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question:Nikhil is a math enthusiast and loves playing with numbers. One day, he encounters a problem where he has to find the sum of all even numbers up to a certain number 'k' (inclusive). He needs your help to compute the sums quickly for multiple values of 'k'. Input The first line consists of a number 'm' which specifies the number of test cases. 1 ≤ m ≤ 100. 'm' lines follow with a number 'k' on each line. 'k' can have values up to 10^18. 1 ≤ k ≤ 10^18. Output For each test case, output the sum of all even numbers up to 'k'. SAMPLE INPUT 2 4 10 SAMPLE OUTPUT 6 30 Explanation Sum of all even numbers up to 4 = 2+4 = 6 Sum of all even numbers up to 10 = 2+4+6+8+10 = 30
answer:def sum_of_evens_up_to_k(k): Computes the sum of all even numbers up to and including k. # Find the largest even number less than or equal to k if k % 2 != 0: k = k - 1 # The largest even number is k, so finding sum of first k/2 even numbers # Sum of first n even numbers is n * (n + 1) n = k // 2 return n * (n + 1) def process_test_cases(m, test_cases): results = [] for k in test_cases: results.append(sum_of_evens_up_to_k(k)) return results
question:Graphs are data structures that consist of a set of vertices and edges connecting pairs of vertices. They can be used to represent a variety of real-world systems such as social networks, computer networks, biological networks, and more. One fundamental problem in the context of graphs is determining the shortest path between vertices. Given a weighted graph where edges have associated weights, Dijkstra's algorithm can be used to find the shortest path from a starting vertex to all other vertices in the graph. The graph is represented by adjacency lists for each vertex, containing pairs of adjacent vertex and edge weight. For a vertex u with adjacent vertex v and edge weight w, the adjacency list will contain the pair (v, w). The pseudocode for Dijkstra's algorithm is as follows: 1 Dijkstra(graph, start) 2 for each vertex v in graph.vertices 3 dist[v] = infinity 4 prev[v] = undefined 5 dist[start] = 0 6 Q = graph.vertices 7 8 while Q is not empty 9 u = vertex in Q with min dist[u] 10 remove u from Q 11 12 for each neighbor (v, w) of u 13 alt = dist[u] + w 14 if alt < dist[v] 15 dist[v] = alt 16 prev[v] = u 17 18 return dist, prev Given a weighted undirected graph and a starting vertex, your task is to implement Dijkstra's algorithm to find the shortest paths from the starting vertex to all other vertices. Input: - The first line contains two integers n and e, representing the number of vertices and edges in the graph, respectively. (1 leq n leq 1000, 0 leq e leq 10000) - The next e lines each contain three integers u, v, and w indicating an edge between vertices u and v with weight w. (1 leq u, v leq n, 1 leq w leq 1000) - The last line contains an integer s, the starting vertex. (1 leq s leq n) Output: - One line for each vertex v (in ascending order of vertex index) except the starting vertex s, showing the length of the shortest path from s to v. If there is no path from s to v, output "INF". Example: Input: 5 6 1 2 2 1 3 4 2 3 1 2 4 7 3 5 3 4 5 1 1 Output: 2 3 INF 6
answer:import heapq def dijkstra(n, edges, start): # Create adjacency list for the graph graph = {i: [] for i in range(1, n + 1)} for u, v, w in edges: graph[u].append((v, w)) graph[v].append((u, w)) # Undirected graph # Distance and previous vertex arrays dist = {i: float('inf') for i in range(1, n + 1)} dist[start] = 0 prev = {i: None for i in range(1, n + 1)} # Priority queue to hold the vertices to explore pq = [(0, start)] # (distance, vertex) while pq: current_dist, u = heapq.heappop(pq) if current_dist > dist[u]: continue for v, weight in graph[u]: alt = current_dist + weight if alt < dist[v]: dist[v] = alt prev[v] = u heapq.heappush(pq, (alt, v)) result = [] for i in range(1, n + 1): if i != start: result.append(str(dist[i]) if dist[i] != float('inf') else "INF") return result
question:Given a string representing a sentence, rearrange the letters of the words in the sentence in alphabetical order. Input Format A single line of input containing a sentence, text{S}. Constraints 0 < text{len(S)} < 1000 The string consists of alphabetic characters and spaces only. Output Format Print the rearranged sentence with each word's letters sorted in alphabetical order. Sample Input hello world Sample Output ehllo dlorw
answer:def rearrange_sentence(sentence): Returns the sentence with each word's letters sorted in alphabetical order. words = sentence.split() sorted_words = [''.join(sorted(word)) for word in words] return ' '.join(sorted_words)
question:A software company is developing a system to manage their inventory of software packages. Each package has a unique ID, a price, and a list of dependencies. A dependency indicates that a package cannot be installed unless another specific package is also installed. Each package is listed in a JSON-like format with the following keys: - "id": a unique string identifying the package. - "price": an integer representing the cost of the package in dollars. - "dependencies": a list of package IDs that this package depends on. Your task is to determine the total cost of installing a specific package, considering the cost of any packages it depends on directly or indirectly. If a package has multiple dependencies, all must be installed. Input: The first line of the input contains an integer T, denoting the number of test cases. Each test case contains the following: - A single line containing a package ID, identifying the package for which the total installation cost needs to be calculated. - A list of package descriptions. Each description is a JSON-like object as described above. Output: For each test case, print the total cost of installing the specified package and all its dependencies. Constraints: - The number of packages (unique IDs) will be between 1 and 1,000 elements, inclusive. - Packages will not have circular dependencies. - Each "price" value will be between 1 and 10,000, inclusive. Assume valid input, where all dependencies listed are also present in the given list of packages. SAMPLE INPUT: 2 B [ {"id": "A", "price": 50, "dependencies": []}, {"id": "B", "price": 30, "dependencies": ["A"]}, {"id": "C", "price": 40, "dependencies": ["B"]} ] A [ {"id": "A", "price": 50, "dependencies": []}, {"id": "B", "price": 30, "dependencies": ["A"]}, {"id": "C", "price": 40, "dependencies": ["B"]} ] SAMPLE OUTPUT: 80 50 Explanation: Test Case #1: To install package "B", we must first install its dependency "A". The cost of package "A" is 50, and the cost of "B" is 30, making the total installation cost 50+30=80. Test Case #2: To install package "A" itself, there are no dependencies, so the total installation cost is simply the cost of "A", which is 50.
answer:def total_installation_cost(target_id, packages): Calculate the total cost of installing a package including its dependencies. :param target_id: The id of the package to install. :param packages: A list of package descriptions in JSON-like format. :return: The total installation cost. package_map = {pkg['id']: pkg for pkg in packages} def get_cost(pkg_id): pkg = package_map[pkg_id] cost = pkg['price'] for dep_id in pkg['dependencies']: cost += get_cost(dep_id) return cost return get_cost(target_id)