WO2007061264A1  Reducing method of shortest path searching area and calculating method of minimal expecting load and method of searching shortest path  Google Patents
Reducing method of shortest path searching area and calculating method of minimal expecting load and method of searching shortest path Download PDFInfo
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 WO2007061264A1 WO2007061264A1 PCT/KR2006/005003 KR2006005003W WO2007061264A1 WO 2007061264 A1 WO2007061264 A1 WO 2007061264A1 KR 2006005003 W KR2006005003 W KR 2006005003W WO 2007061264 A1 WO2007061264 A1 WO 2007061264A1
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 G—PHYSICS
 G01—MEASURING; TESTING
 G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
 G01C21/00—Navigation; Navigational instruments not provided for in preceding groups
 G01C21/26—Navigation; Navigational instruments not provided for in preceding groups specially adapted for navigation in a road network
 G01C21/34—Route searching; Route guidance
 G01C21/3453—Special cost functions, i.e. other than distance or default speed limit of road segments

 G—PHYSICS
 G01—MEASURING; TESTING
 G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
 G01C21/00—Navigation; Navigational instruments not provided for in preceding groups
 G01C21/26—Navigation; Navigational instruments not provided for in preceding groups specially adapted for navigation in a road network
 G01C21/34—Route searching; Route guidance
 G01C21/3446—Details of route searching algorithms, e.g. Dijkstra, A*, arcflags, using precalculated routes
Abstract
Description
Description
REDUCEING METHOD OF SHORTEST PATH SEARCHING AREA AND CALCULATING METHOD OF MINIMAL EXPECTING LOAD AND METHOD OF SEARCHING
SHORTEST PATH Technical Field
[1] The present invention relates to a method of limiting a search area for searching for a shortest path on a complex network in a further speedy way, and a method of calculating a minimum expected cost and a method of searching for a shortest distance using the same. More specifically, the present invention relates to a method of reducing a search area using a minimum basic unit of spatialdistancebased true cost and searching for a shortest path within the reduced search area again using the minimum basic unit of spatialdistancebased true cost. Background Art
[2] The methods of searching for a shortest path are used in a variety of fields, such as car navigation system, communication networks, game software, artificial intelligence, or the like. Representative algorithms among the search algorithms known up to date include the Dijkstra's algorithm, Floyd algorithm, A* algorithm, and the like.
[3] The Dijkstra's algorithm is a method of selecting by comparison a path having a minimum time distance among directly connected paths starting from a starting node regardless of directionality toward a destination node. However, it has a problem in that searching time is considerably extended in a network of a quite large scale.
[4] The Floyd algorithm also examines and compares all possible links from a starting node to a destination node to find a shortest path, and thus has a problem in that searching time is considerably prolonged as is the same with the Dijkstra's algorithm.
[5] The A* algorithm is a representative of heuristic algorithms, which deduces a shortest path using a minimum cost estimator of residual paths from a current node to a destination node. That is, the algorithm searches for a shortest path using real distance from a starting node to the current node and estimated distance from the current node to the destination node (generally, a straightline distance), and thus has directionality. Therefore, the A* algorithm is advantageous over the Dijkstra's algorithm or the Floyd algorithm in that searching time is reduced. However, it is disadvantageous in that the minimum cost estimator of residual paths cannot be easily determined, and further, if the minimum cost estimate is mistakenly selected, it may not get a solution. Furthermore, if a straight distance is simply adopted as an estimator, searching can be ineffective depending on the structure of a network.
[6] In addition, existing searching methods are inefficiently performed in that after a search is completed, a shortest path is provided through performing an inverse search (restoration) function from the destination node to the starting node using connection relations between nodes. Disclosure of Invention Technical Problem
[7] Accordingly, the present invention has been made in order to solve the above problems, and it is an object of the present invention to provide a method of searching for a shortest path from a starting node to a destination node in a correct and speedy way in any network configuration, the method comprising the steps of constructing a temporary shortest path having directly connected nodes while minimizing the sum of spatial distances between the starting node and the destination node, obtaining a minimum basic unit of spatialdistancebased true cost by dividing a true cost, such as expense, time, real distance, or the like, by a spatial distance, reducing a search area to a range having a spatial distance that is smaller than a value of the true cost of the temporary shortest path divided by the minimum basic unit of spatialdistancebased true cost, and calculating a minimum expected cost using the minimum basic unit of spatialdistancebased true cost and spatial distances within the reduced search range, without searching for entire nodes by comparing the minimum expected cost with the true cost of the temporary shortest path.
