Observer of changes in the forest of the shortest paths on dynamic graphs of transport networks

The purpose of the work is the development of basic data structures, speed-efficient and memoryefficient algorithms for tracking changes in predefined decisions about sets of shortest paths on transport networks, notifications about which are received by autonomous coordinated transport agents with centralized or collective control. A characteristic feature of transport operations is the independence and asynchrony of the emergence of perturbations of optimal solutions, as well as the lack of global influence of individual perturbations on the set of all processes on the network. This clearly determines the feasibility of realizing the idea of reoptimizing existing solutions in real time as information is received about disturbances in the structure and parameters of the transport network, various restrictions on the use of existing shortest paths. In contrast to the classical problems of finding shortest paths on static or dynamic graphs, it is proposed to supplement the set of situations controlled by the observer by taking into account the associations of shortest path trees with agents that actually use such paths. This will improve the responsiveness of agent notification processes for timely switching to a new path. The space of search states is a dynamically generated bipartite sparse graph of the transport network, represented by a list of arcs. The basic algorithm for finding the shortest paths uses Dijkstra's scheme, but implements a bootstrapping method to generate the search result. The compactness of the representation of the observed forest of shortest paths is achieved by mapping individual trees of such a forest onto the projection of tree vertices in memory, where the position of each vertex corresponds to the distance from the tree root. The proposed version of the construction of the search procedure is based on the mechanisms existing in database management systems for creating different relational representations of the physical data model. This eliminates the need to solve technological problems of complexing heterogeneous models of dynamic transport networks, memory allocation. As a result, the specification of various rules for the logistics of transport operations is simplified, since such operations in terms of object-oriented models are easily determined by polymorphic classes of transitions between nodes of the transport network.

Dynamic Algorithms for Visibility Polygons in Simple Polygons

We devise the following dynamic algorithms for both maintaining as well as querying for the visibility and weak visibility polygons amid vertex insertions and deletions to the simple polygon. A fully-dynamic algorithm for maintaining the visibility polygon of a fixed point located interior to the simple polygon amid vertex insertions and deletions to the simple polygon. The time complexity to update the visibility polygon of a point [Formula: see text] due to the insertion (resp. deletion) of vertex [Formula: see text] to (resp. from) the current simple polygon is expressed in terms of the number of combinatorial changes needed to the visibility polygon of [Formula: see text] due to the insertion (resp. deletion) of [Formula: see text]. An output-sensitive query algorithm to answer the visibility polygon query corresponding to any point [Formula: see text] in [Formula: see text] amid vertex insertions and deletions to the simple polygon. If [Formula: see text] is not exterior to the current simple polygon, then the visibility polygon of [Formula: see text] is computed. Otherwise, our algorithm outputs the visibility polygon corresponding to the exterior visibility of [Formula: see text]. An incremental algorithm to maintain the weak visibility polygon of a fixed-line segment located interior to the simple polygon amid vertex insertions to the simple polygon. The time complexity to update the weak visibility polygon of a line segment [Formula: see text] due to the insertion of vertex [Formula: see text] to the current simple polygon is expressed in terms of the sum of the number of combinatorial updates needed to the geodesic shortest path trees rooted at [Formula: see text] and [Formula: see text] due to the insertion of [Formula: see text]. An output-sensitive algorithm to compute the weak visibility polygon corresponding to any query line segment located interior to the simple polygon amid both the vertex insertions and deletions to the simple polygon. Each of these algorithms requires preprocessing the initial simple polygon. And, the algorithms that maintain the visibility polygon (resp. weak visibility polygon) compute the visibility polygon (resp. weak visibility polygon) with respect to the initial simple polygon during the preprocessing phase.

A Graph-Intersection-Based Algorithm to Determine Maximum Lifetime Communication Topologies for Cognitive Radio Ad Hoc Networks

The authors propose a generic graph intersection-based benchmarking algorithm to arrive at upper bounds for the lifetimes of any communication topology that spans the entire network of secondary user (SU) nodes in a cognitive radio ad hoc network wherein the SUs attempt to access the licensed channels that are not in use. At any time, instant t when we need a stable communication topology spanning the entire network, the authors look for the largest value of k such that the intersection of the static SU graphs from time instants t to t+k, defined as the mobile graph Gt...t+k(SU) = Gt(SU) ∩ Gt+1(SU) ∩ .... ∩ Gt+k(SU), is connected and Gt...t+k+1(SU) is not connected. The authors repeat the above procedure for the entire network session to determine the sequence of longest-living instances of the mobile graphs and the corresponding instances of the topology of interest such that the number of topology transitions is the global minimum. They prove the theoretical correctness of the algorithm and study its effectiveness by implementing it to determine a sequence of maximum lifetime shortest path trees.

A Graph-Intersection-Based Algorithm to Determine Maximum Lifetime Communication Topologies for Cognitive Radio Ad Hoc Networks

We propose a generic graph intersection-based benchmarking algorithm to arrive at upper bounds for the lifetimes of any communication topology that spans the entire network of secondary user (SU) nodes in a cognitive radio ad hoc network wherein the SUs attempt to access the licensed channels that are not in use. At any time instant t when we need a stable communication topology spanning the entire network, we look for the largest value of k such that the intersection of the static SU graphs from time instants t to t+k, defined as the mobile graph Gt...t+k(SU) = Gt(SU) n Gt+1(SU) n .... n Gt+k(SU), is connected and Gt...t+k+1(SU) is not connected. We repeat the above procedure for the entire network session to determine the sequence of longest-living instances of the mobile graphs and the corresponding instances of the topology of interest such that the number of topology transitions is the global minimum. We prove the theoretical correctness of the algorithm and study its effectiveness by implementing it to determine a sequence of maximum lifetime shortest path trees.

Comparative visualizations of transport networks in Calgary using shortest-path trees

Three maps are generated to visually compare the structure of transport networks, differences in travel times, and critical travel pathways for three travel modes in Calgary, Canada. The maps also highlight how the fractal-like structure of these urban transport networks are visually similar to natural phenomena.