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question:In a distant kingdom, there's a peculiar road network connecting its many cities. Each city has a bidirectional road to every other city. However, the cost to build and maintain these roads varies depending on the pair of cities. The kingdom's ruler has decided to construct a minimal-cost network that still allows travel between any pair of cities, directly or indirectly. You are tasked with finding the minimal total cost required to connect all the cities such that there is a path between any two cities. This is essentially a problem of finding the Minimum Spanning Tree (MST) for the given set of cities. The first line of the input contains an integer N (2 ≤ N ≤ 1000), representing the number of cities. The next N lines each contain N integers. The j-th integer on the i-th line, C[i][j], represents the cost to build a road between city i and city j (1 ≤ C[i][j] ≤ 10^6). It is guaranteed that C[i][j] = C[j][i] and C[i][i] = 0. Output a single integer, the minimal total cost required to construct the road network. # Example Input: ``` 3 0 1 3 1 0 2 3 2 0 ``` Output: ``` 3 ``` # Explanation For the given example, the optimal way to connect all cities with minimal cost is: - Connect city 1 to city 2 with a cost of 1 - Connect city 2 to city 3 with a cost of 2 Thus, the minimal total cost is 1 + 2 = 3.
answer:def find_mst_cost(n, cost_matrix): def find(parent, i): if parent[i] == i: return i else: return find(parent, parent[i]) def union(parent, rank, x, y): root_x = find(parent, x) root_y = find(parent, y) if root_x != root_y: if rank[root_x] > rank[root_y]: parent[root_y] = root_x elif rank[root_x] < rank[root_y]: parent[root_x] = root_y else: parent[root_y] = root_x rank[root_x] += 1 edges = [(cost_matrix[i][j], i, j) for i in range(n) for j in range(i + 1, n)] edges.sort() parent = [i for i in range(n)] rank = [0] * n mst_cost = 0 for cost, u, v in edges: if find(parent, u) != find(parent, v): union(parent, rank, u, v) mst_cost += cost return mst_cost def minimal_total_cost(n, cost_matrix): return find_mst_cost(n, cost_matrix)
question:You are given a string s of length n, which consists of lowercase English letters. You can choose up to k positions in the string and change the letter at each of those positions to any other lowercase English letter. You need to determine the length of the longest substring of repeating characters that can be achieved with at most k changes. Write a function that takes in the string s, the value of n, and the integer k, and returns the length of the longest substring of repeating characters that can be obtained by changing at most k characters. The first line contains an integer n (1 ≤ n ≤ 100,000), the length of the string. The second line contains an integer k (0 ≤ k ≤ n), the maximum number of characters that can be changed. The third line contains the string s, which consists of lowercase English letters. Output a single integer, the length of the longest substring of repeating characters that can be obtained by changing at most k characters. Example: Input: 6 2 aabccd Output: 4 Explanation: By changing the two 'c's to 'a', you get the string "aaaaccd", which contains a substring "aaaa" of length 4.
answer:def longest_substring_with_k_changes(n, k, s): Returns the length of the longest substring of repeating characters that can be obtained by changing at most k characters. from collections import defaultdict max_len = 0 max_count = 0 start = 0 count = defaultdict(int) for end in range(n): count[s[end]] += 1 max_count = max(max_count, count[s[end]]) while end - start + 1 - max_count > k: count[s[start]] -= 1 start += 1 max_len = max(max_len, end - start + 1) return max_len
question:Petra is fascinated by geometric patterns and has been playing around with various grid-based games. Recently, she came up with a new game on a grid and would like to challenge you with the following problem: You are given a 2D grid of size N times M, initially filled with zeros. Petra will provide you with a list of commands, each describing a rectangular region on the grid that needs to be incremented. Specifically, each command is of the form r1, c1, r2, c2 which describes a rectangle where all the elements inside this rectangle should be incremented by one (inclusive). Here, r1 and r2 are the starting and ending row indices, and c1 and c2 are the starting and ending column indices of the rectangle. After all the commands have been processed, your task is to return the final state of the grid. You will be provided the following inputs: - The first line contains two integers N and M (1 leq N, M leq 100) representing the number of rows and columns in the grid, respectively. - The second line contains an integer K (1 leq K leq 3000) representing the number of commands. - Each of the next K lines contains four integers r1, c1, r2, and c2 (1 leq r1 leq r2 leq N, 1 leq c1 leq c2 leq M) describing a command to increment all elements within the rectangle specified by these coordinates. Output the resulting grid after all commands have been applied. Example: Input: ``` 3 4 2 1 1 2 2 2 2 3 3 ``` Output: ``` 1 1 0 0 1 2 1 0 0 1 1 0 ``` Note: In the provided example, the grid size is 3x4. The first command increments the subgrid from (1, 1) to (2, 2). The second command increments the subgrid from (2, 2) to (3, 3). Consequently, at the end, the grid should be displayed as shown in the output.
answer:def update_grid(N, M, commands): Updates the grid of size N x M according to the given commands and returns the final grid. Parameters: N (int): number of rows in the grid M (int): number of columns in the grid commands (list of tuples): list of commands, each represented as (r1, c1, r2, c2) Returns: List[List[int]]: the resulting grid after all commands have been applied # Initialize the grid with zeros grid = [[0] * M for _ in range(N)] # Apply each command for r1, c1, r2, c2 in commands: for i in range(r1 - 1, r2): for j in range(c1 - 1, c2): grid[i][j] += 1 return grid def process_input_and_update_grid(N, M, K, command_list): return update_grid(N, M, command_list)
question:You are given a connected undirected graph with N nodes and M edges. Each edge has a weight, denoted by W_i. You need to process Q queries. In each query, you are given an integer K, and you need to determine the length of the shortest path from node 1 to node N where all the edges in the path have weights at most K. If there is no such path, return -1 for that query. # Input The first line contains three integers N, M, and Q (2 leq N leq 1000, 1 leq M leq 5000, 1 leq Q leq 1000) — the number of nodes, the number of edges, and the number of queries, respectively. Each of the next M lines contains three integers U_i, V_i, and W_i (1 leq U_i, V_i leq N, 1 leq W_i leq 10^6) — an edge between nodes U_i and V_i with weight W_i. Each of the next Q lines contains a single integer K (1 leq K leq 10^6) — the maximum allowed weight of the edges in the path. # Output For each query, print the length of the shortest path from node 1 to node N where all the edges in the path have weights at most K. If there is no such path, print -1. # Example Input ``` 4 5 3 1 2 5 2 3 1 3 4 1 1 4 10 2 4 5 2 6 10 ``` Output ``` -1 7 7 ```
answer:import heapq def dijkstra(graph, N, max_weight): distances = [float('inf')] * (N + 1) distances[1] = 0 min_heap = [(0, 1)] while min_heap: current_distance, current_node = heapq.heappop(min_heap) if current_distance > distances[current_node]: continue for neighbor, weight in graph[current_node]: if weight > max_weight: continue distance = current_distance + weight if distance < distances[neighbor]: distances[neighbor] = distance heapq.heappush(min_heap, (distance, neighbor)) return distances[N] if distances[N] != float('inf') else -1 def process_queries(N, M, edges, queries): graph = [[] for _ in range(N + 1)] for u, v, w in edges: graph[u].append((v, w)) graph[v].append((u, w)) results = [] for k in queries: result = dijkstra(graph, N, k) results.append(result) return results