|
| 1 | +# Time: O((|E| + |V|) * log|V|) = O(|E| * log|V|), |
| 2 | +# if we can further to use Fibonacci heap, it would be O(|E| + |V| * log|V|) |
| 3 | +# Space: O(|E| + |V|) = O(|E|) |
| 4 | + |
| 5 | +# Starting with an undirected graph (the "original graph") |
| 6 | +# with nodes from 0 to N-1, subdivisions are made to some of the edges. |
| 7 | +# The graph is given as follows: edges[k] is a list of integer pairs |
| 8 | +# (i, j, n) such that (i, j) is an edge of the original graph, |
| 9 | +# |
| 10 | +# and n is the total number of new nodes on that edge. |
| 11 | +# |
| 12 | +# Then, the edge (i, j) is deleted from the original graph, |
| 13 | +# n new nodes (x_1, x_2, ..., x_n) are added to the original graph, |
| 14 | +# |
| 15 | +# and n+1 new edges (i, x_1), (x_1, x_2), (x_2, x_3), ..., (x_{n-1}, x_n), (x_n, j) |
| 16 | +# are added to the original graph. |
| 17 | +# |
| 18 | +# Now, you start at node 0 from the original graph, and in each move, |
| 19 | +# you travel along one edge. |
| 20 | +# |
| 21 | +# Return how many nodes you can reach in at most M moves. |
| 22 | +# |
| 23 | +# Example 1: |
| 24 | +# |
| 25 | +# Input: edges = [[0,1,10],[0,2,1],[1,2,2]], M = 6, N = 3 |
| 26 | +# Output: 13 |
| 27 | +# Explanation: |
| 28 | +# The nodes that are reachable in the final graph after M = 6 moves are indicated below. |
| 29 | +# |
| 30 | +# Example 2: |
| 31 | +# |
| 32 | +# Input: edges = [[0,1,4],[1,2,6],[0,2,8],[1,3,1]], M = 10, N = 4 |
| 33 | +# Output: 23 |
| 34 | +# |
| 35 | +# Note: |
| 36 | +# - 0 <= edges.length <= 10000 |
| 37 | +# - 0 <= edges[i][0] < edges[i][1] < N |
| 38 | +# - There does not exist any i != j for which |
| 39 | +# edges[i][0] == edges[j][0] and edges[i][1] == edges[j][1]. |
| 40 | +# - The original graph has no parallel edges. |
| 41 | +# - 0 <= edges[i][2] <= 10000 |
| 42 | +# - 0 <= M <= 10^9 |
| 43 | +# - 1 <= N <= 3000 |
| 44 | + |
| 45 | +import collections |
| 46 | + |
| 47 | +class Solution(object): |
| 48 | + def reachableNodes(self, edges, M, N): |
| 49 | + """ |
| 50 | + :type edges: List[List[int]] |
| 51 | + :type M: int |
| 52 | + :type N: int |
| 53 | + :rtype: int |
| 54 | + """ |
| 55 | + graph = collections.defaultdict(dict) |
| 56 | + for u, v, w in edges: |
| 57 | + graph[u][v] = graph[v][u] = w |
| 58 | + |
| 59 | + min_heap = [(0, 0)] |
| 60 | + best = collections.defaultdict(lambda: float("inf")) |
| 61 | + best[0] = 0 |
| 62 | + count = collections.defaultdict(lambda: collections.defaultdict(int)) |
| 63 | + result = 0 |
| 64 | + while min_heap: |
| 65 | + curr_total, node = heapq.heappop(min_heap) # O(|V|*log|V|) in total |
| 66 | + if best[node] < curr_total: |
| 67 | + continue |
| 68 | + result += 1 |
| 69 | + for nei, weight in graph[node].iteritems(): |
| 70 | + count[node][nei] = min(weight, |
| 71 | + M-curr_total) |
| 72 | + next_total = curr_total+weight+1 |
| 73 | + if next_total <= M and next_total < best[nei]: |
| 74 | + heapq.heappush(min_heap, (next_total, nei)) # binary heap O(|E|*log|V|) in total |
| 75 | + # Fibonacci heap O(|E|) in total |
| 76 | + best[nei] = next_total |
| 77 | + for u, v, w in edges: |
| 78 | + result += min(w, count[u][v]+count[v][u]) |
| 79 | + return result |
0 commit comments