Non-decreasing Subsequences

class Solution {
public:
    vector<vector<int>> getAllIncreasingSubsequences(vector<int>& elements) {
        int total = elements.size();
        set<vector<int>> allSubsequences;
        for (int i = 1; i < (1 << total); i++) {
            vector<int> subsequence;
            for (int j = 0; j < total; j++) {
                if ((i >> j) & 1) {
                    subsequence.push_back(elements[j]);
                }
            }
            if (subsequence.size() > 1) {
                bool isValid = true;
                for (int k = 0; k < subsequence.size() - 1; k++) {
                    isValid &= subsequence[k] <= subsequence[k + 1];
                }
                if (isValid) {
                    allSubsequences.insert(subsequence);
                }
            }
        }
        return vector<vector<int>>(allSubsequences.begin(), allSubsequences.end());
    }
};

This C++ solution addresses the problem of finding all non-decreasing subsequences of a given integer array, where each subsequence consists of at least two elements. By leveraging bitwise operations and set data structures, the solution efficiently generates and checks each possible subsequence for the non-decreasing order requirement.

• Use a set<vector<int>> to store unique subsequences, ensuring that no duplicate subsequences are counted.
• Iterate through all possible combinations of array indices using a binary representation.
• For each combination represented by a number, loop through the array elements and use bitwise operations to check if an element should be included in the current subsequence.
• After constructing each candidate subsequence, a validation step checks if each element in the subsequence is less than or equal to the next element, confirming the non-decreasing order.
• If valid, add the subsequence to the set.
• Convert and return the unique subsequences from the set as a vector of vectors.

Such an approach is useful for ensuring optimal performance while handling unique order-sensitive subsequences, making it a robust solution for problems involving subsequence order validation and generation. With this code, efficiently handle larger datasets by minimizing redundancy and unnecessary validations.

class Solution {
    public List<List<Integer>> generateIncreasingSubsequences(int[] arr) {
        int length = arr.length;
        Set<List<Integer>> resSet = new HashSet<>();
        for (int mask = 1; mask < (1 << length); mask++) {
            List<Integer> subseq = new ArrayList<>();
            for (int j = 0; j < length; j++) {
                if ((mask & (1 << j)) != 0) {
                    subseq.add(arr[j]);
                }
            }
            if (subseq.size() >= 2) {
                boolean isInOrder = true;
                for (int k = 0; k < subseq.size() - 1; k++) {
                    isInOrder &= subseq.get(k) <= subseq.get(k + 1);
                }
                if (isInOrder) {
                    resSet.add(subseq);
                }
            }
        }
        return new ArrayList<>(resSet);
    }
}

The provided Java code outlines a solution for finding all non-decreasing subsequences of length two or more from a given array.

• The function generateIncreasingSubsequences accepts an array of integers and returns a list of lists, each representing a non-decreasing subsequence.
• The solution utilizes a HashSet named resSet to store unique subsequences and thereby avoid duplications.
• It employs bitwise operations to generate all possible subsequences of the input array, iterating through these using an outer loop where a mask represents each subset of array elements.
• An inner loop then examines each bit of the mask to determine which elements are included in the current subsequence.
• The subseq.size() >= 2 condition ensures that only subsequences of at least two elements are considered.
• The boolean isInOrder checks if the elements in subseq are in non-decreasing order.
• Only subsequences that remain in non-decreasing order are added to the resSet.
• Finally, the function converts the HashSet to an ArrayList before returning, providing the requested list of non-decreasing subsequences.

This method effectively explores all potential combinations and enforces the non-decreasing and minimum size constraints to produce the desired output. The use of bit masking simplifies the generation of all subsets while ensuring efficient computation and memory utilization by avoiding storage of duplicate subsequences through HashSet.

class Solution:
    def generateSubsequences(self, values: List[int]) -> List[List[int]]:
        size = len(values)
        subsequences = set()
        for mask in range(1, 1 << size):
            candidate = [values[idx] for idx in range(size) if (mask >> idx) & 1]
            if len(candidate) >= 2 and all(candidate[i] <= candidate[i + 1] for i in range(len(candidate) - 1)):
                subsequences.add(tuple(candidate))
        return subsequences

This Python 3 solution addresses the problem of generating all possible non-decreasing subsequences from a given list of integers. The program is structured within a class named Solution, and contains a single function generateSubsequences, which accepts a list of integers, values.

• The overall approach utilizes bit manipulation to explore all potential subsequences.
• A set, subsequences, is used to store each valid subsequence, ensuring that duplicate sequences are not added.

To elucidate the mechanism:

1. Calculate the number of elements in values.
2. Iterate over each possible subset of values using a bitmask ranging from 1 to 2^size - 1.
3. For each bitmask, generate a subsequence by including elements that correspond to set bits in the bitmask.
4. Check if this generated subsequence is non-decreasing. To be valid, each element must be less than or equal to the subsequent element.
5. Convert the subsequence to a tuple and add to the subsequences set to maintain uniqueness.

• Return subsequences as the output, providing all unique non-decreasing subsequences extracted from the input list.

This method efficiently determines valid subsequences without repeatedly checking the entire dataset, leveraging the power of bit manipulation and combinatorics. Ensure the input list is not empty to avoid unnecessary computation and optimize performance.