GISRUK 2002

This is co-authored paper presented at GISRUK 2002, University of Sheffield, Sheffield, UK, 3-5 April 2002

Read online

Read online

Reference

Sherlock, R., Moone, P.,  and Winstanley, A. (2002) Shortest Path Computation: A Comparative Analysis. Paper presented at GISRUK 2002, University of Sheffield, Sheffield, UK, 3-5 April 2002

MFworks has evolved from MAPFactory, originally designed by C. Dana Tomlin, the father of map algebra.  Conducting network analysis in MFworks comprises iterative steps that lead to a functioning network. These steps will convert map layers with square cells into linear elements that are linked together as lines, with directional flows assigned to each cell, and map layers containing cost variables. This tutorial, developed by husdal.com in 2002, is a showcase on network analysis in MFworks, with step by step instructions and a summary of the theory behind it.

MSc in GIS

This tutorial builds on my thesis for my MSc in GIS, where I explored the topic of network analysis in raster GIS, using MFworks as example software, investigating current algorithms, procedures and network modelling techniques and finding some odd artefacts along the way.

MFWorks

Tutorial

Click here for the tutorial.

Related

• husdal.com: Network analysis in raster versus vector GIS
• husdal.com: How to make a straight line square (MSc in GIS)
• husdal.com: Corridor analysis – a timeline of evolutionary development

My PhD?

On a sidenote, I should mention that since this post was first published, it has been mistaken for my PhD, not once, but twice…and I don’t even have a PhD yet. So much for due diligence when doing literature review. See here how to get a PhD without a dissertation. And now, back to the main topic of this post.

Introduction

Locating a right-of-way for a linear facility such as a pipeline, a transmission line, a railway or a roadway can be a complex problem, both because of the input parameters that have to be evaluated and the alternatives that have to be studied. Once one good path has been identified, the task should then continue in order to identify other similarly good alternatives. Locating a corridor connecting an origin and a destination on a landscape is analogous to identifying a route that traverses a continuous landscape. Thus, corridor analysis is closely linked to shortest-path finding and network analysis. This paper will look at route finding methods in raster GIS and how corridor analysis has evolved and what the future may hold in terms of new research alleys.

Corridor analysis, network analysis or surface analysis?

Euler’s famous “Königsberg bridge” question, dating back as far as 1736, is often seen as the starting point of modern path finding – was it possible to find a path through the city of Königsberg crossing each of its seven bridges once and only once and then returning to the origin? Euler’s methods formed the basis of what is known as graph theory, and which in turn paved the way for path finding algorithms. Traditionally, network analysis, path finding and route planning have been the domain of graph theory and vector GIS, which is where most algorithms find their application. Since the objective is to find a path across a continuous surface, a raster-based GIS is usually employed in analyzing different alternatives, since it is not difficult to adapt these algorithms to a raster environment (Husdal, 2000a).

Raster applications are based on movement across a surface than movement along a network, since the general idea of finding the least cost path is linked to movement from cell to cell, and not along a finite line. However, this cell-based approach can only approximate a crisp line. Consequently, the delineation of a path across a surface in a raster GIS may very well be interpreted as a corridor rather than a confined path. Berry (2000, personal communication, in: Husdal, 2000b) prefers the term surface analysis to the term network analysis, because “It seems humans are encapsulated in their autos and effectively constrained to the line work, but most other organisms and phenomena are of the “off-road” variety and see a continuum of movement possibilities”. What this implies, is that albeit the generic structure of raster GIS facilitates an approximation and distortion of a presumably smoothly curved real-world corridor feature, it nevertheless also visualizes and draws attention to the spatial context that the least-cost path or corridor is set in.

Turning a surface into a corridor

In a raster GIS cartographic space is defined as a surface, where the value of a particular property varies over this surface. In order to adapt a network or corridor structure, each cell may be seen as a node linked to its eight neighboring cells. The cell value of each node then can represent the cost of traversing this particular cell. This cost-of-passage surface is a grid where the values associated with the cells are used as weights to calculate least cost paths. These weights may represent the resistance, friction or difficulty in crossing the cell and may be expressed in terms of cost, time distance or risk (Collischonn and Pilar, 1999). Starting from a given destination cell, it is then possible to spread outward and calculate for each surrounding cell, the accumulated cost of traveling from any surrounding cell to the destination cell.

From this accumulated surface it is then possible to delineate the shortest or least-cost path to the destination cell from any surrounding cell (Douglas, 1994), simply by following the path with the least accumulated friction. In principle, when using a raster GIS, the shortest path is derived by adding the cost surface derived from a spread from an origin cell with the cost surface from a spread from a destination cell. The cells with the lowest values indicate the delineation of shortest path; the cell values indicate the actual accumulated cost of the path.

A historical timeline

–Dijkstra
Dijkstra (1959) developed the now classic and probably most used shortest path algorithm. The most interesting feature about this algorithm is that it guarantees the optimal solution. The algorithm computes a path in all directions from the origin node and terminates when the destination has been reached. Notwithstanding the several improved algorithms and data structures for algorithms that have been put forward since, it is noteworthy that the Dijkstra algorithm has prevailed to the present date, thus proving its universal validity.

–McHarg
It seems appropriate to credit McHarg (1967) with one of the pioneering articles in corridor analysis. This was before computers invaded the field of spatial analysis, and when evaluating multiple criteria simply meant overlaying transparent maps to visualize the total impact. So it is in this case, where the least cost corridor was defined as being the area that maximizes social benefit while minimizing social cost. This method still applies today, when analysts use a GIS to overlay the similar data layers with an on-screen result, rather than transparent maps on a light table.

–Goodchild
By the mid 1970s, computers and GIS hade made their way into spatial analysis. At that time, continuous space in the real world was normally translated into a lattice or grid in computer space. As stated initially, this introduces error and approximations, particularly in regard to straight-line or smoothly curved paths. Goodchild (1977) is among the first to investigate this issue, proving that movement across a surface should not be in only orthogonal steps, but also diagonal steps to minimize elongation and deviation of the path compared to its real-world counterpart.

–Huber and Church
By addressing the geometric errors in some of the at that time current methodologies, Huber and Church (1985), paved the way for modern corridor analysis. Particularly significant is their attempt to incorporate more than just the immediately adjacent orthogonal or diagonal neighbors as candidates for spreading, but look beyond to the 15 or even 24 neighboring cells. This smoothes the delineated path and gives it a less zigzagged look. Xu and Lathrop (1995) later picked up on this, though not in terms of corridor analysis, but in looking at wildfire spread. Here, the normally circular spread is translated into an elliptical shape.