Technical Solution
[8] In order to accomplish the above object of the invention, according to one aspect of the invention, there is provided a method of reducing a search area (the number of nodes), comprising the steps of: obtaining a minimum basic unit of spatial distancebased true cost, such as expense, time, real distance, or the like, of each link in the entire shortest path search area (the entire network); constructing a temporary shortest path that is directly connected to a starting node and a destination node with a minimum spatial distance and calculating true cost of the temporary shortest path; and applying the minimum basic unit of spatialdistancebased true cost to the true cost of the temporary shortest path.
[9] According to another aspect of the invention, there is provided a method of calculating a minimum expected cost, comprising the steps of: obtaining a minimum basic unit of spatialdistancebased true cost; searching for a node that is directly connected to an immediate previous search node and not included both in a search completed node set and a temporary shortest path among the nodes connected between a starting node and a destination node; and adding true cost (from the starting node) to a search node and the value of spatial distance from the search node to the destination node multiplied by the minimum basic unit of spatialdistancebased true cost. [10] According to another aspect of the invention, there is provided a method of searching for a shortest path comprising the steps of: obtaining a minimum basic unit of spatialdistancebased true cost within the reduced search area described above or the entire network; calculating a minimum expected cost using the obtained minimum basic unit of spatialdistancebased true cost; and iteratively comparing the minimum expected cost with the true cost of the temporary shortest path.
Advantageous Effects
[11] As described above, the present invention is effective in that the search area for a shortest path is minimized and a shortest path to the destination node is correctly searched in a speedy way as well. Furthermore, since a basic unit based on spatial distance is used, searching can be performed not only for the time distance, but also for any kind of costs (loads) of a path, such as real distance, expense, or the like. If the travel time is used as the cost, the present invention can be applied to the change of costs (loads) according to the change of time interval, so that it is effective in that a dynamic shortest path search is allowed. Brief Description of the Drawings
[12] Further objects and advantages of the invention can be more fully understood from the following detailed description taken in conjunction with the accompanying drawings in which:
[13] FIGS. 1 to 4 are a flowchart illustrating a method of searching for a shortest path using a minimum basic unit of spatialdistancebased time distance according to an embodiment of the invention;
[14] FIGS. 5 and 6 are a flowchart illustrating a method of constructing a temporary shortest path having a minimum spatial distance according to an embodiment of the invention; and
[15] FIG. 7 is a view showing a virtual network for explaining the present invention.
Mode for the Invention
[16] Hereinafter, the method of searching for a shortest path using the method of limiting a search area according to the present invention will be described in detail with reference to the accompanying drawings.
[17] The embodiment described below is implemented through a general computer system, a car navigation system, or the like, which comprises, as publicized, a command input unit, such as a keyboard, a mouse, a touchpad, or the like, for selecting a starting and a destination nodes, a memory unit for recording path information of a plurality of nodes, an arithmetic unit for searching for a shortest path from the starting node to the destination node using the information in the memory unit according to inputted operational control signals. The configuration of such systems is apparent to those skilled in the art that details thereof will not be described. The embodiment described below relates to the process of determining a starting node and a destination node through the command input unit and operating the arithmetic unit with reference to the content of the memory unit, so that the arithmetic unit, i.e., the subject of the operation, will not be described below.
[18] FIGS. 1 to 4 are a flowchart illustrating a method of searching for a shortest path using a minimum basic unit of spatialdistancebased time distance according to an embodiment of the invention. In the embodiment of the present invention, time distance, i.e., actual travel time, is used as a true cost.
