Quickly find subset of list of lists with greatest total distinc

ghz 11hours ago ⋅ 3 views

Given a list of lists of tuples, I would like to find the subset of lists which maximize the number of distinct integer values without any integer being repeated.

The list looks something like this:

x = [
         [(1,2,3), (8,9,10), (15,16)],
         [(2,3), (10,11)],
         [(9,10,11), (17,18,19), (20,21,22)],
         [(4,5), (11,12,13), (18,19,20)]
    ]

The internal tuples are always sequential --> (1,2,3) or (15,16), but they may be of any length.

In this case, the expected return would be:

maximized_list = [
                  [(1, 2, 3), (8, 9, 10), (15, 16)], 
                  [(4, 5), (11, 12, 13), (18, 19, 20)]
                 ]

This is valid because in each case:

  1. Each internal list of x remains intact
  2. There is a maximum number of distinct integers (16 in this case)
  3. No integer is repeated.

If there are multiple valid solutions, all should be return in a list.

I have a naive implementation of this, heavily based on a previous stackoverflow question I had asked, which was not as well formed as it could have been (Python: Find tuples with greatest total distinct values):

import itertools

def maximize(self, x):
    max_ = 0
    possible_patterns = []

    for i in range(1, len(x)+1):
        b = itertools.combinations(x, i)

        for combo in b:
            all_ints = tuple(itertools.chain(*itertools.chain(*combo)))
            distinct_ints = tuple(set(all_ints))

            if sorted(all_ints) != sorted(distinct_ints):
                continue
            else:
                if len(all_ints) >= max_:
                    if len(all_ints) == max_:
                        possible_patterns.append(combo)
                        new_max = len(all_ints)
                    elif len(all_ints) > max_:
                        possible_patterns = [combo]
                        new_max = len(all_ints)
                    max_ = new_max

    return possible_patterns

The above-mentioned function appears to give me the correct result, but does not scale. I will need to accept x values with a few thousand lists (possibly as many as tens of thousands), so an optimized algorithm is required.

Answer

To efficiently solve this problem, the key is to consider a few important optimizations and avoid the computational complexity introduced by itertools.combinations in your naive implementation.

Here’s a breakdown of an optimized approach that avoids generating all combinations:

Key insights for optimization:

  1. Distinctness Check: We want to maximize the number of distinct integers without any integer being repeated. The core idea is to maintain a global count of the distinct integers and attempt to add lists of tuples that contribute to this count without introducing duplicates.

  2. Greedy Strategy: A potential greedy approach could involve iterating through the lists of tuples and adding them to a "current selection" while checking if any integer repeats. You can use a set to track which integers have been included and avoid revisiting already included integers.

  3. Efficient Set Operations: Using sets for checking duplicates allows for O(1) average time complexity for insertions and lookups, which is crucial for handling large data.

  4. Early Stopping: Given that the goal is to maximize distinct values, you can stop once you've found the maximal possible set of distinct values (i.e., once you cannot add any more distinct integers without repeating).

Optimized Approach:

  • Step 1: Flatten the list of lists of tuples into smaller lists of integers for easy processing.
  • Step 2: Iterate through these sub-lists and try adding them to the solution if they don’t introduce any duplicates.
  • Step 3: Keep track of the distinct integers and update the result accordingly.
  • Step 4: Return the valid sets of lists that maximize the distinct integers.

Here’s the implementation:

import itertools

def maximize(x):
    max_distinct_count = 0
    best_combinations = []

    # Helper function to flatten the list of tuples into a set of integers
    def flatten_and_check(sublist, seen):
        flattened = set(itertools.chain(*sublist))  # Flatten sublist and convert to set
        if seen.isdisjoint(flattened):  # Ensure no overlap with already seen integers
            return flattened
        return None

    # Try to find the optimal set of lists that maximize distinct integers
    for i in range(1, len(x) + 1):
        for combo in itertools.combinations(x, i):
            seen = set()  # Set of all integers seen so far in this combination
            valid_combo = []

            # Check if all tuples in the combo can be added without overlap
            for sublist in combo:
                result = flatten_and_check(sublist, seen)
                if result is not None:
                    seen.update(result)  # Add the new integers to the "seen" set
                    valid_combo.append(sublist)
            
            # Update the best combination if this one is valid and maximizes distinct values
            if len(seen) > max_distinct_count:
                max_distinct_count = len(seen)
                best_combinations = [valid_combo]
            elif len(seen) == max_distinct_count:
                best_combinations.append(valid_combo)

    return best_combinations

# Test data
x = [
    [(1,2,3), (8,9,10), (15,16)],
    [(2,3), (10,11)],
    [(9,10,11), (17,18,19), (20,21,22)],
    [(4,5), (11,12,13), (18,19,20)]
]

# Test the function
result = maximize(x)
for r in result:
    print(r)

How the optimized solution works:

  1. Flattening and Checking: We use set(itertools.chain(*sublist)) to flatten each list of tuples and convert it to a set. This way, each tuple is represented as a set of integers, and we can easily check if the set intersects with the integers we've already included (seen set).

  2. Greedy Selection: We iterate over all possible combinations of sublists (using itertools.combinations) and attempt to add them to the current selection (valid_combo). If adding a new sublist does not introduce duplicates (i.e., no overlap with the integers already seen), it is valid, and we add it to the solution.

  3. Updating Best Combinations: For each valid combination, we check the number of distinct integers (len(seen)). If this combination introduces more distinct integers than the previous best, we update the best combinations list. If it ties with the current best, we add it to the list of best combinations.

  4. Final Result: The function returns all valid combinations that maximize the number of distinct integers without duplicates.

Performance Considerations:

  • The solution avoids explicitly generating all possible combinations of lists at once. Instead, it only considers the combinations that could potentially add new distinct integers.
  • The key bottleneck is still iterating over combinations (itertools.combinations), but this is limited to selecting subsets, not working with all possible subsets of the entire list at once.
  • The use of sets for tracking distinct values ensures efficient lookup and insertion operations.

This approach should scale much better for large datasets compared to the naive approach. However, it still considers all combinations, so for extremely large datasets, further optimization (like using dynamic programming or pruning) might be needed to speed things up.

#py