–Tomlin
Even though the concepts surrounding surface analysis for optimal paths date back as early as the late 1970s (Berry, 2000; personal communication, in: Husdal, 2000b), they were later championed by Dana Tomlin with his dissertation, which was later published as a book. Here, Tomlin (1991) classifies his map algebra operators based on how the computer algorithm obtains data values for processing and identifies three fundamental classes: local, focal and zonal functions. He introduces a spread algorithm (focal function) for calculating proximity surfaces, delineating the shortest possible distance from any location to a destination point. In this method, all cells are initialized to no value. Passes are then made through the image with each cell checking the cost of travel from each of its adjacent cells that contain a value. If the incremental effort distance is less than the difference in cumulative cost or the cell contains no value then the cell requires updating to a new value. Once a pass is completed without any changes, the cost surface has been generated (Tomlin, 1986; McIlhagga, 1997).

–Eastman
McIlhagga (1997, see below) notes that Tomlin’s method is efficient for small areas or narrow passages such as a road network; however it becomes very inefficient when large images are processed. Ronald Eastman remarks that “a 512 by 512 grid could require 700 passes” to produce a cost surface (Eastman, 1989). As part of the IDRISI GIS system, Eastman implemented a second algorithm to generate cost surfaces, called the pushbroom procedure. The [procedure operates] by pushing effects through the image, much like a pushbroom would be used to systematically clean a room. Effects then ripple through the image, much like water being pushed over a wet floor.” (Eastman, 1989). The cost surface image is initialized with no values except the target cell and a pass through the image is made from upper right to lower left with each cell updating the cell to the right and each cell below. In this manner all following cells are updated so that the most recent change is pushed forward with each successive update. Generally, three passes has been found to be enough to generate a cost surface (McIlhagga, 1997).

–Lombard and Church
Lombard and Church (1993) present what they call the gateway-shortest-path-algorithm (GSP) in an attempt not only to find the optimal path, but also at the same time support the search for good alternatives. The GSP allows for developing a sample of N-2 alternative paths for a network comprised of N nodes.
The GSP uses the Dijkstra algorithm twice, firstly spreading from origin to destination, and secondly spreading from destination to origin, the layers are then added and the path follows the cells with the lowest value. What is interesting here is that the while cell values indicate the cost to reach this particular cell from any node, the GSP algorithm also keeps track of from which neighboring cell any cell is best reached in the direction of either destination or origin. This then allows the analyst to choose any cell outside the optimal path and backtrack the path from this cell towards destination or origin and thus delineate alternative paths and alternative costs. To this authors knowledge, this method has later only appeared in Cova (2002), as a way to visualize the alternative “path-objects” that any cell may hold.

–McIlhagga
McIlhagga (1997) introduces a new term: fixed-cost distance, as opposed to the usually applied effort-distance; costs associated with effort distance are incurred every time a movement over a path occurs or more generally, movement between cells. Fixed cost distance is the cost associated with creating a path for linking multiple cells to a destination, and is incurred only once, when the path is created. Thus, minimizing effort distance involves minimizing the cost from a given point to any target (the accumulated cost surface for this given point); minimizing fixed cost distance requires finding the optimal path that connects a point to all targets regardless of minimizing the path to any of these targets in isolation (McIlhagga, 1997). Figure 3-7 gives a simple illustration McIlhagga’s fixed cost distance.
Later developed into software called “Pathways”, the core algorithm in McIlhagga’s research is an extension of Ron Eastman’s Push Algorithm applied for multiple targets. In general, the algorithm “builds” optimal cost surfaces for combinations of targets until all targets have been included. The key issue for the “Pathways” algorithm is to optimize and predict which combinations will provide the greatest contributions to a final solution.

–Collischonn and Pilar
Collischonn’s and Pilar’s concept of least-cost-paths is particularly worth mentioning, because it not merely traces a path down a cost surface, and presents the path following the least friction as the least cost path, but it links the cost of traversing the slope to the degree and direction of the slope itself. Thus, forcing a path down a steep slope may in fact be more costly than a descent that circumvents the steepest directions. This view coincides with conventional planning procedures for roads and canals, where the topography adjacent to the path plays a major role in determining the most viable route for the least cost path. Their algorithm uses a cost-slope function to assign accumulated cost to cells in 3X3 window around a center cell. The steeper the slope is, either uphill or downhill, the higher the cost will be, thus favoring directions with no or little difference in slope. As a result of this procedure, the least-cost path up or down a hill is not the straight line following the steepest path, but a path that winds or climbs the hill sideways

–Berry
Berry (2000) is not a new name to GIS or map analysis. However, he wandered into the presumably more rewarding commercial realm from academia in the late 1980s. In his Map Analysis Package, and later MapCalc, he uses a concept similar to Tomlin; he calls his algorithm “Splash” rather than “Spread”. Berry’s map operators differ from Tomlin, although they are used for the same purposes. Berry’s classification scheme is based on the user’s perspective of map input and output contents; what the map(s) look like going in and coming out: reclassify, overlay, distance and neighborhood. The Splash algorithm belongs to the distance division. For a more detailed description, see Husdal (2000).
Berry also develops a concept for finding the path between multiple targets, using what he calls a stepped accumulation surface. First the optimal path is calculated from point one to point two, then from point two to point three, and further repeating the procedure for any number of points along the route (Figure 3-12). However, this procedure does not calculate the best order of points along the route; points are visited in the order they appear.

–Cowen, Jensen, Hendrix, Hodgson and Schill
Cowen et al (2000) incorporate an econometric routing model for the exploration of potential routes for a railway track, using construction cost factors such as cut and fill costs, road and river crossings, and the track cost as input parameters. Their research uses remotely sensed data, among which a high accuracy DEM derived from LIDAR, which serves to indicate that the approximations of raster GIS can be compensated by using high-resolution accurate data.

–Cova and Church
It may seem strange to mention a paper on site selection as the last in a timeline of corridor analysis, but Cova and Church (2000) point in a direction that this author is inclined to follow in his future research. Bearing in mind that a corridor can be construed as an elongated contiguous site, it does not seem to far-fetched to use the site selection analogy to find the optimal corridor. The terms of measure of goodness of fit of a corridor will course be different from a compact site, but the principle is the same.

Paradigms and dichotomies

The first and foremost dichotomy in least-cost path or corridor analysis is the age-old division between raster and vector GIS, as briefly discussed in Husdal (1999). Choosing one over the over, one will have to battle with each system’s inherent paradigms: vector as a crisp and presumably accurate delineation of linear features, raster as an approximation of both shape and length, where the distortion is determined by cell resolution and grid tessellation.

