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Authors:
Anany Levitin
Chapter:
Coping With The Limitations Of Algorithm Power
Exercise:
12.1 Exercise
Question:7 | ISBN:9780132316811 | Edition: 3

Question

Generate all permutations of {1, 2, 3, 4} by backtracking.

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Answer

  • Start with an empty permutation and an empty set of unused numbers: [], {1, 2, 3, 4} Choose a number from the unused set: 1 Add the chosen number to the current permutation: [1] Recursively generate permutations with the remaining unused numbers: [1], {2, 3, 4} Backtrack by removing the chosen number from the current permutation and adding it back to the unused set: [], {1, 2, 3, 4}

  • Start with an empty permutation and an empty set of unused numbers: [], {1, 2, 3, 4} Choose a number from the unused set: 2 Add the chosen number to the current permutation: [2] Recursively generate permutations with the remaining unused numbers: [2], {1, 3, 4} Backtrack by removing the chosen number from the current permutation and adding it back to the unused set: [], {1, 2, 3, 4}

  • Start with an empty permutation and an empty set of unused numbers: [], {1, 2, 3, 4} Choose a number from the unused set: 3 Add the chosen number to the current permutation: [3] Recursively generate permutations with the remaining unused numbers: [3], {1, 2, 4} Backtrack by removing the chosen number from the current permutation and adding it back to the unused set: [], {1, 2, 3, 4}

  • Start with an empty permutation and an empty set of unused numbers: [], {1, 2, 3, 4} Choose a number from the unused set: 4 Add the chosen number to the current permutation: [4] Recursively generate permutations with the remaining unused numbers: [4], {1, 2, 3} Backtrack by removing the chosen number from the current permutation and adding it back to the unused set: [], {1, 2, 3, 4}

  • Start with an empty permutation and an empty set of unused numbers: [], {1, 2, 3, 4} Choose a number from the unused set: 1 Add the chosen number to the current permutation: [1] Recursively generate permutations with the remaining unused numbers: [1], {2, 3, 4} Choose a number from the unused set: 2 Add the chosen number to the current permutation: [1, 2] Recursively generate permutations with the remaining unused numbers: [1, 2], {3, 4} Choose a number from the unused set: 3 Add the chosen number to the current permutation: [1, 2, 3] Recursively generate permutations with the remaining unused numbers: [1, 2, 3], {4} Choose a number from the unused set: 4 Add the chosen number to the current permutation: [1, 2, 3, 4] This is a valid permutation.

  • Continue the process by exploring other branches and combinations until all permutations are generated.

The generated permutations are:

[1, 2, 3, 4]

[1, 2, 4, 3]

[1, 3, 2, 4]

[1, 3, 4, 2]

[1, 4, 3, 2]

[1, 4, 2, 3]

[2, 1, 3, 4]

[2, 1, 4, 3]

[2, 3, 1, 4]

[2, 3, 4, 1]

[2, 4, 3, 1]

[2, 4, 1, 3]

[3, 2, 1, 4]

[3, 2, 4, 1]

[3, 1, 2, 4]

[3, 1, 4, 2]

[3, 4, 1, 2]

[3, 4, 2, 1]

[4, 2, 3, 1]

[4, 2, 1, 3]

[4, 3, 2, 1]

[4, 3, 1, 2]

[4, 1, 3, 2]

[4, 1, 2, 3]

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