This function computes the length of the longest strictly increasing subsequence (LIS) within a given array of integers. An increasing subsequence is a sequence of elements from the original array that are in strictly in...

The `length_of_lis` function utilizes dynamic programming to solve the Longest Increasing Subsequence problem. It defines `dp[i]` as the length of the longest increasing subsequence ending at index `i`. The base case is that every element itself forms an increasing subsequence of length 1, so `dp` is initialized with all ones. The transition involves iterating through all previous elements `j` for each element `i`. If `nums[i]` is greater than `nums[j]`, it means `nums[i]` can extend the increasing subsequence ending at `j`. Therefore, `dp[i]` is updated to be the maximum of its current value and `dp[j] + 1`. The final answer is the maximum value in the `dp` array. The time complexity is O(n^2) due to the nested loops, and the space complexity is O(n) to store the `dp` array. Edge cases like an empty input array are handled by returning 0.

function length_of_lis(nums): if nums is empty: return 0 n = length of nums dp = array of size n, initialized with 1s for i from 0 to n-1: for j from 0 to i-1: if nums[i] > nums[j]: dp[i] = max(dp[i], dp[j] + 1) return m...

This function takes a string as input and reverses the order of the words within it. For example, if the input is 'the sky is blue', the output will be 'blue is sky the'. This is a common string manipulation task used in...

The `reverse_words` function first splits the input string `s` into a list of words using the `split()` method. This method intelligently handles multiple spaces between words and also removes any leading or trailing whitespace. Then, the `reverse()` method is called on the list of words to reverse their order in-place. Finally, the `join()` method is used to concatenate the reversed list of words back into a single string, with each word separated by a single space. The time complexity is O(n), where n is the length of the string, due to splitting and joining. The space complexity is also O(n) in the worst case to store the list of words.

function reverse_words(s): words = split s by spaces, removing empty strings reverse the order of words in the list return join words with a single space

This function finds the minimum element in a rotated sorted array. A rotated sorted array is an array that was originally sorted in ascending order and then had a portion of its elements moved from the end to the beginni...

The algorithm employs a binary search strategy to find the minimum element in a rotated sorted array. The key insight is to compare the middle element with the rightmost element. If `nums[mid] > nums[right]`, it implies that the rotation point (and thus the minimum element) lies in the right half of the array, so we move `left` to `mid + 1`. Otherwise, if `nums[mid] <= nums[right]`, the minimum element must be in the left half (including `mid`), so we set `right` to `mid`. This process continues until `left` equals `right`, at which point `nums[left]` (or `nums[right]`) is the minimum element. The time complexity is O(log n) because of the binary search. The space complexity is O(1) as it uses a constant amount of extra space. Edge cases like an unrotated array or a two-element array are handled by the initial check and the binary search logic itself.

function find_min(nums): n = length of nums if n == 1 or nums[0] < nums[n-1]: return nums[0] left = 0 right = n - 1 while left < right: mid = (left + right) // 2 if nums[mid] > nums[right]: left = mid + 1 else: right = m...

This function finds a peak element in a given array of integers. A peak element is defined as an element that is strictly greater than its adjacent neighbors. If multiple peaks exist, any one of their indices can be retu...

The algorithm efficiently finds a peak element using a modified binary search approach. It first handles edge cases where the peak might be at the beginning or end of the array. The core logic then performs binary search on the interior of the array. At each step, it checks if the middle element is a peak. If not, it moves towards the direction of the larger neighbor, as a peak is guaranteed to exist in that direction. The time complexity is O(log n) due to the binary search. The space complexity is O(1) as it uses a constant amount of extra space. The invariant maintained is that a peak element is guaranteed to exist within the `[left, right]` range during the binary search. Edge cases like single-element arrays or monotonic arrays are explicitly handled.

function find_peak_element(nums): n = length of nums if n == 1: return 0 if nums[0] > nums[1]: return 0 if nums[n-1] > nums[n-2]: return n-1 left = 1 right = n - 2 while left <= right: mid = (left + right) // 2 if nums[m...

