Combinatorial Algorithms

8.1 Definition of Combinatorial Algorithms.

Combinatorial algorithms are algorithms designed to solve problems related to counting, enumerating, and generating different combinatorial structures. These algorithms are used for working with sets, subsets, permutations, combinations, and other combinatorial objects.

Main types of combinatorial problems

• Permutations: All possible ordered combinations of set elements.
• Combinations: All possible unordered subsets of a given size.
• Arrangements: All possible ordered subsets of a given size.
• Partitions: All possible ways to divide a set into subsets.
• Combinatorial objects: Other structures such as graphs, matrices, etc.

Advantages and disadvantages of combinatorial algorithms

Advantages:

• Complete enumeration: These algorithms can list all possible combinations, which is useful for exhaustive search problems.
• Flexibility: They can be adapted to solve a wide variety of problems related to combinatorics.
• Intuitiveness: Many find these algorithms easy to understand and implement due to their recursive approach and backtracking technique.

Disadvantages:

• Combinatorial explosion: As the input size increases, execution time and memory usage can grow dramatically (for example, O(n!)).
• Suboptimality: For a large number of combinations, these algorithms may be inefficient and require improvements or replacement by more advanced techniques (e.g., using dynamic programming or greedy algorithms).

Time and space complexity of combinatorial algorithms

Time complexity:

• Permutations: O(n!), where n is the number of elements in the set.
• Combinations: O(C(n, k)), where C(n, k) is the binomial coefficient, equal to n! / (k! * (n - k)!).
• Arrangements: O(A(n, k)), where A(n, k) is the number of arrangements, equal to n! / (n - k)!.
• Subsets: O(2^n), as each of the n elements can be included or not in a subset.

Space complexity:

Typically O(n) for storing intermediate results during recursive construction, but total memory might be needed to store all results (e.g., O(n * n!) for permutations).

Examples of combinatorial algorithms

8.2 Generating Permutations.

Task:

Find all unique permutations of elements in a given set.

Algorithm:

Use recursion or iterative methods to generate all possible ordered combinations of elements.

def permute(nums):
    result = []

    def backtrack(start):
        if start == len(nums):
            result.append(nums[:])
        for i in range(start, len(nums)):
            nums[start], nums[i] = nums[i], nums[start]
            backtrack(start + 1)
            nums[start], nums[i] = nums[i], nums[start]

    backtrack(0)
    return result

print(permute([1, 2, 3]))

8.3 Generating Combinations

Task:

Find all combinations of a given size from a set.

Algorithm:

Use recursion or iteration to generate all possible subsets of a given size.

def combine(n, k):
    result = []

    def backtrack(start, path):
        if len(path) == k:
            result.append(path[:])
            return
        for i in range(start, n + 1):
            path.append(i)
            backtrack(i + 1, path)
            path.pop()

    backtrack(1, [])
    return result

print(combine(4, 2))
        

8.4 Generating Set Partitions

Task:

Find all possible ways to partition a set into subsets.

Algorithm:

Use recursion to generate all possible partitions of a set.

import copy

def partition_set(s):
    result = []

    def backtrack(part, rest):
        if not rest:
            result.append(copy.deepcopy(part[:]))
            return
        for i in range(len(part)):
            part[i].append(rest[0])
            backtrack(part, rest[1:])
            part[i].pop()
        part.append([rest[0]])
        backtrack(part, rest[1:])
        part.pop()

    backtrack([], list(s))
    return result

print(partition_set({1, 2, 3}))
        
        

Using copy.deepcopy(part[:]) creates a deep copy of the list part. This means that a new, independent list is created that does not reference the original part. Therefore, each element in result will represent a unique partition.

2

Task

Python SELF EN, level 60, lesson 3

Locked

Set Permutations

Set Permutations

Open

2

Task

Python SELF EN, level 60, lesson 3

Locked

All Subsets

All Subsets

Open