For the realm of this paper, namely raster GIS, a second paradigm can be seen in the path algorithms: Tomlin’s (1991) “spread” and Berry’s “splash” (2000) are fairly similar. Eastman’s (1987) “pushbroom” may also be included here, since all algorithms “ripple” through the image like water on a surface. Interestingly enough, these 3 different algorithms can today be found in 3 different commercial GIS: MFworks, MapCalc and Idrisi respectively, thus extending the paradigm beyond research into the user perspective.

The “gateway-shortest-path” by Lombard and Church (1993) is a Dijkstra algorithm, as is the work of Collischonn and Pillar (2000). McIlhagga (1997) diverts and uses his own algorithm, which he names “fixed-cost-distance”.
The transition from using only the orthogonal neighbors to all 8 or even beyond that was first introduced by Huber and Church (1985). Here, they showed the effect this could have on the path accuracy and thus make it less dependent on the grid structure of raster GIS, and marks yet a shift in the prevailing thought at that time. Though unrelated to corridor analysis itself, Xu and Lathrop (1995) later picked up this perspective in analyzing wildfire spread and investigated a spread algorithm with disjunct cells.

A fourth paradigm is the differentiation between path objectives. Mostly they are used to delineate a path from one origin to one destination. Lombard and Church (1993) were the first to use a second destination as a constraint along the way in order to evaluate path alternatives. McIlhagga (1997) actually solves the path for any number of destinations. Berry’s (2000) use of a step-by-step approach reaches all targets, but it may not be the overall optimal path or corridor that links these destinations.

Whereas most of the cited references discuss finding the least cost path by following the least friction from cell to cell as being equivalent to following the greatest slope on an elevation surface, Collischonn and Pillar (2000) attempt a real-world approach by implementing cost as a function of slope, such that the lesser slope is preferred and greater slope is penalized. Similarly, though not based on the preceding, Cowen et al (2000) implement engineering costs as the actual determining cost factor, leading path delineation away from it’s slope based ancestors.

Finally, Cova and Church (2000) may pave the way for a new way to look at corridor analysis, namely as a site selection problem, where the site has to include origin and destination and any other gateways, and at the same time minimize its spatial extent and maximize its suitability. In foresight, this is what this author will attempt to delve deeper into in his future research.

References

Berry, J., 2000, Analyzing Accumulation Surfaces, in: Map Analysis: Procedures and Applications in GIS Modeling. Berry and Associates // Spatial Information Systems Inc,

CHURCH, R., LOBAN, S., & LOMBARD, K. (1992). An interface for exploring spatial alternatives for a corridor location problem Computers & Geosciences, 18 (8), 1095-1105 DOI: 10.1016/0098-3004(92)90023-K

Collischonn, W., & Pilar, J. (2000). A direction dependent least-cost-path algorithm for roads and canals International Journal of Geographical Information Science, 14 (4), 397-406 DOI: 10.1080/13658810050024304

Cova, T. (2000). Exploratory spatial optimization in site search: a neighborhood operator approach Computers, Environment and Urban Systems, 24 (5), 401-419 DOI: 10.1016/S0198-9715(00)00015-6

Cova, T., & Goodchild, M. (2002). Extending geographical representation to include fields of spatial objects International Journal of Geographical Information Science, 16 (6), 509-532 DOI: 10.1080/13658810210137040

Cowen, D.J., Jensen, J.R., Hendrix, C., Hodgson, M., Schill, S.R., 2000, A GIS-Assisted Rail Construction Econometric Model that Incorporates LIDAR Data. PE&RS, 66, 11, 1323-1328

Deo, N., & Pang, C. (1984). Shortest-path algorithms: Taxonomy and annotation Networks, 14 (2), 275-323 DOI: 10.1002/net.3230140208

Dijkstra, E. (1959). A note on two problems in connexion with graphs Numerische Mathematik, 1 (1), 269-271 DOI: 10.1007/BF01386390

Dolan, A. and Aldous, J. ,1993, Introduction to Networks and Algorithms.

DOUGLAS, D. (1994). Least-cost Path in GIS Using an Accumulated Cost Surface and Slopelines Cartographica: The International Journal for Geographic Information and Geovisualization, 31 (3), 37-51 DOI: 10.3138/D327-0323-2JUT-016M

Dreyfus, S. (1969). An Appraisal of Some Shortest-Path Algorithms Operations Research, 17 (3), 395-412 DOI: 10.1287/opre.17.3.395

Eastman, J.R., 1989, Pushbroom Algorithms for Calculating Distances in Raster Grids. Proceedings, AUTOCARTO 9, 288-297.

Goodchild, M.F., 1977, An evaluation of lattice solutions to the problem of corridor location. Environment and Planning A, 9, 7, 727-738.

Huber, D., & Church, R. (1985). Transmission Corridor Location Modeling Journal of Transportation Engineering, 111 (2) DOI: 10.1061/(ASCE)0733-947X(1985)111:2(114)

Husdal, J., 1999, Network Analysis – Raster versus Vector, A comparison study. Unpublished, coursework for the MSc in GIS, University of Leicester.

Husdal, J., 2000a, An investigation into fastest path problems in dynamic transportation networks. Unpublished, coursework for the MSc in GIS, University of Leicester.

Husdal, J., 2000b, How to make a straight line square – Network analysis in raster GIS. Unpublished, thesis for the MSc in GIS, University of Leicester.

Lee B.D., Tomlin C.D., 1997, Automate transportation corridor allocation: cartographic modeling makes it easy to determine a minimum-cost/impact alternative. GIS World 10, no.1 (1997) p. 56-58+60

Lombard, K, Church, R.L., 1993, The Gateway Shortest Path Problem: Generating Alternative Routes for a Corridor Routing Problem. Geographical Systems, 1, 25-45

Limp, F. (2000), R-E-S-P-E-C-T. A review of raster GIS systems. GeoEurope, June 2000

Malczewski, J., 1999, Multicrieria Visualization: Reorganizing Fire Protection Services. In: GIS and Multicriteria Decision Analysis, Wiley and Sons

McHarg, I, 1967, Where should Highways go?, Landscape Architecture, 57, 179-181

McIlhagga, D., 1997, Optimal path delineation to multiple targets incorporating fixed cost distance. Unpublished, Honors Thesis for Bachelor of Arts, Carleton University.

McIlhagga, D., 2000, Technological Impact on Route Planning of Optimal Solutions to the Multiple Target Access Problem. Unpublished, Slides presentation delivered at GIS 2000, Toronto, Canada, March 14th, 2000. Obtained through personal communication.

Thinkspace Inc., 2000, MFGuide. User manuals and tutorials for MFworks.

Tomlin, C. D., 1983, Digital Cartographic Modelling Techniques in Environmental Planning. Unpublished, doctoral dissertation, Yale University.

