Appearance
question:You are given n computers in a network. The network has become very complex with multiple direct connections between pairs of computers. Some connections might introduce a security vulnerability and hence need to be identified. For security reasons, we need to find all the "critical connections" in the network. A connection is critical if its removal causes the network to become disconnected. The network is represented as an undirected graph with nodes labeled from 0 to n-1, and an array `connections` where `connections[i] = [u, v]` represents a connection between nodes `u` and `v`. Write a function `criticalConnections` to find all critical connections in the network. You should return the list of critical connections in any order. # Function Signature ```python def criticalConnections(n: int, connections: List[List[int]]) -> List[List[int]]: ``` # Input - `n`: an integer representing the number of computers in the network (1 ≤ n ≤ 10^5). - `connections`: a list of lists where each inner list contains two integers `u` and `v` (0 ≤ u, v < n). # Output - A list of lists where each inner list represents a critical connection `[u, v]`. # Example ```python n = 4 connections = [[0,1],[1,2],[2,0],[1,3]] print(criticalConnections(n, connections)) # Output: [[1,3]] ``` Explanation: In the given example, there are 4 computers with connections between them as specified. The removal of connection [1,3] will cause the network to be disconnected (either node 1 or node 3 gets isolated), hence it is a critical connection. # Note - There might be multiple critical connections in the network. - You may return the critical connections in any order. - Each connection `[u, v]` appears at most once in the list `connections`. Your solution should use an efficient graph algorithm to ensure it runs within reasonable time limits for large inputs, such as 10^5 nodes and connections.
answer:from typing import List, Dict def criticalConnections(n: int, connections: List[List[int]]) -> List[List[int]]: graph = {i: [] for i in range(n)} for u, v in connections: graph[u].append(v) graph[v].append(u) discovery = [-1] * n lowest = [-1] * n result = [] time = [0] def dfs(node: int, parent: int) -> None: discovery[node] = lowest[node] = time[0] time[0] += 1 for neighbor in graph[node]: if neighbor == parent: continue if discovery[neighbor] == -1: dfs(neighbor, node) lowest[node] = min(lowest[node], lowest[neighbor]) if lowest[neighbor] > discovery[node]: result.append([node, neighbor]) else: lowest[node] = min(lowest[node], discovery[neighbor]) for i in range(n): if discovery[i] == -1: dfs(i, -1) return result
question:You are given an array of n integers. We define a contiguous subarray of the array to be good if the number of distinct integers in the subarray is exactly k. Your task is to find the number of good subarrays. The first line contains two integers n and k (1 ≤ k ≤ n ≤ 2 * 10^5) — the size of the array and the exact number of distinct integers that must be in the subarray. The second line contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9) — the elements of the array. Print one integer — the number of good subarrays. **Example:** Input: ``` 5 2 1 2 1 2 3 ``` Output: ``` 7 ``` Explanation: The good subarrays are [1, 2], [2, 1], [1, 2], [2, 1, 2], [1, 2, 3], [2, 1, 2, 3], [1, 2, 3]. Total count is 7. In the first sample, there are 7 good subarrays in total.
answer:from collections import defaultdict def count_good_subarrays(n, k, arr): Counts the number of good subarrays with exactly k distinct integers. def at_most_k_distinct(arr, k): count = defaultdict(int) i = 0 result = 0 for j in range(len(arr)): if count[arr[j]] == 0: k -= 1 count[arr[j]] += 1 while k < 0: count[arr[i]] -= 1 if count[arr[i]] == 0: k += 1 i += 1 result += j - i + 1 return result # Number of subarrays with at most k distinct integers at_most_k = at_most_k_distinct(arr, k) # Number of subarrays with at most k-1 distinct integers at_most_k_minus_1 = at_most_k_distinct(arr, k - 1) return at_most_k - at_most_k_minus_1
question:You have a round table with n seats numbered from 1 to n in clockwise order. There are n guests, each guest has a preferred seat. If a guest can't sit in their preferred seat, they will take the next available seat in the clockwise direction. Write a program to determine the seating order of the guests such that each guest either sits in their preferred seat or the next available one in a clockwise direction. The first line contains a single positive integer n (1 ≤ n ≤ 1000) — the number of seats and guests. The second line contains n space-separated integers a1, a2, ..., an (1 ≤ ai ≤ n) — the preferred seat for each of the n guests. Print n space-separated integers representing the seating order of the guests, i.e., the seat number each guest will occupy in the order of their guest number. Note: - If multiple guests prefer the same seat, the first guest in input order will take that seat, and the subsequent guests will move to the next available seat found in clockwise order. Examples: Input 3 3 1 2 Output 2 3 1 Input 5 2 2 2 2 2 Output 2 3 4 5 1
answer:def seating_order(n, preferences): Determine the seating order for n guests with their preferred seats. :param n: Number of seats and guests :param preferences: List of integers representing the preferred seat for each guest :return: List of integers representing the seating order for each guest occupied = [False] * n # Track occupied seats result = [0] * n # To store the seat each guest occupies for index, pref in enumerate(preferences): seat = pref - 1 # Convert to 0-based index while occupied[seat]: seat = (seat + 1) % n occupied[seat] = True # Mark the seat as occupied result[index] = seat + 1 # Convert back to 1-based index return result
question:In the magical world of Algoria, mages have a unique way of storing their magical energy. Each mage has a set amount of energy, represented as an integer e (1 ≤ e ≤ 10^9). Mages can split their energy into any number of parts and redistribute it among themselves. However, each part must be an exact power of 2 (i.e., 2^0, 2^1, 2^2, etc.). For instance, a mage with an energy of 9 can split it into 8 + 1. You've been given an integer e representing the energy of a single mage. Your task is to determine the smallest number of parts the mage can split their energy into, such that each part is an exact power of 2. The input consists of a single line containing the integer e. Output the smallest number of parts into which the energy can be split. For example, if the mage has an energy of 13, the optimal split would be 8 + 4 + 1, which sums up to 3 parts. For an energy of 1, the split is just 1, which is 1 part. In the first example, the optimal way to split is into the parts 8, 4, and 1, totaling 3 parts. In the second example, the energy is already in its simplest part which is 1, totaling 1 part.
answer:def count_parts_of_two(e): Returns the smallest number of parts into which the energy can be split, such that each part is an exact power of 2. count = 0 while e > 0: count += e & 1 e >>= 1 return count