This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg) Nanyang Technological University, Singapore. Applied graph‑ theoretic approach for two‑ way to one‑ way road network conversion Tan, Kok Chuen 2005 Tan, K. C. (2005). Applied graph‑ theoretic approach for two‑ way to one‑ way road network conversion. Master’s thesis, Nanyang Technological University, Singapore. https://hdl.handle.net/10356/6468 https://doi.org/10.32657/10356/6468 Nanyang Technological University Downloaded on 10 Oct 2021 00:13:03 SGT _.'._"-,1 .- ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library An Applied Graph-Theoretic Approach for Two-way to One-way Road Network Conversion Tan Kok Chuen School ofMechanical & Aerospace Engineering A Master thesis submitted to Nanyang Technological University in fulfillment ofthe requirement for the degree of Master ofEngineering 2005 T \ ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library Abstract It is not uncommon in the field of traffic engineering for two-way roads to be converted to one-way roads and vice versa. In these situations the traffic engineer would be faced with the problem of selecting direction for each of the one-way roads. This problem is named the one-way road network problem in this thesis. By modelling the road networks as graphs, an algorithm (called 'conversion algorithm' in this thesis) is developed; this enables the search for a global optimal solution for the given road network (based on a theoretical efficiency criterion) by means of enumeration, or exhaustive listing of solutions. The said conversion algorithm is next coupled with the minimum cut graph model which allows further refinement of the theoretical solution(s) obtained based on actual existing traffic conditions. The conversion algorithm together with the minimum cut graph model thus form a model named 'one-way road network model' for solving the above-mentioned problem. A case study is selected and simulated in order to verify the usefulness ofthe one-way road network model. In the one-way road network model, two-way road networks are modelled by undirected graphs and one-way road networks by digraphs. The problem then becomes one of converting an undirected graph to a digraph. The conversion algorithm is developed for obtaining feasible digraphs; this is done by using the device ofbinary number counting to model the directionality of digraph edges so as to enumerate all possible digraphs. For a road network, only strongly connected graphs are "feasible" hence the concept of strong connectivity is incorporated in the conversion algorithm to weed out the "infeasible" digraphs. This is done by inspecting the indegrees and outdegrees of each of the enumerated digraphs. Those with zero indegree or outdegree are not strongly connected and are removed by the conversion algorithm. To arrive at a more efficient one-way road network, the theoretical aspects of network efficiency including junction-traversal delay are explored. Dijkstra's algorithm is modified from calculating the shortest path to calculating the network shortest total travel time. The modified algorithm is incorporated as part ofthe conversion algorithm and used to compare the digraphs enumerated to give the digraph(s) representing the road network with shortest total travel time. ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library With the selected 'theoretically efficient' road networks, actual traffic conditions are then taken into consideration and the various one-way road directions on the network are reversed and tested using a minimum-cut graph model. This allows efficient consideration of not only each direction but combinations of directions from different roads as well. By employing a minimum cut algorithm onto the graph model, the desired directions at critical junctions of the- one-way road system for the case study are decided. This gives an efficient one-way road network layout based on actual traffic conditions for the selected case study. The one-way road network is tested against the original two-way road network by means of computer simulation. Results of the simulation runs showed that the one-way road network performed better than the original two-way road network under given traffic load of the case study. This thus verifies the usefulness of the one-way road network model developed in this thesis. Recommendations for future work include improving the conversion algorithm by graph manipulation techniques. Using heuristic reasoning on the results obtained, patterns such as 'twirls' in the one-way roads can be observed; this suggests that