Longest Increasing Subsequence

class LongestIncreasingSubsequence {
    public int findLISLength(int[] elements) {
        ArrayList<Integer> lisSequence = new ArrayList<>();
        lisSequence.add(elements[0]);

        for (int idx = 1; idx < elements.length; idx++) {
            int currentElement = elements[idx];
            if (currentElement > lisSequence.get(lisSequence.size() - 1)) {
                lisSequence.add(currentElement);
            } else {
                int updateIndex = locatePosition(lisSequence, currentElement);
                lisSequence.set(updateIndex, currentElement);
            }
        }
        
        return lisSequence.size();
    }
    
    private int locatePosition(ArrayList<Integer> sequence, int value) {
        int low = 0;
        int high = sequence.size() - 1;

        while (low < high) {
            int mid = (low + high) / 2;
            if (sequence.get(mid) >= value) {
                high = mid;
            } else {
                low = mid + 1;
            }
        }
        
        return low;
    }
}

The Java solution provided for finding the length of the Longest Increasing Subsequence (LIS) initializes by adding the first element to track the possible subsequences using an ArrayList, lisSequence. It iteratively scans each element in the input array, elements, and determines if the element can extend the current longest subsequence or it requires replacing a value in the lisSequence to maintain the potential for the longest sequence without break.

The primary process involves:

• Comparing the current currentElement to the last element of lisSequence. If currentElement is greater, it appends it directly to lisSequence.
• If currentElement is not greater, a helper method locatePosition finds the correct position in lisSequence to replace an element, ensuring lisSequence remains sorted and thus, possibly a longer increasing subsequence can be formed in later iterations.

The function locatePosition performs a binary search to find the appropriate index to replace a value in lisSequence with a smaller or equal value that still maintains an increasing sequence.

This approach ensures an efficient management of potential longest increasing subsequences, allowing the method to return the size of lisSequence, which corresponds to the length of the longest increasing subsequence in the input array by the end of iterations.

The solution is effective for scenarios requiring efficient calculation of subsequence lengths, especially relevant in applications needing analysis of ordered sequences within data arrays.

class Solution:
    def longest_increasing_subsequence(self, elements: List[int]) -> int:
        lis_container = []
        for value in elements:
            position = bisect_left(lis_container, value)

            if position == len(lis_container):
                lis_container.append(value)
            else:
                lis_container[position] = value

        return len(lis_container)

The provided Python solution implements a method to determine the length of the longest increasing subsequence from a list of integers. The code uses binary search through the bisect_left function from Python's bisect module, optimizing the process. Here's a breakdown of how the solution works:

• An empty list named lis_container is initialized to store the lengths of potential subsequences.
• The for loop iterates over each integer in the input list.
• Inside the loop, bisect_left determines the position at which the current integer (value) can be inserted into lis_container while maintaining the order. This position is either where the value can replace an existing number or be appended to maintain a potential increasing subsequence.
• If the calculated position is equal to the length of lis_container, implying the value is greater than all elements currently in lis_container, the value is appended to the list.
• Otherwise, the element at the determined position in lis_container is replaced with value, keeping the sequence optimally increasing without increasing its length unnecessarily.
• After processing all integers, the length of lis_container represents the length of the longest increasing subsequence.

This efficient approach ensures that the longest increasing subsequence is found with minimized computational overhead, suitable for large datasets.