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🧮combinatorics review

Single-source shortest path

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The single-source shortest path problem involves finding the shortest paths from a given source vertex to all other vertices in a graph. This is a fundamental concept in graph theory and plays a crucial role in various applications, such as network routing and geographic information systems. By determining the most efficient way to reach other nodes, algorithms can optimize connectivity and resource allocation.

5 Must Know Facts For Your Next Test

  1. Single-source shortest path algorithms typically assume non-negative edge weights, as negative weights can complicate the solution process.
  2. Dijkstra's Algorithm is efficient for graphs with non-negative weights, while Bellman-Ford is used when negative weights are present.
  3. The time complexity of Dijkstra's Algorithm can vary based on the data structures used, typically ranging from O(V^2) with simple arrays to O(E + V log V) with priority queues.
  4. In unweighted graphs, the single-source shortest path can be solved using Breadth-First Search (BFS), treating all edges as having equal weight.
  5. The concept of single-source shortest path is widely applicable in real-world scenarios like GPS navigation systems, where finding the quickest route is essential.

Review Questions

  • How does Dijkstra's Algorithm function in solving the single-source shortest path problem?
    • Dijkstra's Algorithm starts by initializing distances from the source vertex to all other vertices as infinity, except for the source itself, which is zero. It then repeatedly selects the vertex with the smallest known distance, explores its neighbors, and updates their distances based on the current vertex. This process continues until all vertices have been visited, ensuring that the shortest paths are found for each vertex relative to the source.
  • What are the key differences between Dijkstra's Algorithm and Bellman-Ford in addressing single-source shortest path problems?
    • Dijkstra's Algorithm works optimally with non-negative edge weights and efficiently finds shortest paths in O(E + V log V) time using priority queues. In contrast, Bellman-Ford can handle negative edge weights and detects negative cycles, but it has a higher time complexity of O(VE). While Dijkstra’s is generally faster in practice for suitable graphs, Bellman-Ford is crucial when negative weights are involved.
  • Evaluate how different types of graphs (weighted vs. unweighted) impact the choice of algorithm for solving single-source shortest paths.
    • The type of graph significantly influences the choice of algorithm for single-source shortest paths. In weighted graphs with non-negative weights, Dijkstra's Algorithm is preferred due to its efficiency and accuracy. However, if negative weights are present, Bellman-Ford becomes necessary despite its slower performance. For unweighted graphs, Breadth-First Search (BFS) provides an optimal solution by treating all edges as equal, demonstrating that understanding graph characteristics is essential for selecting the most effective algorithm.