「GIS思维」用prim最小生成树看武汉的中心地位

武汉号称九省通衢，是全国性甚至全球性的交通枢纽，那么，我们就用prim最小生成树来检验一下武汉的中心地位。

什么是最小生成树

所谓最小生成树，给定n个点，用n-1条边将这些点都连起来，就可以构成一棵生成树，而要使得这n-1条边的距离（权值）总和最小，就是最小生成树。

最小生成树的算法，最常见的是prim算法和Kruskal算法，prim算法是首先选定第一个点，然后取连接这个点的最短边，此时相当于已经有两个选定点了，然后再在剩余的边中选择连接这两个点的最短边，以此类推。而Kruskal算法是先选最短边，然后选次短边，以此类推，直至可以访问所有的节点。概括来讲的话，前者是用点选边，后者是用边选点。

36城实验结果

我们首先选择了全国的36个大城市做实验，最后的结果如下图所示。

从上图来看，武汉在华中地区处于中心地位，和华东、华北和西北地区的联系较为密切，和华南地区的距离也不算远。

400城实验结果

然后我们用全国的四百个城市来做重复实验，发现形状有一些变化，但是结论基本上还是一致的，和武汉联系较为紧密的是华东、华北和华南、西北地区，和西南地区的联系相对来说要少一些。

更多关于最小生成数

参考：最小生成树 - 维基百科，自由的百科全书

实现代码

#coding=utf-8

from arcpy import env,da
from os import path
from math import sqrt

def prim(graph, root):
    nodes = graph.keys()
    nodes.remove(root)

    visited = [root]
    path = []
    next = None

    while nodes:
        distance = float('inf')
        for s in visited:
            for d in graph[s]:
                if d in visited or s == d:
                    continue
                if graph[s][d] < distance:
                    distance = graph[s][d]
                    pre = s
                    next = d
        path.append((pre, next))
        visited.append(next)
        nodes.remove(next)

    return path

if __name__ == '__main__':
    flayer=r'F:\data\202003\WDL\city.shp'
    linkLayer=r'F:\data\202003\WDL\link.shp'
    env.workspace=path.dirname(flayer)

