I have a dataframe of 100,000 records so i tried to do a Parallel processing using the joblib library which works fine with my code below, but my question is can i try the same code with 'apply' and 'lambda' function which seems like very close to my original code with minimum change instead of using the for loop like in my code. Please help
Original Code - Without parallel processing:
df['b1'] = df.text1.apply(lambda x: removeNumbers(x))
With parallel processing:
For the purpose of applying the Joblib's parallel processing i converted to for loop below
df['b1'] = Parallel(n_jobs = -1)(delayed(removeNumbers)(x) for x in df.text1)
I have the following code which I use when I have a large dataframe and want to use parallel computing:
import numpy as np
import pandas as pd
import time
from multiprocessing import Pool, cpu_count
from functools import partial
# Wrapper to time functions (not needed for parallel computing but to show that it works...)
def time_function(func):
def decorated_func(*args, **kwargs):
start = time.perf_counter_ns()
ret = func(*args, **kwargs)
stop = time.perf_counter_ns()
temp = []
temp += [type(a) for a in args]
f = lambda x: f"{x}={type(kwargs[x])}"
temp += list(map(f, kwargs))
print(f"Function {func.__name__}{*temp,}: time elapsed: {(stop - start)*1e-6:.3f} [ms]")
return ret
return decorated_func
# This function splits the data and calls the functions.
def parallelize(data, func, num_of_processes=cpu_count()):
data_split = np.array_split(data, num_of_processes)
p = pool(num_of_processes)
data = pd.concat(p.map(func, data_split))
p.close()
p.join()
return data
# This function is only used for pandas (otherwise the parallelize function was enough)
def run_on_subset(func, data_subset):
return data_subset.apply(func, axis=1)
# This function is maybe redundant, but it keeps the code readable.
def parallelize_on_rows(data, func, num_of_processes=8):
return parallelize(data, partial(run_on_subset, func), num_of_processes)
def sum_two_columns(row):
time.sleep(0.1) # Make it a time consuming function
return row[0] + row[1]
#time_function
def oridnary_apply(df):
return df.apply(sum_two_columns, axis=1)
#time_function
def parallel_apply(df):
return parallelize_on_rows(df, sum_two_columns)
if __name__ == '__main__':
array = np.ones((100, 3))
df = pd.DataFrame(array)
print(f"cpu_count: {cpu_count()}")
oridnary_apply(df)
parallel_apply(df)
print('done')
>>> cpu_count: 12
>>> Function oridnary_apply(<class 'pandas.core.frame.DataFrame'>,): time elapsed: 10860.275 [ms]
>>> Function parallel_apply(<class 'pandas.core.frame.DataFrame'>,): time elapsed: 2170.105 [ms]
>>> done
EDIT:
When a lot of values in your rows are equal then it is also possible to cache the your function. If it is a complex function, that means the execution time is relative long, this is also a way to speed up the apply function for your DataFrame.
https://docs.python.org/3/library/functools.html#functools.lru_cache
Related
Looking for best practice for nested parallel jobs. I couldn't nest dask delayed or futures so I mixed both to get it to work. Is this not recommended? Is there better way to do this? Example:
import dask
from dask.distributed import Client
import random
import time
client = Client()
def rndSeries(x):
time.sleep(1)
return random.sample(range(1, 50), x)
def sqNum(x):
time.sleep(1)
return x**2
def subProcess(li):
results=[]
for i in li:
r = dask.delayed(sqNum)(i)
results.append(r)
return dask.compute(sum(results))[0]
futures=[]
for i in range(10):
x = client.submit(rndSeries,random.randrange(5,10,1))
y = client.submit(subProcess, x)
futures.append(y)
client.gather(futures)
Consider modification of your script to have a deterministic workflow. If you start with 1 worker, you will see that the process completes in 20 seconds (as expected, 2 processes of 1 second + 6 processes of 3 seconds). If you have 2 workers, the execution time will drop to 10 seconds.