[19] <Step 101>
[20] All of the nodes (pi, p2,M. ..., pn) in a search area are inserted into a set
[21] <Step 102>
[22] The minimum value is selected among values of time distance divided by spatial distance of all connected links. The minimum value is determined as a minimum basic unit of spatialdistancebased time distance (Min_Unit) for limiting the search area.
[23] Here, rdistφi, pj) denotes time distance between node pi and node pj, and sdis(pi, pj) denotes spatial distance between node pi and node pj. The spatial distance means a straightline distance, which can be calculated from relative coordinates or absolute coordinates of pi and pj using mathematical expression 1 shown below.
[24] [Mathematical expression 1]
[25]
[26] In mathematical expression 1 described above, (Xi, Yi) is the coordinates of pi, and
(Xj, Yj) is the coordinates of pj. [27] Accordingly, the minimum basic unit of spatialdistance based time distance (
Min_Unit) is determined by selecting the minimum value among the values of rdis(pi, pj) of all paths divided by sdis(pi, pj). [28] Since a basic unit is used in the present invention, not only a time distance shortest path, but also a real distance shortest path, a cost shortest path, or the like can be searched.
[29] <Step 103>
[30] The starting node is set to sp, and the destination node is set to ep.
[31] <Step 104> [32] A temporary shortest path is constructed with directly connected nodes, where the sum of spatial distance from the starting node and spatial distance to the destination node at each of the nodes becomes the minimum, and time distance of the temporary shortest path (MINP_rdis) is calculated.
[33] The process of calculation is shown in detail in FIGS. 5 and 6, where the processing stage T is set to 0, and the starting node sp is determined as a selected node p[0] and stored in the temporary shortest path Tem_MINP[i]. Time distance of the temporary shortest path (MINP_rdis) is set to 0 Step 201.
[34] The processing stage T is incremented by 1 Step 202. A search node p[i] that is directly connected to the immediate previous selected node p[il] and not included in the temporary shortest path Tem_MINP[k] (k=0~il) is selected Step 203, where the sum of spatial distance from the starting node sp and spatial distance to the destination node ep becomes the minimum.
[35] The selected search node is stored in the temporary shortest path Tem_MINP[i], and time distance from the immediate previous search node p[il] to the current search node p[i] is added to the time distance of the temporary shortest path Step 204.
[36] It is determined whether the search node p[i] is the destination node ep Step 205. If the search node is the destination node ep, the time distance of the temporary shortest path is outputted Step 206 and determined as the shortest time distance (MINP_rdis). If the search node is not the destination node ep, the steps from step 202 are repeated.
[37] <Step 105>
[38] The limited spatial distance (Lim_sdis) for reducing the search area can be calculated by dividing the time distance of the temporary shortest path (MINP_rdis) by the minimum basic unit of spatialdistancebased time distance (Min_Unit).
[39] That is, a node where the sum of spatial distance from the starting node sp and spatial distance to the destination node ep is larger than the limited spatial distance ( Lim_sdis) will never be included in the time distance shortest path that is shorter than the temporary shortest path.
[40] <Step 106>
[41] A node set NM of a search area that is reduced on the basis of the starting node sp and the destination node ep is constructed. If it is assumed that the nodes included in node set NM are npl, np2, ..., and npi, node npi included in the set NM is a node where the sum of spatial distance sdis(sp, npi) from the starting node sp and spatial distance sdis(npi, ep) to the destination node ep is smaller than the limited spatial distance (Lim_sdis).
[42] <Step 107>
[43] For the links in the node set NM of a reduced search area, a minimum basic unit of spatialdistancebased time distance (NMIN_Unit) is calculated again in the same manner as step 102.
[44] With the node set NM of a reduced search target area obtained as described above, the minimum basic unit of spatialdistancebased time distance (NMIN_Unit), and the shortest time distance (MINP_rdis) calculated from the temporary shortest path, the shortest path from the starting node sp to the destination node ep is searched for.
