The classical shortest path problem is one of the fundamental problems in graph theory in which the objective is to find a path that minimize the length from a source node to a destination node. In the last few decades, a variety of approach have been introduced to solve such problem. In [27], the author develops a single source multiple target (SSMT) algorithm for the shortest path problem. The authors in [94] propose a pre-computation based approach for both single pair alternative shortest path and all pairs shortest paths processing in spatial network. To handle the variability of travel time, [97] present an algorithm to solve the mean-standard deviation reliable shortest path problem (RSPP) in which they take into account the correlations between the link travel time. [33] propose a mathematical model for the constrained shortest path tour problem (CSPTP) in which the path must cross by a sequence of node in a given order must. The author introduces a Branch & Bound method to solve the model.
[41] formulates a non-linear objective functions for the stochastic shortest path problem using the mean and standard deviation values of the resulting probability distribution and general cost functions. To solve the model, the author applies Lagrangian relaxation (LR) to
find the feasible solution. Considering the travel time uncertainty, [21] extend the classical k-loopless shortest path problem to the stochastic transportation network and introduce the k reliable shortest paths (KRSP). To solve the problem in large-scale networks exactly, the authors propose a deviation path algorithm and to further improve the KRSP finding performance they introduce the A? technique.
The authors in [22] study the travel time uncertainties in a road network(RN)where the goal is to find the most reliable path that maximizes the probability of on-time arrival. They propose a two-stage solution algorithm that exactly solve the problem. The authors in [79] use the historical information and the real-time information of a transportation network and present the dynamic shortest path problem with stochastic disruptions as a discrete time fi- nite horizon Markov decision process (MDP). They develop a hybrid approach based on approximate dynamic programming (ADP) and clustering using deterministic lookahead policy and value function approximation. In [2], the author provides an experimental anal- ysis on how traversal algorithms behave on social networks. Dijkstra algorithm has been used in [89] to generate single source single destination (SSSD) shortest path in large sparse graphs where the goal is to reduce the response time for online queries. [1] present ASAP system that can extremely quickly generate the shortest path for all pair of nodes on large scale networks.
The dynamic single source shortest path problem (SSSPP) has been addressed in [37] via dynamizing Dijkstra algorithm and generate the solution of that problem with com- plexity O(nlg m) for the update time. To reach an emergency area in a minimum time, [64] develop a dynamic emergency routing approach based on Dijkstra algorithm and analytical hierarchical processing (AHP). For safe evacuation in a short time from an emergency area, [75] model the evacuation route then they generate the directed graph based on the distance between nodes and coordinate of nodes. To generate the shortest and safest path, they use Dijkstra algorithm and Dijkstra-ACO.
[9] optimize Dijkstra algorithm using a faster queue design based on binary (or d-ary) heap implementation to accelerate the computation of the shortest paths. [43] introduce a new time-dependent shortest path algorithm for SSSD in multimodal transportation net- work.
Considering the stochastic disruption in a network, the authors in [47] present an emer- gency evacuation model for the SSP and they propose a real-time dynamic navigation algo- rithm (DNA) to solve it. [42] study a network that is stochastic on-time arrival and propose a modification for the reliable routing algorithm. The authors modify the shortest path algorithm to decrease its computation time by choosing particular subset of graph nodes and links. Taking into account the travellers concern on travel time reliability in congested RN, [20] formulate the spatially dependent reliable shortest path problem (SD-RSPP) on a RN of correlated link travel times. The authors propose a link-based multi-criteria A* algorithm (ERSPA* algorithm) to solve the model. To find the optimal path in an uncertain transportation network, [19] formulate three model for the shortest path problem typically name expected value, dependent-chance and chance constrained models. The authors de- velop a simulation based genetic algorithm (SBGA) to solve the former models.
[36] derives the uncertainty distribution of the shortest path length in an uncertain net- work and investigates solutions to the α-shortest path. The author indicates that there is an equivalence relation between the α-shortest path in an uncertain network and the shortest path in a corresponding deterministic network and as so proposes an approach to find the α-shortest path and the equivalence deterministic shortest path. An approach based on GA for the SPRP has been presented in [12]. The authors in [70] indicate that most of the exist- ing approaches that handle the shortest path problem have two main drawbacks: generating the path before movement and re-processing when node and/or link fails. To handle these shortcomings, they propose two novel algorithms typically name Realtime Shortest Path (RSP) and Realtime Shortest Path Plus (R-SP+).
The authors in [98] present a robust shortest path model to hedge against the risk of un- certainty in travel times. To solve the model, they propose an efficient primal approximation method that has to solve the deterministic shortest path problem and the mean-standard de- viation shortest path problem. [96] analyze the dynamic stochastic characteristics and the relationships between the links and nodes of the transportation network. The authors de- fine the probabilistic shortest path concept, propose a mathematical model for the dynamic stochastic KSP and develop a GA to solve the model. A survey on variations of the short- est path problem of different objective functions and existing solution approaches can be found in [87]. Table 4 covers a number of relevant existing efforts to handle the shortest path problem and k-shortest path problem.