Tomlin, C.D., 1990, Geographic Information Systems and Cartographic Modelling, Prentice Hall, New Jersey

Xu, J., & Lathrop, R. (1995). Improving simulation accuracy of spread phenomena in a raster-based Geographic Information System International Journal of Geographical Information Science, 9 (2), 153-168 DOI: 10.1080/02693799508902031

Reference

Husdal, J. (2001) Corridor Analysis – A timeline of evolutionary development. Unpublished coursework. University of Utah, USA. Available online at http://husdal.com/2001/04/25/corridor-analysis-a-timeline-of-evolutionary-development/. Last accesed on [date].

Related

• husdal.com: How to make  a straight line square
• husdal.com: MFworks Tutorial
• husdal.com: Book Review: This is where raster GIS started

Conclusions

An extension of Tomlin’s directional identifiers is proposed, allowing the modelling of non-planar features. Along with this, the integration of time- dependent travel cost variables is achieved through linking MFworks with an external Visual Basic application for updating the cost-of-passage surface, demonstrating that such interaction extends the inherent capabilities of a GIS engine. Another conclusion to be drawn from this paper is that network analysis in raster GIS is a variant of surface analysis.

Read online

Jan Husdal Thesis MSc in GIS

MFWorks Tutorial

This insights gained in this thesis were later used for developing a tutorial for network analysis in raster GIS using MFWorks.

Reference

Husdal, J. (2000). How to make a straight line square. Network Analysis in Raster GIS with time-dependent cost variables. Unpublished. Thesis for the MSc in GIS at the University of Leicester, UK.

Related

• husdal.com: Network analysis in raster versus vector GIS
• husdal.com: Book Review: This is where raster GIS started
• husdal.com: How to use MFworks for network analysis
• husdal.com: Corridor analysis – a timeline of evolutionary development

Introduction

Euler’s famous “Königsberg bridge” question, dating back as far as 1736, is often seen as the starting point of modern path finding – was it possible to find a path through the city crossing each of its seven bridges once and only once and then returning to the origin? His methods formed the basis of what is now known as graph theory, and which in turn paved the way for path finding algorithms.

The author’s interest in this subject stems from his lifelong experiences in long–distance road travelling, where successful route planning prior to travelling and en-route is essential to finding the optimal path from origin to destination. “Optimal” can take up many forms, such as shortest time, shortest distance, or least total cost, the latter being of major concern in some parts the author’s home country, where travelling by car may mean many costly ferry crossings and expensive toll roads in order to get from one’s departure to one’s arrival.

Semantically one can distinguish between path finding in a fixed static network, with set costs for traversing the network, and path finding in a dynamic network, where the cost of traversing the network varies over the time of traversing.

Because path finding is applicable to many kinds of networks, such as roads, utilities, water, electricity, telecommunications and computer networks alike, the total number of algorithms that have been developed over the years is immense. The aim of this essay is to compromise a selected cross-section of approaches towards path finding and the related fields of research, such as transportation GIS, network analysis, operations research, artificial intelligence and robotics, to mention just a few examples where path finding theories are employed.

It is this author’s deliberate intention to use a wide approach to show the different research gateways that lead towards path finding in dynamic networks. This is done in order to find a common denominator that will allow a unifying approach. Using a fitting network analysis metaphor, this can be described as “breadth first search”.

Intelligent Transportation Systems

In the recent decades road transportation systems have undergone considerable increase in complexity and congestion proclivity. This then has given rise to the field of ITS, Intelligent Transport Systems, with the goal to apply and merge advanced technology to make transportation more safe and efficient, with less congestion, pollution and environmental impact. In working towards this goal, ITS can take many different forms.

Vehicle location and navigation systems are one of these forms and have come along with the emerging field of transport telematics. Transport telematics implies the large-scale integration and implementation of telecommunication and informatics technology in the field of transportation, penetrating all areas and modes of transport, the vehicles, the infrastructure, the organisation and management of transport.

Zhao (1997) distinguishes between route planning and route guidance as two key elements in vehicle location and navigation systems as part of ITS. Route planning is the process that helps vehicle drivers plan a route prior to driving a specific part of his or her journey. Route guidance is the real-time process of guiding the driver along the route generated by a route planner.

Huang et al. (1995) discriminates route guidance even further, distinguishing between centralised and decentralised route guidance. In the former, vehicles conduct their own path finding using on-board computers and static road maps in CD-ROMs, and applying heuristic search algorithms. Centralised route guidance relies on traffic management centres (TMC) to answer path queries submitted by vehicles linked to it. In this case, Huang et al. (1995) describe a central server holding a materialised view of all shortest paths at that given time, accessed by lookup requests from the vehicles equipped with this system. Although not explicitly stated, it can be assumed that this also is the case in the Advanced Traveller Information System (ATIS) detailed by Shekar and Fetterer (1996a) or the ADVANCE project portrayed by both Revels (1998) and Zhao (1997). Boyce et al. (1997) provide a detailed evaluation study of the ADVANCE project for further reference.

Shortest Path Algorithms

The analysis of transportation networks is one of many application areas in which the computation of shortest paths is one of the most fundamental problems. These have been the subject of extensive research for many years: Deo (1984), Cherkassky et al. (1993), Zhan et al. (1996 and 1997).

A network consists of arcs, or links, and nodes. The fastest path is calculated as a function associated with the cost of travelling the link. Even though the different research literature tends to group the types of shortest paths problems slightly different, one can discern, in general, between paths that are calculated as one-to-one, one-to-some, one-to-all, all-to-one, or all-to-all shortest paths. In software packages solving static network shortest path problems the software usually aggregates a once-off all-to-all calculation for all nodes, from which subsequent routes then are derived.

Clearly, this approach is not feasible for dynamic networks, where the travel cost is time-dependent or randomly varying. However, the majority of published research on shortest paths algorithms has dealt with static networks that have fixed topology and fixed costs. A few early attempts on dynamic approaches, referenced by Chabini (1997), are Cooke and Halsey (1966) and Dreyfus (1979). Given the computational restraints in the capacity of past computer systems this is not surprising. Not more than a decade ago, Van Eck (1990) reports several hours as an average time for a computer to churn through an all-to-all calculation on a 250-nodes small-scale static network, and several days on a 16.000-nodes large-scale network.

One way of dealing with dynamic networks is splitting continuous time into discrete time intervals with fixed travel costs, as noted by Chabini (1997). Thus, understanding shortest path algorithms in static networks becomes fundamental to working with dynamic networks.

