C语言 如何,何时以及在python中进行矢量化



好的,所以这基本上是我之前的一个问题的后续。我有一些浮点二进制格式的二进制数据。使用C,该过程,但是使用atof()会损失一些精度。我试着通过论坛和其他地方寻找,但我的问题没有解决。因此,我转向了python。快乐啊!程序工作得很好,但是与c相比非常慢。我查找了python的优化,它指向Cython和Weave,但我有一些疑问。如果您遵循我的代码,我很困惑在哪里应用优化的C代码,因为我从numpy对象读取。我的问题是,是否可以在Cython中使用numpy函数读取数据,如果可以,请提供一个小示例。

C代码使用PolSARpro的头文件,和libbmp来创建。bmp文件

作为一个说明,我张贴我的代码。天知道,为了让这些公式起作用,我经历了很多。这样,其他有需要的人也可以给出他们的想法和建议:)

C代码(工作,但atof()失去精度,因此输出的长度稍微偏离)


#include <stdlib.h>
#include <stdio.h>
#include <string.h>
#include <math.h>
#include <polSARpro/bmpfile.c>
#include <polSARpro/graphics.c>
#include <polSARpro/matrix.c>
#include <polSARpro/processing.c>
#include <polSARpro/util.c>
#define METAL_THRESHOLD 5.000000
#define POLARIZATION_FRACTION_THRESHOLD 0.900000
#define PI 3.14159265
#define FOURTHPI PI/4
#define deg2rad PI/180
#define rad2deg 180./PI
/*double PI = 3.14159265;
double FOURTHPI = PI / 4;
double deg2rad = PI / 180;
double rad2deg = 180.0 / PI;*/
FILE *L1,*PF,*SPF;
FILE *txt;
FILE *finalLocations;
long i=0,loop_end;
int lig,col;
float l1,pf,spf;
long pos;
int Nlig,Ncol;
float *bufferout;
float *bufferin_L1,*bufferin_L2;
float valueL1,valuePF,xx;
float sizeGridX, sizeGridY, startX, startY;
float posX,posY;
int ZONE;
char Heading[10];
char setZone[15];
int p[4][2];
int degree, minute, second;
void UTM2LL(int ReferenceEllipsoid, double UTMNorthing, double UTMEasting, char* UTMZone, double *Lat, double *Long)
{
//converts UTM coords to lat/long.  Equations from USGS Bulletin 1532
//East Longitudes are positive, West longitudes are negative.
//North latitudes are positive, South latitudes are negative
//Lat and Long are in decimal degrees.
    //Written by Chuck Gantz- chuck.gantz@globalstar.com
    double k0 = 0.9996;
    double a = 6378137;
    double eccSquared =  0.00669438;
    double eccPrimeSquared;
    double e1 = (1-sqrt(1-eccSquared))/(1+sqrt(1-eccSquared));
    double N1, T1, C1, R1, D, M;
    double LongOrigin;
    double mu, phi1, phi1Rad;
    double x, y;
    int ZoneNumber;
    char* ZoneLetter;
    int NorthernHemisphere; //1 for northern hemispher, 0 for southern
    x = UTMEasting - 500000.0; //remove 500,000 meter offset for longitude
    y = UTMNorthing;
    ZoneNumber = strtoul(UTMZone, &ZoneLetter, 10);
    if((*ZoneLetter - 'N') >= 0)
        NorthernHemisphere = 1;//point is in northern hemisphere
    else
    {
        NorthernHemisphere = 0;//point is in southern hemisphere
        y -= 10000000.0;//remove 10,000,000 meter offset used for southern hemisphere
    }
    LongOrigin = (ZoneNumber - 1)*6 - 180 + 3;  //+3 puts origin in middle of zone
    eccPrimeSquared = (eccSquared)/(1-eccSquared);
    M = y / k0;
    mu = M/(a*(1-eccSquared/4-3*eccSquared*eccSquared/64-5*eccSquared*eccSquared*eccSquared/256));
    phi1Rad = mu    + (3*e1/2-27*e1*e1*e1/32)*sin(2*mu)
                + (21*e1*e1/16-55*e1*e1*e1*e1/32)*sin(4*mu)
                +(151*e1*e1*e1/96)*sin(6*mu);
    phi1 = phi1Rad*rad2deg;
    N1 = a/sqrt(1-eccSquared*sin(phi1Rad)*sin(phi1Rad));
    T1 = tan(phi1Rad)*tan(phi1Rad);
    C1 = eccPrimeSquared*cos(phi1Rad)*cos(phi1Rad);
    R1 = a*(1-eccSquared)/pow(1-eccSquared*sin(phi1Rad)*sin(phi1Rad), 1.5);