[45] <Step 108>
[46] The processing stage T is set to 0, and time distance c[i] up to the processing stage
T is set to 0. The starting node sp is stored in the searchcompleted node set FN[i] of stage T and the search node np[i] and the temporary shortest path node (Tem_MINP [i]) are set to the starting node sp.
[47] <Step 109>
[48] The processing stage T is incremented by 1 (i <= i+1).
[49] <Step l lO>
[50] A node that is directly connected to the immediate previous search node np[il] and not included in the searchcompleted node set FN[i] and the temporary shortest path node (Tem_MINP[k] (k=0~il)) is searched for and determined as the search node np
[i].
[51] <Step l l l>
[52] If a search node np[i] matching to the conditions is not found, the process goes to step 117.
[53] <Step l l2>
[54] The searched node np[i] is stored in the searchcompleted node set FN[i], and time distance rdis(np[il], np[i]) from the immediate previous search node np[il] to the current search node np[i] is added to time distance c[il] from the starting node to the immediate previous search node np[il], thereby updating the time distance c[i].
[55] Minimally expected time distance (c_dis) is a distance to the destination node that can be expected in minimum in the current processing stage T which is calculated by adding the updated time distance c[i] and spatial distance sdis(np[i], ep) from the search node np[i] to the destination node ep multiplied by the minimum basic unit of spatialdistancebased time distance (NMIN_Unit).
[56] c_dis = c[i] + sdis(np[i] , ep) x NMINJJnit
[57] Here, c_dis denotes minimally expected time distance, c[i] denotes time distance constructed from the starting node sp to the search node np[i] of stage T, sdis(np[i], ep) denotes spatial distance from the search node of stage T to the destination node ep , and NMIN_Unit denotes the minimum basic unit of spatialdistancebased time distance.
[58] <Step l l3>
[59] If the minimally expected time distance (c_dis) is larger than the shortest time distance (MINP_rdis), the process goes to step 110. [60] <Step l l4>
[61] If the minimally expected time distance (c_dis) is smaller than the shortest time distance (MINP_rdis), the search node (np[i]) is stored in the temporary shortest path node (Tem_MINP[i]). [62] <Step l l5>
[63] If the search node (np[i]) is not the destination node (ep) in step 114, the process goes to step 109. [64] <Step l l6>
[65] If the search node (np[i]) is the destination node (ep) in step 114, time distance c[i] up to the current stage is determined as the shortest time distance (MINP_rdis), the current stage T is determined as the last stage Fi of the shortest path, the temporary shortest path (Temp_MINP[i]) from stage 0 to the current stage Fi is converted to the shortest path MINP[i], and the process goes to step 110. [66] <Step l l7>
[67] All the nodes in the searchcompleted node set FN[i] in the current stage T are removed (a null set) (FN[i] = { }), and the stage is decremented by 1 (i <= i1). [68] <Step l l8>
[69] If stage Tis not 0, the process goes to step 110.
[70] <Step l l9>
[71] The shortest path set (MINP[k] (k=0~Fi)) is outputted, and the shortest time distance (MINP_rdis) is outputted. [72] If the travel time of a path is given according to an update interval (10 minutes, 5 minutes, or the like), the steps from step 103 are repeated within the same time update interval. When the update interval is over, the steps from step 102 are repeated, and thus the shortest travel time path can be efficiently searched for even in a dynamic manner.
[73] Embodiment
[74] FIG. 7 is a view of a virtual network showing node numbers, connection relations between nodes, and time distances. [75] The node set M of the search area in the virtual network of FIG. 7 is shown below
(step 101).
[76] M = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}
[77] Table 1 shows spatial distances calculated from the relative coordinates of the nodes in the virtual network. [78] [Table 1]
[80] Table 2 shows timedistance values of directly connected paths divided by spatial distance.
[81] [Table 2] [82]
[83] The minimum value among the values of timedistance divided by spatialdistance is 0.8929 of the path connecting node 4 to node 1 and the path connecting node 3 to node 2, which becomes the minimum basic unit of spatialdistancebased time distance for limiting the search area (step 102).
