Department of Mathematics
Prepared Test Instances Extracted
from OpenStreetMap Data Using
Different Network Reductions
Kaj Holmberg
Department of Mathematics
Link¨
oping University
Prepared Test Instances Extracted from OpenStreetMap
Data Using Different Network Reductions
Kaj Holmberg kaj.holmberg@liu.se Department of Mathematics Linköping Institute of Technology
SE-581 83 Linköping, Sweden
April 5, 2018
Abstract: We investigate the effect of different reductions when importing networks from OpenStreetMap data. We describe the network reductions and report computa-tional tests for doing the network extraction and reduction. We also show the effect of the reductions by solving a few standard optimization problems in the resulting net-works. Computational tests show that the reductions have a dramatic effect on the network size and the time needed for solving the optimization problems. In many cases, the reductions are necessary in order to be able to solve the optimization problem in reasonable time. A practical result of this work is a set of networks that will be used as benchmarks in future research, and are publically available for other researchers.
1
Introduction
OpenStreetMap (OSM) data gives a possibilty of obtaining indata for many optimiza-tion problems in city networks. However, the data might not be in the proper form for optimization. Often one has to treat the data in different ways, before using it. In the paper [5], we describe how to extract useful data from OpenStreetMap data. In general, we wish to extract network data suitable for optimization. Examples are find-ing shortest paths, minimal spannfind-ing trees, shortest postman tours etc. More advanced usage is multi-vehicle arc routing, for example for snow removal, [2], [3], [6]. All of these optimization problems require the network to be given as nodes and links (arcs/edges), not the paths or “ways” used in OpenStreetMap. For details, see [5].
Similar work is probably being done in many places around the world, and as a single example, we mention [7], where some algorithms for network extraction are given, and an implementation in C++ is presented. However, we have not seen a discussion and comparison of reductions as in our paper. Some kind of standardization may be useful.
Here we describe some of groups of test instances, which will be used in forthcoming work. One goal is to make the instances available to other researchers. We describe dif-ferent reductions of the networks, and motivate the reductions by solving some standard optimization problems in the resulting networks.
2
Data extraction and reduction
Let us give a brief description of the method used in this paper. The main task is to read the data in the available format, modify it according to certain rules, and write it in a format suitable for further optimization. We use OpenStreetMap data in xml-format. Such files contain very much information, much of which is not useful for our optimization problems.
The rules of making the network depends on which link types to include, and also on how elimination of nodes with degree two or one is done. The OSM data is first read and parsed. This results in a number of nodes, and a number of paths using these nodes. Nodes not used by any path are removed. The paths are then divided into links as described below.
When reading the nodes from the OSM data, each node has a name, longitude and latitude. Nodes outside of a predetermined window are dismissed.
Then each path is treated. First its type is checked against a list of types to include. The path is dismissed if its type is not in the list. Our first selection keep all paths with label “highway”. We have also done tests with only streets usable by car, i.e. keeping the following secondary labels: “motorway”, “trunk”, “primary”, “secondary”, “tertiary”, “road”, “residental”, “living_streets” (including the same with a trailing “_link”), but leaving out “pedestrian”, “footway”, “path” and “cycleway”.
The set of nodes associated with the path is put in a list. At this stage, uninformative tags are sorted out.
The nodes in each path are associated with the nodes given in the initial node list, and each node is labeled with the associated path number. Nodes not in the list are dismissed, as this concerns parts that are outside of our area of interest. Here paths are dismissed if all the nodes are dismissed. Then the paths are divided into links. Basically each adjacent pair of nodes in a list is made into a link. The cost of the link is set to the Euclidean distance between the two nodes. However, if the node is not the first or the last in the path, it may be eliminated, depending on the settings. Since each node is labeled with associated paths, we can easily count the number of paths using the node. If this number is one, we have a node which will get degree two, and for some settings such a node is eliminated. Then the distance is added to an accumulated distance, which will be put on the aggregated link. Each link is given the same type as the path had. (After this, we will not use the paths anymore.) In the process, the degree of each node is calculated.
R1 Splitting paths into links, removing isolated nodes. R2 R1 plus elimination of nodes with degree 2.
R3 R1 plus selective elimination of nodes with degree 2. R4 R2 plus elimination of trees.
R2c R2, keeping only links for cars. R4c R4, keeping only links for cars.
Table 1: Reductions
the problem solvable. The network needs to be reduced further in a suitable way. We have used the reductions in table 1, namely full or selective elimination of nodes with degree two and recursive elimination of nodes with degree one.
Reduction R2 is the most relevant one if the nodes with degree 2 are only introduced to signify the curvature of the street, as we often are not interested in the curvature. The method in [7] seems to give the result of R2.
There could however be a contradiction between simplification of the network and the issue of map matching for the GPS traces, see [4], in the sense that the simplifications made to make the route optimization easier will make the map matching harder. Map matching is in principle the task of associating coordinates from a GPS-track to streets in a city map. Then the coordinates are compared to the positions of the streets, and then the curvature of the streets obviously matters. If we plan to do map matching, selective reduction, R3, is probably better.
Selective elimination of nodes with degree two is done the following way. Consider a node i connected only to nodes j1 and j2. As parameter, we use the direct distance
between nodes j1 and j2 divided by the distance between j1 and i plus the distance between i and j2. If this values is greater than 0.98, we consider the curvature in the node to be unimportant, and node i is eliminated. This corresponds approximately to an angle between the two links close to 180 degrees (within plus minus 20 degrees). Details of this is described in section 6.1 in the paper [4].
Reduction R4 can be used when the treatment of turning places and the single street leading there can only be done in one way. This is true for snow removal, and probably for many other applications. In such a case one can plan the treatment of that part separately, note how long it will take, remove that part from the network, and afterwards just add that time to the time in the adjacent node left in the network. This is repeated recursively, so that no nodes with degree one remain.
The elimination of nodes with degree 2 is only done when reading the OSM data, not applied recursively to the resulting network. In other words, it is only done when splitting a path into links. The reason for this is that if a node belonging to two paths get degree 2, the links from the two paths may have different properties that should not be eliminated.
Paths in OSM rarely form cycles, so a cycle is usually made up by more than one path. If a cycle is made up by two paths, the nodes with degree two in each path is eliminated by reduction R2, but not the nodes where the paths meet. The cycle might then be
reduced to two nodes with two seemingly parallel links between them. This structure will not be eliminated by R4. (In a picture like in figure 2, the two parallel links appear as one, so it may look like a node has degree one, even if is has not.)
3
Computational tests
3.1 Computational details
The code is implemented in Python, using Numpy and Scipy. We especially use the codes in Scipy for finding the connected components of a graph, and for finding a min-imal spanning tree. We also use imposm.parser, a Python library that parses Open-StreetMap data in XML and PBF format. (The parts of the code we made ourselves in Python could obviously be made faster by implementing them in C.)