    D = x/(N1*k0);
    *Lat = phi1Rad - (N1*tan(phi1Rad)/R1)*(D*D/2-(5+3*T1+10*C1-4*C1*C1-9*eccPrimeSquared)*D*D*D*D/24
                    +(61+90*T1+298*C1+45*T1*T1-252*eccPrimeSquared-3*C1*C1)*D*D*D*D*D*D/720);
    *Lat = *Lat * rad2deg;
    *Long = (D-(1+2*T1+C1)*D*D*D/6+(5-2*C1+28*T1-3*C1*C1+8*eccPrimeSquared+24*T1*T1)
                    *D*D*D*D*D/120)/cos(phi1Rad);
    *Long = LongOrigin + *Long * rad2deg;
}
void convertToDegree(float decimal)
{
    int negative = decimal < 0;
    decimal = abs(decimal);
    minute = (decimal * 3600/ 60);
    second = fmodf((decimal * 3600),60);
    degree = minute / 60;
    minute = minute % 60;
    if (negative)
    {
        if (degree > 0)
            degree = -degree;
        else if (minute > 0)
            minute = -minute;
        else
            second = -second;
    }
}
void readConfig(int *Row, int *Col)
{
    char tmp[70];
    int i=0;
    FILE *fp = fopen("config.txt","r");
    if(fp == NULL)
    {
        perror("Config.txt");
        exit(1);
    }
    while(!feof(fp))
    {
        fgets(tmp,70,fp);
        if (i==1)
            *Row = atoi(tmp);
        if(i==4)
            *Col = atoi(tmp);
        i++;
    }
    fclose(fp);
}
void readHDR(float *gridX,float *gridY,float *startXPos,float *startYPos)
{
    FILE *fp = fopen("PF.bin.hdr","r");
    int i=0;
    char tmp[255];
    char junk[255];
    memset(tmp,0X00,sizeof(tmp));
    memset(junk,0X00,sizeof(junk));
    if(fp==NULL)
    {
        perror("Please locate or create PF.bin.hdr");
        exit(0);
    }
    while(!feof(fp))
    {
        if(i==13)
            break;
        fgets(tmp,255,fp);
        i++;
    }
    fclose(fp);
    strcpy(junk,strtok(tmp,","));
    strtok(NULL,",");
    strtok(NULL,",");
    strcpy(tmp,strtok(NULL,","));
    //puts(tmp);
    *startXPos = atof(tmp);
    strcpy(tmp,strtok(NULL,","));
    //puts(tmp);
    *startYPos = atof(tmp);
    strcpy(tmp,strtok(NULL,","));
    //puts(tmp);
    *gridX = atof(tmp);
    strcpy(tmp,strtok(NULL,","));
    //puts(tmp);
    *gridY = atof(tmp);
    strcpy(tmp,strtok(NULL,","));
    ZONE = atoi(tmp);
    strcpy(tmp,strtok(NULL,","));
    strcpy(Heading,tmp);
}
int main()
{
    bmpfile_t *bmp;
    double Lat;
    double Long;
    int i;
    rgb_pixel_t pixelMetal = {128, 64, 0, 0};
    rgb_pixel_t pixelOthers = {128, 64, 0, 0};
    readConfig(&Nlig,&Ncol);
    readHDR(&sizeGridX,&sizeGridY,&startX,&startY);
    //startX = startX - (double) 0.012000;
    //startY = startY + (double)0.111000;
    printf("Enter the rectangle's top-left and bottom-right region of interest points as: x yn");
    for(i=0;i<2;i++)
    {
        printf("Enter point %d::t",i+1);
        scanf("%d %d",&p[i][0], &p[i][1]);
    }
    printf("Grid Size(X,Y)::( %f,%f ), Start Positions(X,Y)::( %f, %f ), ZONE::%d, Heading:: %snn",sizeGridX,sizeGridY,startX,startY,ZONE,Heading);
    pixelMetal.red = 255;
    pixelMetal.blue = 010;
    pixelMetal.green = 010;
    pixelOthers.red = 8;
    pixelOthers.blue = 8;
    pixelOthers.green = 8;
    L1 = fopen("l1.bin","rb");
    PF =fopen("PF.bin","rb");
    SPF = fopen("SPF_L1.bin","wb");
    //txt = fopen("locations(UTM).txt","w");