import dask
from dask.distributed import Client, LocalCluster
import time
import numpy as np
cluster = LocalCluster(n_workers=1, threads_per_worker=1)
client = Client(cluster)
# if inside jupyter split the code below into a new cell
# to see accurate timing
%%time
def rndSeries(x):
time.sleep(1)
return np.random.rand()
def sqNum(x):
time.sleep(3)
return 1
def subProcess(li):
results=[]
li = [1,2,3]
for i in li:
r = dask.delayed(sqNum)(i)
results.append(r)
return dask.compute(sum(results))[0]
futures=[]
for i in range(2):
x = client.submit(rndSeries, np.random.rand())
y = client.submit(subProcess, x)
futures.append(y)
client.gather(futures)
What happens if you have 6 workers? Execution time is now 4 seconds (the lowest possible for this task), so it seems that the only drawback of dask.compute() inside a future version is that it forces the results of delayeds to be on a single worker. This is probably OK in many cases, however, if the combined resource requirements of all delayed tasks exceed resources of a single worker, then the best way to proceed is to submit tasks from tasks: https://distributed.dask.org/en/latest/task-launch.html
I've been having an issue where pool.map leaves processes even after pool.terminate is called. I've looked for solutions but they all seems to have some other issue like recursively calling the map function or another process that interferes with the multiprocessing.
So my code imports 2 NETCDF files and processes the data in them using different calculations. These take up a lot of time (several 6400x6400 arrays) so I tried to multi process my code. The multiprocessing works and the first time I run my code it takes 2.5 minutes (down from 8), but every time my code finishes running the memory usage by Spyder never goes back down and it leaves extra python processes in the Windows task manager. My code looks like this:
import numpy as np
import netCDF4
import math
from math import sin, cos
import logging
from multiprocessing.pool import Pool
import time
start=time.time()
format = "%(asctime)s: %(message)s"
logging.basicConfig(format=format, level=logging.INFO, datefmt="%H:%M:%S")
logging.info("Here we go!")
path = "DATAPATH"
geopath = "DATAPATH"
f = netCDF4.Dataset(path)
f.set_auto_maskandscale(False)
f2 = netCDF4.Dataset(geopath)
i5lut=f.groups['observation_data'].variables['I05_brightness_temperature_lut'][:]
i4lut=f.groups['observation_data'].variables['I05_brightness_temperature_lut'][:]
I5= f.groups['observation_data'].variables['I05'][:]
I4= f.groups['observation_data'].variables['I04'][:]
I5=i5lut[I5]
I4=i4lut[I4]
I4Quality= f.groups['observation_data'].variables['I04_quality_flags'][:]
I5Quality= f.groups['observation_data'].variables['I05_quality_flags'][:]
I3= f.groups['observation_data'].variables['I03']
I2= f.groups['observation_data'].variables['I02']
I1= f.groups['observation_data'].variables['I01']
I1.set_auto_scale(True)
I2.set_auto_scale(True)
I3.set_auto_scale(True)
I1=I1[:]
I2=I2[:]
I3=I3[:]
lats = f2.groups['geolocation_data'].variables['latitude'][:]
lons = f2.groups['geolocation_data'].variables['longitude'][:]
solarZen = f2.groups['geolocation_data'].variables['solar_zenith'][:]
sensorZen= solarZen = f2.groups['geolocation_data'].variables['sensor_zenith'][:]
solarAz = f2.groups['geolocation_data'].variables['solar_azimuth'][:]
sensorAz= solarZen = f2.groups['geolocation_data'].variables['sensor_azimuth'][:]
def kernMe(i, j, band):
if i<250 or j<250:
return -1
else:
return np.mean(band[i-250:i+250:1,j-250:j+250:1])
def thread_me(arr):
start1=arr[0]
end1=arr[1]
start2=arr[2]
end2=arr[3]
logging.info("Im starting at: %d to %d, %d to %d" %(start1, end1, start2, end2))
points = []
avg = np.mean(I4)
for i in range(start1,end1):
for j in range (start2,end2):
if solarZen[i,j]>=90:
if not (I5[i,j]<265 and I4[i,j]<295):#
