessern. Eine optimale Lösung hat also jedes R _i (mit Ausnahme des einschränkenden letzten) als einen der Datenwerte. Danach habe ich mich ein wenig in der Mathematik verzettelt, aber es scheint ein vernünftiger Ansatz zu sein, den ursprünglichen R _i als jeder (n/k)-te Datenwert (einfache Perzentile), dann wird iterativ geprüft, ob das Verschieben von R_i auf den vorherigen oder nächsten Wert die Punktzahl verbessert und somit den Fehler verringert. (Die q _i R _i scheint einfacher zu sein, da man die Daten lesen, Wiederholungen zählen und q aktualisieren kann. _i , R _i indem nur ein einziger Daten-/Zählpunkt betrachtet wird. Sie brauchen nur ein Array von 10.000 Datenzählungen zu speichern, egal wie groß die Daten sind).

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 diesem Beispiel könnten wir versuchen, die 3 oder die 8 in eine 7 zu ändern. Wenn wir zum Beispiel die 3 auf 7 erhöhen, dann sehen wir, dass es zwei 3en in den ursprünglichen Daten gibt, also werden die ersten beiden Qs zu 1+2, 3-2 - es stellt sich heraus, dass dies die Summe(QR)) verringert. Ich bin sicher, dass es intelligentere Muster gibt, um zu erkennen, welche Änderungen in der QR-Tabelle machbar sind, aber dies scheint praktikabel.

Answer 3

Wenn die Verteilung annähernd zufällig ist und die Auswahl (n) groß genug ist, verschwendet man im Allgemeinen Zeit mit dem Versuch, für reale Zeitkosten zu optimieren, um abnehmende Verbesserungen in % gegenüber den erwarteten Durchschnittswerten zu erzielen. Die schnellste Durchschnittslösung besteht darin, das untere k-1 an das untere Ende der Intervalle M/(k-1) zu setzen, wobei M die niedrigste obere Grenze - die größte untere Grenze (d. h. M = maximal mögliche Anzahl - 0) und das letzte k bei M+1 liegt. Man bräuchte die Ordnung k (das Beste, was wir mit den Informationen in diesem Problem tun können), um diese Werte herauszufinden. Die Feststellung, was ich gerade getan habe, ist natürlich kein Beweis.

Mein Standpunkt ist folgender. Die obige Diskussion ist eine Vereinfachung, die meiner Meinung nach für eine große Klasse von Mengen sehr praktisch ist. Auf der anderen Seite ist es einfach, jeden möglichen Fehler für alle Permutationen zu berechnen und dann die kleinste auszuwählen. Die dafür benötigte Zeit macht diese Lösung in vielen Fällen unpraktikabel. Die Tatsache, dass der Fragesteller mehr als nur die direkteste und genaueste (unlösbare) Antwort erwartet, lässt vieles offen. Wir können von hier aus bis in alle Ewigkeit versuchen, alle möglichen Eigenschaften entlang des unendlichen Lösungsraums für alle möglichen Permutationen (oder Kombinationen) von n Zahlen und alle k Werte zu quantifizieren.

Answer 4

Die Integrität dieses Programms wurde teilweise durch eine modifizierte Version des Programms bestätigt, die aquí um Daten zu erzeugen, die mit den Ergebnisse unabhängig von @mhum erhalten.

Sie findet die genau minimale(r) Fehlerwert(e) und entsprechende R-Werte für einen gegebenen Datensatz und k-Wert(e).

/************************************************************
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;
}

Einige Beispiele für die erzielten Ergebnisse:

 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}

Die Ausgabe verlangsamte sich auf dem PC zu sehr, bevor sie 100 Zeilen (von 375 möglichen) erreichte.

Answer 5

Dieser Algorithmus liefert zwar keine exakten Antworten, aber er ermöglicht es, sehr große Datensätze viel schneller zu verarbeiten als die memoisierter Algorithmus und mit Ergebnissen, die in der Regel sehr nahe an der genauen Antwort liegen. Der Algorithmus muss noch verbessert werden, um mögliche Endlosschleifen in einigen Fällen zu verhindern, aber das ist kein Problem, da Code hinzugefügt werden kann, um dies zu verhindern. In der Zwischenzeit glänzt der Algorithmus mit Datensätzen, die zu groß sind, um von anderen, im Allgemeinen vorzuziehenden Algorithmen bewältigt zu werden (wie z. B. die zuvor eingereichte memoisierte Algorithmusantwort). Bei fast 10.000 Stichproben aus dem Bereich [1-100000] wird beispielsweise k=500 in Sekunden auf einem alten PC berechnet, während die memo-Version für ein viel kleineres k=90 auf einem viel kleineren Datensatz der Größe 375 über eine Stunde benötigen würde. Für diese Art von zusätzlicher Leistung ist der Verzicht auf die absolut niedrigste Fehlersumme ein sehr geringer Preis, der zu zahlen ist. [Ich habe die Qualität der Ergebnisse nicht abgeleitet, aber alle Vergleiche, die mit Datenwerten gemacht wurden, bei denen memo mithalten konnte, ergaben nicht viel mehr als 10% schlechtere Ergebnisse, wenn überhaupt.]

