1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/examples/dijkstra.py Tue Feb 22 18:58:21 2011 +0100
1.3 @@ -0,0 +1,55 @@
1.4 +"""
1.5 +Name: dijkstra - Dijkstra for the Single Source Shortest Path problem
1.6 +
1.7 +Description: dijkstra applies the Dijkstra algorithm for the Single Source Shortest Path problem.
1.8 +
1.9 +Author: Eugen Sawin <sawine@me73.com>
1.10 +"""
1.11 +
1.12 +def main():
1.13 + args = parse_arguments()
1.14 + s = args.source
1.15 + raw_graph = args.graph
1.16 + graph = [e.strip(' ,(').split(',') for e in raw_graph.split(')')]
1.17 + graph = [(e[0].strip(), e[1].strip(), int(e[2].strip())) for e in graph if len(e) == 3]
1.18 + distances = dijkstra(s, create_graph(graph))
1.19 + for (v, d) in distances.iteritems():
1.20 + print '%s: %i' % (v, d)
1.21 +
1.22 +def create_graph(desc):
1.23 + vertices = set([e[0] for e in desc])
1.24 + vertices = vertices.union([e[1] for e in desc])
1.25 + edges = dict(zip([(v1, v2) for (v1, v2, _) in desc], [c for (_, _, c) in desc]))
1.26 + return (vertices, edges)
1.27 +
1.28 +from heapq import *
1.29 +
1.30 +def dijkstra(s, (vertices, edges)):
1.31 + inf = sum(edges.itervalues()) + 1
1.32 + dist = dict([(v, inf) for v in vertices])
1.33 + dist[s] = 0
1.34 + u = []
1.35 + for (v, d) in dist.iteritems():
1.36 + heappush(u, (d, v))
1.37 + active = vertices
1.38 + while len(u) and len(active):
1.39 + (_, node) = heappop(u)
1.40 + if node in active:
1.41 + active.remove(node)
1.42 + for (v1, v2) in [e for e in edges.iterkeys() if e[0] == node]:
1.43 + c = edges[(v1, v2)]
1.44 + if dist[v2] > dist[v1] + c:
1.45 + dist[v2] = dist[v1] + c
1.46 + heappush(u, (dist[v2], v2))
1.47 + return dist
1.48 +
1.49 +from argparse import ArgumentParser
1.50 +
1.51 +def parse_arguments():
1.52 + parser = ArgumentParser(description='Applies the Dijkstra algorithm for the single source shortest path problem.')
1.53 + parser.add_argument('graph', help='sequence of (u, v, c(u,v))')
1.54 + parser.add_argument('source', help='source vertex')
1.55 + return parser.parse_args()
1.56 +
1.57 +if __name__ == "__main__":
1.58 + main()
2.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
2.2 +++ b/examples/perfect_hashing.py Tue Feb 22 18:58:21 2011 +0100
2.3 @@ -0,0 +1,73 @@
2.4 +"""
2.5 +Name: perfect_hashing - applies the two-level scheme to build a perfect hash function
2.6 +
2.7 +Description: Perfect Hashing builds a perfect hash function.
2.8 +
2.9 +Author: Eugen Sawin <sawine@me73.com>
2.10 +"""
2.11 +
2.12 +def main():
2.13 + args = parse_args()
2.14 + N = args.n
2.15 + k = args.k
2.16 + s = [int(e) for e in args.s.split(',')]
2.17 + perfect_hashing(N, len(s), k, s)
2.18 +
2.19 +def hkx(N, n, k, x):
2.20 + return ((k*x) % N) % n
2.21 +
2.22 +def find_ki(N, mi, Wi):
2.23 + def injective():
2.24 + hx = []
2.25 + for x in Wi:
2.26 + thx = hkx(N, mi, ki, x)
2.27 + if thx in hx:
2.28 + return False
2.29 + else:
2.30 + hx.append(thx)
2.31 + return True
2.32 +
2.33 + ki = 1
2.34 + while not injective():
2.35 + ki += 1
2.36 + return ki
2.37 +
2.38 +def perfect_hashing(N, n, k, xi):
2.39 + hx = {}
2.40 + for x in xi:
2.41 + hx[(k, x)] = hkx(N, n, k, x)
2.42 + print "h%i(%i) = (%i * %i mod %i) mod %i = %i" % (k, x, k, x, N, n, hx[(k,x)])
2.43 + ml = []
2.44 + sl = []
2.45 + kl = []
2.46 + for i in xrange(n):
2.47 + Wi = [thx[0][1] for thx in hx.iteritems() if thx[1] == i]
2.48 + bi = len(Wi)
2.49 + mi = 2 * bi * (bi - 1) + 1
2.50 + si = sum(ml)
2.51 + ki = find_ki(N, mi, Wi)
2.52 + ml.append(mi)
2.53 + sl.append(si)
2.54 + kl.append(ki)
2.55 + print "\nW%i = " % i, Wi
2.56 + print "=> b%i = %i" % (i, bi), "m%i = %i" % (i, mi), "s%i = %i" % (i, si), "k%i = %i" % (i, ki)
2.57 + if bi > 1:
2.58 + print "=> hk%i = (%i * x mod %i) mod %i" % (i, ki, N, mi)
2.59 +
2.60 + print
2.61 + for x in xi:
2.62 + i = hx[(k, x)]
2.63 + posx = sl[i] + hkx(N, ml[i], kl[i], x)
2.64 + print "x = %i => pos(%i) = %i" % (x, x, posx)
2.65 +
2.66 +from argparse import ArgumentParser
2.67 +
2.68 +def parse_args():
2.69 + parser = ArgumentParser(description='Builds a perfect hash function.')
