运筹系列81：LKH代码分析

1. 基本数据结构

基础的node定义在LKH.h中

用于2-level tree的segment定义如下：

LKH可以使用3种数据结构，默认是2-level tree：

2-level tree的flip操作（即2-opt算子），在Flip_SL.c中，特殊的地方在于flip过程中涉及到重新分配segment的操作，其逻辑如下，其中P开头的是node对应的segment编号。

2. 读取问题数据

2.1 关键词清单

读数据代码在ReadProblem.c中，必须包含两部分：specification part和data part。specification part支持的关键词包括:

* NAME : e
 * Identifies the data file.
 *
 * TYPE : 。可以是非对称tsp问题，带时间窗的问题，经典VRP问题，非对称VRP问题，pickup-and-delivery TSP问题，开放VRP问题，。
 * Specifies the type of data. Possible types are
 * TSP          Data for a symmetric traveling salesman problem
 * ATSP         Data for an asymmetric traveling salesman problem
 * SOP          Data for a sequence ordering problem
 * HCP          Hamiltonian cycle problem data
 * HPP          Hamiltonian path problem data (not available in TSPLIB)
 * TSPTW        Data for a TSP instance with time windows
 * CCVRP        Data for a cumulative capacitated vehicle routing problem
 * CVRP         Data for a symmetric capacitated vehicle routing problem
 * ACVRP        Data for an asymmetric capacitated vehicle routing problem
 * ADCVRP       Data for an asymmetric distance constrained vehicle
 *              routing problem
 * CVRPTW       Data for a capacitated vehicle routing problem with
 *              time windows
 * VRPMPD       Data for a mixed pickup and delivery problem with backhauls
 * 1-PDTSP      Data for a one-commodity pickup-and-delivery traveling
 *              salesman problem
 * MLP          Data for a minimum latency problem
 * m-PDTSP      Data for a mulity-commodity pickup-and-delivery traveling
 *              salesman problem
 * m1-PDTSP     Data for a mulity-commodity one-to-one pickup-and-delivery
 *              traveling salesman problem
 * OVRP         Data for an open vehicle routing problem
 * PDTSP        Data for a pickup and delivery traveling salesman problem
 * PDTSPF       Data for a pickup and delivery traveling salesman problem
 *              with FIFO loading
 * PDTSPL       Data for a pickup and delivery traveling salesman problem
 *              with LIFO loading
 * PDPTW        Data for a pickup and delivery problem with time windows
 * RCTVRP       Data for a risk-constrained cash-in-transit vehicle
 *              routing problem
 * RCTVRPTW     Data for a risk-constrained cash-in-transit vehicle
 *              routing problem with time windows
 * TRP          Data for a traveling repairman problem
 * TSPDL        Dara for a traveling salesman problem with draft limits
 * TSPPD        Data for a pickup and delivery travling salesman problem
 * TSTSP        Data for a Steiner traveling salesman problem
 * VRPB         Data for a vehicle routing problem with backhauls
 * VRPBTW       Data for a vehicle routing problem with backhauls and
 *              time windows
 * CTSP         Data for a colored traveling salesman problem
 *
 * COMMENT : 
 * Additional comments (usually the name of the contributor or the creator of
 * the problem instance is given here).
 *
 * DIMENSION : < integer>
 * The number of nodes.
 *
 * CAPACITY : 
 * Specifies the truck capacity in a CVRP.
 *
 * DISTANCE : 
 * The maximum length allowed for each route in a CVRP.
 *
 * EDGE_WEIGHT_TYPE : 
 * Specifies how the edge weights (or distances) are given. The values are:
 * ATT          Special distance function for problem att48 and att532
 * CEIL_2D      Weights are Euclidean distances in 2-D rounded up
 * CEIL_3D      Weights are Euclidean distances in 3-D rounded up
 * EUC_2D       Weights are Euclidean distances in 2-D
 * EUC_3D       Weights are Euclidean distances in 3-D
 * EXACT_2D     Weights are EUC_2D distances (SCALE = 1000 as default)
 * EXACT_3D     Weights are EUC_3D distances (SCALE = 1000 as default)
 * EXPLICIT     Weights are listed explicitly in the corresponding section
 * FLOOR_2D     Weights are Euclidean distances in 2-D rounded down
 * FLOOR_3D     Weights are Euclidean distances in 3-D rounded down
 * GEO          Weights are geographical distances in kilometers (TSPLIB)