[84] Then, the starting node is set to node 5 (sp=5), and the destination node is set to node 9 (ep=9) (step 103). [85] The temporary shortest path and the shortest time distance, where the sum of spatial distances from the starting node (node 5) to the destination node (node 9) becomes the minimum, are obtained as described below (step 104).
[86] The nodes directly connected to the starting node (node 5) are nodes 2, 6, 7, 10, and 11. Among these nodes, the node where the sum of spatial distance from the starting node (node 5) and spatial distance to the destination node (node 9) becomes the minimum is node 7 (the sum of spatial distances is total 5.16). Then, the shortest time distance MINP_rdis is 3.8 of the time distance (rdis(5, 7)) from node 5 to node 7, and the temporary shortest path becomes Tem_MINP[0]=5 and Tem_MINP[l]=7.
[87] Since node 7 is not the destination node, the search is performed again at node 7. The nodes directly connected to node 7 and not included in the temporary shortest path are nodes 6, 9, and 10. Among these nodes, the node where the sum of spatial distance from the starting node (node 5) and spatial distance to the destination node (node 9) becomes the minimum is node 9 (the sum of spatial distances is total 5.10). The temporary shortest path becomes Tem_MINP[0]=5, Tem_MINP[l]=7, and Tem_MINP[2]=9, and the shortest time distance MINP_rdis becomes 3.8 + 3.2 = 7.0.
[88] The limited spatial distance (Lim_sdis) for reducing the search area is MINP_rdis/ MinJJnit, i.e., 7.0 / 0.8929 = 7.84 (step 105). [89] The node set NM within the limited spatial distance includes nodes where the sum of spatial distance from the starting node sp and spatial distance to the destination node ep is smaller than the limited spatial distance 7.84 (step 106).
[90] NM ={5, 6, 7, 8, 9, 10, 11, 13} [91] For reference, Table 3 shows the nodes that are included in the node set NM, among the entire nodes, within the limited spatial distance.
[92] [Table 3] [93]
[94] The minimum basic unit of spatialdistance based time distance NMIN_Unit for searching for the shortest path obtained from the set NM in the same manner as step 2 is 1.025 (step 107).
[95] The search stage T is set to 0, the time distance at the starting node c[0] is set to 0, the searchcompleted node set FN[O] is set to 5 (sp), the temporary shortest path Tem_MINP[0] is set to 5 (sp), and the search node np[0] is set to 5 (sp) (step 108). [96] The stage Tis incremented by 1 (step 109).
[97] i = i + 1 (i = 1)
[98] The nodes directly connected to node 5, i.e., np[0], not included in the temporary shortest path node (Tem_MINP[0]=5), and not included in the searchcompleted node set (FN[0]={5}) are nodes 6, 7, 10, and 11. Among these nodes, node 6 is selected (an arbitrary node can be selected) (step 110). [99] np[l] = 6
[100] The selected node 6 is inserted in the searchcompleted node set (FN[1]={6}), and the time distance c[l] and the minimally expected time distance (c_dis) are calculated as shown below (step 112).
[101] c[l] = c[0] + rdis(np[0], np[l]) = 0 + rdis(5, 6) = 0 + 4.1 = 4.1
[102] c_dis = c[l] + sdis(np[l], ep) NMINJJnit = 4.1 + sdis(6, 9) 1.025 = 7.34
[103] Since the minimally expected time distance 7.34 is larger than the shortest time distance 7.0, the process goes to step 110 (step 113). [104] The nodes directly connected to node 5, i.e., np[0], not included in the temporary shortest path node (Tem_MINP[0]=5), and not included in the searchcompleted node set (FN[1]={6}) are nodes 7, 10, and 11. Among these nodes, node 7 is selected (step
110).
[105] np[l] = 7
[106] The selected node 7 is inserted in the searchcompleted node set (FN[1]={6, 7}), and the time distance c[l] and the minimally expected time distance (c_dis) are calculated as shown below (step 112).
