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question:Vera is organizing a competition, and she has decided to use a simple yet interesting game involving arrays. Each participant is given an array of integers, and the goal is to transform the array so that each element in the new array is the product of the integers at the remaining positions in the original array. Specifically, given an array of integers `a` of length `n`, you need to construct an array `b` such that `b[i]` is equal to the product of all elements of array `a` except `a[i]`. The rules: 1. You cannot use the division operation. 2. Your solution should have time complexity better than O(n^2). Input: - The first line contains an integer `n` (1 ≤ n ≤ 10^5), the number of elements in the array. - The second line contains `n` space-separated integers `a1, a2, ..., an` (-10 ≤ ai ≤ 10, ai ≠ 0) representing the array `a`. Output: - Output `n` space-separated integers representing the array `b`. Example: Input: ``` 4 1 2 3 4 ``` Output: ``` 24 12 8 6 ``` Explanation: - For `b[0]`: The product of all elements except `a[0]` which is `1`, so `2 * 3 * 4 = 24`. - For `b[1]`: The product of all elements except `a[1]` which is `2`, so `1 * 3 * 4 = 12`. - For `b[2]`: The product of all elements except `a[2]` which is `3`, so `1 * 2 * 4 = 8`. - For `b[3]`: The product of all elements except `a[3]` which is `4`, so `1 * 2 * 3 = 6`.
answer:def product_except_self(nums): Returns an array such that each element at index i is the product of all numbers in the array except nums[i]. n = len(nums) if n == 0: return [] # Initialize two arrays for left and right products left_products = [1] * n right_products = [1] * n output = [1] * n # Calculate left products for i in range(1, n): left_products[i] = left_products[i - 1] * nums[i - 1] # Calculate right products for i in range(n - 2, -1, -1): right_products[i] = right_products[i + 1] * nums[i + 1] # Calculate the output array for i in range(n): output[i] = left_products[i] * right_products[i] return output
question:You are part of the cybersecurity team at a software company. Your task is to analyze logs to find potential security breaches. Each log entry consists of a timestamp and a user ID that indicates which user accessed the system. You have to determine the intervals during which a user was logged in the system. A user is considered logged in if there is no more than a specified interval duration between consecutive log entries for that user. You are given n log entries and a maximum interval duration in seconds. Your goal is to count the number of intervals during which each user was logged in. The logs are provided in non-decreasing order of timestamps. The first line of the input contains two integers n and d (1 ≤ n, d ≤ 100,000) — the number of log entries and the maximum interval duration in seconds. The next n lines each contain two integers ti and ui (1 ≤ ti ≤ 10^9, 1 ≤ ui ≤ 100,000) — the timestamp of the i-th log entry and the user ID of the i-th log entry, respectively. Output the number of login intervals for each user sorted by user ID. # Example **Input:** ``` 10 300 1 1 100 1 400 1 550 2 600 2 700 2 1000 1 1300 1 1400 1 1800 1 ``` **Output:** ``` 1 2 2 1 ``` **Explanation:** For user 1, there are two intervals: (1, 100) and (400, 1400). The third log entry of user 1 which occurs at 1800 is isolated and hence forms a new interval. Hence, the output is 2 intervals for user 1. For user 2, all log entries are within 300 seconds of each other, thus all entries form a single interval. Hence, the output is 1 interval for user 2.
answer:def count_login_intervals(n, d, logs): Returns the number of login intervals for each user based on the logs and maximum allowable interval duration d. The logs are sorted based on the user IDs and presented as a list of tuples (timestamp, user ID). from collections import defaultdict user_intervals = defaultdict(int) last_login_time = defaultdict(lambda: -1) for timestamp, user_id in logs: if last_login_time[user_id] == -1 or timestamp - last_login_time[user_id] > d: user_intervals[user_id] += 1 last_login_time[user_id] = timestamp result = sorted(user_intervals.items()) return result
question:A farmer has a rectangular farm divided into n times m grid cells. Some of these cells have trees planted in them, while others are empty. The farmer wants to build a rectangular barn on his farm such that none of the cells where the barn is built have trees. The task is to determine the area of the largest possible rectangular barn that can be built on the farm without overlapping any tree cells. The first line contains two integers n and m (1 le n, m le 10^3) — the dimensions of the farm. Each of the next n lines contains m characters. The j-th character in the i-th line is either '.' indicating an empty cell, or 'T' indicating a cell with a tree. Output a single integer — the area of the largest possible rectangular barn that can be built on the farm. # Example Input: ``` 4 5 ..... .T..T ..... ...T. ``` Output: ``` 6 ``` In this example, the largest rectangle without trees has an area of 6.
answer:def max_histogram_area(histogram): stack = [] max_area = 0 index = 0 while index < len(histogram): if not stack or histogram[stack[-1]] <= histogram[index]: stack.append(index) index += 1 else: top_of_stack = stack.pop() current_area = (histogram[top_of_stack] * ((index - stack[-1] - 1) if stack else index)) max_area = max(max_area, current_area) while stack: top_of_stack = stack.pop() current_area = (histogram[top_of_stack] * ((index - stack[-1] - 1) if stack else index)) max_area = max(max_area, current_area) return max_area def largest_rectangle_area(grid): if not grid or not grid[0]: return 0 n = len(grid) m = len(grid[0]) heights = [0] * m max_area = 0 for i in range(n): for j in range(m): if grid[i][j] == '.': heights[j] += 1 else: heights[j] = 0 max_area = max(max_area, max_histogram_area(heights)) return max_area def largest_rectangle_possible_barn(n, m, grid): return largest_rectangle_area(grid)
question:Alice has a collection of n unique and differently-sized stones, each with a positive integer weight. She wants to distribute these stones into two groups such that the absolute difference between the total weight of the stones in the first group and the total weight of the stones in the second group is minimized. Your task is to determine the minimum possible absolute difference. The first line of input contains an integer n (1 ≤ n ≤ 20) — the number of stones. The second line contains n space-separated integers w1, w2, ..., wn (1 ≤ wi ≤ 1000) — the weights of the stones. Output a single integer — the minimum possible absolute difference between the total weight of the stones in the two groups. **Example:** Input: ``` 5 1 2 3 4 5 ``` Output: ``` 1 ``` Explanation: Alice can divide the stones into two groups with weights 1+2+3=6 and 4+5=9. The absolute difference is |6-9|=3, which is suboptimal. However, dividing them into groups with weights 1+4=5 and 2+3+5=10 gives an absolute difference of |5-10|=5, which is better. Finally, dividing them into 1+2+4=7 and 3+5=8 provides the minimal difference of |7-8|=1.
answer:def min_weight_diff(n, weights): Returns the minimum possible absolute difference between the total weight of the stones in two groups. total_sum = sum(weights) dp = [False] * (total_sum + 1) dp[0] = True for weight in weights: for j in range(total_sum, weight - 1, -1): dp[j] = dp[j] or dp[j - weight] min_diff = float('inf') for s1 in range(total_sum // 2 + 1): if dp[s1]: s2 = total_sum - s1 min_diff = min(min_diff, abs(s1 - s2)) return min_diff