This algorithm merges two sorted arrays, `arr2` into `arr1`, in-place. It assumes that `arr1` has enough capacity to hold all elements from both arrays. This is a common problem in memory-constrained environments or when...

The `merge_sorted_arrays_in_place` function uses three pointers: `p1` for the last element of the initialized part of `arr1`, `p2` for the last element of `arr2`, and `p_merged` for the last position in `arr1` where the merged element will be placed. The merging is done from right to left to ensure that elements from `arr1` are not overwritten before they are compared and moved. In each step of the main `while` loop, the larger element between `arr1[p1]` and `arr2[p2]` is placed at `arr1[p_merged]`, and the corresponding pointer is decremented. If `arr2` still has elements after `arr1` is exhausted (`p1 < 0`), the remaining elements of `arr2` are copied to the beginning of `arr1`. The time complexity is O(m + n) because each element from both arrays is examined and moved at most once. The space complexity is O(1) because the merge is performed in-place, using only a few extra variables for pointers.

function merge_sorted_arrays_in_place(array1, count1, array2, count2): pointer1 = count1 - 1 pointer2 = count2 - 1 merge_pointer = count1 + count2 - 1 while pointer1 is greater than or equal to 0 AND pointer2 is greater...

This function reverses the order of words in a given string. It's a common text processing task used in natural language processing, data cleaning, or for simple string manipulation exercises. For example, if you have a...

The `reverse_words` function first handles the edge case of an empty input string. It then uses the `split()` method, which by default splits on whitespace and discards empty strings resulting from multiple spaces, effectively giving us a list of actual words. The `reverse()` method is then called on this list to reverse the order of the words in place. Finally, `join()` is used with a single space as a separator to reconstruct the string with the words in reversed order. The time complexity is dominated by the split and join operations, which are typically O(n) where n is the length of the string. The space complexity is O(n) to store the list of words.

function reverse_words(string s): if s is empty: return "" words = split s by whitespace, removing empty results reverse the list of words return join words with a single space

This algorithm finds the second smallest element in an unsorted array of numbers. It iterates through the array, keeping track of the smallest and second smallest elements encountered so far. This is useful in scenarios...

The `find_second_smallest` function initializes `smallest` and `second_smallest` to positive infinity. It then iterates through the input array `arr`. If the current number `num` is smaller than `smallest`, it means we've found a new smallest element; the old `smallest` becomes the new `second_smallest`, and `num` becomes the new `smallest`. If `num` is not smaller than `smallest` but is smaller than `second_smallest` and not equal to `smallest` (to handle duplicates), then `num` becomes the new `second_smallest`. The time complexity is O(n) because we iterate through the array once. The space complexity is O(1) as we only use a few extra variables. Edge cases include empty arrays, arrays with one element, and arrays where all elements are the same, which are handled by initial checks and the final return condition.

function find_second_smallest(array): if array is empty or has less than 2 elements: return None smallest = infinity second_smallest = infinity for each number in array: if number < smallest: second_smallest = smallest s...

This algorithm implements binary search recursively to find the index of a target element in a sorted array. Binary search is a highly efficient searching algorithm that works by repeatedly dividing the search interval i...

The recursive `binary_search_recursive` function operates on a sorted array `arr` within a defined range `[low, high]`. The primary base case is when `high < low`, indicating the search interval is empty and the element is not found, returning -1. Otherwise, it calculates the middle index `mid`. If `arr[mid]` equals the target `x`, the index `mid` is returned. If `arr[mid]` is greater than `x`, the search continues recursively on the left half (`low` to `mid - 1`). If `arr[mid]` is less than `x`, the search continues on the right half (`mid + 1` to `high`). The time complexity is O(log n) due to the halving of the search space in each recursive call. The space complexity is O(log n) due to the recursion depth on the call stack. Correctness relies on the array being sorted and the proper adjustment of `low` and `high` indices in recursive calls to ensure termination and coverage of all possibilities.

function binary_search_recursive(array, low_index, high_index, target_element): if high_index is less than low_index: return -1 // Element not found middle_index = floor((low_index + high_index) / 2) if array[middle_inde...