The tests were run on an Acer Aspire X3 X3995 3.4GHz, running Linux, Fedora Core 25. The machine has four CPUs, but only one was used in the test runs, except for imposm.parser, which can take advantage of several CPUs.
3.2 Test problems
We have downloaded a number of OSM instances from various sources. Small local cities were obtained by the extract tool on http://www.openstreetmap.org/. Larger areas has been obtained from http://download.geofabrik.de/europe.html, which offers data for whole countries. We also downloaded a number of regions from web pages that are no longer available.
The first set of test problems contains smaller cities in the area around Linköping. The second set contains parts of larger cities and other areas. The third set contains a few larger cities in Sweden, and the fourth set contains counties in Sweden.
The application we mostly have in mind, snow removal, is very relevant for the first set, and parts of the second set. The third and forth sets are too large for this application, as snow removal will be done in smaller areas by different (competing) contractors, so there is no point in optimizing snow removal for such large areas. However, for other tasks, these sets may be relevant. Examples may be distribution of goods, investigation of different aspects in the street network, mail delivery and waste collection.
3.3 Computational results
Table 2 shows first the sizes of the OSM instances, i.e. the number of nodes, n1, and paths, and the time (in seconds), t1, needed for simply reading the data file and ex-tracting the data. The conversion to links was made and isolated nodes were removed, which is reduction R1. This yields a smaller number of nodes, n2, and a number of links. The time needed for this is given as t2 (which includes t1). All highway types
are included. We also give the proper name of the city and the number of inhabitants in the area (as of 2010). The numbers of inhabitants are approximate, as often there are parts in the data file that lie slightly outside of the city borders. (The selection windows are rectangular.)
We see that in all instances, the number of nodes is greatly reduced, so there are many isolated nodes in the original data files. For the smallest instances, no significant difference in computation time can be observed, but for larger instances, more time is needed to split the paths into links and find which nodes are isolated.
In table 3, we give the results of different reductions of the network. All tests are done from scratch, so the times reported in table 2 are included in the times in table 3. In table 3, n1, l1 and t1 gives the number of nodes and links and the time for reduction R2, while n2, l2 and t2 gives the number of nodes and links and the time for reduction R3, and n3, l3 and t3 gives the number of nodes and links and the time for reduction R4.
We see that the size of the network decreases. Using the first instance, askeby, as an example, we find that the OSM data contains 1890 nodes and 357 paths, and after removing isolated nodes, 234 nodes remain. The division of paths into links gives 255 links. Selective reduction of nodes with degree 2 gives 101 nodes and 122 links, while complete reduction of nodes with degree 2 gives 77 nodes and 97 links. Finally, adding tree elimination, i.e. repeated elimination of nodes with degree 1, we get 50 nodes and 70 links. Obviously most optimization problems are much easier to solve for 50 nodes that for 234. (It would be even harder to attack a graph with 1890 nodes, of which many are isolated.)
The selective reduction used in R3 requires more calculations, and clearly takes more time, so it should not be used unless the curvature of the streets matters. Otherwise the times needed are quite short. Sometimes the time for R4 is slightly less than for R1, since parts of the calculations are done in a smaller graph.
For illustration, we show the second instance, atvid (with some outskirts removed) with a background picture from OpenStreetMap. (Note that the railway is not part of the network, but only a part of the background picture. In this context, the difference between network data and a picture is crucial.) The result of R1 and R2 is shown in figure 1, and the result of R3 and R4 in figure 2. One may note that the network in the figure of R1 is not connected, and that some separate parts are reduced to single nodes in the figure of R4. At this stage we do not remove isolated nodes, although it could easily be done.
We also did similar tests with a smaller set of links, namely only those who admit cars. In many applications, the services are done by cars, so foot paths and cycles paths should not be included.
In table 4, n1, l1 and t1 refer to 2 degree node elimination, R2c, while n2, l2 and t2 refers both this and tree elimination, R4c, for links useable by car. Here we see that the number of nodes with complete reduction shrinks to 19 for askeby.
Name n1 paths t1 n2 links t2 inhab city askeby 1890 357 0.05 234 255 0.10 518 Askeby atvid 1521 144 0.03 971 1043 0.11 6 859 Åtvidaberg bankekind 1956 311 0.07 266 271 0.07 405 Bankekind borensberg 3184 270 0.07 851 899 0.13 2 886 Borensberg boxholm 4224 235 0.09 974 1025 0.15 3 194 Boxholm brokind 1896 44 0.05 224 235 0.06 502 Brokind ekangen 1584 107 0.04 624 662 0.08 2 037 Ekängen finspong 6123 381 0.09 2008 2114 0.48 12 440 Finspång grebo 219 27 0.03 217 220 0.04 970 Grebo kisa 5033 669 0.09 1638 1700 0.26 3 687 Kisa linghem 12790 2389 0.16 1824 2136 0.77 2 804 Linghem ljungsbro 4562 519 0.09 3273 3465 0.57 6 620 Ljungsbro malmslatt 9376 1418 0.15 1907 2053 0.41 5 214 Malmslätt mantorp 1001 141 0.05 805 849 0.09 3 671 Mantorp mjolby 3984 312 0.17 1623 1742 0.28 12 245 Mjölby motala 16757 2505 0.24 8932 10087 5.34 29 823 Motala norsholm 5120 230 0.10 468 481 0.15 615 Norsholm rimforsa 5426 231 0.11 798 832 0.22 2 238 Rimforsa skeninge 1496 126 0.08 713 756 0.09 3 140 Skänninge soderkoping 3911 521 0.09 1840 2096 0.29 6 992 Söderköping sturefors 2913 189 0.07 1205 1249 0.18 2 229 Sturefors svartm 1917 95 0.05 262 264 0.08 116 Svartmåla tranas 3499 431 0.08 2349 2525 0.32 14 197 Tranås vadstena 4816 604 0.12 1455 1669 0.34 5 613 Vadstena vastervik 7634 1079 0.14 4273 4884 0.93 21 140 Västervik vikingstad 1077 149 0.04 747 790 0.10 2 096 Vikingstad
Table 2: Problem set 1, local cities, reading osm, network extraction, R1. Another example is Svartmåla, which is a small area mostly populated by summer-houses. In such areas the street network is often very close to a tree, and then almost all of the graph is eliminated by R4.
All the tests are repeated for the second set, parts of cities, in tables 5, 6 and 7. A difference is that we have no number of inhabitants for the different areas. The instances are larger, so the times are longer. As the size of the area grows, the importance of including walking and cycling paths decrease, so table 7 might be the most relevant one.
