🤖 AI Summary
Establishing the universal optimality—i.e., matching the information-theoretic lower bound for *all* graph instances—of Dijkstra’s algorithm in terms of comparison complexity and running time on arbitrary graphs.
Method: The authors introduce a novel heap structure with the *working-set property*: extracting the minimum element costs only logarithmically in the number of *recently inserted* elements, not the total heap size. This heap is integrated into Dijkstra’s algorithm to achieve instance-optimal comparison complexity for the distance-sorting problem.
Contribution/Results: The resulting algorithm attains the minimal possible number of comparisons for *every* input graph, while preserving worst-case asymptotic optimality comparable to Fibonacci heaps. This work pioneers the systematic application of *universal optimality* to sequential graph algorithms, combining working-set heap design, fine-grained comparison complexity analysis, and coupling-based lower-bound arguments—thereby transcending traditional worst-case modeling and establishing a new theoretical benchmark for shortest-path algorithms.
📝 Abstract
This paper proves that Dijkstra's shortest-path algorithm is universally optimal in both its running time and number of comparisons when combined with a sufficiently efficient heap data structure. Universal optimality is a powerful beyond-worst-case performance guarantee for graph algorithms that informally states that a single algorithm performs as well as possible for every single graph topology. We give the first application of this notion to any sequential algorithm. We design a new heap data structure with a working-set property guaranteeing that the heap takes advantage of locality in heap operations. Our heap matches the optimal (worst-case) bounds of Fibonacci heaps but also provides the beyond-worst-case guarantee that the cost of extracting the minimum element is merely logarithmic in the number of elements inserted after it instead of logarithmic in the number of all elements in the heap. This makes the extraction of recently added elements cheaper. We prove that our working-set property guarantees universal optimality for the problem of ordering vertices by their distance from the source vertex: The sequence of heap operations generated by any run of Dijkstra's algorithm on a fixed graph possesses enough locality that one can couple the number of comparisons performed by any heap with our working-set bound to the minimum number of comparisons required to solve the distance ordering problem on this graph for a worst-case choice of arc lengths.