    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'毕节地区',(105.298588795011,27.2902150834259)],['安顺地区',(105.954417123889,26.2592523787149)],['黔东南苗族侗族自治州',(107.97,26.58)],['黔南布依族苗族自治州',(107.528402705737,26.2606161960738)],['曲靖市',(103.794700623,25.4961662292)],['玉溪市',(102.533233643,24.3549728394)],['昭通地区',(103.723511771968,27.3440838602468)],['楚雄彝族自治州',(101.534412480503,25.0517735653404)],['红河哈尼族彝族自治州',(102.42755057757,23.3744887383648)],['文山壮族苗族自治州',(104.222568991094,23.4059942936117)],['思茅市',(100.983555102975,22.7924979843596)],['西双版纳傣族自治州',(100.803446824556,22.0136012547642)],['大理白族自治州',(100.236519300444,25.5976049054519)],['保山地区',(99.1772732858179,25.139038793266)],['德宏傣族景颇族自治州',(98.5913593561141,24.4380107027581)],['丽江地区',(100.25,26.86)],['怒江傈僳族自治州',(98.8632883813579,25.8237074176578)],['迪庆藏族自治州',(99.7095299901396,27.8251846836433)],['临沧地区',(100.095440420148,23.8904685562785)],['昌都地区',(97.179583594086,31.147346549327)],['山南地区',(91.778675138519,29.2430269392492)],['日喀则地区',(88.8937030348255,29.2756578225115)],['那曲地区',(92.0573384981749,31.4824383884547)],['阿里地区',(80.1127769219265,32.5068660176333)],['林芝地区',(94.3680582871326,29.6540421769515)],['铜川市',(109.057281494,35.0754585266)],['宝鸡市',(107.13835907,34.3822860718)],['咸阳市',(108.710128784,34.3372154236)],['渭南市',(109.500839233,34.5015258789)],['延安市',(109.471282959,36.5938720703)],['汉中市',(107.034393311,33.0781478882)],['安康地区',(109.035601082657,32.6905127705737)],['商洛地区',(109.924417881364,33.8786338522077)],['榆林地区',(109.77,38.3)],['嘉峪关市',(98.2747116089,39.8026542664)],['金昌市',(102.165748596,38.4951934814)],['白银市',(104.183776855,36.5394172668)],['天水市',(105.715240479,34.5842666626)],['酒泉地区',(98.52,39.75)],['张掖地区',(100.456411474056,38.9320660070049)],['武威地区',(102.644554340369,37.9343778081581)],['定西地区',(104.632420083063,35.5868329265619)],['陇南地区',(104.928574970712,33.4066202299513)],['平凉地区',(106.671442348278,35.5492320504635)],['庆阳地区',(107.649385695954,35.715215983562)],['临夏回族自治州',(103.216390565297,35.6075621835031)],['甘南藏族自治州',(102.917584688258,34.9891399099682)],['海东地区',(102.12,36.5)],['海北藏族自治州',(100.907434321456,36.9606628241298)],['黄南藏族自治州',(102.022428278076,35.5258045865155)],['海南藏族自治州',(100.626621144459,36.2921024798988)],['果洛藏族自治州',(100.251591978796,34.4771938664709)],['玉树藏族自治州',(97.0131807627733,33.0109795851427)],['海西蒙古族藏族自治州',(97.3762991160191,37.3827504625274)],['石嘴山市',(106.38205719,39.0242805481)],['吴忠市',(106.199111938,37.985496521)],['固原地区',(106.248577426071,36.0216172580109)],['中卫市',(105.691300667357,37.4980023546252)],['克拉玛依市',(84.863609314,45.5965118408)],['吐鲁番地区',(89.1972972460079,42.9569848487712)],['哈密地区',(93.521215540285,42.8258243681126)],['昌吉回族自治州',(87.3150016244744,44.0168541599199)],['博尔塔拉蒙古自治州',(82.0729147582743,44.9121964134647)],['巴音郭楞蒙古自治州',(86.1517138653326,41.7702873304504)],['阿克苏地区',(80.266943484735,41.1750298600775)],['克孜勒苏柯尔克孜自治州',(76.174308676212,39.7204712048707)],['喀什地区',(75.9963905563975,39.4760967486448)],['和田地区',(79.92,37.12)],['伊犁哈萨克自治州',(81.3305377475322,43.9227231374921)],['塔城地区',(82.9872355184064,46.7509482543734)],['阿勒泰地区',(88.1479262037353,47.8507278701019)],['昭通市',(103.714927673,27.3422794342)],['昌都市',(97.1742,31.1491)],['海东市',(102.11071,36.50824)],['三沙市',(112.34502,16.83704)],['山南市',(91.7781,29.24366)],['吐鲁番市',(89.11,42.54)],['佳木斯市',(130.362319946,46.8136634827)],['儋州市',(109.34,19.31)],['铜仁市',(109.192680359,27.7221660614)],['日喀则市',(87.2341,29.3048)],['固原市',(106.278587341,36.0132598877)],['哈密市',(93.5215990269072,42.8252994627921)],['毕节市',(105.282417297,27.3062953949)],['普洱市',(100.972758458885,22.830950790286)],['那曲市',(92.0579351526946,31.4841315524952)],['龙岩市',(117.030387878,25.109703064)],['安顺市',(105.926071167,26.2442588806)],['保山市',(99.1687240601,25.1168022156)],['林芝市',(94.3682289651794,29.6547599913432)],['酒泉市',(98.5111160278,39.7449684143)],['乌兰察布市',(113.12,40.98)],['丽江市',(100.23425293,26.8766651154)],['安康市',(109.025787354,32.6898651123)],['张掖市',(100.450286865,38.9350585938)],['鄂尔多斯市',(109.8,39.62)],['淮安市',(119.14111,33.502789)],['临沧市',(100.087806702,23.8799304962)],['商洛市',(109.93,33.87)],['武威市',(102.633460999,37.9269104004)],['巴彦淖尔市',(107.393779959703,40.7492309808487)],['定西市',(104.61856842,35.5752372742)],['娄底市',(111.993896484,27.7413349152)],['榆林市',(109.757446289,38.2972755432)],['陇南市',(104.92,33.4)],['资阳市',(104.65019,30.122671)],['平凉市',(106.683067322,35.5355186462)],['滨州市',(118.021766663,37.3678131104)],['庆阳市',(107.63,35.73)],['达州市',(107.5,31.22)],['驻马店市',(114.035690308,32.9790420532)],['雅安市',(102.982696533,29.9822940826)],['亳州市',(115.7709,33.879292)],['菏泽市',(115.445762634,35.2485389709)],['巴中市',(106.75476,31.849014)],['眉山市',(103.83146,30.050497)],['石河子市',(86.0868856719329,44.311976478063)],['仙桃市',(113.449609358564,30.3335877511464)],['五家渠市',(87.5499366322961,44.172444518901)],['潜江市',(112.905474090816,30.4083579324189)],['文昌市',(110.4,18.8)],['济源市',(112.57,35.08)],['阿拉尔市',(81.2873542253954,40.553263894706)],['天门市',(113.172409166328,30.6696218300995)],['神农架林区',(110.6825248504,31.7504960112464)],['琼海市',(110.48054452595,19.2642540199176)],['五指山市',(109.523540320706,1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    with da.UpdateCursor(flayer,["OID@"]) as cursor:
        for row in cursor:
            cursor.deleteRow()

    with da.UpdateCursor(linkLayer,["OID@"]) as cursor:
        for row in cursor:
            cursor.deleteRow()

    with da.InsertCursor(flayer,["NAME","SHAPE@XY"]) as cursor:
        for data in datas:
            cursor.insertRow(data)

    graph_dict=dict()
    for i in range(len(datas)):
        graph_dict[i]=dict()
        for j in range(len(datas)):
            graph_dict[i][j]=sqrt((datas[i][1][0]-datas[j][1][0])**2+(datas[i][1][1]-datas[j][1][1])**2)

    path = prim(graph_dict, 0)

    count=0
    with da.InsertCursor(linkLayer,["SHAPE@JSON"]) as cursor:
        for link in path:
            count+=1
            print count,str([datas[link[0]][1], datas[link[1]][1]]).replace('(','[').replace(')',']')
            cursor.insertRow(['{"paths":[' + str([datas[link[0]][1], datas[link[1]][1]]).replace('(','[').replace(')',']') + '],"spatialReference":{"wkid":4326,"latestWkid":4326}}'])