2.70 + parser.add_argument('n', type=int, help='size of the universe U')
2.71 + parser.add_argument('k', type=int, help='parameter k of the hash function')
2.72 + parser.add_argument('s', help='comma-separated values defining the given set S')
2.73 + return parser.parse_args()
2.74 +
2.75 +if __name__ == "__main__":
2.76 + main()
3.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
3.2 +++ b/examples/prime.py Tue Feb 22 18:58:21 2011 +0100
3.3 @@ -0,0 +1,74 @@
3.4 +isProbPrime = True
3.5 +
3.6 +import sys
3.7 +from math import log
3.8 +from random import randint
3.9 +
3.10 +def main():
3.11 + args = parse_args()
3.12 + n = args.n
3.13 + if args.s:
3.14 + if safe_is_prime(n):
3.15 + print "%i is most probably a prime number." % n
3.16 + else:
3.17 + print "%i is not a prime number." % n
3.18 + return
3.19 + elif args.a:
3.20 + a = args.a
3.21 + else:
3.22 + a = randint(2, n-1)
3.23 + if is_prime(n, a):
3.24 + print "%i is probably a prime." % n
3.25 + else:
3.26 + print "%i is not a prime." % n
3.27 +
3.28 +def safe_is_prime(n):
3.29 + aset = set()
3.30 + while len(aset) < int(log(n)):
3.31 + aset.add(randint(2, n-1))
3.32 + for a in aset:
3.33 + if not is_prime(n, a):
3.34 + return False
3.35 + return True
3.36 +
3.37 +def is_prime(n, a=None):
3.38 + if not a:
3.39 + a = randint(2, n-1)
3.40 + global isProbPrime
3.41 + isProbPrime = True
3.42 + #print "Prime Test for %i with a = %i:" % (n, a)
3.43 + r = power(a, n-1, n)
3.44 + if r != 1 or not isProbPrime:
3.45 + #print "=> %i is not a prime." % n
3.46 + return False
3.47 + else:
3.48 + #print "=> %i is probably a prime." % n
3.49 + return True
3.50 +
3.51 +def power(a, p, n):
3.52 + global isProbPrime
3.53 + if p == 0:
3.54 + return 1
3.55 + x = power(a, p/2, n)
3.56 + r = (x*x)%n
3.57 + #print "power(%i, %i, %i)" % (a, p, n)
3.58 + #print "result = %i" % r
3.59 + if r == 1 and x != 1 and x != n-1:
3.60 + isProbPrime = False
3.61 + #print "isProbPrime = False"
3.62 + if p%2 == 1:
3.63 + r = (a*r)%n
3.64 + #print "updated result = %i" % r
3.65 + return r
3.66 +
3.67 +from argparse import ArgumentParser
3.68 +
3.69 +def parse_args():
3.70 + parser = ArgumentParser(description='Applies the randomised prime number test.')
3.71 + parser.add_argument('n', type=int, help='number to be tested')
3.72 + parser.add_argument('-a', type=int, help='provide a yourself')
3.73 + parser.add_argument('-s', action='store_true', help='multiple tests to minimise probability of failure')
3.74 + return parser.parse_args()
3.75 +
3.76 +if __name__ == "__main__":
3.77 + main()
4.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
4.2 +++ b/examples/quicksort.py Tue Feb 22 18:58:21 2011 +0100
4.3 @@ -0,0 +1,52 @@
4.4 +"""
4.5 +Name: quicksort - Quicksort for arrays of unsorted integers
4.6 +
4.7 +Description: quicksort applies the Quicksort using static or randomised pivot element selection on an unsorted array of integers.
