Incremental Topological Sort and Cycle Detection in  Expected Total Time

Bernstein, Aaron; Chechik, Shiri

doi:10.1137/1.9781611975031.2

Cited by 19 publications

(43 citation statements)

References 19 publications

(26 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Initially, each node v ∈ V is sampled with probability Θ(log n/τ ). Bernstein and Chechik [BC18] maintain a partition of the node-set V into subsets {V i,j }, where a node v belongs to a subset V i,j iff it has exactly i ancestors and j descendants among the sampled nodes. A total order ≺ * is defined on the subsets {V i,j }, where V i,j ≺ * V i ′ ,j ′ iff either i < i ′ or {i = i ′ and j > j ′ }.…”

Section: Perspectivementioning

confidence: 99%

“…The time taken to implement this forward search is determined by the number of nodes x that are explored during this search. Bernstein and Chechik [BC18] now introduce a crucial notion of τ -related pairs of nodes (see Section 2.1 for details), and show that for every node x explored during the forward search we get a newly created τ -related pair (x, u). Next, they prove an upper bound of O(nτ ) on the total number of such pairs that can appear throughout the duration of the algorithm.…”

Section: Perspectivementioning

confidence: 99%

“…Throughout the paper, we assume that the maximum degree of a node in G is at most O(1) times the average degree. It was observed in [BC18] that this assumption is without any loss of generality. We say that a node x ∈ V is an ancestor of another node y ∈ V iff there is a directed path from x to y in G. We let A(y) ⊆ V denote the set of all ancestors of y ∈ V .…”

Section: Preliminariesmentioning

confidence: 99%

“…The following definition becomes relevant in light of this observation. Following the framework developed in [BC18], we will maintain a partition of the node-set V into subsets {V i,j } and the subgraphs {G i,j = (V i,j , E i,j )} induced by these subsets of nodes. At a high level, this partition serves as a useful proxy for determining if a given ordered pair of nodes is τ -related.…”

Section: Definition A2 [Bc18]mentioning

confidence: 99%

“…We now state three lemmas that will be crucially used in our algorithm for incremental cycle detection. Although these lemmas were derived in [BC18], for the sake of completeness we briefly describe their proofs here. Lemma A.5 states that the graph G contains a cycle iff some subgraph G i,j contains a cycle.…”

Section: Definition A2 [Bc18]mentioning

confidence: 99%

See 4 more Smart Citations

An Improved Algorithm for Incremental Cycle Detection and Topological Ordering in Sparse Graphs