    finalLocations = fopen("locationsROI.txt","w");
    if(L1==NULL || PF==NULL || SPF==NULL || finalLocations == NULL)
    {
        perror("Error in opening files!");
        return -1;
    }
    fseek(L1,0,SEEK_END);
    pos = ftell(L1);
    loop_end = pos;
    printf("L1.bin contains::t%ld elementsn",pos);
    fseek(PF,0,SEEK_END);
    pos = ftell(PF);
    printf("PF.bin contains::t%ld elementsn",pos);
    fseek(L1,0,SEEK_SET);
    fseek(PF,0,SEEK_SET);
    bmp = bmp_create(Ncol,Nlig,8); //width * height
    bufferin_L1 = vector_float(Ncol);
    bufferin_L2 = vector_float(Ncol);
    bufferout = vector_float(Ncol);
    printf("Resources Allocated. Beginning...n");
    for (lig = 0; lig < Nlig; lig++) /* rows */
    {
        if (lig%(int)(Nlig/20) == 0)
        {
            printf("%fr", 100. * lig / (Nlig - 1));
            fflush(stdout);
        }
        fread(&bufferin_L1[0], sizeof(float), Ncol, L1);
        fread(&bufferin_L2[0], sizeof(float), Ncol, PF);
        for (col = 0; col < Ncol; col++) /* columns */
        {
            valueL1 = bufferin_L1[col];
            valuePF = bufferin_L2[col];
            if(valueL1 >= METAL_THRESHOLD && valuePF >= POLARIZATION_FRACTION_THRESHOLD)
            {
                if(col >= p[0][0] && col <= p[1][0] && lig >= p[0][1] && lig <= p[1][1])
                {
                    xx = fabs(valueL1 + valuePF);
                    bmp_set_pixel(bmp,col,lig,pixelMetal);
                    posX = startX + (sizeGridX * col);
                    posY = startY - (sizeGridY * lig);
                    //fprintf(txt,"%f %f %d %sn",posX,posY,ZONE,Heading);
                    sprintf(setZone,"%d",ZONE);
                    if(strstr(Heading,"Nor")!=NULL)
                        strcat(setZone,"N");
                    else
                        strcat(setZone,"S");
                    UTM2LL(23, posY, posX, setZone, &Lat, &Long); // 23 for WGS-84
                    convertToDegree(Lat);
                    //fprintf(finalLocations,"UTM:: %.2fE %.2fN , Decimal: %f %f , Degree: %d %d %d, ",posX,posY,Lat,Long,degree,minute,second);
                    //fprintf(finalLocations,"%.2fE,%.2fN,%f,%f ,%d,%d,%d,",posX,posY,Lat,Long,degree,minute,second);
                    fprintf(finalLocations,"%.2f,%.2f,%f,%f ,%d,%d,%d,",posX,posY,Lat,Long,degree,minute,second);
                    convertToDegree(Long);
                    fprintf(finalLocations,"%d,%d,%dn",degree,minute,second);
                }
                else
                {
                    xx = fabs(valueL1) ;
                    bmp_set_pixel(bmp,col,lig,pixelOthers);
                }
            }
            else
            {
                xx = fabs(valueL1) ;
                bmp_set_pixel(bmp,col,lig,pixelOthers);
            }
            bufferout[col] = xx;
        }
        fwrite(&bufferout[0], sizeof(float), Ncol, SPF);
    }
    free_vector_float(bufferout);
    fclose(L1);
    fclose(PF);
    fclose(SPF);
    //fclose(txt);
    fclose(finalLocations);
    printf("n----------Writing BMP File!----------n");
    bmp_save(bmp,"SPF_L1(ROI).bmp");
    bmp_destroy(bmp);
    printf("nDone!n");
}