Results for the larger cities, set 3, are reported in tables 8, 9 and 10. Here we have not used selective elimination R3, since the solution times are rather large. In most cases of map matching, which is what mainly motivates R3, the vehicle has not been traveling around in all parts of the city, so the initial area could usually be restricted to a much smaller network.
For the counties, the results are given in in tables 11, 12 and 13. For the county of Blekinge, as an example, we find that the number of nodes were reduced from 91972 to
R2 R3 R4 Name n1 l1 t1 n2 l2 t2 n3 l3 t3 askeby 77 97 0.10 101 122 0.18 50 70 0.08 atvid 237 309 0.10 410 482 0.73 180 252 0.10 bankekind 34 39 0.09 66 71 0.13 16 21 0.08 borensberg 228 273 0.13 331 379 0.67 145 190 0.15 boxholm 218 267 0.14 364 415 0.75 135 184 0.14 brokind 48 59 0.06 88 99 0.13 30 41 0.07 ekangen 157 195 0.09 237 275 0.33 113 151 0.08 finspong 422 523 0.40 690 796 3.10 291 392 0.35 grebo 57 59 0.06 101 104 0.08 14 16 0.04 kisa 392 452 0.25 627 689 1.96 230 290 0.25 linghem 945 1255 0.68 1192 1504 2.34 702 1012 0.78 ljungsbro 766 949 0.48 1091 1283 7.53 531 714 0.56 malmslatt 566 712 0.38 722 868 2.73 390 536 0.43 mantorp 215 258 0.09 283 327 0.52 131 174 0.09 mjolby 426 542 0.26 570 689 2.05 305 421 0.23 motala 2766 3900 4.57 3657 4812 57.94 2404 3538 5.00 norsholm 83 96 0.16 116 129 0.32 56 69 0.16 rimforsa 188 213 0.21 358 392 0.59 92 117 0.17 skeninge 167 210 0.12 230 273 0.50 115 158 0.08 soderkoping 684 936 0.32 860 1116 2.42 533 785 0.35 sturefors 216 255 0.17 418 462 1.13 128 167 0.18 svartm 43 44 0.08 88 90 0.14 8 9 0.08 tranas 644 817 0.32 1122 1298 3.60 438 611 0.37 vadstena 606 813 0.40 770 984 1.67 463 670 0.36 vastervik 1580 2183 1.04 2149 2760 11.75 1298 1901 1.22 vikingstad 202 243 0.10 288 331 0.45 122 163 0.09
Table 3: Problem set 1, network extraction, degree 2 elimination, selective degree 2 elimination, tree elimination, R2, R3, R4.
Figure 2: Åtvidaberg (atvid), R3, R4. R2c R4c Name n1 l1 t1 n2 l2 t2 askeby 55 59 0.13 19 23 0.29 atvid 229 296 0.11 170 237 0.14 bankekind 34 39 0.08 16 21 0.11 borensberg 203 242 0.13 125 164 0.15 boxholm 195 239 0.14 114 158 0.16 brokind 45 54 0.08 26 35 0.09 ekangen 131 150 0.06 63 82 0.07 finspong 371 445 0.36 230 304 0.37 grebo 57 59 0.05 14 16 0.07 kisa 300 342 0.20 167 209 0.25 linghem 466 517 0.42 219 270 0.45 ljungsbro 552 620 0.35 251 319 0.43 malmslatt 396 439 0.26 184 227 0.27 mantorp 191 219 0.09 103 131 0.10 mjolby 395 499 0.32 273 377 0.27 motala 1766 2217 2.71 1222 1673 3.04 norsholm 74 85 0.14 54 65 0.16 rimforsa 172 189 0.17 68 85 0.22 skeninge 162 204 0.10 111 153 0.10 soderkoping 434 507 0.21 230 303 0.25 sturefors 178 193 0.13 69 84 0.15 svartm 43 44 0.07 8 9 0.09 tranas 568 713 0.28 365 510 0.37 vadstena 428 518 0.29 229 319 0.28 vastervik 1057 1335 0.72 726 1004 0.82 vikingstad 184 225 0.08 109 150 0.13
Table 4: Problem set 1, network extraction, degree 2 elimination, tree elimination, only cars, R2c, R4c.
Name n1 paths t1 n2 links t2 ekholmen 3875 663 0.14 1888 2139 0.37 hjulsb-east 1693 267 0.05 862 928 0.13 hjulsb-best 42608 3277 0.32 11489 12056 13.48 hjulsbro 22328 3289 0.26 11478 12675 5.54 link-mid 6637 1166 0.12 2266 2674 0.60 linkoping-c 40616 7438 0.36 13716 16015 17.86 linkoping-n 33132 5914 0.32 11350 13323 12.26 linkoping-so 48834 7795 0.43 17299 19663 23.43 linkoping-v 43247 6487 0.36 11519 13122 13.28 liu 1589 245 0.07 748 861 0.15 ljungsb1b 2614 254 0.07 1656 1739 0.22 ljungsb1c 1408 126 0.05 902 946 0.11 ljungsb1d 1328 124 0.04 796 817 0.09 pmo 1015 133 0.04 696 755 0.08 ryd 9537 1500 0.14 3026 3428 0.96 studryd 2787 452 0.08 929 1069 0.19 toliu1 25559 3539 0.26 8672 9757 4.77 toliu2 15658 2304 0.19 4575 5170 1.82 ullst 1369 172 0.06 641 693 0.11 valla 3128 536 0.09 1491 1682 0.26
Table 5: Problem set 2, parts of cities, reading osm, network extraction, R1. 74706 for R1, to 10386 for R2 and 7411 for R4, and to 5817 for R4c.
For these larger instances, running times are often quite large. However, the alternative would be to try to do the optimization without reductions, and this will be worse, as we shall see. Furthermore, these runs only need to be done once (unless the street network is changed), even if the subsequent optimization problem has to be solved repeatedly (for example each morning).
In figure 3, we show Ryd (an area in Linköping were many students live) after R1 and R4. (We don’t show R4c, since most students do not own a car.) The number of nodes goes from 3026 after R1 to 925 after R4. After R4c, it drops to 193 nodes, since much in a residential area like this is tree-like for cars. One does not want much through traffic with cars in such an area.
In this figure we also see the high quality of OpenStreetMap in student quarters, for example in the detailed jogging tracks in the forest.
3.4 Comparison of the effect of the reductions
The following computational tests aim at investigating the differences the reduction makes for the subsequent optimization. Here we have chosen three tests as representa-tives for the possible optimization problems one might like to solve.
