Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance

Abstract

We present two on-line algorithms for maintaining a topological order of a directed n-vertex acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm handles m arc additions in O(m 3/2 ) time. For sparse graphs (m/n = O(1)), this bound improves the best previous bound by a logarithmic factor, and is tight to within a constant factor among algorithms satisfying a natural locality property. Our second algorithm handles an arbitrary sequence of arc additions in O(n 5/2 ) time. For sufficiently dense graphs, this bound improves the best previous bound by a polynomial factor. Our bound may be far from tight: we show that the algorithm can take Ω(n 2 2 √ 2 lg n ) time by relating its performance to a generalization of the k-levels problem of combinatorial geometry. A completely different algorithm running in Θ(n 2 log n) time was given recently by Bender, Fineman, and Gilbert. We extend both of our algorithms to the maintenance of strong components, without affecting the asymptotic time bounds.

Incremental Graph Computations

“…Proposed in [44], the notion was studied for graph algorithms in [17,38,39]. A number of incremental algorithms have been developed for graphs [12,17,26,28,32,[38][39][40][41][42]46] (see [16] for a survey). However, their costs are typically studied in terms of amortized analysis for averaged operation time of a sequence of unit updates to G, not in the size of changes that is inherent to the incremental problem itself.…”

“…We implemented the following algorithms, all in Java. (1) Incremental algorithms (a) IncKWS (Section 4.2), IncRPQ (Section 5.2), IncSCC (Section 5.3) and IncISO (see Appendix); (b) IncKWSn, IncRPQ n , IncSCCn and IncISOn, which process unit updates in batch ∆G one by one by calling their algorithms for unit updates developed in this work; (c) DynSCC that combines the incremental algorithm in [26] to process insertions and decremental algorithm in [32] Experimental results. We next report our findings.…”

A Multi-Threading Algorithm to Detect and Remove Cycles in Vertex- and Arc-Weighted Digraph

“…The first for loop (lines 3-7) finds the cycle weight. The second for loop (lines [8][9][10][11][12][13][14][15][16][17][18][19][20] decreases all of the arc weights of C by w C . If the weight of an arc becomes 0, delete this arc (line 11).…”

“…After calling Algorithm 3, the arc 3, 4 is deleted. Repeat the above process, cycles (5,6,7,10) and (1,4,5) are detected and removed, and then the graph is cycle-free. In one word, Algorithm 4 can find 3 cycles in Figure 1.…”

Study on a Multi-Hierarchy Topological Sort Algorithm for Automatic Calculation

“…Katriel and Bodlaender studied online algorithms to maintain a topological ordering of a directed acyclic graph, it is optimal that the algorithm was implemented to run in O(n log n) time on trees [13]. Haeupler et al [14] proposed the two algorithms for maintaining topological order, detecting the cycle when an arc was added and maintaining the strong component by extending the new algorithms. Liu [15] found a sorting algorithm, which was different from Kahn or DFS algorithm and owned low complexity.…”