4.8 +
4.9 +Author: Eugen Sawin <sawine@me73.com>
4.10 +"""
4.11 +
4.12 +def main():
4.13 + args = parse_arguments()
4.14 + array = [int(e.strip()) for e in args.array.split(',')]
4.15 + random = args.r
4.16 + print quicksort(array, random)
4.17 +
4.18 +from random import randrange
4.19 +
4.20 +def quicksort(array, random=False):
4.21 + if len(array) < 2:
4.22 + return array
4.23 + if random:
4.24 + p = randrange(len(array))
4.25 + pivot = array[p]
4.26 + array[p] = array[-1]
4.27 + array[-1] = pivot
4.28 + else:
4.29 + pivot = array[-1]
4.30 + left, right = partition(array, pivot)
4.31 + left = quicksort(left, random)
4.32 + right = quicksort(right, random)
4.33 + print left + [pivot] + right
4.34 + return left + [pivot] + right
4.35 +
4.36 +def partition(array, pivot):
4.37 + left = []
4.38 + right = []
4.39 + for e in array[:-1]:
4.40 + if e < pivot:
4.41 + left.append(e)
4.42 + else:
4.43 + right.append(e)
4.44 + return left, right
4.45 +
4.46 +from argparse import ArgumentParser
4.47 +
4.48 +def parse_arguments():
4.49 + parser = ArgumentParser(description='Applies Quicksort on an unsorted array of integers.')
4.50 + parser.add_argument('array', help='comma-separated values')
4.51 + parser.add_argument('-r', action='store_true', help='randomised pivot selection')
4.52 + return parser.parse_args()
4.53 +
4.54 +if __name__ == "__main__":
4.55 + main()
5.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
5.2 +++ b/examples/rsa.py Tue Feb 22 18:58:21 2011 +0100
5.3 @@ -0,0 +1,63 @@
5.4 +"""
5.5 +Name: rsa - RSA encryption and decryption
5.6 +Description: RSA-based encryption and decryption of messages.
5.7 +
5.8 +Author: Eugen Sawin <sawine@me73.com>
5.9 +"""
5.10 +MAX_PRIME = 120
5.11 +
5.12 +from random import choice
5.13 +
5.14 +def main():
5.15 + args = parse_arguments()
5.16 + if args.e:
5.17 + m, e, n = args.e
5.18 + print encrypt(m, e, n)
5.19 + elif args.d:
5.20 + c, d, n = args.d
5.21 + print decrypt(c, d, n)
5.22 + else:
5.23 + print create_keys()
5.24 +
5.25 +from prime import safe_is_prime as isprime
5.26 +
5.27 +def create_keys():
5.28 + primes = [p for p in xrange(3, MAX_PRIME) if isprime(p)]
5.29 + p = choice(primes)
5.30 + q = choice(primes)
5.31 + n = p * q
5.32 + pn = (p-1)*(q-1)
5.33 + e = choice(range(2, pn))
5.34 + (ggt, d, k) =[int(i) for i in extended_euclid(e, pn)]
5.35 + while ggt != 1 or d < 0:
5.36 + e = choice(range(2, pn))
5.37 + (ggt, d, k) = [int(i) for i in extended_euclid(e, pn)]
5.38 + return ((e, n), (d, n))
5.39 +
5.40 +def encrypt(m, e, n):
5.41 + c = m**e%n
5.42 + return c
5.43 +
5.44 +def decrypt(c, d, n):
5.45 + m = c**d%n
5.46 + return m
5.47 +
5.48 +from math import floor
5.49 +
5.50 +def extended_euclid(a, b):
5.51 + if b == 0:
5.52 + return (a, 1, 0)
5.53 + (ds, ss, ts) = extended_euclid(b, a%b)
5.54 + (d, s, t) = (ds, ts, ss - floor(a/b)*ts)
5.55 + return (d, s, t)
5.56 +
5.57 +from argparse import ArgumentParser
5.58 +
5.59 +def parse_arguments():
5.60 + parser = ArgumentParser(description='Applies the Dijkstra algorithm for the single source shortest path problem.')