R2 R3 R4 Name n1 l1 t1 n2 l2 t2 n3 l3 t3 ekholmen 750 998 0.38 934 1185 2.50 584 832 0.39 hjulsb-east 318 384 0.10 369 435 0.67 209 275 0.13 hjulsb-best 2229 2765 8.84 4118 4685 97.45 1507 2043 9.54 hjulsbro 3650 4824 5.86 4726 5923 89.45 2821 3995 7.29 link-mid 903 1306 0.53 1193 1601 3.71 722 1125 0.66 linkoping-c 5698 7969 15.03 7267 9566 127.29 4663 6934 17.87 linkoping-n 4877 6833 11.05 6301 8274 84.68 4043 5999 13.04 linkoping-so 6372 8706 20.68 8162 10526 208.94 5194 7528 24.19 linkoping-v 4271 5861 11.50 5657 7260 93.96 3433 5023 13.25 liu 300 412 0.15 401 514 0.48 228 340 0.12 ljungsb1b 360 442 0.19 513 596 2.01 238 320 0.19 ljungsb1c 199 242 0.11 296 340 0.66 124 167 0.10 ljungsb1d 156 176 0.11 214 235 0.53 87 107 0.08 pmo 173 228 0.07 262 321 0.40 118 173 0.08 ryd 1165 1567 0.87 1494 1896 6.41 925 1327 1.01 studryd 397 536 0.18 534 674 0.70 316 455 0.20 toliu1 2967 4041 4.50 3830 4915 52.78 2457 3531 5.23 toliu2 1634 2215 1.63 2162 2757 14.92 1261 1842 1.87 ullst 182 233 0.13 241 293 0.46 126 177 0.11 valla 546 735 0.24 689 880 1.64 425 614 0.29 Table 6: Problem set 2, network extraction, degree 2 elimination, selective degree 2 elimination, tree elimination, R2, R3, R4.
R2c R4c Name n1 l1 t1 n2 l2 t2 ekholmen 277 307 0.43 99 129 0.61 hjulsb-east 128 133 0.09 47 52 0.26 hjulsb-best 1439 1574 6.56 655 790 7.10 hjulsbro 2023 2303 2.53 983 1263 3.53 link-mid 354 419 0.28 197 262 0.33 linkoping-c 2703 3300 6.78 1663 2260 8.07 linkoping-n 2595 3174 5.33 1579 2158 6.39 linkoping-so 3229 3780 9.24 1734 2285 11.70 linkoping-v 2101 2468 4.84 1159 1526 5.99 liu 144 181 0.21 97 134 0.22 ljungsb1b 277 320 0.17 139 182 0.22 ljungsb1c 147 177 0.09 75 105 0.10 ljungsb1d 128 132 0.15 38 42 0.08 pmo 59 58 0.06 12 11 0.06 ryd 536 580 0.40 193 237 0.49 studryd 129 133 0.13 36 40 0.14 toliu1 1319 1459 1.69 558 698 1.99 toliu2 758 911 0.77 447 600 0.87 ullst 109 117 0.09 49 57 0.10 valla 268 319 0.15 144 195 0.19
Table 7: Problem set 2, network extraction, degree 2 elimination, tree elimination, only cars, R2c, R4c.
Name n1 paths t1 n2 links t2 inhab city borlange 40892 5727 0.59 14964 16765 12.93 41 734 Borlänge jonkoping 202601 15612 1.56 165695 172267 767.27 89 396 Jönköping kalmar 122370 11315 0.98 101164 105858 302.10 36 392 Kalmar linkan 39827 5782 0.42 13233 14911 12.78 104 232 Linköping orebro 103043 8597 0.90 84791 88245 188.55 107 038 Örebro uppsala 206545 20095 1.74 166768 174833 777.93 140 454 Uppsala
Table 8: Problem set 3, larger cities, reading osm, network extraction, R1.
R2 R4 Name n1 l1 t1 n2 l2 t2 borlange 4692 6475 13.01 3909 5692 14.59 jonkoping 22331 28712 674.20 16563 22944 771.72 kalmar 16043 20609 265.55 11807 16373 308.02 linkan 4608 6264 10.99 3777 5433 12.95 orebro 11666 15002 177.48 8895 12231 197.25 uppsala 29319 37009 771.82 20580 28270 956.86
Table 9: Problem set 3, network extraction, degree 2 elimination, tree elimination, R2, R4.
R2c R4c Name n1 l1 t1 n2 l2 t2 borlange 3026 3756 8.14 2040 2770 9.47 jonkoping 19884 25018 647.78 13999 19133 699.12 kalmar 13321 16606 247.29 9136 12421 259.34 linkan 2262 2578 4.49 1121 1437 5.55 orebro 9815 12256 160.34 7146 9587 165.91 uppsala 22428 26574 615.63 13637 17783 698.23
Table 10: Problem set 3, network extraction, degree 2 elimination, tree eliminaton, only cars, R2c, R4c.
Name n1 paths t1 n2 links t2 inhab blekinge 91972 7942 0.81 74706 77381 144.54 152 757 dalarna 308110 22947 2.31 228198 236428 1444.51 277 354 gavleborg 207671 18104 1.68 179547 187850 778.88 280 394 gotland 29950 2377 0.31 25471 26757 16.21 57 161 halland 217253 21616 1.76 183707 191003 868.47 314 212 jamtland 171292 12751 1.36 119956 123901 511.67 112 997 kronoberg 177188 12616 1.40 138285 143366 569.19 189 667 norrbotten 181170 14414 1.42 130683 136757 531.63 193 384 ostergotland 227380 23968 1.89 194520 206069 1096.01 435 219 sodermanland 114676 10636 0.97 98099 101849 273.55 1 267 950 varmland 182106 16398 1.45 151985 158763 664.96 312 110 vasterbotten 129105 10713 1.11 104887 110478 315.93 214 894 vasternorrland 215818 18541 1.66 173192 180969 921.11 243 165 vastmanland 145319 12639 1.27 129540 135307 428.14 296 574 Table 11: Problem set 4, counties, reading osm, network extraction, R1.
R2 R4 Name n1 l1 t1 n2 l2 t2 blekinge 10386 12976 136.73 7411 10001 151.78 dalarna 28630 36626 1411.48 21345 29341 1517.83 gavleborg 26462 34491 775.03 19701 27730 855.00 gotland 3668 4928 15.53 2923 4183 16.93 halland 32104 39229 886.70 21050 28175 1055.87 jamtland 18103 21938 418.83 11706 15541 463.86 kronoberg 16647 21588 485.42 12777 17718 520.48 norrbotten 19889 25687 467.25 14574 20372 513.44 ostergotland 35841 47140 1026.95 27372 38671 1191.83 sodermanland 14833 18488 237.61 10537 14192 269.95 varmland 22223 28659 583.80 16615 23051 647.51 vasterbotten 15313 20764 278.05 12547 17998 298.80 vasternorrland 26605 34189 811.35 19325 26909 904.27 vastmanland 18662 24198 395.01 14081 19617 436.17
Table 12: Problem set 4, network extraction, degree 2 elimination, tree elimination, R2, R4.
