#882

Hard

auto_awesomeGraph plus Heap (Priority Queue)

LeetCode Problem Workspace

Reachable Nodes In Subdivided Graph

The Reachable Nodes In Subdivided Graph problem requires efficiently finding the reachable nodes using graph traversal and heap-based optimization.

graph heap priority queue shortest path

Problem Statement

You are given an undirected graph with n nodes, labeled from 0 to n-1, and a list of edges where each edge connects two nodes. The edges also contain a count indicating how many new nodes will be inserted along the edge, subdividing it into chains of nodes. A count of 0 means the edge is not subdivided, while a positive count means the edge is subdivided into multiple new nodes connected by new edges.

You are also given a maximum number of moves, maxMoves. Your goal is to determine how many nodes can be reached starting from node 0, considering that each move traverses one edge or chain of subdivided edges. The graph is represented by a list of edges, and you must calculate the maximum number of reachable nodes within the given move limit.

Examples

Example 1

Input: edges = [[0,1,10],[0,2,1],[1,2,2]], maxMoves = 6, n = 3

Output: 13

The edge subdivisions are shown in the image above. The nodes that are reachable are highlighted in yellow.

Example 2

Input: edges = [[0,1,4],[1,2,6],[0,2,8],[1,3,1]], maxMoves = 10, n = 4

Output: 23

Example details omitted.

Example 3

Input: edges = [[1,2,4],[1,4,5],[1,3,1],[2,3,4],[3,4,5]], maxMoves = 17, n = 5

Output: 1

Node 0 is disconnected from the rest of the graph, so only node 0 is reachable.

Constraints

0 <= edges.length <= min(n * (n - 1) / 2, 104)
edges[i].length == 3
0 <= ui < vi < n
There are no multiple edges in the graph.
0 <= cnti <= 104
0 <= maxMoves <= 109
1 <= n <= 3000

Solution Approach

Graph Transformation and Representation

Start by transforming the graph as described. Each edge is subdivided, and each new node created by an edge subdivision should be represented in the graph. This transforms the graph into a more complex structure that includes all intermediate nodes.

Priority Queue for Shortest Path Search

Utilize a priority queue (min-heap) to perform a shortest path search. The priority queue helps track the shortest available path from the starting node (node 0) to any other node, ensuring that nodes are explored efficiently within the move limit.

Efficient Reachability Calculation

Using the priority queue, traverse the graph while keeping track of the number of moves taken. If a node can be reached within the move limit, count it as reachable. The goal is to maximize the number of reachable nodes while adhering to the move constraint.

Complexity Analysis

Metric	Value
Time	O(E \log N)
Space	O(E)

The time complexity of the solution is O(E \log N), where E is the number of edges and N is the number of nodes, due to the priority queue operations. The space complexity is O(E), as the graph is represented by a list of edges, and each edge may include additional nodes resulting from edge subdivisions.

What Interviewers Usually Probe

Understanding graph traversal techniques and shortest path algorithms is key to solving this problem effectively.
Look for familiarity with heaps or priority queues, as they are essential for managing the node traversal efficiently.
The ability to handle edge transformations and correctly manage graph subdivisions will signal strong problem-solving skills.

Common Pitfalls or Variants

Common pitfalls

Mismanaging edge subdivisions can lead to incorrect graph representations, making it hard to track reachable nodes.
Overcomplicating the algorithm or using inefficient traversal techniques can result in suboptimal performance, especially with larger graphs.
Failure to properly account for the move limit and correctly implementing the priority queue could lead to incorrect answers.

Follow-up variants

Increase the size of the graph and test how well the algorithm scales for larger input sizes.
Add restrictions on the maximum number of new nodes that can be added for each edge, impacting the complexity of the transformation.
Change the traversal rule to consider different starting nodes, altering the reachability calculation.

FAQ

What is the key pattern in the Reachable Nodes In Subdivided Graph problem?

The key pattern involves graph transformations combined with a heap-based priority queue to efficiently solve the shortest path problem within a given number of moves.

How do I handle edge subdivisions in this problem?

You must transform each edge into a chain of new nodes, creating additional edges between them. The number of new nodes depends on the 'cnti' value for each edge.

What is the best way to manage move limits in this problem?

Use a priority queue to efficiently track the shortest path to each node, ensuring that the number of moves does not exceed the given limit.

How does a priority queue help solve this problem?

A priority queue helps by efficiently finding the shortest available path from the starting node to other nodes, minimizing the number of moves used to reach each node.

Can I use BFS or DFS for this problem instead of a priority queue?

While BFS or DFS can be used in simpler cases, a priority queue is more efficient for handling the shortest path problem within a move constraint, especially with graph transformations.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def reachableNodes(self, edges: List[List[int]], maxMoves: int, n: int) -> int:
        g = defaultdict(list)
        for u, v, cnt in edges:
            g[u].append((v, cnt + 1))
            g[v].append((u, cnt + 1))
        q = [(0, 0)]
        dist = [0] + [inf] * n
        while q:
            d, u = heappop(q)
            for v, cnt in g[u]:
                if (t := d + cnt) < dist[v]:
                    dist[v] = t
                    q.append((t, v))
        ans = sum(d <= maxMoves for d in dist)
        for u, v, cnt in edges:
            a = min(cnt, max(0, maxMoves - dist[u]))
            b = min(cnt, max(0, maxMoves - dist[v]))
            ans += min(cnt, a + b)
        return ans

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