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question:def longest_zigzag_subsequence(arr: List[int]) -> int: Given a list of integers, return the length of the longest zigzag subsequence. A zigzag subsequence is defined as one where consecutive elements alternately increase and decrease. >>> longest_zigzag_subsequence([1, 7, 4, 9, 2, 5]) 6 >>> longest_zigzag_subsequence([1, 4, 7, 2, 5]) 4 >>> longest_zigzag_subsequence([10, 10, 10]) 1 >>> longest_zigzag_subsequence([42]) 1 >>> longest_zigzag_subsequence([1, 3, 2, 4, 3, 5, 4]) 7 >>> longest_zigzag_subsequence([1, 2, 3, 4, 5, 6, 7]) 2 >>> longest_zigzag_subsequence([7, 6, 5, 4, 3, 2, 1]) 2 >>> longest_zigzag_subsequence([1, 1, 1, 1, 1]) 1 >>> longest_zigzag_subsequence([3, 1, 2, 5, 3, 5, 6, 2, 3, 8]) 7

answer:def longest_zigzag_subsequence(arr): n = len(arr) if n == 0: return 0 # Initialize the tables up = [1] * n down = [1] * n # Fill the tables for i in range(1, n): for j in range(0, i): if arr[i] > arr[j]: up[i] = max(up[i], down[j] + 1) elif arr[i] < arr[j]: down[i] = max(down[i], up[j] + 1) # The length of the longest zigzag subsequence return max(max(up), max(down)) # Example usage: # arr = [1, 7, 4, 9, 2, 5] # print(longest_zigzag_subsequence(arr)) # Output: 6

question:def three_sum_exists(arr: list[int], target: int) -> str: Determines if there exist three distinct elements in the array that add up to the target sum. >>> three_sum_exists([1, 4, 45, 6, 10, 8], 22) "YES" >>> three_sum_exists([1, 2, 3, 4], 10) "NO" >>> three_sum_exists([0, -1, 2, -3, 1], 0) "YES" >>> three_sum_exists([1, 2, 3, 5, 7], 10) "YES" >>> three_sum_exists([1, -2, 1, 0, 5], 2) "YES" >>> three_sum_exists(list(range(-50000, 50000)), -3) "YES"

answer:def three_sum_exists(arr, target): Determines if there exist three distinct elements in the array that add up to the target sum. Parameters: arr (list): List of integers. target (int): The target sum to find. Returns: str: 'YES' if such elements exist, otherwise 'NO'. arr.sort() n = len(arr) for i in range(n - 2): left, right = i + 1, n - 1 while left < right: current_sum = arr[i] + arr[left] + arr[right] if current_sum == target: return "YES" elif current_sum < target: left += 1 else: right -= 1 return "NO"

question:def minPops(grid: List[List[int]]) -> int: Determines the minimum number of 'pop' operations required to reduce all values in the grid to zero. >>> minPops([[1, 1, 1], [1, 1, 1], [1, 1, 1]]) 1 >>> minPops([[2, 2, 2], [2, 2, 2], [2, 2, 2]]) 4 >>> minPops([[3]]) 3 >>> minPops([[0, 0, 0], [0, 0, 0], [0, 0, 0]]) 0 >>> minPops([[3, 1, 2], [1, 4, 1], [2, 1, 3]]) 4 >>> minPops([[10, 2, 3], [100, 2, 6], [5, 6, 10]]) 100

answer:def minPops(grid): Determines the minimum number of 'pop' operations required to reduce all values in the grid to zero. :param grid: List[List[int]] - a 2D grid of non-negative integers :returns: int - the minimum number of 'pop' operations required n = len(grid) m = len(grid[0]) if grid else 0 if n == 0 or m == 0: return 0 max_pops = 0 for i in range(n): for j in range(m): max_pops = max(max_pops, grid[i][j]) return max_pops

question:def min_votes_to_win(test_cases): Determine the minimum number of votes a candidate needs to secure the first place or be tied in first place. Args: test_cases (List[List[int]]): A list of test cases, each containing a list of integers representing the votes each candidate received. Returns: List[int]: A list of integers representing the minimum number of additional votes needed for each test case to secure the first place or be tied in first place. Example: >>> min_votes_to_win([[10, 20, 15]]) [11] >>> min_votes_to_win([[5, 5, 5, 5]]) [1] >>> min_votes_to_win([[0, 0, 1, 0]]) [2] >>> min_votes_to_win([[100, 0, 1000, 50]]) [1001] >>> min_votes_to_win([[1]]) [1] def read_input_and_calculate(input_string): Read input string and calculate the results. Args: input_string (str): Input string containing the number of test cases followed by the votes each candidate received. Returns: List[int]: A list of integers representing the minimum number of additional votes needed for each test case to secure the first place or be tied in first place. Example: >>> input_string = "2n3n10 20 15n4n5 5 5 5" >>> read_input_and_calculate(input_string) [11, 1] >>> input_string = "1n4n0 0 1 0" >>> read_input_and_calculate(input_string) [2] >>> input_string = "1n4n100 0 1000 50" >>> read_input_and_calculate(input_string) [1001]

answer:def min_votes_to_win(test_cases): results = [] for votes in test_cases: max_votes = max(votes) min_votes_needed = max_votes + 1 - min(votes) results.append(min_votes_needed) return results def read_input_and_calculate(input_string): input_list = input_string.strip().split() index = 0 T = int(input_list[index]) index += 1 test_cases = [] for _ in range(T): N = int(input_list[index]) index += 1 votes = list(map(int, input_list[index:index + N])) index += N test_cases.append(votes) return min_votes_to_win(test_cases)