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Below is a master-level programming assignment question completed by our expert, followed by a concise, production-ready solution and explanation.
Master-Level Question:
Design and implement an algorithm to compute the minimum cost path in a directed acyclic graph (DAG) with up to 10^5 nodes and 2×10^5 edges, where each edge has a non-negative weight. The algorithm must run in O(V+E) time and return both the minimum distance from a given source to all nodes and the predecessor list for path reconstruction.
Solution:
Because the graph is a DAG, we can leverage topological sorting to relax edges in linear time. First, compute a topological order using Kahn’s algorithm or DFS. Initialize distances with infinity, set the source distance to zero, and maintain a predecessor array. Then, process nodes in topological order, relaxing outgoing edges: for each edge (u, v, w), if dist[u] + w dist[v], update dist[v] and set pred[v] = u. This guarantees each edge is considered once, achieving O(V+E).
Pseudocode:
topo = TopologicalSort(G)
for each v in V: dist[v] = ∞, pred[v] = NIL
dist[source] = 0
for u in topo:
for each (u, v, w) in Adj[u]:
Return dist, pred.
Correctness follows from the property that in a topological order, all edges go from earlier to later vertices, so when u is processed, dist[u] is final. The predecessor array enables reconstruction of the shortest path by backtracking from any target node to the source. This approach scales to large DAGs and is optimal under the given constraints.
Students who request help with programming assignment at an advanced level benefit from such domain-specific strategies, rigorous complexity analysis, and implementation-ready guidance. Our experts ensure that solutions are not only correct but also aligned with academic standards, including clear documentation and edge-case handling.
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