if I4[i,j]>320 and I4Quality[i,j]==0:
points.append([lons[i,j],lats[i,j], 1])
elif I4[i,j]>300 and I5[i,j]-I4[i,j]>10:
points.append([lons[i,j],lats[i,j], 2])
elif I4[i,j] == 367 and I4Quality ==9:
points.append([lons[i,j],lats[i,j, 3]])
else:
if not ((I1[i,j]>I2[i,j]>I3[i,j]) or (I5[i,j]<265 or (I1[i,j]+I2[i,j]>0.9 and I5[i,j]<295) or
(I1[i,j]+I2[i,j]>0.7 and I5[i,j]<285))):
if not (I1[i,j]+I2[i,j] > 0.6 and I5[i,j]<285 and I3[i,j]>0.3 and I3[i,j]>I2[i,j] and I2[i,j]>0.25 and I4[i,j]<=335):
thetaG= (cos(sensorZen[i,j]*(math.pi/180))*cos(solarZen[i,j]*(math.pi/180)))-(sin(sensorZen[i,j]*(math.pi/180))*sin(solarZen[i,j]*(math.pi/180))*cos(sensorAz[i,j]*(math.pi/180)))
thetaG= math.acos(thetaG)*(180/math.pi)
if not ((thetaG<15 and I1[i,j]+I2[i,j]>0.35) or (thetaG<25 and I1[i,j]+I2[i,j]>0.4)):
if math.floor(I4[i,j])==367 and I4Quality[i,j]==9 and I5>290 and I5Quality[i,j]==0 and (I1[i,j]+I2[i,j])>0.7:
points.append([lons[i,j],lats[i,j, 4]])
elif I4[i,j]-I5[i,j]>25 or True:
kern = kernMe(i, j, I4)
if kern!=-1 or True:
BT4M = max(325, kern)
kern = min(330, BT4M)
if I4[i,j]> kern and I4[i,j]>avg:
points.append([lons[i,j],lats[i,j], 5])
return points
if __name__ == '__main__':
#Separate the arrays into 1616*1600 chunks for multi processing
#TODO: make this automatic, not hardcoded
arg=[[0,1616,0,1600],[0,1616,1600,3200],[0,1616,3200,4800],[0,1616,4800,6400],
[1616,3232,0,1600],[1616,3232,1600,3200],[1616,3232,3200,4800],[1616,3232,4800,6400],
[3232,4848,0,1600],[3232,4848,1600,3200],[3232,4848,3200,4800],[3232,4848,4800,6400],
[4848,6464,0,1600],[4848,6464,1600,3200],[4848,6464,3200,4800],[4848,6464,4800,6400]]
print(arg)
p=Pool(processes = 4)
output= p.map(thread_me, arg)
p.close()
p.join()
print(output)
f.close()
f2.close()
logging.info("Aaaand we're here!")
print(str((time.time()-start)/60))
p.terminate()
I use both p.close and p. terminate because I thought it would help (it doesn't). All of my code runs and produces the expected output but I have to manually end the lingering processes using the task manager. Any ideas as to
what's causing this?
I think I put all the relevant information here, if you need more I'll edit with the requests
Thanks in advance.
Ok, here is my problem: I have a nested for loop in my program which runs on a single core. Since the program spend over 99% of run time in this nested for loop I would like to parallelize it. Right now I have to wait 9 days for the computation to finish. I tried to implement a parallel for loop by using the multiprocessing library. But I only find very basic examples and can not transfer them to my problem. Here are the nested loops with random data:
import numpy as np
dist_n = 100
nrm = np.linspace(1,10,dist_n)
data_Y = 11000
data_I = 90000
I = np.random.randn(data_I, 1000)
Y = np.random.randn(data_Y, 1000)
dist = np.zeros((data_I, dist_n)
for t in range(data_Y):
for i in range(data_I):
d = np.abs(I[i] - Y[t])
for p in range(dist_n):
dist[i,p] = np.sum(d**nrm[p])/nrm[p]
print(dist)
Please give me some advise how to make it parallel.
There's a small overhead with initiating a process (50ms+ depending on data size) so it's generally best to MP the largest block of code possible. From your comment it sounds like each loop of t is independent so we should be free to parallelize this.
When python creates a new process you get a copy of the main process so you have available all your global data but when each process writes the data, it writes to it's own local copy. This means dist[i,p] won't be available to the main process unless you explicitly pass it back with a return (which will have some overhead). In your situation, if each process writes dist[i,p] to a file then you should be fine, just don't try to write to the same file unless you implement some type of mutex access control.