 *              Coordinates are given in the form DDD.MM where DDD are the
 *              degrees and MM the minutes
 * GEOM         Weights are geographical distances in meters (used for the
 *              world TSP). Coordinates are given in decimal form
 * GEO_MEEUS    Weights are geographical distances in kilometers, computed
 *              according to Meeus' formula.  Coordinates are given in the
 *              form DDD.MM where DDD are the degrees and MM the minutes
 * GEOM_MEEUS   Weights are geographical distances, computed according to
 *              Meeus' formula. Coordinates are given in decimal form
 * MAN_2D       Weights are Manhattan distances in 2-D
 * MAN_3D       Weights are Manhattan distances in 3-D
 * MAX_2D       Weights are maximum distances in 2-D
 * MAX_3D       Weights are maximum distances in 3-D
 * TOR_2D       Wirghes are toroidal distances in 2-D
 * TOR_3D       Wirghes are toroidal distances in 3-D
 * XRAY1        Distance function for crystallography problems (Version 1)
 * XRAY2        Distance function for crystallography problems (Version 2)
 * SPECIAL      There is a special distance function implemented in
 *              the Distance_SPECIAL function.
 *
 * EDGE-WEIGHT_FORMAT : 
 * Describes the format of the edge weights if they are given explicitly.
 * The values are
 * FUNCTION         Weights are given by a function (see above)
 * FULL_MATRIX      Weights are given by a full matrix
 * UPPER_ROW        Upper triangular matrix
 *                      (row-wise without diagonal entries)
 * LOWER_ROW        Lower triangular matrix
 *                      (row-wise without diagonal entries)
 * UPPER_DIAG_ROW   Upper triangular matrix
 *                      (row-wise including diagonal entries)
 * LOWER_DIAG_ROW   Lower triangular matrix
 *                      (row-wise including diagonal entries)
 * UPPER_COL        Upper triangular matrix
 *                      (column-wise without diagonal entries)
 * LOWER_COL        Lower triangular matrix
 *                      (column-wise without diagonal entries)
 * UPPER_DIAG_COL   Upper triangular matrix
 *                      (column-wise including diagonal entries)
 * LOWER_DIAG_COL   Lower triangular matrix
 *                      (column-wise including diagonal entries)
 *
 * EDGE_DATA_FORMAT : 
 * Describes the format in which the edges of a graph are given, if the
 * graph is not complete. The values are
 * EDGE_LIST    The graph is given by an edge list
 * ADJ_LIST     The graph is given by an adjacency list
 *
 * NODE_COORD_TYPE : 
 * Specifies whether the coordinates are associated with each node
 * (which, for example may be used for either graphical display or
 * distance computations.
 * The values are
 * TWOD_COORDS      Nodes are specified by coordinates in 2-D
 * THREED_COORDS    Nodes are specified by coordinates in 3-D
 * NO_COORDS        The nodes do not have associated coordinates
 * The default value is NO_COORDS. In the current implementation, however,
 * the value has no significance.
 *
 * DISPLAY_DATA_TYPE : 
 * Specifies how a graphical display of the nodes can be obtained.
 * The values are
 * COORD_DISPLAY    Display is generated from the node coordinates
 * TWOD_DISPLAY     Explicit coordinates in 2-D are given
 * NO_DISPLAY       No graphical display is possible
 *
 * The default value is COORD_DISPLAY if node coordinates are specifies and
 * NO_DISPLAY otherwise. In the current implementation, however, the value
 * has no significance.
 *
 * DEMAND_DIMENSION : 
 * The number of objects in an m1-PDTSP.
 *
 * GRID_SIZE : 
 * The grid size for toroidal instances.
 * Default: 1000000.0
 *
 * RISK_THRESHOLD : 
 * The maximum risk alllowed for each route in an RCTVRP or RCTVRPTW instance.
 *
 * SALESMEN : 
 * VEHICLES : 
 * The number of vehicles/salesmen in a CVRP.
 *
 * SCALE : 
 * Scale factor. Distances are multiplied by this factor.
 *
 * SERVICE_TIME : 
 * Same service time for all nodes.
 *
 * EOF
 * Terminates input data. The entry is optional.
 *

