A tight analysis of the Katriel–Bodlaender algorithm for online topological ordering

Liu, Hsiao-Fei; Chao, Kun-Mao

doi:10.1016/j.tcs.2007.08.009

Cited by 9 publications

(10 citation statements)

References 10 publications

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“…Hence, the naive approach in which after every update we simply recompute everything from scratch yields a O(nm + m 2 ) total update time. The problems of the incremental cycle detection and topological order have been extensively studied in the last three decades [3,19,21,15,18,2,1,16,8,4,5,6]. Marchetti-Spaccamela et al [19] obtained algorithms for these problems in O(nm) total update time.…”

Section: Introductionmentioning

confidence: 99%

“…Katriel and Bodlaender [15] later gave algorithms with improved bounds of O(min{m 3/2 log n, m 3/2 + n 2 log n}). Afterward, Liu and Chao [18] improved the bound to O(m 3/2 + mn 1/2 log n), and Kavitha and Mathew [16] gave another algorithm with a total update time bound of O(m 3/2 +nm 1/2 log n). See [8] for further discussion on these problems.…”

Section: Introductionmentioning

confidence: 99%

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Incremental Topological Sort and Cycle Detection in Expected Total Time

Bernstein

Chechik

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the incremental cycle detection problem edges are inserted to a directed graph (initially empty) and the algorithm has to report once a directed cycle is formed in the graph. A closely related problem to the incremental cycle detection is that of the incremental topological sort problem, in which edges are inserted to an acyclic graph and the algorithm has to maintain a valid topological sort on the vertices at all times.Both incremental cycle detection and incremental topological sort have a long history. The state of the art is a recent breakthrough of Bender, Fineman, Gilbert and Tarjan [TALG 2016], with two different algorithms with respective total update times ofÕ(n 2 ) and O(m · min{m 1/2 , n 2/3 }). The two algorithms work for both incremental cycle detection and incremental topological sort.In this paper we introduce a novel technique that allows us to improve upon the state of the art for a wide range of graph sparsity. Our algorithms has a total expected update time ofÕ(m √ n) for both the incremental cycle detection and the topological sort problems.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Incremental Topological Sort and Cycle Detection in Expected Total Time

Bernstein

Chechik

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…They also showed significantly better bounds for graphs of bounded treewidth. Liu and Chao [13] gave a tighter analysis of the Katriel-Bodlaender algorithm, showing that it runs in O(m 3/2 + mn 1/2 log n) time. Kavitha and Mathew [12] gave a slightly better variant taking O(m 3/2 + m 1/2 n log n).…”

Section: Introductionmentioning

confidence: 99%

A New Approach to Incremental Topological Ordering

Bender¹,

Fineman²,

Gilbert³

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Let G = (V, E) be a directed acyclic graph (dag) with n = |V | and m = |E|. We say that a total ordering ≺ on vertices V is a topological ordering if for every edge (u, v) ∈ E, we have u ≺ v. In this paper, we consider the problem of maintaining a topological ordering subject to dynamic changes to the underlying graph. That is, we begin with an empty graph G = (V, / 0) consisting of n nodes. The adversary adds m edges to the graph G, one edge at a time. Throughout this process, we maintain an online topological ordering of the graph G.In this paper, we present a new algorithm that has a total cost of O(n 2 log n) for maintaining the topological ordering throughout all the edge additions. At the heart of our algorithm is a new approach for maintaining the ordering. Instead of attempting to place the nodes in an ordered list, we assign each node a label that is consistent with the ordering, and yet can be updated efficiently as edges are inserted. When the graph is dense, our algorithm is more efficient than existing algorithms. By way of contrast, the best known prior algorithms achieve only O(min(m 1.5 , n 2.5 )) cost.

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“…There has been much recent work on incremental cycle detection, topological ordering, and strong component maintenance [1,2,3,4,8,9,12,13,16,17,18,21,22]. For a thorough discussion of this work see [9]; here we discuss the heretofore best results and others related to our work.…”

mentioning

confidence: 89%

A New Approach to Incremental Cycle Detection and Related Problems

Bender

Fineman

Gilbert

et al. 2015

ACM Trans. Algorithms

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Abstract. We consider the problem of detecting a cycle in a directed graph that grows by arc insertions, and the related problems of maintaining a topological order and the strong components of such a graph. For these problems we give two algorithms, one suited to sparse graphs, the other to dense graphs. The former takes O(min{m 1/2 , n 2/3 }m) time to insert m arcs into an n-vertex graph; the latter takes O(n 2 log n) time. Our sparse algorithm is substantially simpler than a previous O(m 3/2 )-time algorithm; it is also faster on graphs of sufficient density. The time bound of our dense algorithm beats the previously best time bound of O(n 5/2 ) for dense graphs. Our algorithms rely for their efficiency on vertex numberings weakly consistent with topological order: we allow ties. Bounds on the size of the numbers give bounds on running time.

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A tight analysis of the Katriel–Bodlaender algorithm for online topological ordering

Cited by 9 publications

References 10 publications

Incremental Topological Sort and Cycle Detection in Expected Total Time

Incremental Topological Sort and Cycle Detection in Expected Total Time

A New Approach to Incremental Topological Ordering

A New Approach to Incremental Cycle Detection and Related Problems

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