/************************************************************
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

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69,11714,51146,37615,88888,44783,94305,70233,19140,88140,54117,69619,93691,63860,64113,38961,85274,83126,16814,59257,35115,47597,69520,86248,82673,58660,20880,41290,12433,47215,34085,89592,19558,40045,76312,14199,74436,86458,70215,91487,71433,842,52231,56386,50702,87931,25493,45389,77371,50364,58718,16481,47618,59975,34313,10748,46240,47191,67951,40516,61168,65880,38343,23913,43060,82008,62035,75518,40273,83224,93027,21877,77435,88307,71632,1290,632,50128,9736,88768,37080,45712,4640,24781,93644,30882,33452,46119,42674,37702,89906,3306,6481,4953,12024,59540,87372,48001,2808,86196,55303,4471,18788,16408,59537,96648,92719,33499,6722,48955,33862,37879,13396,51670,43929,3939,48079,7524,30503,53480,5113,51939,94524,21433,17288,99724,16560,84499,11638,87050,60679,30010,88386,23716,28451,49719,22809,4109,25706,42582,55513,37422,47324,48847,53170,43576,84234,70617,40713,83624,15968,10641,63638,32104,65516,91641,5415,11173,5659,26470,28816,5998,33061,37595,42178,89808,43363,5269,23750,61805,51709,85293,88466,97116,4958,53628,55383,7265,38854,41885,40104,76385,44247,11543,30538,2550,62201,6462,67803,58608,73861,24575,92339,66125,11831,69331,66295,92341,93284,77231,44467,36331,47688,61093,39930,11186,17004,50702,88419,87123,4120,75947,73915,56934,53118,9829,22476,31866,85915,78365,50359,87479,80483,43982,24880,11468,69964,23984,20071,95228,63841,65270,25149,25133,58416,90652,74823,57118,96185,80077,2593,71310,39144,50045,38367,99790,79818,24103,43742,15546,60521,37955,28865,40358,85080,82134,23407,26816,91597,55285,34872,40824,81081,96452,96514,19764,63241,8287,7009,94878,93697,19786,50498,812,16033,1477,5958,72682,24719,2862,24640,75466,72096,62615,59825,51657,88987,31905,7150,1124,89274,93786,93492,12603,77640,93879,50029,38429,46416,14540,60096,99854,53701,21337,14776,23149,88425,87612,14118,73275,95677,39736,3861,38363,43532,86976,15608,57283,51214,31728,86864,3893,27587,60312,59186,75516,88981,54955,51563,28305,65703,39850,31114,26250,3584,46705,77618,3806,65896,68972,86255,44699,33004,84586,3957,78670,42706,9693,82971,71501,40885,48798,242,43439,36694,26933,51184,3285,42256,95213,33537,49816,36308,90626,951,40183,83542,11529,28352,57885,13323,48035,12880,7877,91954,93393,52484,52896,23149,50824,21749,49257,22391,26697,86114,57421,78006,81506,93889,24402,6114,92508,14331,62855,31552,80177,74401,2284,59442,18156,14048,16802,43234,64081,10157,79349,27857,18695,43181,37814,81532,19176,12963,37409,62303,51145,82240,82460,24582,65136,51998,26422,3929,13113,49884,43757,5622,55423,97518,40118,86731,92631,27205,90926,96293,52068,23709,65996,26947,96154,90337,7128,61897,4087,1991,23993,72488,86899,79374,11324,65148,88418,54336,12508,45647,16415,24710,13391,49148,95397,52338,72425,10023,68818,69355,49133,6885,23866,96638,15108,4489,91949,27222,5337,23033,81313,53666,77066,51356,802,88481,73212,23401,9683,58821,34532,78650,75983,37081,45008,45518,28358,73269,62559,78852,33061,22244,40088,55471,93262,69180,69656,21966,99573,56305,78275,32777,50891,53474,49154,20607,2148,90109,39213,57031,23313,61937,36049,86539,44626,86424,72189,79772,35617,2010,10973,83124,61716,63266,76741,70735,42465,12545,50465,55141,56235,27373,11852,54391,62949,35903,11676,42076,94570,98170,21346,17823,34684,94320,5241,17876,22221,86826,60898,6228,79471,82826,96582,25270,22077,78881,45064,40919,62442,72087,63729,34985,31142,15176,70720,52435,18755,39650,40171,90443,81261,36559,81823,67630,78532,56197,16870,77809,5654,18834,96386,3089,20120,95531,2744,78053,81987,16283,43645,86248,80595,11559,18234,59452,81379,53882,15148,5439,15665,98296,52359,40524,34081,
  //+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;
}