Shortest‐path methods: Complexity, interrelations and new propositions

Abstract

We present a new unified approach for shortest-path problems. Based on this approach, we develop a computational method which consists of determining shortest paths on a finite sequence of partial graphs defined as the "growth of the original graph." We show that the proposed method allows us to interpret within the same framework several different well-known algorithms, such as those of D'Esopo-Pape, Dijkstra, and Dial, and leads to a uniform analysis of their computational complexity. We also stress the existing parallelism between the proposed method and the matrix-multiplication methods of Floyd-Warshall, and Dantzig.

Average-case complexity of single-source shortest-paths algorithms: lower and upper bounds

“…There are many more ways to maintain Q and select nodes from it (see [6,27] for an overview). For example, the algorithms of Pallottino [54], Goldberg and Radzik [34], and Glover et al [29][30][31] subdivide Q into two sets Q 1 and Q 2 each of which is implemented as a list. Intuitively, Q 1 represents the "more promising" candidate nodes.…”

“…Average-case time Bellman-Ford algorithm [3,21] Ω(n 4/3−ε ) Pallottino's Incremental Graph algorithm [54] Ω(n 4/3−ε ) Basic Topological Ordering algorithm [34] Ω(n 4/3−ε ) Threshold algorithm [29][30][31] Ω(n · log n/ log log n) ABI-Dijkstra [6] Ω(n · log n/ log log n) ∆-Stepping [48] Ω(n · √ log n/ log log n) Finally, we present a general method to construct sparse input graphs with random edge weights for which several label-correcting SSSP algorithms require superlinear averagecase running-time: we consider the "Bellman-Ford algorithm" [3,21], "Pallottino's Incremental Graph algorithm" [54], the "Threshold approach" by Glover et al [29][30][31], the basic version of the "Topological Ordering SSSP algorithm" by Goldberg and Radzik [34], the "Approximate Bucket implementation" of Dijkstra's algorithm (ABI-Dijkstra) [6], and its refinement, the "∆-Stepping algorithm" [48]. The obtained lower bounds are summarized in Fig.…”

“…In the following we briefly consider two examples: the incremental graph algorithm of Pallottino [54] and the topological ordering algorithm of Goldberg and Radzik [34].…”

“…Pallottino's algorithm [54]-PAL for short-maintains two FIFO queues Q 1 and Q 2 . Labeled nodes that have been scanned at least once are stored in Q 1 whereas labeled nodes that have never been scanned are maintained in Q 2 .…”

Parallel implementation of a transportation network model

“…These include the Bellman-Ford-Moore algorithm [5,26,53], algorithms due to Moore [53], Dijkstra [22], Pallottino [58] and the D'Esopo-Pape algorithm, as described and tested by Pape [61] while being credited to D'Esopo by Pollock et al [62]. A number of reviews exist on shortest path algorithms in general, for example see, amongst others, Deo and…”