以及Python代码::


# -*- coding: utf-8 -*-
"""
Created on Wed Apr 10 10:29:18 2013
@author: Binayaka
"""
import numpy as Num;
import math;
import array;
class readConfiguration(object):  
    def __init__(self,x):
        self.readConfig(x);
    def readConfig(self,x):
        try:
            crs = open(x,'r');
            srs = open('config.txt','r');
        except IOError:
            print "Files missing!";
        else:
            rows = crs.readlines();                
            values = rows[12].split(',');
            rows = srs.readlines();                
            self.startX = float(values[3]);
            self.startY = float(values[4]);
            self.gridSizeX = float(values[5]);
            self.gridSizeY = float(values[6]);
            self.Zone = int(values[7]);
            self.Hemisphere = values[8];
            self.NRows = int(rows[1].strip());
            self.NCols = int(rows[4].strip());
            self.MetalThreshold = 5.000000;
            self.PFThreshold = 0.900000;
            self.rad2deg = 180/math.pi;
            self.deg2rad = math.pi/180;
            self.FOURTHPI = math.pi/4;
            crs.close();
            srs.close();
    def decdeg2dms(dd):
        negative = dd < 0;
        dd = abs(dd);
        minutes,seconds = divmod(dd*3600,60);
        degrees,minutes = divmod(minutes,60);
        if negative:
            if degrees > 0:
                degrees = -degrees;
            elif minutes > 0:
                    minutes = -minutes;
        else:
            seconds = -seconds;
        return (degrees,minutes,seconds);
    def UTM2LL(self,UTMEasting, UTMNorthing):
        k0 = 0.9996;
        a = 6378137;
        eccSquared =  0.00669438;
        e1 = (1-math.sqrt(1-eccSquared))/(1+math.sqrt(1-eccSquared));
        x = UTMEasting - 500000.0;#remove 500,000 meter offset for longitude
        y = UTMNorthing;
        if self.Hemisphere == "North":
            self.Hemi = 1;
        else:
            self.Hemi = -1;
            y -= 10000000.0;
        LongOrigin = (self.Zone - 1)*6 - 180 + 3;
        eccPrimeSquared = (eccSquared)/(1-eccSquared);
        M = y / k0;
        mu = M/(a*(1-eccSquared/4-3*eccSquared*eccSquared/64-5*eccSquared*eccSquared*eccSquared/256));
        phi1Rad = mu    + (3*e1/2-27*e1*e1*e1/32)*math.sin(2*mu) + (21*e1*e1/16-55*e1*e1*e1*e1/32)*math.sin(4*mu) +(151*e1*e1*e1/96)*math.sin(6*mu);
        #phi1 = phi1Rad*self.rad2deg;
        N1 = a/math.sqrt(1-eccSquared*math.sin(phi1Rad)*math.sin(phi1Rad));
        T1 = math.tan(phi1Rad)*math.tan(phi1Rad);
        C1 = eccPrimeSquared*math.cos(phi1Rad)*math.cos(phi1Rad);
        R1 = a*(1-eccSquared)/pow(1-eccSquared*math.sin(phi1Rad)*math.sin(phi1Rad), 1.5);
        D = x/(N1*k0);
        self.Lat = phi1Rad - (N1*math.tan(phi1Rad)/R1)*(D*D/2-(5+3*T1+10*C1-4*C1*C1-9*eccPrimeSquared)*D*D*D*D/24   +(61+90*T1+298*C1+45*T1*T1-252*eccPrimeSquared-3*C1*C1)*D*D*D*D*D*D/720);
        self.Lat = self.Lat * self.rad2deg;
        self.Long = (D-(1+2*T1+C1)*D*D*D/6+(5-2*C1+28*T1-3*C1*C1+8*eccPrimeSquared+24*T1*T1)*D*D*D*D*D/120)/math.cos(phi1Rad);
        self.Long = LongOrigin + self.Long * self.rad2deg;   
    def printConfiguration(self):        
        """ Just to check whether our reading was correct """
        print "Metal Threshold:t" + str(self.MetalThreshold);        
        print "PF    Threshold:t" + str(self.PFThreshold);
        print "Start         X:t" + str(self.startX);
        print "Start         Y:t" + str(self.startY);
        print "Grid size(X)   :t" + str(self.gridSizeX);
        print "Grid size(Y)   :t" + str(self.gridSizeY);
    def createROIfile(self,ROIFilename):
        firstPoint = raw_input('Enter topLeft point coordt').split();
        secondPoint = raw_input('Enter bottomRight point coordt').split();
        try:
            L1 = open('l1.bin','rb');
            PF = open('PF.bin','rb');
            SPF = open('pySPF_L1.bin','wb');
            targetFilename = open(ROIFilename,'w');
        except IOError:
            print "Files Missing!";
        else:
            L1.seek(0,2);
            elementsL1 = L1.tell();
            L1.seek(0,0);
            PF.seek(0,2);
            elementsPF = PF.tell();
            PF.seek(0,0);
            print "L1.bin containst" + str(elementsL1) + " elements";
            print "PF.bin containst" + str(elementsPF) + " elements";
            binvaluesL1 = array.array('f');
            binvaluesPF = array.array('f');
            binvaluesSPF = array.array('f');            
            for row in range(0,self.NRows):                                
                binvaluesL1.read(L1,self.NCols);
                binvaluesPF.read(PF,self.NCols);
                dataL1 = Num.array(binvaluesL1, dtype=Num.float);
                dataPF = Num.array(binvaluesPF, dtype=Num.float);
                dataSPF = dataL1 + dataPF;
                binvaluesSPF.fromlist(Num.array(dataSPF).tolist());                                
                for col in range(0,self.NCols):                     
                    if(dataL1[col] >= self.MetalThreshold and dataPF[col] >= self.PFThreshold):
                        if(col >= int(firstPoint[0]) and col <= int(secondPoint[0]) and row >= int(firstPoint[1]) and row <= int(secondPoint[1])):
                            posX = self.startX + (self.gridSizeX * col);                         
                            posY = self.startY - (self.gridSizeY * row);
                            self.UTM2LL(posY,posX);
                            tmp1 = self.decdeg2dms(posY);
                            tmp2 = self.decdeg2dms(posX);
                            strTarget = "Decimal Degree:: " + str(posX) + "E " + str(posY) + "N t Lat long:: " + str(tmp1) + " " + str(tmp2) + "n";
                            targetFilename.write(strTarget);
                binvaluesSPF.tofile(SPF);
            L1.close();
            PF.close();
            SPF.close();
            targetFilename.close();
            print "Done!";
dimensions = readConfiguration('PF.bin.hdr');
dimensions.printConfiguration();
dimensions.createROIfile('testPythonROI.txt');