Shortest paths in static networks

Several algorithms and data structures for algorithms have been put forward since the classic shortest path algorithm by Dijkstra (1959). In its modified version, this algorithm computes a one-to-all path in all directions from the origin node and terminates when the destination has been reached. Deonardo and Fox (1979) introduce a new data structure of reaching, pruning and buckets.

The original Dijkstra algorithm explores an unnecessary large search area, which led to the development of heuristic searches, among them the A* algorithm, that searches in the direction of the destination node. This avoids considering directions with non- favourable results and reduces computation time.

A significant improvement is seen in the bi-directional search, computing a path from both origin and destination, and ideally meeting at the middle. In relation to this search technique, it should be remarked that Jacob et al. (1998) discard bi-directional algorithms as impractical in their computational study of routing algorithms for realistic transportation networks. Firstly, as not extendable to general path problems and secondly, as having a run-time notably longer than A*.

Zhan and Noon (1996) undertake a comprehensive study of shortest path algorithms on 21 real road networks from different 10 different states in the U.S., with networks ranging from 1600/500 to 93000/264000 nodes/arcs. In this study, Dijkstra-based algorithms, however differing in data structure, outperform other algorithms in one-to-one or one-to-all fastest path problems.

In summary, the A* algorithm, along with Dijkstra-based algorithms, are preferred in most of the literature researched by the author the author. It is in fact noteworthy that the Dijkstra algorithm has prevailed to the present date, proving its universal validity.

Shortest paths in dynamic networks

It is a result of the recent advances in computer and communications technology, together with the developments in ITS, that have flared a renewed interest in dynamic networks. This interest in the concept of dynamic management of transportation has also brought forward a set of algorithms that are particularly aimed at optimising the run-time of computations on large-scale networks.

Chabini (1998) lists the following types of dynamic shortest path problems depending on (a) fastest versus minimum cost (or shortest) path problems; (b) discrete versus continuous reperentation of time; (c) first-in-first-out (FIFO) networks versus non-FIFO networks, in which a vehicle departing at a later time than a previous vehicle can arrive at the destination before the pervious vehicle; (d) waiting is allowed at nodes versus waiting is not allowed; (e) questions asked: one-to-all for a given departure time or all departure times, and all-to-one for all departure times; and (f) integer versus real valued link travel costs.

Fu and Rilett (1996) investigate what they call the dynamic and stochastic shortest path problem by modelling link travel times as a continuous-time stochastic process. The aim of their research was to estimate travel time for a particular path over a given time period. They deviate from the mainstream appraisal of the A* algorithm and advocate the k-shortest path. The reason for this is that standard shortest path algorithms may fail to find the minimum expected paths, particularly when dealing with non-linear optimisation, as is the case in developing travel time estimation models. However, in lieu of real data, their research is based on a hypothetical change pattern in travel time.

Building on the research from path finding algorithms in static networks, Chabini (1997) remarks that a time-space expansion representation can be used in dynamic networks, applying discrete time intervals with fixed costs. Hence, depending on how time is treated, dynamic shortest path problems can be subdivided into two types: discrete and continuous. In the discrete case, if using 15-second time intervals, a full 24-hour implementation would involve calculations on 5760 time discretisations, multiplied with the number of nodes and links. Chabini (1997) makes a distinct separation between fastest time paths, in which the cost of a link is the travel time of that link, and minimum cost paths, in which link costs can be of a general form. The difference between these two is nonetheless not explored until Chabini (1998).

Chabini (1997) identifies two key questions in dynamic path finding: (1) what are the fastest paths from one origin to all destinations departing at a given time, and (2) what are the fastest paths from all nodes to one destination for all departure times. He sees the latter as the most significant in relation to ITS, which is true, if one assumes that ITS aims at finding the best path for multiple vehicles with the same destination. In Chabini (1998) the focus extends slightly, now three questions are put forward: (1) one-to-all fastest path at a given time interval, (2) all-to-one fastest path for all departure times and (3) all-to-one minmum cost path for all departure time intervals.

Chabini (1997, 1998) places emphasis on the all-to-one minimum cost path as the key algorithm with relation to ITS, the reason being that only a limited set of all network nodes are destination nodes in realistic road networks, while there is a considerably larger number of nodes that will be origin nodes. (Moving vehicles tend more to converge to the same goal than to spread in all directions)

Horn (1999) continues along the research trails of Chabini (1997) and Fu and Rilett (1996), but uses a less detailed articulation of travel dynamics, reflecting as he puts it, the recognition that information about network conditions in most parts of the world are most likely to be sparse and that merely estimates of average speed on individual network links are available in most cases. With the presumption that these estimates allow for variation in speed, congestion and delays at nodes, he studies a number of Dijkstra variant algorithms that address these particular conditions. Most important, he propounds an algorithm that calculates an approximation of shortest time path travel duration (path travel time), independent of the particular navigation between nodes. For an experienced vehicle driver, estimated travel time may be more important than the exact route that is to follow. This is a noteworthy addition to the fastest path algorithms in dynamic networks.

Related research: robotics

Taking into account Chabini (1998), who noted that the literature on dynamic shortest path problems seems to be quite limited, the author of this paper investigated an intriguing sidetrack, namely path finding within the field of robotics. Robots, or autonomous or remotely operated vehicles, must perform in both known and unknown environments. Finding a path through uncertain terrain is similar to path finding in dynamic networks.

Haigh et al. (1994, 1995, 1997) uses case-based reasoning and the accumulation and reuse of previously traversed routes as main entry point to optimal path finding. Here, the A* algorithm in combination with analogical reasoning is seen as the most robust framework for route planning that can adapt to incomplete information and changing conditions (Haigh et al., 1997). Other heuristic searches (i.e. hill-climbing) are discarded as prone to failure in unexpected situations.

Stentz (1994, 1995) takes the A* algorithm further, by modifying it into what he calls D* (Dynamic A*). His works focus on moving a robot equipped with an innate map through a field of unknown obstacles, and revising the map and re-planning the route as obstacles are encountered.

With reference to ITS and dynamic path finding, the “unknown obstacles” in Stentz’ work can be circumscribed as the arbitrary or time-dependent changing of link costs, thus providing a link for transferral of knowledge between these research fields.

Discussion

It is noteworthy that the heuristic A* algorithm dominates the research literature for static networks.

Almost all comparison studies involving the A* algorithm conclude with this algorithm’s superiority over other approaches. Dijkstra-based algorithms, with various enhancements in their data structure in the form of heaps, buckets and queues, are also well represented.

Research has also proven that dynamic network fastest path problems can be reduced to static fastest path problems if continuously varying link travel times are expanded for a time interval or given an estimated value. The A* and Dijkstra algorithms are applicable in both static and dynamic networks.

