Book ChapterDOI
Faster Algorithms for Incremental Topological Ordering
Bernhard Haeupler,Telikepalli Kavitha,Rogers Mathew,Sercan Sen,Robert E. Tarjan +4 more
- pp 421-433
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TLDR
This work presents two online algorithms for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created, using a deterministic and a randomized method.Abstract:
We present two online algorithms for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created Our first algorithm takes O(m1/2) amortized time per arc and our second algorithm takes O(n25/m) amortized time per arc, where nis the number of vertices and mis the total number of arcs For sparse graphs, our O(m1/2) bound improves the best previous bound by a factor of lognand is tight to within a constant factor for a natural class of algorithms that includes all the existing ones Our main insight is that the two-way search method of previous algorithms does not require an ordered search, but can be more general, allowing us to avoid the use of heaps (priority queues) Instead, the deterministic version of our algorithm uses (approximate) median-finding; the randomized version of our algorithm uses uniform random sampling For dense graphs, our O(n25/m) bound improves the best previously published bound by a factor of n1/4and a recent bound obtained independently of our work by a factor of logn Our main insight is that graph search is wasteful when the graph is dense and can be avoided by searching the topological order space instead Our algorithms extend to the maintenance of strong components, in the same asymptotic time boundsread more
Citations
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Journal ArticleDOI
Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance
TL;DR: In this article, the authors present two online 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.
Journal ArticleDOI
A New Approach to Incremental Cycle Detection and Related Problems
TL;DR: This work considers 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, and gives two algorithms, one suited to sparse graphs, the other to dense graphs.
Proceedings Article
Angelic hierarchical planning: optimal and online algorithms
TL;DR: The Angelic Hierarchical A* algorithm is described, which generates provably optimal plans, and its advantages over alternative algorithms are shown, one of the first algorithms to do hierarchical lookahead in an online setting.
Proceedings ArticleDOI
A new approach to incremental topological ordering
TL;DR: A new algorithm is presented that has a total cost of O(n2logn) for maintaining the topological ordering throughout all the edge additions of the graph G, and which is more efficient than existing algorithms.
Proceedings ArticleDOI
Applications of forbidden 0-1 matrices to search tree and path compression-based data structures
TL;DR: This paper improves, reprove, and simplify several theorems on the performance of data structures based on path compression and search trees, and presents the first asymptotically sharp bound on the length of arbitrary path compressions on arbitrary trees.
References
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The Art of Computer Programming
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Journal ArticleDOI
Depth-First Search and Linear Graph Algorithms
TL;DR: The value of depth-first search or “backtracking” as a technique for solving problems is illustrated by two examples of an improved version of an algorithm for finding the strongly connected components of a directed graph.
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