需要优化的是Python代码,因为NRows和NCols的值可以并且确实达到数千的数量级。

一些一般性评论:

  1. 使用python,出于多种原因,最好坚持使用PEP8。Python程序员对可读性特别挑剔,基本上普遍遵守社区编码指南(PEP8)。避免驼峰式大小写,每行不超过80列,去掉分号,并且可以随意忽略这些会降低可读性的指导原则。

  2. 如果您使用numpy,则不需要在这里使用内置的array类型。

  3. 使用投影库。指定你使用的基准面和椭球体,否则坐标(东/北或晚/长)完全没有意义。

  4. 不要用一个大的类来容纳不相关的东西。只有几个函数是没有错的。除非有必要,否则不需要将其放入类中。

  5. 对numpy数组使用矢量化操作。

下面是你的性能瓶颈:

    for row in range(0,self.NRows):                                
        binvaluesL1.read(L1,self.NCols);
        binvaluesPF.read(PF,self.NCols);
        dataL1 = Num.array(binvaluesL1, dtype=Num.float);
        dataPF = Num.array(binvaluesPF, dtype=Num.float);
        dataSPF = dataL1 + dataPF;
        binvaluesSPF.fromlist(Num.array(dataSPF).tolist());                                
        for col in range(0,self.NCols):                     
            if(dataL1[col] >= self.MetalThreshold and dataPF[col] >= self.PFThreshold):
                if(col >= int(firstPoint[0]) and col <= int(secondPoint[0]) and row >= int(firstPoint[1]) and row <= int(secondPoint[1])):
                    posX = self.startX + (self.gridSizeX * col);                         
                    posY = self.startY - (self.gridSizeY * row);
                    self.UTM2LL(posY,posX);
                    tmp1 = self.decdeg2dms(posY);
                    tmp2 = self.decdeg2dms(posX);
                    strTarget = "Decimal Degree:: " + str(posX) + "E " + str(posY) + "N t Lat long:: " + str(tmp1) + " " + str(tmp2) + "n";
                    targetFilename.write(strTarget);
        binvaluesSPF.tofile(SPF);
你最大的问题之一是你读取数据的方式。你不断地把东西作为一个东西读入,然后把它转换成一个列表,然后把它转换成一个numpy数组。完全没有必要绕过所有这些障碍。Numpy会像array一样为你解包二进制浮点数。

