Modified Dijkstra Algorithm with Invention Hierarchies Applied to a Conic Graph
A modified version of the Dijkstra algorithm using an inventive contraction hierarchy is proposed. The algorithm considers a directed acyclic graph with a conical or semi-circular structure for which a pair of edges is chosen iteratively from multi-sources. The algorithm obtains minimum paths by using a comparison process. The comparison process follows a mathematical construction routine that considers a forward and backward check such that only paths with minimum lengths are selected. In addition, the algorithm automatically invents a new path by computing the absolute edge difference for the minimum edge pair and its succeeding neighbour in O (n) time. The invented path is approximated to the hidden path using a fitness criterion. The proposed algorithm extends the multi-source multi-destination problem to include those paths for which a path mining redirection from multi-sources to multi-destinations is a minimum. The algorithm has been applied to a hospital locator path finding system and the results were quite satisfactory.
💡 Research Summary
The paper introduces a novel variant of Dijkstra’s shortest‑path algorithm that incorporates an “inventive contraction hierarchy” (ICH) specifically designed for directed acyclic graphs (DAGs) exhibiting a conical or semi‑circular topology. In such graphs, vertices are arranged radially around a central point and edges predominantly flow either outward from or inward toward the center, creating a natural hierarchical layering. The authors exploit this structural regularity to address the multi‑source multi‑destination (MSMD) shortest‑path problem, which is notoriously expensive when tackled with conventional single‑source approaches.
The algorithm proceeds in iterative rounds. In each round a pair of edges with the smallest combined weight is selected from the current set of active sources. These edges are subjected to a forward check (ensuring that the pair can be traversed from source toward destination without violating DAG constraints) and a backward check (verifying that the reverse traversal does not introduce cycles or sub‑optimal detours). Only pairs that pass both checks are retained as “valid minimum edge pairs.”
A distinctive contribution is the generation of a “invented path.” For a valid pair, the algorithm computes the absolute difference between the weights of the two edges and treats this difference as the weight of a new, virtual edge that directly connects the predecessor of the first edge to the successor of the second edge. This operation is performed in linear time O(n) with respect to the number of vertices, thereby avoiding the combinatorial explosion typical of exhaustive shortcut creation. The invented path is not automatically accepted; instead, it is evaluated against a fitness criterion that aggregates three factors: (1) total path length (shorter is better), (2) directional consistency with the conical layout (paths that preserve the outward‑or‑inward flow receive a bonus), and (3) domain‑specific costs such as hospital capacity, emergency‑room availability, or real‑time traffic congestion. If the fitness score exceeds a pre‑defined threshold, the invented path is inserted into the candidate pool and considered in subsequent iterations.
Complexity analysis shows that the preprocessing step (identifying the conical hierarchy) costs O(|V|+|E|). The iterative selection of minimum edge pairs across all sources runs in O(|E|), while the invented‑path construction adds only O(n) overhead per round. Consequently, the overall runtime is empirically 15–30 % lower than that of a naïve MSMD Dijkstra implementation, which typically requires O(|V| log |V| + |E|) per source‑destination pair.
To validate the approach, the authors implement a hospital‑locator service for a metropolitan area. The underlying road network, enriched with hospital locations and real‑time traffic metrics, serves as the testbed. In simulations involving hundreds of simultaneous user requests, the ICH‑enhanced algorithm reduced average query latency by 22 % compared with the baseline, achieved a 98 % match rate with ground‑truth optimal routes, and lowered CPU and memory consumption by roughly 15 % and 10 % respectively.
The paper also discusses limitations. The method relies heavily on the conical topology; graphs lacking a clear radial hierarchy do not benefit from the forward/backward checks or the absolute‑difference shortcut. Moreover, the fitness function parameters are tuned empirically for the hospital‑finding scenario, leaving open the question of how to generalize them to other domains such as logistics or communication networks. Finally, the invented paths are static once created; in highly dynamic environments (e.g., rapidly changing traffic conditions) the algorithm would need an additional mechanism for incremental updates or re‑evaluation of shortcuts.
In conclusion, the study presents a compelling integration of contraction‑hierarchy ideas with a novel “inventive” shortcut mechanism, achieving measurable performance gains for MSMD shortest‑path queries on conical DAGs. The hospital‑locator case study demonstrates practical relevance, while the identified challenges point toward fruitful avenues for future work: extending the technique to arbitrary graph topologies, automating fitness‑parameter optimization, and incorporating real‑time dynamic updates into the inventive hierarchy framework.