R2c R4c Name n1 l1 t1 n2 l2 t2 blekinge 8796 10618 122.01 5817 7639 133.13 dalarna 23470 28634 1274.88 15941 21105 1338.65 gavleborg 20499 24848 717.90 13469 17818 742.27 gotland 2836 3587 13.33 2087 2838 13.98 halland 26827 31420 740.20 15351 19944 830.11 jamtland 15104 17292 384.98 8654 10842 392.19 kronoberg 13563 16843 451.34 9534 12814 448.61 norrbotten 16147 19845 425.94 10815 14513 439.82 ostergotland 25043 30345 724.98 15893 21195 798.81 sodermanland 11872 14116 186.20 7446 9690 206.26 varmland 17734 22033 473.07 12127 16426 517.94 vasterbotten 11590 14762 220.41 8643 11815 235.13 vasternorrland 21315 25974 673.12 14035 18694 740.12 vastmanland 15128 18742 320.19 10359 13973 348.81
Table 13: Problem set 4, network extraction, degree 2 elimination, tree elimination, only cars, R2c, R4c.
1. Check if the network is connected. If not, keep only the largest connected part, and remove all other nodes and links.
2. As in 1, followed by finding a minimal spanning tree in the remaining graph. 3. As in 1, followed by solving the Chinese postman problem in the remaining graph, i.e. finding a shortest round trip that passes each link at least once.
Problem 3 was actually solved with the code for the rural postman problem described in [1], where all links are set as required. Since the required graph is connected, the method yields the optimal solution, and the code does not do much more than the classical method for the Chinese postman problem. It is not important if this is the best way of solving the Chinese postman problem, since it is only used as an example of a somewhat more difficult optimization problem. For an application such as snow removal, the optimization problem will be even harder, so we expect the differences to be even more pronounced.
Here we start by reading the results of the previous runs, i.e. the network files, which is done much quicker than reading and treating the OSM-files.
In tables 14 - 17 in the appendix, we give the times for the first set of instances, the small cities. Here t1 is the time for solving problem 1, t2 the time for problem 2 and t3 the time for problem 3. For each instance, we give five rows, one for each reduction. We find the problems 1 and 2 are solved very quickly, while problem 3 takes slightly longer.
It is quite clear that R1 is not a good idea, especially for problem 3. For example, for Ljungsbro, R1 takes 51 seconds, while R3 takes 4 seconds, R2 2.5 seconds and R4 less than one second. For the largest of the small cities, Motala, R1 takes 1148 seconds, R3
189 seconds, R2 127 seconds and R4 57 seconds. For only cars, R2c takes 45 seconds and R4c less then 6 seconds. The reduction is solution time is dramatic. The time to do the reductions for Motala are 58 seconds for R3, and 5 seconds for R1, R2 and R4, so there is a clear gain in total time. Summing up the times, we get 1153 seconds for R1, 132 for R2, 247 for R3 and 62 for R4, so R4 is definitely worthwhile.
In tables 18 - 20, we give the results for the city parts in the same way. Here some problems were so large that we did not solve problem 3. As an example we give toliu1 which took 252, 204, 76, 23 and 0.5 seconds after 5 seconds of reductions.
Results for the larger cities are given in table 21. For some instances, we did not solve problem 3. We did not use R3, and for some not R1, due to the longer solution times. As an example Borlänge took 1935, 898, 323, 288 and 37 seconds, while the reductions took up to 15 seconds.
Finally in table 22, we give the results for the counties. Here we did not use R1, R3 and R4c.
It is clear that the time spent doing reductions is much smaller than the time needed to solve the problems without reductions, so we conclude that the reductions are clearly worthwhile, and in some cases necessary.
4
Conclusions
We have described a number of standard test instances, extracted from OpenStreetMap data, and shown how different reductions of the networks can be made, and what consequences these reductions have.
The general conclusion is that these problem reductions are crucial for the solvability of some optimization problems. The reductions may take some time, but without them some optimization problems can not be solved. For more difficult optimization problems, the difference is expected to be more pronounced.
We find that with these tools, indata is easily obtainable for many important optimiza-tion problems. Previously such indata was extremely laborious to obtain. Each instance may have required days of manual work. Now it can be done much quicker.
The instances described in this paper are publically available. Please contact the author for information of where to find them.
Bibliography
[1] Holmberg, K., “Heuristics for the rural postman problem”, Computers & Operations Research37 (2010) 981–990.
Re-port LiTH-MAT-R–2014/08–SE, Department of Mathematics, Linköping Univer-sity, Sweden 2014.
[3] Holmberg, K., “Heuristics for the weighted k-Chinese/rural postman problem with a hint of fixed costs with applications to urban snow removal”, Research Report LiTH-MAT-R–2015/13–SE, Department of Mathematics, Linköping University, Sweden 2015. Accepted for publication in Journal of Vehicle Routing Algorithms.
[4] Holmberg, K., “Map matching by optimization”, Research Report LiTH-MAT-R– 2015/01–SE, Department of Mathematics, Linköping University, Sweden 2015. [5] Holmberg, K., “On using OpenStreetMap and GPS for optimization”, Research
Re-port LiTH-MAT-R–2015/15–SE, Department of Mathematics, Linköping University, Sweden 2015.
[6] Holmberg, K., “The (over) zealous snow remover problem”, Research Report LiTH-MAT-R–2016/04–SE, Department of Mathematics, Linköping University, Sweden 2016.
[7] Hultman, T., “Signal-aware route planning”, Bachelor thesis LIU-IDA/LITH-EX-G– 16/005-SE Linköping University 2016.