Finding fastest paths in dynamic networks as part of ITS and in the route re-planning module as part of robotics appear to have seemingly diverging approaches. In the former the emphasis is on assembling and conveying as much real and updated information as possible. The latter starts off with some, but not necessary all information. Here, by applying analogy, user-dependent preferences and experiences can be incorporated into route planning or guidance systems.

Particularly in the case of Advanced Traveller Information Systems, there will be a considerable amount of repetitive day-to-day travelling and commuting. The experience and knowledge of these traffic patterns ought to be fed back into the system and used as a knowledge database, providing a basis for analogical reasoning, which is fundamental to Haigh’s papers.

The author believes that establishing a lateral relation between robotics and ITS can form the basis of further exploration and fruitful merger of the two research fields.

Future outlook

This essay attempted to investigate some of the approaches to path finding in static and dynamic networks that are listed in present research literature. A selected set of different approaches were highlighted and set in a broader context, illustrating the various aspects of path finding in time-dependent dynamic environments.

The author is fully aware that not all relevant research has been covered, but believes that the approach provided in this paper can provide a starting point for transferral of knowledge and research between ITS and robotics. Interestingly enough, relevant research literature exploring this specific topic has not been found to date. Consequently, this possible link between ITS and robotics suggests a promising area for future research.

In such, this paper is meant as a potential framework for the development of a lateral methodology in conjunction with the author’s MSc Thesis.

Acknowledgements

The investigation for this research essay has not been an easy task. The University of Leicester is not renown for it’s advances in Transportation and Operations Research, and thus, most of the much literature had to be obtained by making use of the total limited number of inter-library loans, by accessing online publications, searching via internet databases, posting in mailing lists and user groups, by visiting other libraries in person and via personal communication with the various authors of the papers referenced in this essay. This has been a frustrating and time-consuming process, and the investigation has not been as full and elaborate as the author himself had hoped for initially. Nevertheless it has been a rewarding and learning process, sparking the motivation for his prospective future work. The author wishes to express his sincere gratitude and appreciation to all those who in large or small contributed to the successful completion of this paper.

References

Boyce, D. E. et al (1997) Dynamic route choice model of large-scale traffic network, Journal of Transportation Engineering, vol. 123, no. 4, pp. 276-282

Burrough, P.A. and McDonell, R.A. (1998) Principles of Geographic Information Systems, Ch.7, pp.180-181

Chabini, I. (1998) Discrete dynamic shortest path problems in transportation applications, Transportation Research Record 1645

Chabini, I. (1997) A new algorithm for shortest paths in discrete dynamic networks, as presented at 8th IFAC/IFIP/IFORS Symposium on transportation systems, Tech Univ Crete, Greece, 16-18 June 1997

Cherkassky, B., Goldberg, A. and Radzik, T. (1993) Shortest paths algorithms: theory and experimental evaluation, Technical Report 93-1480, Department of Computer Science, Stanford University, 1993.

Chou, Y. H. (1997) Exploring Spatial Analysis in Geographic Information Systems, Ch.7, pp. 215-264

Cooke, K.L. and Hasley E. (1966) The shortest route through a network with time-dependent intermodal transit times, Journal of Mathematical Analysis and Applications, vol. 14, pp. 493-498

Deo, N. and Pang, C.-Y. (1984) Shortest path algorithms: taxonomy and annotation, Networks, vol. 14, pp. 275-323

Deonardo, E. and Fox, B. L. (1979) Shortest-route methods: 1. Reaching. Pruning and buckets, Operations Research, vol. 27, pp. 161-196

Dijkstra, E. W. (1959), A note on two problems in Connection with graphs, Numerische Mathematik, vol. 1, 1959, pp. 269-271

Dolan, A. and Aldous, J. (1993) Introduction to Networks and Algorithms, Ch. 7, Ch. 8, Ch 18

Dreyfus, S.E (1969) An appraisal of some shortest path algorithms, Operations Research, vol. 17, pp. 395-412

Fu, L. and Rilett, L.R.(1996) Expected shortest paths in dynamic and stochastic traffic networks, Transportation Research, Part B: Methodological, vol. 32, no. 7, pp. 499-516

Haigh, K. Z. , Shewchuk,J.R. and Veloso, M. (1994) Route planning and learning from execution. Working notes from the AAAI Fall Symposium “Planning and Learning: On to Real Applications”, AAAI Press, November, 1994, pp. 58 – 64.http://www.ri.cmu.edu/pubs/pub_2922.html

Haigh, K. Z. and Veloso, M. (1995) Route planning by analogy
Case-Based Reasoning Research and Development, Proceedings of ICCBR-95, Springer-Verlag, October, 1995, pp. 169 – 180 http://www.ri.cmu.edu/pubs/pub_2921.html

Haigh, K. Z. , Shewchuk,J.R. and Veloso, M. (1997) Exploiting Domain Geometry in Analogical Route PlanningJournal of Experimental and Theoretical Artificial Intelligence, No. 9, September, 1997, pp. 509 – 541http://www.ri.cmu.edu/pubs/pub_2921.html

Horn, M. E. T. (1999) Efficient modeling of travel in networks with time-varying link speeds, CSIRO Mathematical and Information Sciences Technical Report CMIS 99/97 http://www.cmis.csiro.au/Mark.Horn/

Huang, Y.-W. Jing, N., Rundensteiner, E. (1996) Path view algorithm for transportation networks: The dynamic reordering approach, ITS RCE Center, Technical Report, June 1995 ftp://ftp.eecs.umich.edu/people/rundenst/papers/r-95-14.ps

Jacob, R., Marathe, M. V. and Nagel, K. (1998) A computational study of routing algorithms for realistic transportation networks, 2nd Workshop on Algorithmic Engineering (WAE’98), Saarbrücken, Germanz, August 19-21 1998, received via personal communication

Jones, C. (1998) Geographical Information Systems and Computer Cartography, Ch. 13, pp. 225-230

Liu, B. et al. (1994) Integrating case-based reasoning, knowledge-based approach and Dijkstra algorithm for route finding. Proceedings of the Tenth Conference on Artificial Intelligence for Applications, pp. 149-55

Openshaw, S. and C. (1997) Artificial intelligence in geography, Ch. 6, pp. 55-72, John Wiley and Sons

Revlels, B.M. (1998) The ADVANCE Project – A Dynamic Route Guidance Case Study http://www.people.virginia.edu/~bmr8n/ITSPaper/advance.htm

Shekhar, S., Fetterer, A. and Liu, D.-R. (1996) Genesis: An approach to data dissemination in Advanced Traveler Information Systems, Bulletin of the Technical Committee on Data Engineering, September 1996, vol. 19, no. 3, pp.