#!/usr/bin/python
import time
import multiprocessing as mp
import numpy as np
data_Y = 11 #11000
data_I = 90 #90000
dist_n = 100
nrm = np.linspace(1,10,dist_n)
I = np.random.randn(data_I, 1000)
Y = np.random.randn(data_Y, 1000)
dist = np.zeros((data_I, dist_n))
def worker(t):
st = time.time()
for i in range(data_I):
d = np.abs(I[i] - Y[t])
for p in range(dist_n):
dist[i,p] = np.sum(d**nrm[p])/nrm[p]
# Here - each worker opens a different file and writes to it
print 'Worker time %4.3f mS' % (1000.*(time.time()-st))
if 1: # single threaded
st = time.time()
for x in map(worker, range(data_Y)):
pass
print 'Single-process total time is %4.3f seconds' % (time.time()-st)
print
if 1: # multi-threaded
pool = mp.Pool(28) # try 2X num procs and inc/dec until cpu maxed
st = time.time()
for x in pool.imap_unordered(worker, range(data_Y)):
pass
print 'Multiprocess total time is %4.3f seconds' % (time.time()-st)
print
If you re-increase the size of data_Y/data_I again, the speed-up should increase up to the theoretical limit.
This question already has an answer here:
How can I improve CPU utilization when using the multiprocessing module?
(1 answer)
Closed 8 years ago.
Using multiprocessing, I tried to parallelize a function but I have no performance improvement:
from MMTK import *
from MMTK.Trajectory import Trajectory, TrajectoryOutput, SnapshotGenerator
from MMTK.Proteins import Protein, PeptideChain
import numpy as np
filename = 'traj_prot_nojump.nc'
trajectory = Trajectory(None, filename)
def calpha_2dmap_mult(trajectory = trajectory, t = range(0,len(trajectory))):
dist = []
universe = trajectory.universe
proteins = universe.objectList(Protein)
chain = proteins[0][0]
traj = trajectory[t]
dt = 1000 # calculate distance every 1000 steps
for n, step in enumerate(traj):
if n % dt == 0:
universe.setConfiguration(step['configuration'])
for i in np.arange(len(chain)-1):
for j in np.arange(len(chain)-1):
dist.append(universe.distance(chain[i].peptide.C_alpha,
chain[j].peptide.C_alpha))
return(dist)
c0 = time.time()
dist1 = calpha_2dmap_mult(trajectory, range(0,11001))
c1 = time.time() - c0
print(c1)
# Multiprocessing
from multiprocessing import Pool, cpu_count
pool = Pool(processes=4)
c0 = time.time()
dist_pool = [pool.apply(calpha_2dmap_mult, args=(trajectory, t,)) for t in
[range(0,2001), range(3000,5001), range(6000,8001),
range(9000,11001)]]
c1 = time.time() - c0
print(c1)
The time spent to calculate the distances is the 'same' without (70.1s) or with multiprocessing (70.2s)! I was maybe not expecting an improvement of a factor 4 but I was at least expecting some improvements!
Is someone knows what I did wrong?
Pool.apply is a blocking operation:
[Pool.apply is the] equivalent of the apply() built-in function. It blocks until the result is ready, so apply_async() is better suited for performing work in parallel ..
In this case Pool.map is likely more appropriate for collecting the results; the map itself blocks but the sequence elements / transformations are processed in parallel.
It addition to using partial application (or manual realization of such), also consider expanding the data itself. It's the same cat in a different skin.
data = ((trajectory, r) for r in [range(0,2001), ..])
result = pool.map(.., data)
This can in turn be expanded:
def apply_data(d):
return calpha_2dmap_mult(*d)
result = pool.map(apply_data, data)
The function (or simple argument-expanded proxy of such of such) will need to be written to accept a single argument but all the data is now mapped as a single unit.
Suppose I have a large in memory numpy array, I have a function func that takes in this giant array as input (together with some other parameters). func with different parameters can be run in parallel. For example:
def func(arr, param):
# do stuff to arr, param
# build array arr
pool = Pool(processes = 6)
results = [pool.apply_async(func, [arr, param]) for param in all_params]
output = [res.get() for res in results]
If I use multiprocessing library, then that giant array will be copied for multiple times into different processes.
Is there a way to let different processes share the same array? This array object is read-only and will never be modified.
What's more complicated, if arr is not an array, but an arbitrary python object, is there a way to share it?
