by Spurthi Chaganti S Dayakar Reddy Graph Indexing of Road Networks for Shortest Path Queries with Label Restrictions Michael Rice, Vassilis J. Tsotras, University of California, Riverside by Spurthi Chaganti S Dayakar Reddy

Problem Definition & Importance Widespread usage of GPS Technologies and Maps Applications No support of dynamic constraints in existing applications Supporting Dynamic Constraints over large data sets Enhancing the efficiency compared to the existing applications. How to avoid toll gates, ferries while retrieving the shortest path? Personalization and Logistics and Commercial Transportation

Why is the problem hard? Supporting dynamic constraints involves either explicit re-computation of the graph index online as the weights (or cost functions) of the edges (roads) change or the query algorithm must make increasingly limited use of the information available in the static graph index based on the dynamic changes.

Proposed Approach There are 2 concepts which are explained and used as the backbone in the paper. Kleene Language Constrained Shortest Path Contraction Hierarchies with Label Restrictions Language Constrained Shortest Path (LCSP) is shortest path whose edge labels must satisfy specified formal language constraint over a fixed alphabet ∑.

Let G = (V, E,𝜔, ∑, l) be a directed graph, where V is the set of vertices in G, E is the set of edges in G, 𝜔: E → R+ is a function mapping edges in G to a positive, real-valued weight, ∑ is a finite alphabet used for labeling of edges in G, and l: E->∑ is a function mapping edges in G to a label in ∑. Let Ps,t = (e1, e2, · · · , ek) be any path in G from some vertex s ∈ V to some vertex t ∈ V , such that e1 = (s, v1) ∈ E, ek = (vk−1, t) ∈ E, and for 1 < i < k, ei = (vi−1, vi) ∈ E. Let 𝜔(Ps, t) =Σ 1 ≤ I ≤ k 𝜔(ei) be the total weight of all edges in Ps,t. Let l (Ps,t) = l (e1) l (e2) · · · l (ek) be the concatenation of the labels of all edges in Ps,t. Given any formal language L ⊆ ∑*, a language constrained shortest path is a path P’s,t in G such that l(P’s,t) ∈ L and ∀ Ps,t in G where l(P’s,t) ∈ L, 𝜔(P’s,t) ≤ 𝜔(Ps,t).

CH graph indexing technique supports static point-to-point shortest path queries very efficiently. The vertices in the graph G are given an absolute ordering based on their importance which is defined through the bijective function𝜙: V → {1, … ,|V |}). In the preprocessing stage, the nodes (vertices) are contracted one at a time based on the importance given by the bijective function 𝜙.

Kleene language is a Kleene closure of any subset of Σ i. e Kleene language is a Kleene closure of any subset of Σ i.e.,∀ A∈Σ, a Kleene Language over alphabet A can be defined by L(A*). Alphabet A defines the set of allowable labels that can appear on the shortest path in KLCSP problem. Contraction Techniques Witness Paths Shortcut Edges Bidirectional Dijkstras algorithm over upward and downward Edges Limitations with CH when constraints are involved?

Authors Contributions Contraction hierarchies with label restriction Revised bidirectional dijkstras method Multi edge support Optimisations

Validation Methodology The KLCSP and CHLR are constructed on already proven and existing algorithms whose research work has already been published. Also, the author gave mathematical proofs extending the already existing proofs. Along with theoretical proofs an experimental setup has been run on the continent-wide graph dataset of North America (this includes only the US and Canada), represented by a total of 21, 133, 774 nodes and 52, 523, 592 edges. 6, 779, 795 edges support one or more labels in this dataset, with 0.21 labels per edge, on average.

Changes if I were to rewrite Frequency of addition/deletion/modification of the constraints Best path calculation instead of the shortest path. Example: Travel b/n A and B, when time is the factor. Traffic conditions inclusion in path calculation and prediction of traffic condition.

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