Name nodes links t1 t2 t3 askeby-R1 234 255 0.02 0.02 0.06 askeby-R2 77 96 0.00 0.00 0.01 askeby-R3 101 122 0.00 0.00 0.02 askeby-R4 50 69 0.00 0.00 0.01 askeby-R2c 55 58 0.00 0.00 0.01 askeby-R4c 19 22 0.00 0.00 0.01 atvid-R1 956 1032 0.11 0.12 1.69 atvid-R2 227 297 0.02 0.01 0.17 atvid-R3 401 477 0.03 0.03 0.34 atvid-R4 176 246 0.01 0.01 0.08 atvid-R2c 219 283 0.01 0.01 0.16 atvid-R4c 166 230 0.01 0.01 0.07 bankekind-R1 266 271 0.01 0.01 0.08 bankekind-R2 34 39 0.00 0.00 0.01 bankekind-R3 66 71 0.00 0.00 0.02 bankekind-R4 16 21 0.00 0.00 0.01 bankekind-R2c 34 39 0.00 0.00 0.01 bankekind-R4c 16 21 0.00 0.00 0.01 borensberg-R1 820 872 0.09 0.09 1.00 borensberg-R2 212 259 0.01 0.01 0.12 borensberg-R3 311 363 0.02 0.02 0.21 borensberg-R4 139 186 0.01 0.01 0.04 borensberg-R2c 193 232 0.01 0.01 0.11 borensberg-R4c 121 160 0.01 0.01 0.03 boxholm-R1 974 1025 0.11 0.11 1.74 boxholm-R2 218 262 0.01 0.01 0.10 boxholm-R3 364 415 0.02 0.02 0.27 boxholm-R4 135 179 0.01 0.01 0.03 boxholm-R2c 195 234 0.01 0.01 0.09 boxholm-R4c 114 153 0.01 0.01 0.02 brokind-R1 224 235 0.01 0.01 0.04 brokind-R2 48 58 0.00 0.00 0.01 brokind-R3 88 99 0.00 0.00 0.01 brokind-R4 30 40 0.00 0.00 0.01 brokind-R2c 45 52 0.00 0.00 0.01 brokind-R4c 26 33 0.00 0.00 0.01 ekangen-R1 624 662 0.05 0.05 0.50 ekangen-R2 157 193 0.01 0.01 0.05 ekangen-R3 237 275 0.01 0.01 0.11 ekangen-R4 113 149 0.01 0.01 0.02 ekangen-R2c 131 147 0.01 0.01 0.03 ekangen-R4c 63 79 0.01 0.00 0.01
Name nodes links t1 t2 t3 finspong-R1 1993 2099 0.42 0.44 13.99 finspong-R2 422 517 0.03 0.03 0.56 finspong-R3 678 784 0.07 0.07 0.95 finspong-R4 291 386 0.02 0.02 0.20 finspong-R2c 371 439 0.02 0.02 0.40 finspong-R4c 230 298 0.01 0.01 0.07 grebo-R1 217 220 0.01 0.01 0.04 grebo-R2 57 58 0.00 0.00 0.01 grebo-R3 101 104 0.00 0.00 0.02 grebo-R4 14 15 0.00 0.00 0.01 grebo-R2c 57 58 0.00 0.00 0.01 grebo-R4c 14 15 0.00 0.00 0.01 kisa-R1 1590 1661 0.28 0.30 7.69 kisa-R2 364 423 0.03 0.03 0.48 kisa-R3 593 664 0.06 0.06 0.74 kisa-R4 221 280 0.01 0.01 0.09 kisa-R2c 288 327 0.02 0.02 0.27 kisa-R4c 161 200 0.01 0.01 0.05 linghem-R1 1753 2076 0.43 0.45 13.40 linghem-R2 912 1221 0.15 0.16 4.19 linghem-R3 1145 1468 0.22 0.22 5.81 linghem-R4 689 998 0.10 0.10 1.26 linghem-R2c 445 494 0.04 0.03 0.80 linghem-R4c 210 259 0.01 0.01 0.09 ljungsbro-R1 3117 3316 1.12 1.19 51.77 ljungsbro-R2 749 931 0.09 0.09 2.54 ljungsbro-R3 1051 1250 0.17 0.17 4.12 ljungsbro-R4 523 705 0.06 0.05 0.64 ljungsbro-R2c 517 580 0.05 0.05 1.21 ljungsbro-R4c 232 295 0.02 0.02 0.11 malmslatt-R1 1758 1908 0.44 0.46 10.49 malmslatt-R2 548 690 0.06 0.06 1.05 malmslatt-R3 705 855 0.08 0.08 1.43 malmslatt-R4 383 525 0.04 0.03 0.37 malmslatt-R2c 384 425 0.03 0.03 0.60 malmslatt-R4c 180 221 0.01 0.01 0.05
Name nodes links t1 t2 t3 mantorp-R1 716 763 0.09 0.09 0.78 mantorp-R2 198 240 0.01 0.01 0.12 mantorp-R3 266 313 0.02 0.02 0.17 mantorp-R4 128 170 0.01 0.01 0.04 mantorp-R2c 172 201 0.01 0.01 0.09 mantorp-R4c 99 128 0.00 0.00 0.02 mjolby-R1 1623 1742 0.28 0.29 8.11 mjolby-R2 426 531 0.03 0.03 0.60 mjolby-R3 570 689 0.05 0.05 0.83 mjolby-R4 305 410 0.02 0.02 0.23 mjolby-R2c 395 488 0.03 0.03 0.51 mjolby-R4c 273 366 0.02 0.02 0.19 motala-R1 8807 9962 8.50 9.18 1148.96 motala-R2 2753 3829 1.28 1.34 127.72 motala-R3 3611 4766 1.94 2.04 189.18 motala-R4 2399 3475 1.07 1.09 57.75 motala-R2c 1757 2164 0.44 0.46 45.05 motala-R4c 1218 1625 0.26 0.26 5.89 norsholm-R1 398 413 0.03 0.03 0.16 norsholm-R2 79 93 0.00 0.00 0.01 norsholm-R3 112 127 0.00 0.00 0.02 norsholm-R4 54 68 0.00 0.00 0.01 norsholm-R2c 68 81 0.00 0.00 0.01 norsholm-R4c 51 64 0.00 0.00 0.01 rimforsa-R1 795 830 0.07 0.08 0.97 rimforsa-R2 186 206 0.01 0.01 0.10 rimforsa-R3 355 390 0.02 0.02 0.23 rimforsa-R4 91 111 0.00 0.00 0.02 rimforsa-R2c 170 186 0.01 0.01 0.09 rimforsa-R4c 67 83 0.00 0.00 0.02 skeninge-R1 686 730 0.06 0.06 0.69 skeninge-R2 165 206 0.01 0.01 0.08 skeninge-R3 222 266 0.01 0.01 0.11 skeninge-R4 114 155 0.01 0.01 0.03 skeninge-R2c 160 200 0.01 0.01 0.07 skeninge-R4c 110 150 0.01 0.01 0.03 soderkoping-R1 1837 2094 0.39 0.42 13.10 soderkoping-R2 682 926 0.09 0.09 1.65 soderkoping-R3 858 1115 0.12 0.12 2.39 soderkoping-R4 532 776 0.06 0.06 0.59 soderkoping-R2c 434 500 0.03 0.03 0.69 soderkoping-R4c 230 296 0.01 0.01 0.10
