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Solving Shortest Paths Efficiently on Nearly Acyclic Directed Graphs
"... Shortest path problems can be solved very efficiently when a directed graph is nearly acyclic. Earlier results defined a graph decomposition, now called the 1dominator set, which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associate ..."
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Shortest path problems can be solved very efficiently when a directed graph is nearly acyclic. Earlier results defined a graph decomposition, now called the 1dominator set, which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associated trigger vertex. In this framework, a specialised shortest path algorithm only spends deletemin operations on trigger vertices, thereby making the computation of shortest paths through nontrigger vertices easier. A previously presented algorithm computed the 1dominator set in O(mn) worstcase time, which allowed it to be integrated as part of an O(mn + nr log r) time allpairs algorithm. Here m and n respectively denote the number of edges and vertices in the graph, while r denotes the number of trigger vertices. A new algorithm presented in this paper computes the 1dominator set in just O(m) time. This can be integrated as part of the O(m+r log r) time spent solving singlesource, improving on the value of r obtained by the earlier treedecomposition singlesource algorithm. In addition, a new bidirectional form of 1dominator set is presented, which further improves the value of r by defining acyclic structures in both directions over edges in the graph. The bidirectional 1dominator set can similarly be computed in O(m) time and included as part of the O(m + r log r) time spent computing singlesource. This paper also presents a new allpairs algorithm under the more general framework where r is defined as the size of any predetermined feedback vertex set of the graph, improving the previous allpairs time complexity from O(mn + nr 2) to O(mn + r 3).
Violation Semirings in Optimality Theory
"... This paper provides a brief algebraic characterization of constraint violations in Optimality Theory (OT). I show that if violations are taken to be multisets over a fixed basis set Con then the merge operator on multisets and a ‘min ’ operation expressed in terms of harmonic inequality provide a se ..."
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This paper provides a brief algebraic characterization of constraint violations in Optimality Theory (OT). I show that if violations are taken to be multisets over a fixed basis set Con then the merge operator on multisets and a ‘min ’ operation expressed in terms of harmonic inequality provide a semiring over violation profiles. This semiring allows standard optimization algorithms to be used for OT grammars with weighted finitestate constraints in which the weights are violationmultisets. Most usefully, because multisets are unordered, the merge operation is commutative and thus it is possible to give a single graph representation of the entire class of grammars (i.e. rankings) for a given constraint set. This allows a neat factorization of the optimization problem that isolates the main source of complexity into a single constant γ denoting the size of the graph representation of the whole constraint set. I show that the computational cost of optimization is linear in the length of the underlying form with the multiplicative constant γ. This perspective thus makes it straightforward to evaluate the complexity of optimization for different constraint sets. 1
Efficient Algorithms for Solving Shortest Paths on Nearly Acyclic Directed Graphs
 Research and Practice in Information Technology
, 2005
"... This paper presents new algorithms for computing shortest paths in a nearly acyclic directed graph G = (V, E). The new algorithms improve on the worstcase running time of previous algorithms. Such algorithms use the concept of a 1dominator set. A 1dominator set divides the graph into a unique col ..."
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This paper presents new algorithms for computing shortest paths in a nearly acyclic directed graph G = (V, E). The new algorithms improve on the worstcase running time of previous algorithms. Such algorithms use the concept of a 1dominator set. A 1dominator set divides the graph into a unique collection of acyclic subgraphs, where each acyclic subgraph is dominated by a single associated trigger vertex. The previous time for computing a 1dominator set is improved from O(mn) to O(m), where m = E and n = V. Efficient shortest...
Entropy as Computational Complexity
"... Abstract. If the given problem instance is partially solved, we want to minimize our effort to solve the problem using that information. In this paper we introduce the measure of entropy, H(S), for uncertainty in partially solved input data S(X) = (X1,..., Xk), where X is the entire data set, and e ..."
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Abstract. If the given problem instance is partially solved, we want to minimize our effort to solve the problem using that information. In this paper we introduce the measure of entropy, H(S), for uncertainty in partially solved input data S(X) = (X1,..., Xk), where X is the entire data set, and each Xi is already solved. We propose a generic algorithm that merges Xi’s repeatedly, and finishes when k becomes 1. We use the entropy measure to analyze three example problems, sorting, shortest paths and minimum spanning trees. For sorting Xi is an ascending run, and for minimum spanning trees, Xi is interpreted as a partially obtained minimum spanning tree for a subgraph. For shortest paths, Xi is an acyclic part in the given graph. When k is small, the graph can be regarded as nearly acyclic. The entropy measure, H(S), is defined by regarding pi = Xi/X  as a probability measure, that is, H(S) = −nΣ k i=1pi log pi, where n = Σ k i=1Xi. We show that we can sort the input data S(X) in O(H(S)) time, and that we can complete the minimum cost spanning tree in O(m + H(S)) time, where m in the number of edges. Then we solve the shortest path problem in O(m + H(S)) time. Finally we define dual entropy on the partitioning process, whereby we give the time bounds on a generic quicksort and the shortest path problem for another kind of nearly acyclic graphs.
All Pairs Shortest Paths Algorithms
, 1999
"... There are many algorithms for the all pairs shortest path problem, depending on variations of the problem. The simplest version takes only the size of vertex set as a parameter. As additional parameters, other problems specify the number of edges and/or the maximum value of edge costs. In this ..."
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There are many algorithms for the all pairs shortest path problem, depending on variations of the problem. The simplest version takes only the size of vertex set as a parameter. As additional parameters, other problems specify the number of edges and/or the maximum value of edge costs. In this
Improved Shortest Path Algorithms for Nearly Acyclic Directed Graphs Abstract
"... This paper presents new algorithms for computing single source shortest paths (SSSPs) in a nearly acyclic directed graph G. The first part introduces higherorder decomposition. This decomposition is an extension of the technique of strongly connected component (sccomponent) decomposition. The seco ..."
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This paper presents new algorithms for computing single source shortest paths (SSSPs) in a nearly acyclic directed graph G. The first part introduces higherorder decomposition. This decomposition is an extension of the technique of strongly connected component (sccomponent) decomposition. The second part presents a new method for measuring acyclicity based on modifications to two existing methods. In the new method, we decompose the graph into a 1dominator set, which is a set of acyclic subgraphs where each subgraph is dominated by one trigger vertex. Meanwhile we compute sccomponents of a degenerated graph derived from triggers. Using this preprocessing, a new SSSP algorithm has O(m + rlogl) time complexity, where r is the size of the 1dominator set, and l is the size of the largest sccomponent. In the third part, we modify the concept of a 1dominator set to that of a 12dominator set. Each of acyclic subgraphs obtained by the 12dominator decomposition are dominated by one or two trigger vertices cooperatively. Such subgraphs are potentially larger than those decomposed by the 1dominator set. Thus fewer trigger vertices are needed to cover the graph.