[107] c[l] = c[0] + rdis(np[0] , np[l]) = 0 + rdis(5 , 7) = 0 + 3.8 = 3.8
[108] c_dis = c[l] + sdis(np[l], ep) x NMINJJnit = 3.8 + sdis(7, 9) x 1.025 = 5.44
[109] Since the minimally expected time distance 5.44 is smaller than the shortest time distance 7.0 (step 113), the search node is stored in the temporary shortest path (step
114).
[110] Tem_MINP[l] = 7
[111] Since the search node 7 is not the destination node 9, the process goes to step 109
(step 115).
[112] The stage T is incremented by 1 (step 109).
[113] i = i + l (i = 2)
[114] The nodes directly connected to node 7, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[l]=7), and not included in the searchcompleted node set (FN[2]={ }) are nodes 6, 9, and 10. Among these nodes, node 6 is selected (step 110). [115] np[2] = 6 [116] The selected node 6 is inserted in the searchcompleted node set (FN[2]={6}), and the time distance c[2] and the minimally expected time distance (c_dis) are calculated as shown below (step 112).
[117] c[2] = c[l] + rdis(np[l], np[2]) = 3.8 + rdis(7,6) = 3.8 + 1.5 = 5.3
[118] c_dis = c[2] + sdis(np[2], ep) xNMINJJnit = 5.3 + 3.16 x 1.025 = 8.54
[119] Since the minimally expected time distance 8.54 is larger than the shortest time distance 7.0, the process goes to step 110 (step 113). [120] The nodes directly connected to node 7, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[l]=7), and not included in the searchcompleted node set (FN[2]={6}) are nodes 9 and 10. Among these nodes, node
9 is selected (step 110). [121] np[2] = 9
[122] The selected node 9 is inserted in the searchcompleted node set (FN[2]={6, 9}), and the time distance c[2] and the minimally expected time distance c_dis are calculated as shown below (step 112).
[123] c[2] = c[l] + rdis(np[l], np[2]) = 3.8 + rdis(7,9) = 3.8 + 3.2 = 7.0
[124] c_dis = c[2] + sdis(np[2], ep) xNMINJJnit = 7.0 + (0 x 1.025) = 7.0
[125] Since the minimally expected time distance 7.0 is equal to the shortest time distance
7.0 (step 113), the search node is stored in the temporary shortest path (step 114). [126] Tem_MINP[2] = 9
[127] Since search node 9 is the destination node (step 115), the shortest time distance (
MINP_rdis) becomes 7.0, i.e., the time distance c[2]. Since the current stage T is 2, the final stage Fi of the shortest path becomes 2. The temporary shortest path from stages 0 to Fi is converted to the shortest path (step 116), and the process goes to step
110.
[128] MINP[O] = Tem_MINP[0] =5
[129] MINP[I] = Tem_MINP[l] =7
[ 130] MINP[2] = Tem_MINP[2] =9
[131] The node directly connected to node 7, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[l]=7) up to the immediate previous stage, and not included in the searchcompleted node set (FN[2]={6, 9}) is node 10 (step 110). [132] np[2] = 10
[133] Node 10 is inserted in the searchcompleted node set (FN[2]={6, 9, 10}), and the time distance c[2] and the minimally expected time distance (c_dis) are calculated as shown below (step 112).
[134] c[2] = c[l] + rdis(np[l], np[2]) = 3.8 + rdis(7,10) = 3.8 + 2.1 = 5.9
[135] c_dis = c[2] + sdis(np[2], ep) xNMINJJnit = 5.9 + (1.41 x 1.025) = 7.35 [136] Since the minimally expected time distance 7.35 is larger than the shortest time distance 7.0, the process goes to step 110 (step 113). [137] A node directly connected to node 7, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[l]=7) up to the immediate previous stage, and not included in the searchcompleted node set (FN[2]={6, 9, 10}) does not exist, and thus the process goes to step 117 (steps 110 and 111). [138] The searchcompleted node set in the current stage (i = 2) is set to a null set (FN [2]
={ }), and the stage is decremented by 1 (step 117). [139] i = i  l (i = l)
[140] Since stage Tis not 0, the process goes to step 110 (step 118).
