Time-Dependent Alternative Route Planning

We present a new method for computing a set of alternative origin-to-destination routes in road networks with an underlying time-dependent metric. The resulting set is aggregated in the form of a time-dependent alternative graph and is characterized by minimum route overlap, small stretch factor, small size and low complexity. To our knowledge, this is the first work that deals with the time-dependent setting in the framework of alternative routes. Based on preprocessed minimum travel-time information between a small set of nodes and all other nodes in the graph, our algorithm carries out a collection phase for candidate alternative routes, followed by a pruning phase that cautiously discards uninteresting or low-quality routes from the candidate set. Our experimental evaluation on real time-dependent road networks demonstrates that the new algorithm performs much better (by one or two orders of magnitude) than existing baseline approaches. In particular, the entire alternative graph can be computed in less than 0.384sec for the road network of Germany, and in less than 1.24sec for that of Europe. Our approach provides also "quick-and-dirty" results of decent quality, in about 1/300 of the above mentioned query times for continental-size instances

All-Pairs Min-Cut in Sparse Networks

Algorithms are presented for the all-pairs min-cut problem in bounded treewidth, planar, and sparse networks. The approach used is to preprocess the input n-vertex network so that afterward, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after an Onlog Ž n. preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for Ž 2 such networks the all-pairs min-cut problem can be solved in time On.. This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, �, of the input network. The parameter � varies between 1 and �Ž. n; the algorithms perform well when � � on. Ž. The value Ž 2 of a min-cut can be found in time On� � log �. and all-pairs min-cut can be Ž 2 4 solved in time On � � log �. for sparse networks. The corresponding runnin

On-Line and Dynamic Shortest Paths Through Graph Decompositions

We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only $O(\log n)$ time, where $n$ is the number of vertices of the digraph. We also describe the first parallel algorithms for solving the dynamic version of the shortest path problem. Our results can be extended to hold for digraphs of genus $o(n)$

Parallel Max Cut Approximations

Given a graph with positive integer edge weights one may ask whether there exists an edge cut whose weight is bigger than a given number. This problem is NP-complete. We present here an approximation algorithm in NC which provides tight upper bounds to the proportion of edge cuts whose size is bigger than a given number. Our technique is based on the methods to convert randomized parallel algorithms into deterministic ones introduced by Karp and Wigderson. The basic idea of those methods is to replace an exponentially large sample space by one of polynomial size. In this work, we prove the interesting result that the statistical distance of random variables of the small sample space is bigger than the statistical distance of corresponding variables of the exponentially large space, which is the space of all edge cuts taken equiprobably

Quickest Paths: Faster Algorithms and Dynamization

Given a network $N=(V,E,{c},{l})$, where $G=(V,E)$, $|V|=n$ and $|E|=m$, is a directed graph, ${c}(e) \u3e 0$ is the capacity and ${l}(e) \ge 0$ is the lead time (or delay) for each edge $e\in E$, the quickest path problem is to find a path for a given source--destination pair such that the total lead time plus the inverse of the minimum edge capacity of the path is minimal. The problem has applications to fast data transmissions in communication networks. The best previous algorithm for the single--pair quickest path problem runs in time $O(r m+r n \log n)$, where $r$ is the number of distinct capacities of $N$ \cite{ROS}. In this paper, we present algorithms for general, sparse and planar networks that have significantly lower running times. For general networks, we show that the time complexity can be reduced to $O(r^{\ast} m+r^{\ast} n \log n)$, where $r^{\ast}$ is at most the number of capacities greater than the capacity of the shortest (with respect to lead time) path in $N$. For sparse networks, we present an algorithm with time complexity $O(n \log n + r^{\ast} n + r^{\ast} \tilde{\gamma} \log \tilde{\gamma})$, where $\tilde{\gamma}$ is a topological measure of $N$. Since for sparse networks $\tilde{\gamma}$ ranges from $1$ up to $\Theta(n)$, this constitutes an improvement over the previously known bound of $O(r n \log n)$ in all cases that $\tilde{\gamma}=o(n)$. For planar networks, the complexity becomes $O(n \log n + n\log^3 \tilde{\gamma}+ r^{\ast} \tilde{\gamma})$. Similar improvements are obtained for the all--pairs quickest path problem. We also give the first algorithm for solving the dynamic quickest path problem

Engineering Graph-Based Models for Dynamic Timetable Information Systems

Many efforts have been done in the last years to model public transport timetables in order to find optimal routes. The proposed models can be classified into two types: those representing the timetable as an array, and those representing it as a graph. The array-based models have been shown to be very effective in terms of query time, while the graph-based models usually answer queries by computing shortest paths, and hence they are suitable to be used in combination with speed-up techniques developed for road networks. In this paper, we focus on the dynamic behavior of graph-based models considering the case where transportation systems are subject to delays with respect to the given timetable. We make three contributions: (i) we give a simplified and optimized update routine for the well-known time-expanded model along with an engineered query algorithm; (ii) we propose a new graph-based model tailored for handling dynamic updates; (iii) we assess the effectiveness of the proposed models and algorithms by an experimental study, which shows that both models require negligible update time and a query time which is comparable to that required by some array-based models

The Societal Impact of Algorithms in Transport Optimization 1

We review some recent advances for solving core algorithmic problems encountered in public transportation systems. We show that efficient algorithms can make a great difference both in efficiency and in optimality, thus contributing significantly to improving the quality and service-efficiency of public transportation systems. 1