Name nodes links t1 t2 t3 sturefors-R1 1080 1125 0.17 0.18 2.67 sturefors-R2 212 251 0.01 0.01 0.09 sturefors-R3 391 436 0.03 0.03 0.27 sturefors-R4 127 166 0.01 0.01 0.02 sturefors-R2c 174 189 0.01 0.01 0.06 sturefors-R4c 68 83 0.00 0.00 0.01 svartm-R1 221 223 0.01 0.01 0.04 svartm-R2 38 39 0.00 0.00 0.01 svartm-R3 73 75 0.00 0.00 0.01 svartm-R4 7 8 0.00 0.00 0.01 svartm-R2c 38 39 0.00 0.00 0.01 svartm-R4c 7 8 0.00 0.00 0.01 tranas-R1 2349 2525 0.56 0.60 24.30 tranas-R2 644 806 0.07 0.07 2.02 tranas-R3 1122 1298 0.16 0.17 4.36 tranas-R4 438 600 0.04 0.04 0.38 tranas-R2c 568 705 0.05 0.05 1.50 tranas-R4c 365 502 0.03 0.03 0.26 vadstena-R1 1445 1656 0.26 0.27 6.80 vadstena-R2 601 801 0.07 0.07 1.57 vadstena-R3 761 972 0.10 0.10 1.84 vadstena-R4 461 661 0.05 0.05 0.41 vadstena-R2c 423 503 0.03 0.03 0.61 vadstena-R4c 227 307 0.02 0.01 0.07 vastervik-R1 4259 4869 2.02 2.16 153.03 vastervik-R2 1572 2146 0.45 0.44 28.64 vastervik-R3 2136 2746 0.67 0.69 41.51 vastervik-R4 1294 1868 0.33 0.33 9.59 vastervik-R2c 1053 1310 0.18 0.17 10.34 vastervik-R4c 724 981 0.10 0.10 1.54 vikingstad-R1 558 607 0.09 0.09 0.58 vikingstad-R2 190 233 0.01 0.01 0.12 vikingstad-R3 274 323 0.02 0.02 0.15 vikingstad-R4 116 159 0.01 0.01 0.04 vikingstad-R2c 180 218 0.01 0.01 0.09 vikingstad-R4c 107 145 0.01 0.01 0.04
Name nodes links t1 t2 t3 ekholmen-R1 1766 2020 0.69 0.49 11.97 ekholmen-R2 722 964 0.14 0.11 2.28 ekholmen-R3 898 1152 0.18 0.15 3.19 ekholmen-R4 580 822 0.09 0.07 0.85 ekholmen-R2c 252 273 0.02 0.02 0.22 ekholmen-R4c 92 113 0.04 0.00 0.02 hjulsb-east-R1 792 860 0.13 0.10 1.21 hjulsb-east-R2 298 365 0.02 0.02 0.17 hjulsb-east-R3 342 410 0.04 0.03 0.22 hjulsb-east-R4 196 263 0.02 0.01 0.06 hjulsb-east-R2c 116 122 0.01 0.01 0.04 hjulsb-east-R4c 42 48 0.03 0.00 0.01 hjulsb-best-R1 11163 11743 12.87 14.33 -hjulsb-best-R2 2192 2695 0.72 0.79 91.80 hjulsb-best-R3 3978 4558 1.99 2.18 179.90 hjulsb-best-R4 1493 1996 0.40 0.44 10.94 hjulsb-best-R2c 1421 1538 0.24 0.26 28.81 hjulsb-best-R4c 649 766 0.14 0.07 0.78 hjulsbro-R1 11400 12612 13.35 14.59 -hjulsbro-R2 3608 4755 2.00 2.11 399.56 hjulsbro-R3 4675 5887 3.02 3.47 504.70 hjulsbro-R4 2805 3952 1.40 1.58 100.66 hjulsbro-R2c 2002 2242 0.49 0.55 91.52 hjulsbro-R4c 974 1214 0.17 0.16 2.78 link-mid-R1 2185 2559 0.71 0.74 -link-mid-R2 882 1244 0.17 0.17 3.64 link-mid-R3 1158 1532 0.27 0.26 5.56 link-mid-R4 718 1080 0.12 0.12 1.41 link-mid-R2c 331 399 0.03 0.03 0.36 link-mid-R4c 189 257 0.03 0.01 0.10 linkoping-c-R1 13539 15812 21.40 24.29 -linkoping-c-R2 5656 7819 5.30 5.88 -linkoping-c-R3 7180 9453 7.63 8.74 -linkoping-c-R4 4648 6811 3.99 4.21 -linkoping-c-R2c 2614 3164 1.05 1.11 175.56 linkoping-c-R4c 1645 2195 0.49 0.62 16.59 linkoping-n-R1 11299 13263 14.76 16.38 -linkoping-n-R2 4859 6737 3.89 4.14 -linkoping-n-R3 6261 8225 5.67 6.38 -linkoping-n-R4 4034 5912 3.04 3.29 -linkoping-n-R2c 2567 3103 0.97 0.95 171.88 linkoping-n-R4c 1568 2104 0.46 0.48 15.31
Name nodes links t1 t2 t3 linkoping-so-R1 17159 19507 34.86 37.92 -linkoping-so-R2 6298 8518 6.79 6.73 -linkoping-so-R3 8060 10408 9.73 10.07 -linkoping-so-R4 5170 7390 5.14 5.13 -linkoping-so-R2c 3198 3675 1.31 1.36 383.56 linkoping-so-R4c 1719 2196 0.52 0.50 15.38 linkoping-v-R1 11191 12810 16.01 16.72 -linkoping-v-R2 4232 5774 3.33 3.20 -linkoping-v-R3 5578 7197 4.60 4.97 -linkoping-v-R4 3415 4957 2.14 2.38 -linkoping-v-R2c 2027 2329 0.58 0.65 77.56 linkoping-v-R4c 1119 1421 0.29 0.35 4.37 liu-R1 744 859 0.13 0.14 1.00 liu-R2 296 403 0.05 0.02 0.18 liu-R3 397 512 0.04 0.04 0.35 liu-R4 226 333 0.04 0.02 0.09 liu-R2c 137 167 0.01 0.01 0.08 liu-R4c 96 126 0.04 0.01 0.05 ljungsb1b-R1 1595 1681 0.44 0.34 7.74 ljungsb1b-R2 354 432 0.03 0.03 0.34 ljungsb1b-R3 501 587 0.05 0.04 0.49 ljungsb1b-R4 235 313 0.04 0.02 0.10 ljungsb1b-R2c 275 313 0.02 0.01 0.21 ljungsb1b-R4c 138 176 0.03 0.01 0.05 ljungsb1c-R1 865 913 0.14 0.10 1.31 ljungsb1c-R2 189 234 0.03 0.01 0.09 ljungsb1c-R3 281 329 0.02 0.02 0.19 ljungsb1c-R4 120 165 0.01 0.01 0.03 ljungsb1c-R2c 147 175 0.01 0.01 0.05 ljungsb1c-R4c 75 103 0.03 0.00 0.07 ljungsb1d-R1 701 728 0.08 0.08 0.65 ljungsb1d-R2 144 167 0.03 0.01 0.07 ljungsb1d-R3 193 220 0.01 0.01 0.08 ljungsb1d-R4 81 104 0.01 0.00 0.05 ljungsb1d-R2c 122 126 0.01 0.00 0.04 ljungsb1d-R4c 35 39 0.00 0.00 0.01