[141] The nodes directly connected to node 5, i.e., np[0], not included in the temporary shortest path node (Tem_MINP[0]=5) up to the immediate previous stage, and not included in the searchcompleted node set (FN[1]={6, 7}) are nodes 10 and 11. Among these nodes, node 10 is selected (step 110). [142] np[l] = 10
[143] The selected node 10 is inserted in the searchcompleted node set (FN[1]={6, 7,
10}), and the time distance c[l] and the minimally expected time distance (c_dis) are calculated as shown below (step 112).
[144] c[l] = c[0] + rdis(np[0] , np[l]) = 0 + rdis(5, 10) = 0 + 4.1 = 4.1
[145] c_dis = c[l] + sdis(np[l], ep) xNMINJJnit = 4.1 + 1.41 xl.025 = 5.55
[146] Since the minimally expected time distance 5.55 is smaller than the shortest time distance 7.0, the search node is stored in the temporary shortest path (step 114). [147] Tem_MINP[l] = 10
[148] Since the search node is not the destination node, the stage is incremented by 1
(step 109).
[149] i = i + l (i = 2)
[150] The nodes directly connected to node 10, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[ I]=IO) up to the immediate previous stage, and not included in the searchcompleted node set (FN[2]={ }) are nodes 7, 9 and 13. Among these nodes, node 7 is selected (step 110). [151] np[2] = 7
[152] The selected node 7 is inserted in the searchcompleted node set (FN[2]={7}), and the time distance c[2] and the minimally expected time distance (c_dis) are calculated as shown below (step 112).
[153] c[2] = c[l] + rdis(np[l], np[2]) = 4.1 + rdis(10, 7) = 4.1 + 2.1 = 6.2
[154] c_dis = c[2] + sdis(np[2], ep) xNMINJJnit = 6.2 + 2.00 x 1.025 = 8.25
[155] Since the minimally expected time distance 8.25 is larger than the shortest time distance 7.0, the process goes to step 110 (step 113). [156] The nodes directly connected to node 10, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[ I]=IO) up to the immediate previous stage, and not included in the searchcompleted node set (FN[2]={7}) are nodes 9 and 13. Among these nodes, node 9 is selected (step 110).
[157] np[2] = 9
[158] The selected node 9 is inserted in the searchcompleted node set (FN[2]={7, 9}), and the time distance c[2] and the minimally expected time distance c_dis are calculated as shown below (step 112).
[159] c[2] = c[l] + rdis(np[l], np[2]) = 4.1 + rdis(10, 9) = 4.1 + 2.8 = 6.9
[160] c_dis = c[2] + sdis(np[2], ep)xNMIN_Unit = 6.9 + 0 x 1.025 = 6.9
[161] Since the minimally expected time distance 6.9 is smaller than the shortest time distance 7.0, the search node is stored in the temporary shortest path (step 114).
[162] Tem_MINP[2] = 9
[163] Since the search node 9 is the destination node (step 115), the shortest time distance
(MINP_rdis) is converted to 6.9, i.e., the time distance c[2]. Since the current stage T is 2, the final stage Fi of the shortest path becomes 2. The temporary shortest path from stages 0 to Fi is converted to the shortest path (step 116), and the process goes to step 110.
[164] MINP[O] = Tem_MINP[0] =5
[165] MINP[I] = Tem_MINP[l] =10
[166] MINP[2] = Tem_MINP[2] =9
[167] The node directly connected to node 10, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[ I]=IO) up to the immediate previous stage, and not included in the searchcompleted node set (FN[2]={7, 9}) is node 13 (step 110).
[168] np[2] = 13
[169] Node 13 is inserted in the searchcompleted node set (FN[2]={7, 9, 13}), and the time distance c[2] and the minimally expected time distance c_dis are calculated as shown below (step 112).
