Compare sorting algorithms' performance
You are encouraged to solve this task according to the task description, using any language you may know.
Measure a relative performance of sorting algorithms implementations.
Plot execution time vs. input sequence length dependencies for various implementation of sorting algorithm and different input sequence types (example figures).
Consider three type of input sequences:
- ones: sequence of all 1's. Example: {1, 1, 1, 1, 1}
- range: ascending sequence, i.e. already sorted. Example: {1, 2, 3, 10, 15}
- shuffled range: sequence with elements randomly distributed. Example: {5, 3, 9, 6, 8}
Consider at least two different sorting functions (different algorithms or/and different implementation of the same algorithm). For example, consider Bubble Sort, Insertion sort, Quicksort or/and implementations of Quicksort with different pivot selection mechanisms. Where possible, use existing implementations.
Preliminary subtask:
- Bubble Sort, Insertion sort, Quicksort, Radix sort, Shell sort
- Query Performance
- Write float arrays to a text file
- Plot x, y arrays
- Polynomial Fitting
General steps:
- Define sorting routines to be considered.
- Define appropriate sequence generators and write timings.
- Plot timings.
- What conclusions about relative performance of the sorting routines could be made based on the plots?
Python
Examples of sorting routines
def builtinsort(x): x.sort()
def partition(seq, pivot): low, middle, up = [], [], [] for x in seq: if x < pivot: low.append(x) elif x == pivot: middle.append(x) else: up.append(x) return low, middle, up import random def qsortranpart(seq): size = len(seq) if size < 2: return seq low, middle, up = partition(seq, seq[random.randrange(size)]) return qsortranpart(low) + middle + qsortranpart(up)
Sequence generators
def ones(n): return [1]*n
def reversedrange(n): x = range(n) x.reverse() return x
def shuffledrange(n): x = range(n) random.shuffle(x) return x
Write timings
def write_timings(npoints=10, maxN=10**4, sort_functions=(builtinsort,insertion_sort, qsort), sequence_creators = (ones, range, shuffledrange)): Ns = range(2, maxN, maxN//npoints) for sort in sort_functions: for make_seq in sequence_creators: Ts = map(lambda n: usec(sort, (make_seq(n),)), Ns) writedat('%s-%s-%d-%d.xy' % (sort.__name__, make_seq.__name__, len(Ns), max(Ns)), Ns, Ts)
Where writedat() is defined in the Write float arrays to a text file, usec() - Query Performance, insertion_sort() - Insertion sort, qsort - Quicksort subtasks, correspondingly.
Plot timings
import operator import numpy, pylab def plotdd(dictplotdict): """See ``plot_timings()`` below.""" symbols = ('o', '^', 'v', '<', '>', 's', '+', 'x', 'D', 'd', '1', '2', '3', '4', 'h', 'H', 'p', '|', '_') colors = map(None, 'bgrcmyk') # split string on distinct characters for npoints, plotdict in dictplotdict.iteritems(): for ttle, lst in plotdict.iteritems(): pylab.hold(False) for i, (label, polynom, x, y) in enumerate(sorted(lst,key=operator.itemgetter(0))): pylab.plot(x, y, colors[i % len(colors)] + symbols[i % len(symbols)], label='%s %s' % (polynom, label)) pylab.hold(True) y = numpy.polyval(polynom, x) pylab.plot(x, y, colors[i % len(colors)], label= '_nolegend_') pylab.legend(loc='upper left') pylab.xlabel(polynom.variable) pylab.ylabel('log2( time in microseconds )') pylab.title(ttle, verticalalignment='bottom') figname = '_%(npoints)03d%(ttle)s' % vars() pylab.savefig(figname+'.png') pylab.savefig(figname+'.pdf') print figname
See Plot x, y arrays and Polynomial Fitting subtasks for a basic usage of pylab.plot() and numpy.polyfit().
import collections, itertools, glob, re import numpy def plot_timings(): makedict = lambda: collections.defaultdict(lambda: collections.defaultdict(list)) df = makedict() ds = makedict() # populate plot dictionaries for filename in glob.glob('*.xy'): m = re.match(r'([^-]+)-([^-]+)-(\d+)-(\d+)\.xy', filename) print filename assert m, filename funcname, seqname, npoints, maxN = m.groups() npoints, maxN = int(npoints), int(maxN) a = numpy.fromiter(itertools.imap(float, open(filename).read().split()), dtype='f') Ns = a[::2] # sequences lengths Ts = a[1::2] # corresponding times assert len(Ns) == len(Ts) == npoints assert max(Ns) <= maxN # logsafe = numpy.logical_and(Ns>0, Ts>0) Ts = numpy.log2(Ts[logsafe]) Ns = numpy.log2(Ns[logsafe]) coeffs = numpy.polyfit(Ns, Ts, deg=1) poly = numpy.poly1d(coeffs, variable='log2(N)') # df[npoints][funcname].append((seqname, poly, Ns, Ts)) ds[npoints][seqname].append((funcname, poly, Ns, Ts)) # actual plotting plotdd(df) plotdd(ds) # see ``plotdd()`` above
Figures: log2( time in microseconds ) vs. log2( sequence length )
sort_functions = [ builtinsort, # see implementation above insertion_sort, # see Insertion sort insertion_sort_lowb, # insertion_sort, where sequential search is replaced # by lower_bound() function qsort, # see Quicksort qsortranlc, # qsort with randomly choosen pivot # and the filtering via list comprehension qsortranpart, # qsortranlc with filtering via partition function qsortranpartis, # qsortranpart, where for a small input sequence lengths ] # insertion_sort is called if __name__=="__main__": import sys sys.setrecursionlimit(10000) write_timings(npoints=100, maxN=1024, # 1 <= N <= 2**10 an input sequence length sort_functions=sort_functions, sequence_creators = (ones, range, shuffledrange)) plot_timings()
Executing above script we get belowed figures.
ones
ones.png (143KiB)
builtinsort - O(N) insertion_sort - O(N) qsort - O(N**2) qsortranpart - O(N)
range
range.png (145KiB)
builtinsort - O(N) insertion_sort - O(N) qsort - O(N**2) qsortranpart - O(N*log(N))
shuffled range
shuffledrange.png (152KiB)
builtinsort - O(N) insertion_sort - O(N**4) ??? qsort - O(N*log(N)) qsortranpart - O(N) ???