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Signed-off-by: Christoph M. Wintersteiger <cwinter@microsoft.com>
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Christoph M. Wintersteiger
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############################################
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# Copyright (c) 2012 Ganesh Gopalakrishnan ganesh@cs.utah.edu
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#
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# Check if the given graph has a Hamiltonian cycle.
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#
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# Author: Ganesh Gopalakrishnan ganesh@cs.utah.edu
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############################################
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from z3 import *
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def gencon(gr):
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"""
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Input a graph as an adjacency list, e.g. {0:[1,2], 1:[2], 2:[1,0]}.
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Produces solver to check if the given graph has
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a Hamiltonian cycle. Query the solver using s.check() and if sat,
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then s.model() spells out the cycle. Two example graphs from
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http://en.wikipedia.org/wiki/Hamiltonian_path are tested.
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=======================================================
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Explanation:
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Generate a list of Int vars. Constrain the first Int var ("Node 0") to be 0.
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Pick a node i, and attempt to number all nodes reachable from i to have a
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number one higher (mod L) than assigned to node i (use an Or constraint).
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=======================================================
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"""
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L = len(gr)
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cv = [Int('cv%s'%i) for i in range(L)]
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s = Solver()
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s.add(cv[0]==0)
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for i in range(L):
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s.add(Or([cv[j]==(cv[i]+1)%L for j in gr[i]]))
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return s
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def examples():
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# Example Graphs: The Dodecahedral graph from http://en.wikipedia.org/wiki/Hamiltonian_path
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grdodec = { 0: [1, 4, 5],
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1: [0, 7, 2],
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2: [1, 9, 3],
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3: [2, 11, 4],
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4: [3, 13, 0],
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5: [0, 14, 6],
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6: [5, 16, 7],
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7: [6, 8, 1],
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8: [7, 17, 9],
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9: [8, 10, 2],
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10: [9, 18, 11],
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11: [10, 3, 12],
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12: [11, 19, 13],
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13: [12, 14, 4],
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14: [13, 15, 5],
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15: [14, 16, 19],
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16: [6, 17, 15],
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17: [16, 8, 18],
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18: [10, 19, 17],
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19: [18, 12, 15] }
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import pprint
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pp = pprint.PrettyPrinter(indent=4)
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pp.pprint(grdodec)
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sdodec=gencon(grdodec)
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print(sdodec.check())
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print(sdodec.model())
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# =======================================================
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# See http://en.wikipedia.org/wiki/Hamiltonian_path for the Herschel graph
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# being the smallest possible polyhdral graph that does not have a Hamiltonian
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# cycle.
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#
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grherschel = { 0: [1, 9, 10, 7],
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1: [0, 8, 2],
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2: [1, 9, 3],
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3: [2, 8, 4],
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4: [3, 9, 10, 5],
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5: [4, 8, 6],
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6: [5, 10, 7],
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7: [6, 8, 0],
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8: [1, 3, 5, 7],
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9: [2, 0, 4],
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10: [6, 4, 0] }
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pp.pprint(grherschel)
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sherschel=gencon(grherschel)
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print(sherschel.check())
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# =======================================================
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if __name__ == "__main__":
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examples()
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############################################
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# Copyright (c) 2012 Ganesh Gopalakrishnan ganesh@cs.utah.edu
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#
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# Check if the given graph has a Hamiltonian cycle.
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#
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# Author: Ganesh Gopalakrishnan ganesh@cs.utah.edu
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############################################
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from z3 import *
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def gencon(gr):
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"""
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Input a graph as an adjacency list, e.g. {0:[1,2], 1:[2], 2:[1,0]}.
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Produces solver to check if the given graph has
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a Hamiltonian cycle. Query the solver using s.check() and if sat,
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then s.model() spells out the cycle. Two example graphs from
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http://en.wikipedia.org/wiki/Hamiltonian_path are tested.
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=======================================================
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Explanation:
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Generate a list of Int vars. Constrain the first Int var ("Node 0") to be 0.
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Pick a node i, and attempt to number all nodes reachable from i to have a
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number one higher (mod L) than assigned to node i (use an Or constraint).
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=======================================================
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"""
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L = len(gr)
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cv = [Int('cv%s'%i) for i in range(L)]
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s = Solver()
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s.add(cv[0]==0)
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for i in range(L):
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s.add(Or([cv[j]==(cv[i]+1)%L for j in gr[i]]))
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return s
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def examples():
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# Example Graphs: The Dodecahedral graph from http://en.wikipedia.org/wiki/Hamiltonian_path
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grdodec = { 0: [1, 4, 5],
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1: [0, 7, 2],
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2: [1, 9, 3],
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3: [2, 11, 4],
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4: [3, 13, 0],
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5: [0, 14, 6],
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6: [5, 16, 7],
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7: [6, 8, 1],
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8: [7, 17, 9],
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9: [8, 10, 2],
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10: [9, 18, 11],
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11: [10, 3, 12],
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12: [11, 19, 13],
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13: [12, 14, 4],
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14: [13, 15, 5],
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15: [14, 16, 19],
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16: [6, 17, 15],
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17: [16, 8, 18],
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18: [10, 19, 17],
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19: [18, 12, 15] }
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import pprint
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pp = pprint.PrettyPrinter(indent=4)
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pp.pprint(grdodec)
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sdodec=gencon(grdodec)
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print(sdodec.check())
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print(sdodec.model())
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# =======================================================
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# See http://en.wikipedia.org/wiki/Hamiltonian_path for the Herschel graph
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# being the smallest possible polyhdral graph that does not have a Hamiltonian
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# cycle.
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#
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grherschel = { 0: [1, 9, 10, 7],
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1: [0, 8, 2],
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2: [1, 9, 3],
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3: [2, 8, 4],
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4: [3, 9, 10, 5],
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5: [4, 8, 6],
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6: [5, 10, 7],
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7: [6, 8, 0],
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8: [1, 3, 5, 7],
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9: [2, 0, 4],
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10: [6, 4, 0] }
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pp.pprint(grherschel)
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sherschel=gencon(grherschel)
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print(sherschel.check())
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# =======================================================
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if __name__ == "__main__":
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examples()
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