[170] c[2] = c[l] + rdis(np[l], np[2]) = 4.1 + rdis(10, 13) = 4.1 + 2.0 = 6.1
[171] c_dis = c[2] + sdis(np[2], ep)xNMIN_Unit = 6.1 + 2.24 x 1.025 = 8.40
[172] Since the minimally expected time distance 8.40 is larger than the shortest time distance 6.9, the process goes to step 110 (step 113).
[173] A node directly connected to node 10, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[ I]=IO) up to the immediate previous stage, and not included in the searchcompleted node set (FN[2]={7, 9, 13}) does not exist, and thus the process goes to step 117 (steps 110 and 111).
[174] The searchcompleted node set in the current stage (i = 2) is set to a null set (FN [2] ={ }), and the stage is decremented by 1 (step 117). [175] i = i  l (i = l)
[176] Since stage Tis not 0, the process goes to step 110 (step 118).
[177] The node directly connected to node 5, i.e., np[0], not included in the temporary shortest path node (Tem_MINP[0]=5) up to the immediate previous stage, and not included in the searchcompleted node set (FN[1]={6, 7, 10}) is node 11 (step 110). [178] np[l] = l l
[179] Node 11 is inserted in the searchcompleted node set (FN[1]={6, 7, 10, 11 }), and the time distance c[l] and the minimally expected time distance c_dis are calculated as shown below (step 112).
[180] c[l] = c[0] + rdis(np[0] , np[l]) = 0 + rdis(5 , 11) = 0 + 2.8 = 2.8
[181] c_dis = c[l] + sdis(np[l], ep) xNMINJJnit = 2.8 + 3.61 xl.025 = 6.50
[182] Since the minimally expected time distance(c_dis) 6.50 is smaller than the shortest time distance 6.9, the search node is stored in the temporary shortest path (step 114). [183] Tem_MINP[l] = 11
[184] Since the search node is not the destination node, the stage is incremented by 1
(step 109).
[185] i = i + l (i = 2)
[186] The node directly connected to node 11, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[ I]=IO) up to the immediate previous stage, and not included in the searchcompleted node set (FN[2]={ }) is node
13 (step 110). [187] np[2] = 13
[188] Node 13 is inserted in the searchcompleted node set (FN[2]={ 13}), and the time distance c[l] and the minimally expected time distance c_dis are calculated as shown below (step 112).
[189] c[2]=c[l] + rdis(np[l], np[2])=2.8 + rdis(l l, 13) = 2.8 + 3.6 = 6.4
[190] c_dis = c[2] + sdis(np[l], ep) xNMINJJnit = 6.4 + 2.24 xl.025 = 8.70
[191] Since the minimally expected time distance 8.70 is larger than the shortest time distance 6.9, the process goes to step 110 (step 113). [192] A node directly connected to node 11, i.e., np[l], not included in the temporary shortest path nodes (Tem_MINP[0]=5, Tem_MINP[ I]=IO) up to the immediate previous stage, and not included in the searchcompleted node set (FN[2]={ 13}) does not exist, and thus the process goes to step 117 (steps 110 and 111). [193] The searchcompleted node set in the current stage (i = 2) is set to a null set (FN [2]
={ }), and the stage is decremented by 1 (step 117). [194] i = i  l (i = l)
[195] Since stage Tis not 0, the process goes to step 110 (step 118). [196] A node directly connected to node 5, i.e., np[0], not included in the temporary shortest path node (Tem_MINP[0]=5) up to the immediate previous stage, and not included in the searchcompleted node set (FN[1]={6, 7, 10, 11 }) does not exist, and thus the process goes to step 117 (steps 110 and 111).
[197] The searchcompleted node set in the current stage (i = 1) is set to a null set (FN[I]
={ }), and the stage is decremented by 1 (step 117).
[198] i = i  1 (i = 0)
[199] Since stage i is 0 (step 118), the shortest path MINP[k] (k=0~Fi) and the shortest time distance MINP_rdis are outputted.
[200] The shortest path is MINP[O] = 5, MINP[I] = 10, and MINP[2] = 9, and
[201] the shortest time distance MINP_rdis is 6.9.
[202]
Claims
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