Основные понятия
Dijkstra's algorithm, when combined with a sufficiently efficient heap data structure, is universally optimal for the problem of ordering nodes by their distance from the source.
Аннотация
The paper proves that Dijkstra's algorithm is universally optimal for the distance ordering problem, which requires ordering all vertices by their distance from a given source vertex. This is achieved by combining Dijkstra's algorithm with a new heap data structure that satisfies a "working set" property.
The key insights are:
- The working set property of the heap, which guarantees that the cost of extracting the minimum element is logarithmic in the number of elements inserted after it, is sufficient to ensure the universal optimality of Dijkstra's algorithm.
- The authors construct a new heap data structure that satisfies this working set property, improving upon the Fibonacci heap.
- The universal optimality result reveals a clean interplay between the working set property of the heap and the structure of Dijkstra's algorithm, allowing it to efficiently leverage every structural attribute of the graph.
- The authors also present a variant of Dijkstra's algorithm that is universally optimal with respect to both time complexity and the number of comparisons performed.
The paper opens up new directions for applying (variants of) universal optimality to problems in the standard sequential model of computation.