5.61 + parser.add_argument('-e', metavar=('M', 'K', 'N'), type=int, nargs=3, help='sequence of (u, v, c(u,v))')
5.62 + parser.add_argument('-d', metavar=('M', 'K', 'N'), type=int, nargs=3, help='source vertex')
5.63 + return parser.parse_args()
5.64 +
5.65 +if __name__ == "__main__":
5.66 + main()
6.1 --- a/exzerpt.aux Thu Jan 20 00:56:20 2011 +0100
6.2 +++ b/exzerpt.aux Tue Feb 22 18:58:21 2011 +0100
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8.1 --- a/exzerpt.tex Thu Jan 20 00:56:20 2011 +0100
8.2 +++ b/exzerpt.tex Tue Feb 22 18:58:21 2011 +0100
8.3 @@ -247,4 +247,4 @@
8.4 \label{fig:landau_sym}
8.5 \end{figure}
8.6 \end{appendix}
8.7 -\end{document}
8.8 \ No newline at end of file
8.9 +\end{document}
9.1 --- a/exzerpt.toc Thu Jan 20 00:56:20 2011 +0100
9.2 +++ b/exzerpt.toc Tue Feb 22 18:58:21 2011 +0100
9.3 @@ -1,34 +1,34 @@
9.4 -\select@language {ngerman}
9.5 -\contentsline {section}{\numberline {1}Einleitung}{2}{section.1}
9.6 -\contentsline {subsection}{\numberline {1.1}Themen:}{2}{subsection.1.1}
9.7 -\contentsline {subsection}{\numberline {1.2}Problem-/Anwendungsbereiche:}{2}{subsection.1.2}
9.8 -\contentsline {subsection}{\numberline {1.3}Literatur:}{2}{subsection.1.3}
9.9 -\contentsline {section}{\numberline {2}Divide and Conquer}{3}{section.2}
9.10 -\contentsline {subsection}{\numberline {2.1}Formulierung des Divide and Conquer Prinzips}{3}{subsection.2.1}
9.11 -\contentsline {subsection}{\numberline {2.2}Quicksort}{3}{subsection.2.2}
9.12 -\contentsline {subsubsection}{\numberline {2.2.1}Analyse}{3}{subsubsection.2.2.1}
9.13 -\contentsline {subsection}{\numberline {2.3}N\"achste Paare}{3}{subsection.2.3}
9.14 -\contentsline {subsubsection}{\numberline {2.3.1}Algorithmus}{3}{subsubsection.2.3.1}
9.15 -\contentsline {subsubsection}{\numberline {2.3.2}Analyse}{4}{subsubsection.2.3.2}
9.16 -\contentsline {subsection}{\numberline {2.4}Segmentschnitt}{4}{subsection.2.4}
9.17 -\contentsline {subsubsection}{\numberline {2.4.1}Algorithmus}{5}{subsubsection.2.4.1}
9.18 -\contentsline {subsubsection}{\numberline {2.4.2}Analyse}{5}{subsubsection.2.4.2}
9.19 -\contentsline {subsection}{\numberline {2.5}Polynomprodukt und Fast Fourier-Transformation}{5}{subsection.2.5}
9.20 -\contentsline {section}{\numberline {3}Randomisierung}{6}{section.3}
9.21 -\contentsline {subsection}{\numberline {3.1}Hashing}{6}{subsection.3.1}
9.22 -\contentsline {section}{\numberline {4}Amortisierte Analyse}{7}{section.4}
9.23 -\contentsline {subsection}{\numberline {4.1}Binomial Heaps}{7}{subsection.4.1}
9.24 -\contentsline {subsection}{\numberline {4.2}Fibonacci Heaps}{7}{subsection.4.2}
9.25 -\contentsline {subsection}{\numberline {4.3}Union Find}{7}{subsection.4.3}
9.26 -\contentsline {section}{\numberline {5}Greedy}{8}{section.5}
9.27 -\contentsline {subsection}{\numberline {5.1}K\"urzeste (billigste) Wege}{8}{subsection.5.1}
9.28 -\contentsline {section}{\numberline {6}Bin Packing}{9}{section.6}
9.29 -\contentsline {subsection}{\numberline {6.1}Online Verfahren}{9}{subsection.6.1}
9.30 -\contentsline {subsubsection}{\numberline {6.1.1}Next Fit (NF)}{9}{subsubsection.6.1.1}
9.31 -\contentsline {subsubsection}{\numberline {6.1.2}First-Fit (FF)}{9}{subsubsection.6.1.2}
9.32 -\contentsline {subsubsection}{\numberline {6.1.3}Best-Fit (BF)}{10}{subsubsection.6.1.3}
9.33 -\contentsline {subsection}{\numberline {6.2}Offline Verfahren}{10}{subsection.6.2}