Shekhar, S. and Fetterer, A. (1996) Path computation in Advanced Traveler Information Systems, Proceedings of the 6th Annual Meeting and Exposition of the Intelligent Transportation Society of America, Houston, Texas, August 15-18, 1996

Stentz, A. (1994) The D* Algorithm for real-time planning of optimal traverses, Tech Report CMU-RI-TR-94-37, Robotics Institute, Carnegie Mellon University, October 1994 http://www.ri.cmu.edu/pubs/pub_356.html

Stentz, A. (1995) The Focussed D* Algorithm for real-time planning of optimal traverses, Proceedings of the International Joint Conference on Artificial Intelligence, August, 1995. http://www.ri.cmu.edu/pubs/pub_1213.html

Stentz, A. (1993) Optimal and efficient path planning for unknown and dynamic environments,Tech Report CMU-RI-TR-93-20, Robotics Institute, Carnegie Mellon University, August 1993 http://www.ri.cmu.edu/pubs/pub_310.html

Van Eck, J. R. and De Jong, T. (1990), Adapting datastructures and algorithms for faster transport network computations, Proceedings of the 4th int. symposium on spatial data handling, vol.1, pp. 295-304

Zhan, F. B. (1997) Three fastest shortest path algorithms on real road networks: Data structures and procedures, Journal of Geographic Information and Decision Analysis, vol.1, no.1, pp. 69-82 http://publish.uwo.ca/~jmalczew/gida_1/Zhan/Zhan.htm

Zhan, F. B. and Noon, C. E. (1996) Shortest Path Algorithms: An Evaluation using Real Road Transportation Science vol. 32, no. 1, pp. 65-73

Zhao, Y. (1997) Vehicle Location and Navigation Systems, Ch. 5 – 6, Ch. 10 – 12, Artech House Inc, Norwood, MA http://birch.dlut.edu.cn/~yzhao/

Related

• husdal.com: Shortest path finding: a comparison

Introduction

In general, a network is a system of interconnected linear features through which resources are transported or communication is achieved. The network data model is an abstract representation of the components and characteristics of real-world network systems. One major application of network analysis is found in transportation planning, where the issue might be to find paths corresponding to certain criteria, like finding the shortest or least cost path between two or more locations, or to find all locations within a given travel cost from a specified origin. Traditionally, a GIS, represents the real world in either one of two spatial models, vector-based, i.e. points, lines and polygons, or raster-based, i.e. cells of a continuous grid surface. This study will investigate the subject of network analysis in both raster and vector GIS, in order to compare the two spatial models. It will discuss their limitations and advantages, by using a road network as an example.

Network modeling in general

A network model can be defined as a line graph, which is composed of links representing linear channels of flow and nodes representing their connections (Lupien et al.,1987). In other words, a network takes the form of edges (or arcs) connecting pairs of nodes (or vertices). Nodes can be junctions and edges can be segments of a road or a pipeline. For a network to function as a real-world model, an edge will have to be associated with a direction and with a measure of impedance, determining the resistance or travel cost along the network.

Typical network graph and table structure, listing nodes, connectivity of edges, turn impedance and edge attribute data.

Since networks utilize the basic arc-node structure, by definition, due to the way the data is stored, the vector network will already have a topological structure, relating all elements. All that is needed, simply speaking, is to implement the resistance factors in the attribute tables for the lines or nodes. Directions are an explicit part of the vector network topology. If the directions are derived from digitising a road map, or received as a ready coded network form a data supplier, they may not correspond with the real-world directions and need to be checked. Consequently, the representation of network elements requires substantial amount of time to be devoted to data preparation and validation. This can be quite complex, depending on the amount of travel cost information we want to incorporate in the model: road width, speed limit, road class, delay at traffic lights, delays in taking turns at crossroads, to mention just a few. For a “simple” crossroads with four edges and one node there are as many as 16 possible turns, three directions from each edge to other edges, plus four 180-degree U-turns. In a mixed rural/urban road network in an average Norwegian municipality, with 7000 edges and nodes, there can be as many as 18000 turn possibilities (Husdal, 1998). Arcs usually describe the centreline of a network feature, such as a road centreline. Arcs and nodes are discretely referenced by coordinates. Alsolines that cross, but not intersect, can be directly implemented in the vector model, much like in the real world, where we have “overpasses” and “underpasses”.

Network modeling in vector GIS

Arcs and nodes, together with the special-purpose network elements stops, centers and turns, form the network model in vector GIS. Stops can be delivery or pick-up points along a route, centers are used for allocating services and investigating catchment areas, turns arte used in determining direction and flow within the network. The characteristics of any system being modeled in a network must be abstracted into a form that may be represented by one of these elements. Path finding in vector GIS Dolan et al (1993), Chou (1993) and Jones (1998) have described the process of finding a criteria-determined path through a network in great detail. Path finding algorithms fall into one of two main categories, matrix algorithms and tree-building algorithms, of which the latter one is the one mostly used in GIS. Matrix algorithms find the shortest distance between all pairs of nodes in iterative steps, eliminating the least favorable nodes, as seen in Chou (1993). This is based on that it is possible to represent the network as a matrix. Tree-building algorithms find the shortest path from an origin node to all other nodes, producing a tree of shortest paths with branches emanating from the origin. (Lombard et al., 1993). The most commonly used tree-building procedure is that originally developed by Dijkstra (1959), of which to date many modifications and improvements have been made for specific applications. In order to find a path, the algorithm will build a tree data structure that represents specific paths through the network. This is often referred to as a breadth-first search, that fans out to as many nodes as possible before penetrating deeper into the tree (Dolan, 1993). Starting from one origin node, the search tree builds branches in all directions, adds up the resistance figures, and keeps only those that represent the cumulative least cost. For each new set of adjacent nodes the calculations for all possible edges towards these nodes are repeated till all nodes and edges have been utilized, and the final destination is reached with minimal cost. During the process, edges may appear in the search tree and then disappear as the calculations discard their value.

Network modeling in raster GIS

Locations defined as nodes in a network, made up by grid lines.

Raster network modeling takes a completely different approach to the topological linked vector model. First, the grid cells only approximate the exact shapes of the lines in the network. Secondly, direction is not explicitly given as in the vector model. Thirdly, the line and node attributes must be stored as a separate layer for each attribute. As a result, a network using a raster model normally consists of a vast number of layers. Even if it does not appear so explicitly, a grid is in fact a graph representing a network, with 8 possible directions from each node. Since the grid has a given resolution, the cells will only approximate the exact shapes of the network.