graph manipulation techniques identifying digraphs with such patterns can be developed as shortcuts to obtaining theoretically efficient digraphs. Such techniques allow the huge number of infeasible graphs to be removed without having to enumerate all possible digraphs, saving on time and computational resources. A limitation of the minimum-cut graph model caused by the triangle graphs employed can also be further researched. In addition, complexity aspects ofthe one-way road network problem can also be looked into. In the above study, it is assumed that all roads used are unsignalised, that is road junctions are assumed to have no traffic lights; traffic flow consists of only motorised vehicles excluding motorcycles, bicycles and pedestrians; road network is analysed at a macro level, that is to say concepts and variables at the micro level (like driver behaviour, fuel emission, car-following theory) will not be taken into consideration; a fixed traffic demand is used, that is the amount of traffic used for various simulation runs will remain unchanged. 11 ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library Acknowledgements The author would like to extend his sincere gratitude to the following people: Supervisor, A/Prof Lew Sin Chye, for his guidance and contributions throughout this project. A/Prof Wong Yiik Diew, A/Prof Lum Kit Meng and A/Prof Tan Van Weng for advice and help in matters pertaining to traffic assignment, transportation simulation and statistics. Technical and administrative staff from Centre for Advanced Numeric Engineering Simulations, School of Mechanical & Production Engineering and Transportation Laboratory, School ofCivil & Environmental Engineering. Fellow graduate students from Rapid Prototyping Research Laboratory and MAE Graduate Student Club for their support and friendship. Fellow hostelites from Hall of Residence 1, Hall of Residence 2 and Hall of Residence 9 for their care and companionship. Especially to Willi Ching Choon Yong, May Win Naing, Valren Wee Zhi Sheng, Gabriel Lee Yeong Yeow, Lawrence Lim Chee Leong and Andy Chua Loy Ngee for their help in making this journey possible. Last but not least friends, family and loved ones for their love and encouragement. iii ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library T able of Contents ~]J~1r~,<:1r.......•..•.•.. D.O.OD •••••••• OO •••••••••••••••••••••••••••••••••••• ~ •••••••••••••••••••• ~.I ~<:KN"OWLEDGEMENTS .....•....• 5 •••••••••••••••••••••••••••••••••••••• 0 •••••••••• 00 ••• 0 ••111 1r~]JLE OF <:ON1rENT~.•...•. II "' II II II IV LIS1r. OF FIGURES•..•.......•.... 0 •• I) II) a •••••••••••••••••••••••••• D ••••••••••••••••••••••••••••••IX LIST OF T~BLES•..................•...•...•.....••....•.•.•.....•.• o ••••••••••••••••••••••••• ~XII I IN1rRODUC1rION•......•••. II '...... 1 1.1 Background............................................................................. 1 1.2 Objectives... 4 1.3 Scope... 4 1.4 Organisation ofthe Thesis. ....... ... ... ... .... ...... .. ............. .............. ... 5 II LITERA1rURE REVIEW. II c> C> II................... 7 2.1 Overview 0......................................... 7 2.2 Background information... ... 7 2.2.1 Combinatorics...... 7 2.2.2 Graph Theory 00. ....................... 8 2.2.2.1 Graph........................................................................... 9 2.2.2.2 Adjacency matrix............................................................. 9 2.2.2.3 Directed graph................................................................. 10 ~ ~ 2.2.2.4 Indegree and outdegree 0•••••••••••••••••••••• 0.. 11 2.2.2.5 Weighted graph............................................................... 12 2.2.2.6 Minimum spanning tree (MS1)... 13 2.2.2.7 Strong Connectivity. ............................. ........................ .. 14 2.2.2.8 Shortest path algorithm...................................................... 14 2.2.2.9 Shortest total path... .. .. 15 2.2.2.10 Minimum cut and Maximum flow... 16 iv ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library 2.2.3 Application ofgraph theory. ....... ... .......................................... 18 2.2.3.1 Tree building in traffic assignment...................................... 18 2.2.3.2 Minimum-cut graph model in Module Orientation Problem......... 19 2.2.4 Traffic assignment. 24 2.2.4.1 O-D matrix... 25 2.2.4.2 Cost-flow function............................................................ 26 2.2.4.3 Vehicle arrival distribution... ... 27 2.2.4.4 Measurement-of-effectiveness (MOE)..................................... 27 2.2.4.5 Traffic