只做grid = np.fromfile(yourfile, dtype=np.float32).reshape(ncols, nrows)。(循环外)

之后,您的嵌套循环可以很容易地矢量化,并表示只需几行代码。

我将如何编写您的代码。这可能不会按原样运行,因为我无法用您的数据进行测试。然而,它应该给你一些大致的想法。

import numpy as np
import pyproj
def main():
    config = Config('PF.bin.hdr')
    grid1, grid2 = load_data('l1.bin', 'PF.bin', config.nrows, config.ncols)
    spf = grid1 + grid2
    spf.tofile('pySPF_L1.bin')
    easting_aoi, northing_aoi = subset_data(grid1, grid2, config)
    save_selected_region(easting_aoi, northing_aoi, config.zone, 
                        'testPythonROI.txt')
def load_data(filename1, filename2, nrows, ncols):
    """It would really be good to use more descriptive variable names than "L1"
    and "PF".  I have no idea what L1 and PF are, so I'm just calling them
    grid1 and grid2."""
    grid1 = np.fromfile(filename1, dtype=np.float32).reshape(nrows, ncols)
    grid2 = np.fromfile(filename2, dtype=np.float32).reshape(nrows, ncols)
    return grid1, grid2
def subset_data(grid1, grid2, config):
    """Select points that satisfy some threshold criteria (explain??) and are
    within a user-specified rectangular AOI."""
    northing, easting = np.mgrid[:config.nrows, :config.ncols]
    easting = config.xstart + config.xgridsize * easting
    northing = config.ystart + config.ygridsize * northing
    grids = grid1, grid2, easting, northing
    grid1, grid2, easting, northing = [item[config.user_aoi] for item in grids]
    mask = (grid1 >= config.metal_threshold) & (grid2 >= config.pf_threshold)
    return easting[mask], northing[mask]
def save_selected_region(easting, northing, zone, filename):
    """Convert the given eastings and northings (in UTM zone "zone") to 
    lat/long and save to a tab-delimited-text file."""
    lat, lon = utm2geographic(easting, northing, zone)
    data = np.vstack([easting, northing, lat, lon]).T
    with open(filename, 'w') as outfile:
        outfile.write('EastingtNorthingtLatitudetLongituden')
        np.savetxt(outfile, data, delimiter='t')
def utm2geographic(easting, northing, zone):
    """We need to know which datum/ellipsoid the UTM coords are in as well!!!!
    I'm assuming it's a Clark 1866 ellipsoid, based on the numbers in your
    code..."""
    utm = pyproj.Proj(proj='utm', zone=zone, ellip='clrk66')
    geographic = pyproj.Proj(proj='latlong', ellip='clrk66')
    return pyproj.transform(utm, geographic, easting, northing)
class Config(object):
    """Read and store configuration values for (something?)."""
    config_file = 'config.txt'
    def __init__(self, filename):
        """You should add docstrings to clarify what you're expecting 
        "filename" to contain."""
        with open(filename, 'r') as infile:
            crs_values = list(infile)[12].split(',')
        crs_values = [float(item) for item in crs_values]
        self.xstart, self.ystart = crs_values[3:5]
        self.xgridsize, self.ygridsize = crs_values[5:7]
        self.zone = int(crs_values[7])
        with open(self.config_file, 'r') as infile:
            srs_values = list(infile)
        self.nrows, self.ncols = srs_values[1], srs_values[4]
        # It would be good to explain a bit about these (say, units, etc)
        self.metal_threshold = 5.0
        self.pf_threshold = 0.9
        self.user_aoi = self.read_user_aoi()
    def read_user_aoi(self):
        """Get an area of interest of the grids in pixel coordinates."""
        top_left = raw_input('Enter top left indext')
        bottom_right = raw_input('Enter bottom right indext')
        min_i, min_j = [int(item) for item in top_left.split()]
        max_i, max_j = [int(item) for item in bottom_right.split()]
        return slice(min_i, max_i), slice(min_j, max_j)
if __name__ == '__main__':
    main()

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