[EDITED]
I read the answer but I am still a bit confused. Since fork() is copy-on-write, we should not invoke any additional cost when spawning new processes in python multiprocessing library. But the following code suggests there is a huge overhead:
from multiprocessing import Pool, Manager
import numpy as np;
import time
def f(arr):
return len(arr)
t = time.time()
arr = np.arange(10000000)
print "construct array = ", time.time() - t;
pool = Pool(processes = 6)
t = time.time()
res = pool.apply_async(f, [arr,])
res.get()
print "multiprocessing overhead = ", time.time() - t;
output (and by the way, the cost increases as the size of the array increases, so I suspect there is still overhead related to memory copying):
construct array = 0.0178790092468
multiprocessing overhead = 0.252444982529
Why is there such huge overhead, if we didn't copy the array? And what part does the shared memory save me?
If you use an operating system that uses copy-on-write fork() semantics (like any common unix), then as long as you never alter your data structure it will be available to all child processes without taking up additional memory. You will not have to do anything special (except make absolutely sure you don't alter the object).
The most efficient thing you can do for your problem would be to pack your array into an efficient array structure (using numpy or array), place that in shared memory, wrap it with multiprocessing.Array, and pass that to your functions. This answer shows how to do that.
If you want a writeable shared object, then you will need to wrap it with some kind of synchronization or locking. multiprocessing provides two methods of doing this: one using shared memory (suitable for simple values, arrays, or ctypes) or a Manager proxy, where one process holds the memory and a manager arbitrates access to it from other processes (even over a network).
The Manager approach can be used with arbitrary Python objects, but will be slower than the equivalent using shared memory because the objects need to be serialized/deserialized and sent between processes.
There are a wealth of parallel processing libraries and approaches available in Python. multiprocessing is an excellent and well rounded library, but if you have special needs perhaps one of the other approaches may be better.
I run into the same problem and wrote a little shared-memory utility class to work around it.
I'm using multiprocessing.RawArray (lockfree), and also the access to the arrays is not synchronized at all (lockfree), be careful not to shoot your own feet.
With the solution I get speedups by a factor of approx 3 on a quad-core i7.
Here's the code:
Feel free to use and improve it, and please report back any bugs.
'''
Created on 14.05.2013
#author: martin
'''
import multiprocessing
import ctypes
import numpy as np
class SharedNumpyMemManagerError(Exception):
pass
'''
Singleton Pattern
'''
class SharedNumpyMemManager:
_initSize = 1024
_instance = None
def __new__(cls, *args, **kwargs):
if not cls._instance:
cls._instance = super(SharedNumpyMemManager, cls).__new__(
cls, *args, **kwargs)
return cls._instance
def __init__(self):
self.lock = multiprocessing.Lock()
self.cur = 0
self.cnt = 0
self.shared_arrays = [None] * SharedNumpyMemManager._initSize
def __createArray(self, dimensions, ctype=ctypes.c_double):
self.lock.acquire()
# double size if necessary
if (self.cnt >= len(self.shared_arrays)):
self.shared_arrays = self.shared_arrays + [None] * len(self.shared_arrays)
# next handle
self.__getNextFreeHdl()
# create array in shared memory segment
shared_array_base = multiprocessing.RawArray(ctype, np.prod(dimensions))
# convert to numpy array vie ctypeslib
self.shared_arrays[self.cur] = np.ctypeslib.as_array(shared_array_base)
# do a reshape for correct dimensions
# Returns a masked array containing the same data, but with a new shape.
# The result is a view on the original array
self.shared_arrays[self.cur] = self.shared_arrays[self.cnt].reshape(dimensions)
# update cnt
self.cnt += 1
self.lock.release()
# return handle to the shared memory numpy array
return self.cur
def __getNextFreeHdl(self):
orgCur = self.cur
while self.shared_arrays[self.cur] is not None:
self.cur = (self.cur + 1) % len(self.shared_arrays)
if orgCur == self.cur:
raise SharedNumpyMemManagerError('Max Number of Shared Numpy Arrays Exceeded!')