数据部分的格式如下，可以是
NODE_COORD_SECTION：node坐标
EDGE_DATA_SECTION：edge清单（如果是3个整数，最后一个表示weight），临近edge清单（第一个数据是当前点，后面是临近点，以-1结束）
FIXED_EDGES_SECTION：必须出现的edge清单

* NODE_COORD_SECTION :
 * Node coordinates are given in this section. Each line is of the form
 *
 *        
 *
 * if NODE_COORD_TYPE is TWOD_COORDS, or
 *
 *         
 *
 * if NODE_COORD_TYPE is THREED_COORDS. The integers give the number of the
 * respective nodes. The real numbers are the associated coordinates.
 *
 * EDGE_DATA_SECTION :
 * Edges of the graph are specified in either of the two formats allowed in
 * the EDGE_DATA_FORMAT entry. If the type is EDGE_LIST, then the edges are
 * given as a sequence of lines of one of the forms
 *
 *       
 *        
 *
 * each entry giving the terminal nodes of some edge, and if three integers are
 * given, the last one specifies its weight. The list is terminated by a -1.
 * If the type is ADJ_LIST, the section consists of adjacency lists for nodes.
 * The adjacency list of a node x is specified as
 *
 *        ...  -1
 *
 * where the first integer gives the number of node x and the following
 * integers (terminated by -1) the numbers of the nodes adjacent to x.
 * The list of adjacency lists are terminated by an additional -1.
 *
 * FIXED_EDGES_SECTION :
 * In this section, edges are listed that are required to appear in each
 * solution to the problem. The edges to be fixed are given in the form
 * (per line)
 *
 *       
 *
 * meaning that the edge (arc) from the first node to the second node has
 * to be contained in a solution. This section is terminated by a -1.
 *
 * DISPLAY_DATA_SECTION :
 * If DISPLAY_DATA_TYPE is TWOD_DISPLAY, the 2-dimensional coordinates from
 * which a display can be generated are given in the form (per line)
 *
 *        
 *
 * The integers specify the respective nodes and the real numbers give the
 * associated coordinates. The contents of this section, however, has no
 * significance in the current implementation.
 *
 * EDGE_WEIGHT_SECTION :
 * The edge weights are given in the format specifies by the EDGE_WEIGHT_FORMAT
 * entry. At present, all explicit data are integral and is given in one of the
 * (self-explanatory) matrix formats, with explicitly known lengths.
 *
 * TOUR_SECTION :
 * A tour is specified in this section. The tour is given by a list of
 * integers giving the sequence in which the nodes are visited in the tour.
 * The tour is terminated by a -1. Note: In contrast to the TSPLIB format,
 * only one tour can be given in this section. The tour is used to limit
 * the search (the last edge to be excluded in a non-gainful move must not
 * belong to the tour). In addition, the Alpha field of its edges is set to
 * -1.
 *
 * BACKHAUL_SECTION :
 * This section is used for specifying VRPB instances.
 * It contains a list of backhaul nodes. This list is terminated by a -1.
 *
 * CTSP_SET_SECTION :
 * This section is used for specifying CTSP instances.
 * Each entry has the following format:
 * c v1 v2 ... vk -1, where c is the color number (colors are numbered
 * from 1 to SALESMEN), and v1 v2 ... vk are vertices with color c
 * (vertices are numbered from 1 to Dimension).
 *
 * DEMAND_SECTION :
 * The demands of all nodes of a CVRP are given in the form (per line)
 *
 *     
 *
 * The first integer spcifies a node number, the second its demand. The depot
 * nodes must also occcur in this section. Their demands are 0.
 *
 * DEPOT_SECTION :
 * Contains a list of possible alternate depot nodes. This list is terminated
 * by a -1. The current implementation allows only one depot.
 *
 * DRAFT_LIMIT_SECTION :
 * The draft limits of all nodes of a CVRP are give in the form (per line)
 *
 *     
 *
 * The first integer spcifies a node number, the second its draft limit.
 * The depot nodes must also occcur in this section. Their demands are 0.
 *
 * PICKUP_AND_DELIVERY_SECTION :
 * This section is used for specifying specifying pickup-and-delivery
 * instances. Each line is of the form
 *
 *           
 *
 * The first integer gives the number of the node.
 * The second integer gives its demand (ignored for PDTSPF, PDTSPL, VRPMPD
 * and VRPSPD instances).
 * The third and fourth number give the earliest and latest time for the node.
 * The fifth number specifies the service time for the node.
 * The last two integers are used to specify pickup and delivery. For a PDPTW,
 * PDTSP, PDTSPF and PDTSPL instance, the first of these integers gives the
 * index of the pickup sibling, whereas the second integer gives the index of
 * the delivery sibling. For a VRPMPD and VRPSPD instance, the two integers
 * simply give the size of the pickup and delivery for the node.
 *
 * REIQUIRED_NODES_SECTION :
 * Contains a list of required nodes for a Steiner traveling salesman problem.
 * This list is terminated * by a -1.
 *
 * SERVICE_TIME_SECTION :
 * The service times of all nodes of a CVRP are given in the form (per line)
 *
 *     
 *
 * The integer specifies a node number, the real its service time.
 * The depot node must also occur in this section. Its service time is 0.
 *
 * TIME_WINDOW_SECTION :
 * Time windows are given in this section. Each line is of the form
 *
 *        
 *
 * The first integer specifies a node number. The two reals specify
 * earliest and latest arrival time for the node, respectively.

2.2 示例

直接使用点坐标（node_coord_section）：

使用距离矩阵，我们以lower diag row为例，代码如下，

示例如下：

如果想要固定某些边，可以参考下面的代码，在tsp文件中添加fixed edge section

2.3 示例2

上面是通过读写文件的方式来传递问题数据的，下面说下直接通过库文件传递问题数据的方法。
首先在LKHmain.c中添加函数，例如double run_matrix(int* distM, int Dim)，然后将readParameters和readProblem两个函数的内容修改后粘贴进函数中。读取数据的部分修改为从内存中读取：