9.34 -\contentsline {subsubsection}{\numberline {6.2.1}First Fit Decreasing (FFD oder FFNI)}{10}{subsubsection.6.2.1}
9.35 -\contentsline {subsubsection}{\numberline {6.2.2}Best Fit Decreasing (BFD)}{10}{subsubsection.6.2.2}
9.36 -\contentsline {section}{\numberline {7}Dynamische Programmierung}{11}{section.7}
9.37 -\contentsline {section}{\numberline {A}Landau-Symbole}{12}{appendix.A}
9.38 +\select@language {ngerman}
9.39 +\contentsline {section}{\numberline {1}Einleitung}{2}{section.1}
9.40 +\contentsline {subsection}{\numberline {1.1}Themen:}{2}{subsection.1.1}
9.41 +\contentsline {subsection}{\numberline {1.2}Problem-/Anwendungsbereiche:}{2}{subsection.1.2}
9.42 +\contentsline {subsection}{\numberline {1.3}Literatur:}{2}{subsection.1.3}
9.43 +\contentsline {section}{\numberline {2}Divide and Conquer}{3}{section.2}
9.44 +\contentsline {subsection}{\numberline {2.1}Formulierung des Divide and Conquer Prinzips}{3}{subsection.2.1}
9.45 +\contentsline {subsection}{\numberline {2.2}Quicksort}{3}{subsection.2.2}
9.46 +\contentsline {subsubsection}{\numberline {2.2.1}Analyse}{3}{subsubsection.2.2.1}
9.47 +\contentsline {subsection}{\numberline {2.3}N\"achste Paare}{3}{subsection.2.3}
9.48 +\contentsline {subsubsection}{\numberline {2.3.1}Algorithmus}{3}{subsubsection.2.3.1}
9.49 +\contentsline {subsubsection}{\numberline {2.3.2}Analyse}{4}{subsubsection.2.3.2}
9.50 +\contentsline {subsection}{\numberline {2.4}Segmentschnitt}{4}{subsection.2.4}
9.51 +\contentsline {subsubsection}{\numberline {2.4.1}Algorithmus}{4}{subsubsection.2.4.1}
9.52 +\contentsline {subsubsection}{\numberline {2.4.2}Analyse}{5}{subsubsection.2.4.2}
9.53 +\contentsline {subsection}{\numberline {2.5}Polynomprodukt und Fast Fourier-Transformation}{5}{subsection.2.5}
9.54 +\contentsline {section}{\numberline {3}Randomisierung}{6}{section.3}
9.55 +\contentsline {subsection}{\numberline {3.1}Hashing}{6}{subsection.3.1}
9.56 +\contentsline {section}{\numberline {4}Amortisierte Analyse}{7}{section.4}
9.57 +\contentsline {subsection}{\numberline {4.1}Binomial Heaps}{7}{subsection.4.1}
9.58 +\contentsline {subsection}{\numberline {4.2}Fibonacci Heaps}{7}{subsection.4.2}
9.59 +\contentsline {subsection}{\numberline {4.3}Union Find}{7}{subsection.4.3}
9.60 +\contentsline {section}{\numberline {5}Greedy}{8}{section.5}
9.61 +\contentsline {subsection}{\numberline {5.1}K\"urzeste (billigste) Wege}{8}{subsection.5.1}
9.62 +\contentsline {section}{\numberline {6}Bin Packing}{9}{section.6}
9.63 +\contentsline {subsection}{\numberline {6.1}Online Verfahren}{9}{subsection.6.1}
9.64 +\contentsline {subsubsection}{\numberline {6.1.1}Next Fit (NF)}{9}{subsubsection.6.1.1}
9.65 +\contentsline {subsubsection}{\numberline {6.1.2}First-Fit (FF)}{9}{subsubsection.6.1.2}
9.66 +\contentsline {subsubsection}{\numberline {6.1.3}Best-Fit (BF)}{10}{subsubsection.6.1.3}
9.67 +\contentsline {subsection}{\numberline {6.2}Offline Verfahren}{10}{subsection.6.2}
9.68 +\contentsline {subsubsection}{\numberline {6.2.1}First Fit Decreasing (FFD oder FFNI)}{10}{subsubsection.6.2.1}
9.69 +\contentsline {subsubsection}{\numberline {6.2.2}Best Fit Decreasing (BFD)}{10}{subsubsection.6.2.2}
9.70 +\contentsline {section}{\numberline {7}Dynamische Programmierung}{11}{section.7}
9.71 +\contentsline {section}{\numberline {A}Landau-Symbole}{12}{appendix.A}