def __freeArray(self, hdl):
self.lock.acquire()
# set reference to None
if self.shared_arrays[hdl] is not None: # consider multiple calls to free
self.shared_arrays[hdl] = None
self.cnt -= 1
self.lock.release()
def __getArray(self, i):
return self.shared_arrays[i]
#staticmethod
def getInstance():
if not SharedNumpyMemManager._instance:
SharedNumpyMemManager._instance = SharedNumpyMemManager()
return SharedNumpyMemManager._instance
#staticmethod
def createArray(*args, **kwargs):
return SharedNumpyMemManager.getInstance().__createArray(*args, **kwargs)
#staticmethod
def getArray(*args, **kwargs):
return SharedNumpyMemManager.getInstance().__getArray(*args, **kwargs)
#staticmethod
def freeArray(*args, **kwargs):
return SharedNumpyMemManager.getInstance().__freeArray(*args, **kwargs)
# Init Singleton on module load
SharedNumpyMemManager.getInstance()
if __name__ == '__main__':
import timeit
N_PROC = 8
INNER_LOOP = 10000
N = 1000
def propagate(t):
i, shm_hdl, evidence = t
a = SharedNumpyMemManager.getArray(shm_hdl)
for j in range(INNER_LOOP):
a[i] = i
class Parallel_Dummy_PF:
def __init__(self, N):
self.N = N
self.arrayHdl = SharedNumpyMemManager.createArray(self.N, ctype=ctypes.c_double)
self.pool = multiprocessing.Pool(processes=N_PROC)
def update_par(self, evidence):
self.pool.map(propagate, zip(range(self.N), [self.arrayHdl] * self.N, [evidence] * self.N))
def update_seq(self, evidence):
for i in range(self.N):
propagate((i, self.arrayHdl, evidence))
def getArray(self):
return SharedNumpyMemManager.getArray(self.arrayHdl)
def parallelExec():
pf = Parallel_Dummy_PF(N)
print(pf.getArray())
pf.update_par(5)
print(pf.getArray())
def sequentialExec():
pf = Parallel_Dummy_PF(N)
print(pf.getArray())
pf.update_seq(5)
print(pf.getArray())
t1 = timeit.Timer("sequentialExec()", "from __main__ import sequentialExec")
t2 = timeit.Timer("parallelExec()", "from __main__ import parallelExec")
print("Sequential: ", t1.timeit(number=1))
print("Parallel: ", t2.timeit(number=1))
This is the intended use case for Ray, which is a library for parallel and distributed Python. Under the hood, it serializes objects using the Apache Arrow data layout (which is a zero-copy format) and stores them in a shared-memory object store so they can be accessed by multiple processes without creating copies.
The code would look like the following.
import numpy as np
import ray
ray.init()
#ray.remote
def func(array, param):
# Do stuff.
return 1
array = np.ones(10**6)
# Store the array in the shared memory object store once
# so it is not copied multiple times.
array_id = ray.put(array)
result_ids = [func.remote(array_id, i) for i in range(4)]
output = ray.get(result_ids)
If you don't call ray.put then the array will still be stored in shared memory, but that will be done once per invocation of func, which is not what you want.
Note that this will work not only for arrays but also for objects that contain arrays, e.g., dictionaries mapping ints to arrays as below.
You can compare the performance of serialization in Ray versus pickle by running the following in IPython.
import numpy as np
import pickle
import ray
ray.init()
x = {i: np.ones(10**7) for i in range(20)}
# Time Ray.
%time x_id = ray.put(x) # 2.4s
%time new_x = ray.get(x_id) # 0.00073s
# Time pickle.
%time serialized = pickle.dumps(x) # 2.6s
%time deserialized = pickle.loads(serialized) # 1.9s
Serialization with Ray is only slightly faster than pickle, but deserialization is 1000x faster because of the use of shared memory (this number will of course depend on the object).
See the Ray documentation. You can read more about fast serialization using Ray and Arrow. Note I'm one of the Ray developers.
Like Robert Nishihara mentioned, Apache Arrow makes this easy, specifically with the Plasma in-memory object store, which is what Ray is built on.
I made brain-plasma specifically for this reason - fast loading and reloading of big objects in a Flask app. It's a shared-memory object namespace for Apache Arrow-serializable objects, including pickle'd bytestrings generated by pickle.dumps(...).
The key difference with Apache Ray and Plasma is that it keeps track of object IDs for you. Any processes or threads or programs that are running on locally can share the variables' values by calling the name from any Brain object.
$ pip install brain-plasma
$ plasma_store -m 10000000 -s /tmp/plasma
from brain_plasma import Brain
brain = Brain(path='/tmp/plasma/')
brain['a'] = [1]*10000
brain['a']
# >>> [1,1,1,1,...]