Efficient Large-scale Road Inspection Routing Yujie Chen1, Peter Cowling1, Stephen Remde2 and Fiona Polack1 1YCCSA, Computer Science Department, University of York, York, U.K. 2Gaist Solutions Limited, Lancaster, U.K. Keywords: Graph reduction, Routing, Chinese Postman Problem. Abstract: Gaist Solutions Ltd. carries out large scale surveys for UK road inspection. To estimate the total distance that vehicles travel, we model routing as a Chinese Postman Problem. We propose a novel graph reduction ap- proach that dramatically speeds up the calculation of the Chinese Postman Tour for large-scale road networks. Because the analysis of large road-network graphs is now possible, planners can explore the effects of changes to traditional inspection techniques and scheduling. Case studies of road networks from six UK cities and the county of Norfolk are tested. The graph reduction process is also analysed on ten randomly generated road networks with different characteristics, to show its ability and give advice for suitable use. 1 INTRODUCTION in Section 5. Section 6 and 7 compare our results over seven local authority road networks and ten simulated In the UK, local governments maintain roads for pub- data sets. Conclusions are presented in Section 8. lic utility and safety. We can model route inspection as a Chinese Postman Problem (CPP). To estimate the total distance that vehicles travel to inspect all routes, 2 REAL-WORLD ROAD a very large-scale CPP must be solved. Published ap- NETWORKS proaches to deriving a Chinese Postman Tour (CPT) are computationally demanding and do not scale to UK local authority road networks are designated as large graphs. Graphs that represent road networks 1-lane, 2-lane, 3-lane and 4-lane single carriageways are characterised by vertices with low degree (e.g. or 2-lane dual carriageways (with a central reserva- T junctions of degree 3 or crossroads of degree 4). tion). We represent a road network as an undirected Also, roads in residential areas have a strong branched graph G(V,E). Vertices V represent the data collec- structure. By understanding these characteristics, we tion points: some are junctions or dead ends, but oth- propose a graph reduction approach to improve the ers are simply bends in the road. Edges E represent efficiency of routing large-scale real-world road in- the roads that link vertices. spection problem. Our road-inspection routes are for seven large-scale road networks monitored by Gaist In our inspection problems, up to 3 lanes can be Solutions Ltd. in partnership with the UK local coun- monitored in one pass. Figure 1 shows 1-lane, 2-lane cils of Blackpool, Southend, Manchester, Stockport, and 3-lane single carriageways represented by a sin- Halton, Warrington, and the rural county of Norfolk gle undirected edge. Figure 2 shows 4-lane single car- (road lengths of 515, 508, 1,315, 945, 619, 879 and riageways and dual carriageways represented as two, 26,243 kilometres, respectively). We also evaluate parallel undirected edges. A close road is represented our graph reduction approach using simulated net- as a loop (Figure 3), and a cul-de-sac is represented works. as an degree-1 vertex (Figure 4). The road information was originally collected Section 2 introduces road networks and their street by street, but has some errors or omissions. Be- transformation to abstract graphs. Section 3 outlines fore creating the graph representation, we pre-process CPP solutions and their scalability limitations. The and label intersections as vertices, as follows. graph reduction process is introduced in Section 4, then the application of the conventional CPT solution All degree-2 vertices are removed, since they do and of two variations using heuristics are introduced • not represent intersections, and thus have no im- 304 Chen, Y., Cowling, P., Remde, S. and Polack, F. Efficient Large-scale Road Inspection Routing. DOI: 10.5220/0005749203040312 In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 304-312 ISBN: 978-989-758-171-7 Copyright c 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved Efficient Large-scale Road Inspection Routing (a) 1-lane single carriageway (b) 2-lane single carriageway (c) 3-lane single carriageway (d) Graph representation (a) Close (b) Graph representation Figure 1: Graph representation of single carriageway roads. Figure 3: Graph representation of close roads. (a) 4-lane single carriage- (b) 2-lane dual carriageway way (a) Bulb (b)Dead-end (c)Representation Figure 4: Graph representation of cul-de-sacs. (c) Graph representation an Eulerian graph, which must then contain an Euler Figure 2: Graph representation of 4-lane single carriageway and 2-lane dual carriageway. tour. Edmonds and Johnson (1973) provide a widely- used CPP approach that is polynomial on the number of vertices and edges, (and does not scale well to large pact on the construction or distance of inspection graphs) as follows. tours. For all degree-2 vertices, vk, e(vi,vk) and e(vk,v j) are replaced by a single edge, e(vi,v j), with length l((vi,v j)) = l((vi,vk)) + l((vk,v j)). Step 1: From an undirected graph G(V,E), find the shortest path between all pairs of odd-degree vertices. We assume that an intersection has been omitted • if the data indicates that a road terminates close Two algorithms are commonly used in this to another road or road-termination point. The step. The most efficient implementations critical value for “close”, is set at ε = 3 me- of Dijkstra’s algorithm (Dijkstra (1959)), tres, based on our analysis of the original data. a single-source shortest-path algorithm, can So, where the distance from a degree-1 vertex achieve O(m + nlogn), where m is the num- vk to an edge e(vi,v j) is less than ε, the edge ber of edges (Fredman and Tarjan (1987)). e(vi,v j) is replaced by two new edges e(vi,vk) Floyd (1962) and others developed an algo- and e(vk,v j). Or, where the distance between two rithm, now known as the Floyd-Warshall al- degree-1 vertices is smaller than some ε, the ver- gorithm (FW), of complexity O(n3) where n tices are merged. is the number of vertices. Removal of degree-2 vertices significantly reduces Tested on our large-scale sparse graphs, FW system- the size of the graphs, as shown in Table 1. atically outperforms Dijkstra’s algorithm. However, the computational efficiency of FW can itself be sig- nificantly improved by reducing the number of nodes in a graph, e.g. by applying graph reduction. 3 CPP AND THE CHALLENGE OF SCALE Step 2: Find the minimum-cost perfect matching, M, of odd-degree vertices. CPP was formulated by a Chinese mathematician: A mailman has to deliver mail to every road of his ser- The best-known minimum-cost per- vice area before returning to the post office. The prob- fect matching implementation (Gabow lem is to find the shortest walking distance for the (1990)), using the blossom algorithm (Ed- mailman. (Guan (1962)). monds (1965); Edmonds and Johnson (1973)), is also computationally expensive 3.1 Edmonds’ Approach (O(n(m + nlogn))). Kolmogorov (2009) has an implementation which achieves O(n2m). To solve a standard CPP, a CPT is derived by adding Again, the complexity of the algorithm is dependent the smallest possible number of edges to construct on the number of edges and vertices in the graph. 305 ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems Table 1: Number of vertices before and after removal of 2-degree vertices for seven UK local authority road networks. Vertices Blackpool Southend Manchester Stockport Halton Warrington Norfolk Total 26302 22864 45408 44470 29610 24518 549345 less degree-2s 5103 4571 13337 8665 4796 7471 44912 Step 3: Add extra edges to connect all matched Proof. Suppose that there is an Euler tour v w x − − 1 − pairs of vertices through the shortest path in G, to cre- x1 ... xn v w y1 y2 ... yn v, where edge ate an Eulerian graph. (v,−w) is− travelled− − in− the− same− direction− − both times. Then v w v yn yn 1 ... y1 w x1 x2 − − − − − − − − − − − ...xn v is another Euler tour where edge (v,w) is trav- Step 4: Find the CPT: an Euler tour in the Eulerian elled− in both directions. graph. Then, the length, lCPT , of the identified Chi- nese Postman Tour (CPT) is: lCPT = ∑ l(e) + l(M) 4 REDUCTION OF LARGE e E ∈ GRAPHS The best known approach to finding an Euler tour is Hierholzer’s algorithm (Hierholzer and As noted above, the greatest contribution to compu- Wiener (1873)), with time complexity O(m). tational complexity of the algorithms used to derive Another widely applied approach, by Fleury a CPT is the number of vertices and edges. Our (1883), achieves O(m2). graph reduction applies graph contraction techniques as used in graph minor theory (Chartrand and Oeller- mann (1993); Lovasz´ (2006)). Edge contraction is a 3.2 Other Approaches fundamental operation in graph minors which deletes an edge from a graph G and merges the two end Apart from Edmonds’ CPP solution, Laporte (1997) points. Here, we propose a novel graph reduction introduces methods of transforming an arc routing method to decrease the size of the calculations whilst problem into an equivalent TSP. This approach is also maintaining the necessary characteristics of the origi- used by Irnich (2008) to solve a large-scale real-world nal graph to reconstruct a CPT. After the data prepara- postman problem. Heuristics for the TSP can then be tion described in Section 2, each road network is rep- used to solve the transformed CPP.