Bhattacharya

Kulkarni

2020

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

View full text Add to dashboard Cite

We consider the problem of incremental cycle detection and topological ordering in a directed graph G = (V, E) with |V | = n nodes. In this setting, initially the edge-set E of the graph is empty. Subsequently, at each time-step an edge gets inserted into G. After every edge-insertion, we have to report if the current graph contains a cycle, and as long as the graph remains acyclic, we have to maintain a topological ordering of the node-set V . Let m be the total number of edges that get inserted into G. We present a randomized algorithm for this problem withÕ(m 4/3 ) total expected update time.Our result improves theÕ(m · min(m 1/2 , n 2/3 )) total update time bound of [BFGT16; HKMST08; HKMST12; CFKR13]. Furthermore, whenever m = o(n 3/2 ), our result improves upon the recently obtainedÕ(m √ n) total update time bound of [BC18]. We note that if m = Ω(n 3/2 ), then the algorithm of [BFGT16; BFG09; CFKR13], which hasÕ(n 2 ) total update time, beats the performance of theÕ(m √ n) time algorithm of [BC18]. It follows that we improve upon the total update time of the algorithm of [BC18] in the "interesting" range of sparsity where m = o(n 3/2 ).Our result also happens to be the first one that breaks the Ω(n √ m) lower bound of [HKMST08] on the total update time of any local algorithm for a nontrivial range of sparsity. Specifically, the total update time of our algorithm is o(n √ m) whenever m = o(n 6/5 ). From a technical perspective, we obtain our result by combining the algorithm of [BC18] with the balanced search framework of [HKMST12].Result (1): There is an incremental algorithm with total update time ofÕ(n 2 ). This follows from the work of [BFGT16; BFG09; CFKR13]. So the problem is well understood for dense graphs where m = Θ(n 2 ).Result (2): There is an incremental algorithm with total update time ofÕ(m · min(m 1/2 , n 2/3 )). This follows from the work of [BFGT16; HKMST08; HKMST12; CFKR13]. Result (3):There is a randomized incremental algorithm with total expected update time ofÕ(m √ n). This follows from the very recent work of [BC18].Significance of Theorem 1.1. We obtain a randomized incremental algorithm for cycle detection and topological ordering that has an expected total update time ofÕ(m 4/3 ). Prior to this, all incremental algorithms for this problem had a total update time of Ω(n 3/2 ) for sparse graphs with m = Θ(n). Our algorithm breaks this barrier by achieving a bound ofÕ(n 4/3 ) on sparse graphs. More generally, our total update time bound ofÕ(m 4/3 ) outperforms theÕ(m √ n) bound from result (3) as long as m = o(n 3/2 ). Note that if 1 Throughout this paper, we use theÕ(.) notation to hide polylog factors.2 Throughout this paper we assume that m ≥ n. This is because if m = o(n) then many nodes remain isolated (with zero degree) in the final graph, and we can ignore those isolated nodes while analyzing the total update time of the concerned algorithm.3 It is easy to combine two incremental algorithms and get the "best of both worlds". For example, suppose that we want to combine results (1...

show abstract

Section: Perspectivementioning

confidence: 99%

Section: Perspectivementioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Definition A2 [Bc18]mentioning

confidence: 99%

Section: Definition A2 [Bc18]mentioning

confidence: 99%

See 3 more Smart Citations

An Improved Algorithm for Incremental Cycle Detection and Topological Ordering in Sparse Graphs

Bhattacharya

Kulkarni

2020

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

View full text Add to dashboard Cite

show abstract

Efficient Parallel Cycle Search in Large Graphs

Zhu

Yuan

Chen

et al. 2020

Database Systems for Advanced Applications

View full text Add to dashboard Cite

Incremental Dead State Detection in Logarithmic Time

Stanford

Veanes

2023

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Identifying live and dead states in an abstract transition system is a recurring problem in formal verification; for example, it arises in our recent work on efficiently deciding regex constraints in SMT. However, state-of-the-art graph algorithms for maintaining reachability information incrementally (that is, as states are visited and before the entire state space is explored) assume that new edges can be added from any state at any time, whereas in many applications, outgoing edges are added from each state as it is explored. To formalize the latter situation, we propose guided incremental digraphs (GIDs), incremental graphs which support labeling closed states (states which will not receive further outgoing edges). Our main result is that dead state detection in GIDs is solvable in $$O(\log m)$$ O ( log m ) amortized time per edge for m edges, improving upon $$O(\sqrt{m})$$ O ( m ) per edge due to Bender, Fineman, Gilbert, and Tarjan (BFGT) for general incremental directed graphs.We introduce two algorithms for GIDs: one establishing the logarithmic time bound, and a second algorithm to explore a lazy heuristics-based approach. To enable an apples-to-apples experimental comparison, we implemented both algorithms, two simpler baselines, and the state-of-the-art BFGT baseline using a common directed graph interface in Rust. Our evaluation shows 110-530x speedups over BFGT for the largest input graphs over a range of graph classes, random graphs, and graphs arising from regex benchmarks.

show abstract

Incremental Topological Sort and Cycle Detection in Expected Total Time

Cited by 19 publications

References 19 publications

An Improved Algorithm for Incremental Cycle Detection and Topological Ordering in Sparse Graphs

An Improved Algorithm for Incremental Cycle Detection and Topological Ordering in Sparse Graphs

Efficient Parallel Cycle Search in Large Graphs

Incremental Dead State Detection in Logarithmic Time

Contact Info

Product

Resources

About