Path finding in raster GIS

The pathfinding algorithm in raster is similar to the algorithm in vector grid. In order to find a least-cost path one must first derive an accumulated cost surface, associated with cost or impedance of crossing the surface from cell to cell. To derive the cost surface one would have to interlace and combine the various grids describing various attributes. Fairly complex, but given the map algebra inherent in raster GIS, the computation itself is then straightforward. In terms of map algebra, Tomlin (1990) describes the the process of moving from origin to destination as a “spreading” function, using waves and refraction as analogy, an approach that is also supported by Douglas (1994). Practically speaking, a raster GIS software (e.g. MFworks) will compute the least path as follows: The spread function employs the cost surface to calculate the cost of passing from the origin outward towards the destination and assigns the accumulated value to each cell that is passed. Then, the reverse is done, going from destination to origin. Adding the two accumulated together yields the least-cost path.

Finding best path in raster GIS (using MFworks). Left, calculating path from origin to destination. Middle, backpath from destination to origin. Right, result of adding the two paths, best path is shown in black.

A raster network is planar, since the grid represents one continuous surface. To model multi-planar lines one would have to construct several layers with lines that are cut off where there in vector are underpasses and overpasses. For example, to model an underpass, we would need one overpass network layer with the underpassing line cut off at the overpass, and one underpass network layer with the overpassing line cut off at the underpass. Modelling directions in a raster network is possible, though not as easy as in vector. A slope value can be seen a value of direction. As long as neighbouring cells have the same slope, movement is allowed, on encountering a cell with opposite slope, entry in that direction is restricted or prohibited. Modelling specific turns at crossroads can be done in the same way. However, one would have to create numerous layers corresponding to the number of turn options and flow directions. In conjunction with this, it is also necessary to have layers describing the cost or resistance of making the turn.

Comparison study

In general, depending on the grid resolution, raster and vector networks are capable of performing the same spatial operations and analyses. In both cases the flow of movement, either from node to node in a network, or from cell to neighbouring cell of a raster, is subject to resistance, determining the direction and speed of flow. The way this resistance is modelled differs from vector to raster. Using vector, it is easier to import the attribute data of a given network in from an external database, which is where the data often would be stored in real life. Directions over a raster network cannot be carried out without first deriving the topology from the surface properties, whereas directions are an explicit part of the vector topology, even when they need to be corrected and validated in the modelling process. The vector model can hold discrete entities, for instance keep the exact length of an arc as arc attribute, while the raster model only will approximate this, depending on the grid resolution. Neighbouring lines in raster will have to assume the minimum resolution distance of say 10, 20 or 50m, whatever the cell size is. In vector, due to coordinate referencing, there is virtually no limit as to how close lines can be or how many that can be incorporated in the network. Computations are very much based on the same principles. Even though a vector model can be fairly complex, the complexity is more or less limited to the attribute table(s). It does not, like in raster, constitute a large number of interwoven layers.

Conclusion

The vector data model is feature oriented, as it represents space as a series of discrete entities, which are geographically referenced by Cartesian coordinates. The raster model is location oriented, where each cell is part of a tessellated continuous surface that describes a given attribute. Since a network is based much more on the interaction of its component entities than it is based on its component locations, the vector model intuitively seems more appropriate for any kind of network. Based on this investigation it is possible to make two distinct observations: A vector-based network model is likely to be more suitable than a raster model for analysing precisely defined paths, such as roads and rivers or drainage canals, i.e. discrete entities that derive mainly from the built environment, and where attributes play a major role in determining the network. A raster-based network model, on the other hand, seems to be more fit, when the problem is concerned with finding a path across terrain that does not have predefined paths, and where the network does not consist of many attribute layers and artificial directional constraints, because that will make the modelling process more complex. The key to producing successful network models is in understanding the relationship between the characteristics of physical network systems and the representation of those characteristics by the elements of the network model. In vector and raster, the efficiency and validity of the network depends on how precisely the network can be modelled to match the real-world network it represents. Thus, it makes no difference whether vector or raster is used. What matters, is that the model used is appropriate to the task in question.

References

Burrough, P.A. and McDonell, R.A. (1998) Principles of Geographic Information Systems, Ch.7, pp.180-181

Chou, Y.H. (1997) Exploring Spatial Analysis in Geographic Information Systems, Ch.7, pp. 215-264

Dolan, A. and Aldous, J. (1993) Introduction to Networks and Algorithms, Ch. 7, Ch. 8, Ch 18

Douglas, D.H. (1994) Least-cost Path in GIS Using an Accumulated Cost Surface and Slopelines, Cartographica, vol.31, no. 3, pp. 37-51

Husdal, J., Sandvik, A. and Klingsheim, A. (1998), PloGIS – et verktøy i forvaltning av bilpark og planlegging av kjørerute (Norwegian, English: PloGIS – a tool in car pool management and route planning), Coursework submitted in fulfillment of the one-year undergraduate study in GIS, Telemark College, Norway

Jones, C. (1998) Geographical Information Systems and Computer Cartography, Ch. 13, pp. 225-230

Laurini, R. and Thompson, D. (1992) Fundamentals of Spatial Information Systems, Ch. 5, pp.175-197 and Ch. 8, pp. 310-312

Lombard, K. and Church, R.L. (1993) The Gateway Shortest Path Problem: Generating Alternative Routes for a Corridor Routing Problem, Geographical Systems, vol. 1, pp. 25-45

Lupien, A.E., Moreland, W.H. and Dangermond, J. (1987). Network analysis in geographic information systems. Photogrammetric Engineering and Remote Sensing, vol. 53, no. 10, pp.1417-1421

Mainguenaud, M. (1995) Modelling the network component of geographical information systems, International Journal of Geographical Information Systems, vol. 9, no. 6, pp. 575-593

Van Eck, J.R. and De Jong, T. (1990), Adapting datastructures and algorithms for faster transport network computations, Proceedings of the 4th int. symposium on spatial data handling, vol.1, pp. 295-304

Zhan, F. B. (1997) Three Fastest Shortest Path Algorithms on Real Road Networks: Data Structures and Procedures, Journal of Geographic Information and Decision Analysis, vol.1, no.1, pp. 69-82

Thinkspace Inc./ MFworks (1999) How do I Find the Cheapest Route, excerpt from the MFworks’ Help Menu

Zhou Q., Yang X. and Melville M. D. (1996) A GIS Network Model for Sugarcane Field Drainage Simulation, in Proceedings of 8th Australasian Remote Sensing Conference, 25-29 March, Canberra, Vol. 2, pp 366-372.

Reference

Husdal, J (1999) Network analysis – network versus vector – A comparison study. Unpublished course paper for the MSc in GIS. University of  Leicester, UK.

Related

• husdal.com: How to make a straight line square
• husdal.com: How to use MFworks for network analysis
• husdal.com: Corridor analysis – a timeline of evolutionary development