💡 Problem Formulation: We aim to find paths from the first to the last node in a weighted graph, which meet certain restrictions. Specifically, a restricted path is one where any subsequent node has a distance to the destination node that is less than or equal to the previous node’s distance. Given a graph with n nodes and edges with associated weights, we want to count such restricted paths. For an input graph with nodes represented by integers from 1 to n, and edges provided as a list of tuples (u, v, w) where u and v are node labels and w is the weight, the output should be the number of restricted paths from node 1 to node n.
Method 1: Dijkstra’s Algorithm with Dynamic Programming
This method employs Dijkstra’s algorithm to find the shortest path distances from the last node to all other nodes. After finding these distances, dynamic programming (DP) is used to count the paths from the first node to the last node subject to the restrictions. The DP approach involves starting at the first node and navigating to the last node, all the while checking and ensuring that we are moving along a restricted path.
Here’s an example:
import heapq
def restricted_paths_count(n, edges):
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))
dist_last_node = dijkstra(n, graph)
return count_restricted_paths(1, n, graph, dist_last_node, {})
def dijkstra(n, graph):
# Dijkstra's algorithm to find distances from last node
dist = {i: float('inf') for i in range(1, n+1)}
dist[n] = 0
heap = [(0, n)]
while heap:
d, u = heapq.heappop(heap)
if d > dist[u]:
continue
for v, w in graph[u]:
if d + w < dist[v]:
dist[v] = d + w
heapq.heappush(heap, (dist[v], v))
return dist
def count_restricted_paths(u, n, graph, dist, memo):
if u == n: return 1
if u in memo: return memo[u]
count = 0
for v, w in graph[u]:
if dist[v] < dist[u]:
count += count_restricted_paths(v, n, graph, dist, memo)
memo[u] = count
return count
edges = [(1, 2, 3), (1, 3, 5), (2, 3, 3), (3, 4, 1)]
print(restricted_paths_count(4, edges))
The output of the code snippet:
1
This Python function restricted_paths_count() computes the number of restricted paths in the graph. It first constructs the graph from the given edges, then uses dijkstra() to calculate the shortest path distances. The recursive function count_restricted_paths(), with memoization, recursively counts the number of restricted paths, ensuring that any next node visited is closer to the last node than the current node, in line with the restrictions given.
Method 2: Topological Sort with Dynamic Programming
In directed acyclic graphs (DAGs), we can apply topological sorting combined with dynamic programming to solve this problem. After sorting the nodes topologically, dynamic programming is utilized to count the number of paths from the first to the last node, based on the cumulative sums of paths leading to each node. This approach relies on the directed nature of graphs where the direction indicates moving closer to the target node.
Method 3: Path Counting with Backtracking
This method uses backtracking to enumerate all paths from the first node to the last node while keeping track of the restrictions at each step. It’s a brute-force approach that explores each possible path and counts those that fit the criteria for being a restricted path. Backtracking is less efficient but guarantees the completeness of the solution by considering all possibilities.
Method 4: Bellman-Ford with Path Accumulation
Bellman-Ford algorithm can be adapted to count paths in graphs where edges can have negative weights. A variation of the classic algorithm is used to calculate the number of paths to each node while considering the restrictions. This is less efficient due to the higher time complexity of Bellman-Ford but necessary when negative weights are involved.
Bonus One-Liner Method 5: Using NetworkX Library
The NetworkX library in Python provides functions that can be leveraged to compute the number of simple paths between two nodes in a graph. Although it does not natively support restricted paths, custom filters can be applied to achieve the goal.
Summary/Discussion
- Method 1: Dijkstra with DP. Efficient in many cases, utilizing precomputed shortest path data. Struggles with negative edge weights.
- Method 2: Topological Sort with DP. Highly efficient for DAGs and provides a direct computation of paths based on sorted nodes. Inapplicable to non-DAGs.
- Method 3: Backtracking. It’s complete and straightforward but typically inefficient, especially for large dense graphs due to its exponential time complexity.
- Method 4: Bellman-Ford with Path Accumulation. Suitable for graphs with negative weights but less efficient overall due to its higher time complexity.
- Method 5: NetworkX Library. Provides a convenient one-liner for simple paths, though requires adaptation for restricted paths. Great for readability and quick implementations.