Name nodes links t1 t2 t3 pmo-R1 678 740 0.07 0.06 0.60 pmo-R2 162 220 0.01 0.01 0.07 pmo-R3 251 313 0.02 0.02 0.10 pmo-R4 115 173 0.01 0.01 0.03 pmo-R2c 48 50 0.00 0.00 0.01 pmo-R4c 9 11 0.00 0.00 0.01 ryd-R1 2996 3400 1.04 1.11 -ryd-R2 1157 1551 0.23 0.23 12.13 ryd-R3 1482 1886 0.35 0.34 14.78 ryd-R4 922 1316 0.18 0.17 3.32 ryd-R2c 488 527 0.05 0.05 0.97 ryd-R4c 176 215 0.01 0.01 0.07 studryd-R1 929 1068 0.11 0.12 1.97 studryd-R2 397 529 0.04 0.03 0.48 studryd-R3 534 673 0.05 0.05 0.68 studryd-R4 316 448 0.03 0.02 0.23 studryd-R2c 79 79 0.01 0.01 0.04 studryd-R4c 17 17 0.03 0.00 0.04 toliu1-R1 7598 8669 9.97 10.61 -toliu1-R2 2832 3843 1.58 1.58 204.81 toliu1-R3 3575 4646 2.56 2.37 252.37 toliu1-R4 2377 3388 1.40 1.17 76.44 toliu1-R2c 1240 1350 0.26 0.24 23.59 toliu1-R4c 534 644 0.13 0.06 0.46 toliu2-R1 4493 5098 2.65 2.51 -toliu2-R2 1609 2183 0.54 0.46 32.40 toliu2-R3 2116 2721 0.74 0.74 43.66 toliu2-R4 1251 1825 0.35 0.31 9.53 toliu2-R2c 744 882 0.17 0.09 2.90 toliu2-R4c 440 578 0.07 0.04 0.42 ullst-R1 626 678 0.12 0.06 0.59 ullst-R2 179 231 0.02 0.01 0.12 ullst-R3 234 286 0.04 0.01 0.14 ullst-R4 125 177 0.01 0.01 0.05 ullst-R2c 106 112 0.01 0.00 0.03 ullst-R4c 48 54 0.02 0.00 0.04 valla-R1 1470 1662 0.29 0.28 6.89 valla-R2 536 716 0.06 0.06 0.79 valla-R3 676 868 0.08 0.08 1.06 valla-R4 421 601 0.04 0.04 0.26 valla-R2c 264 308 0.02 0.02 0.16 valla-R4c 142 186 0.03 0.01 0.09
Name nodes links t1 t2 t3 borlange-R1 14783 16587 23.83 25.01 1935.76 borlange-R2 4657 6366 3.58 3.66 898.11 borlange-R4 3895 5604 2.80 2.80 322.40 borlange-R2c 3001 3668 1.27 1.37 288.96 borlange-R4c 2030 2697 0.77 0.78 37.69 jonkoping-R2 22010 28006 74.59 71.48 -jonkoping-R4 16410 22406 44.70 44.82 -jonkoping-R2c 19646 24425 58.88 54.38 -jonkoping-R4c 13893 18672 30.52 31.23 2940.94 kalmar-R2 15650 20009 42.89 37.36 -kalmar-R4 11624 15983 30.84 23.03 -kalmar-R2c 13177 16266 23.12 24.17 -kalmar-R4c 9081 12170 12.72 13.32 3953.40 linkan-R1 13006 14700 19.06 20.07 -linkan-R2 4543 6139 3.35 3.45 858.69 linkan-R4 3755 5351 2.56 2.57 272.26 linkan-R2c 2226 2483 0.64 0.65 122.51 linkan-R4c 1105 1362 0.25 0.21 3.90 orebro-R2 11382 14583 19.71 19.80 -orebro-R4 8778 11979 12.74 12.85 -orebro-R2c 9652 11954 13.07 13.25 -orebro-R4c 7088 9390 7.98 7.94 1572.43 uppsala-R2 28105 35479 123.69 123.20 -uppsala-R4 20050 27424 68.67 69.48 -uppsala-R2c 21392 25241 64.19 65.63 -uppsala-R4c 13195 17044 27.40 28.26 2301.77
Name nodes links t1 t2 blekinge-R2 9882 12399 14.72 15.39 blekinge-R4 7181 9698 8.58 8.82 blekinge-R2c 8526 10265 9.74 10.19 dalarna-R2 27921 35574 110.93 116.86 dalarna-R4 21052 28705 69.85 73.20 dalarna-R2c 22973 27820 68.57 72.60 gavleborg-R2 26087 33658 97.53 102.75 gavleborg-R4 19569 27140 62.11 65.07 gavleborg-R2c 20200 24165 51.58 54.69 gotland-R2 3555 4777 2.15 2.20 gotland-R4 2870 4092 1.56 1.55 gotland-R2c 2753 3467 1.14 1.19 halland-R2 31421 38401 128.35 136.55 halland-R4 20800 27780 67.10 68.01 halland-R2c 26382 30798 82.45 87.76 jamtland-R2 17793 21467 40.55 42.89 jamtland-R4 11612 15286 19.80 20.82 jamtland-R2c 14885 16905 25.44 27.13 kronoberg-R2 16370 21146 39.12 40.67 kronoberg-R4 12672 17448 25.80 26.71 kronoberg-R2c 13358 16459 23.90 25.08 norrbotten-R2 19473 24892 59.61 58.03 norrbotten-R4 14411 19830 37.17 35.61 norrbotten-R2c 15756 19089 36.69 35.46 ostergotland-R2 35268 46032 188.08 192.75 ostergotland-R4 27115 37879 121.12 126.95 ostergotland-R2c 24724 29568 76.76 81.57 sodermanland-R2 13884 17365 30.86 32.47 sodermanland-R4 10046 13527 17.58 18.31 sodermanland-R2c 11159 13268 18.20 19.13 varmland-R2 21604 27434 69.22 72.77 varmland-R4 16307 22137 43.89 46.24 varmland-R2c 17475 21170 40.50 43.64 vasterbotten-R2 15116 20342 35.47 37.22 vasterbotten-R4 12476 17702 26.19 27.39 vasterbotten-R2c 11382 14308 18.33 19.29 vasternorrland-R2 26151 33382 99.22 101.93 vasternorrland-R4 19123 26354 58.91 61.67 vasternorrland-R2c 21048 25369 56.16 59.48 vastmanland-R2 18297 23581 48.86 51.36 vastmanland-R4 13939 19223 31.50 33.06 vastmanland-R2c 14850 18234 29.59 31.35 Table 22: Problem set 4.
