从禾略的城市地图谈最佳路径的选择

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&quot;Helvetica Neue&quot;, &quot;PingFang SC&quot;, &quot;Hiragino Sans GB&quot;, &quot;Microsoft YaHei UI&quot;, &quot;Microsoft YaHei&quot;, Arial, sans-serif;box-sizing: border-box !important;overflow-wrap: break-word !important;"><span style="max-width: 100%;font-size: 20px;color: rgb(178, 178, 178);text-transform: uppercase;letter-spacing: 1px;font-family: didot;box-sizing: border-box !important;overflow-wrap: break-word !important;"></span></strong></strong></span></em></span></strong></p></section></section><p style="text-align: center;"><img class="rich_pages" data-ratio="0.18333333333333332" data-s="300,640" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicI8pZcTCS251J40wZibypyKN6xrjetJrILQJHnMwmA0dG1jtYgvUZgow/640?wx_fmt=png" data-type="png" data-w="1080" style="width: 100%;height: auto;" data-backw="562" data-backh="103" /></p><section style="margin-right: 8px;margin-left: 8px;max-width: 100%;box-sizing: border-box;letter-spacing: 0.544px;background-color: rgb(255, 255, 255);text-align: center;line-height: 2em;font-family: -apple-system, BlinkMacSystemFont, &quot;Helvetica Neue&quot;, &quot;PingFang SC&quot;, &quot;Hiragino Sans GB&quot;, &quot;Microsoft YaHei UI&quot;, &quot;Microsoft YaHei&quot;, Arial, sans-serif;overflow-wrap: break-word !important;"><p style="text-align:left;margin: 5px 8px;font-size: 14px;caret-color: rgb(102, 102, 102);color: rgb(102, 102, 102);text-indent: 28px;widows: 1;line-height: 1.5em;box-sizing: border-box;font-family: PingFangSC-Light;"><br /></p></section></section><section data-role="paragraph" style="box-sizing: border-box;"><section style="margin: 5px 8px;max-width: 100%;min-height: 1em;line-height: 1.5em;box-sizing: border-box !important;overflow-wrap: break-word !important;"><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">城市地图作为禾略的核心产品之一，旨在将我们对于地块的判断落入空间网格中，以颜色梯度的形式展现出来，类似于如下的形式：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.6071794871794872" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicfk3Wve8XdPbKn8PibCAskP0G0fwmwDbxYu0Cj1AXCJkicoGD46FQiagtg/640?wx_fmt=png" data-type="png" data-w="975" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">但是这种只给出结论的行为，时常会引起一些非议。无论是在向甲方的汇报的时候，还是在公众号文章的留言板上，经常会有这样的信息反馈：<strong>你们的数据来源可靠吗？你们是怎么判断的，有依据可循吗？</strong></span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">质疑是合理的，人们对于未知的事物总是小心翼翼。但是，要想在汇报过程中或者在评论区去解释合理性，我们只能简单地描述一下方法的定义和执行过程，这类似于以一页PPT的方式简单展示。但这样的回答乏善可陈，可是如果要通篇描述，明显篇幅不够，时间有限。这就是这个专栏存在的意义，无论你是甲方还是粉丝还是感兴趣的同事，<strong>如果你对禾略的方法想要有更细节的了解，这里可能会有你想要的答案。</strong>继上一篇对“熵值法”做了解剖之后，小编会陆续解剖</span><span style="color: rgb(255, 255, 255);font-size: 14px;background-color: rgb(0, 0, 0);"><strong>禾略所使用的经典算法模型</strong></span><span style="font-size: 14px;">，力求通俗易懂，让人信服。</span></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;"><img data-ratio="0.18333333333333332" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicicWW78C7ibysqNpgJAoZovDsLz4cNGws80n1ZDMFDRppBVsSOlDibRCibg/640?wx_fmt=png" data-type="png" data-w="1080" style="box-sizing: border-box;width: 100%;height: auto;" data-backw="562" data-backh="103" /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">回到主题，要判断一个地段的空间价值，<strong>最优路径</strong>是一个重要的考量因素。到商场的最优路径耗时是生活配套资源的重要指标之一，到公交站的最优路径耗时是交通便捷性的重要指标之一。如果知道了这个最优路径，对于空间价值的评价会变得比较轻松。比如如果小明从家出发，在10分钟内能到达的餐饮店的数量为5，很容易判断出小明家附近的餐饮资源是比较丰富的。那么，我们是如何计算这个最优路径呢？</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">解决实际问题时，<strong>抽象建模</strong>是一个很重要的工具，现实生活中的路径错综复杂，针对点对点进行实际路径的分析显然不适用。<strong>网格化处理</strong>是一个非常好的办法，能大幅减少计算量。我们把空间等分的划分成网格，每个网格的边都是等分的路径，如下所示：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.8160291438979964" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicrOm1T1aicBDfaBaKUfovPu9fuJTdibB4X0UTvMY1Lob05PqVp2a5mE3A/640?wx_fmt=png" data-type="png" data-w="549" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">我知道你肯定会质疑：二维网格中等长的路径，在现实生活中可能不等长啊，有些是平路，有些山路，有些甚至是障碍物无法通过，还有红绿灯等等……。但是，咱们之前一直强调，在解决复杂的问题时，我们往往需要先从简单的问题出发，去逐步逼近最终的问题的答案，所以我们还是来先思考当下的这个“简单”问题吧：如何找到小明家到商场的最优路径？</span></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;"><br /></span></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><br /></p><p style="text-indent: 0em;"><img data-ratio="0.18333333333333332" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicHuopRsABnjYhoFQPSP34pFLLD6NhNRbFOVN6rvtrgINXBzRiadspnqQ/640?wx_fmt=png" data-type="png" data-w="1080" style="box-sizing: border-box;width: 100%;height: auto;" data-backw="562" data-backh="103" /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">这个问题早就已经解决了，有一种古老的算法叫</span><span style="color: rgb(255, 255, 255);font-size: 14px;background-color: rgb(0, 0, 0);"><strong><span style="color: rgb(255, 255, 255);font-size: 14px;background-image: initial;background-position: initial;background-size: initial;background-repeat: initial;background-attachment: initial;background-origin: initial;background-clip: initial;">贪心最佳优先搜索(Greedy Best First Search)</span></strong></span><span style="font-size: 14px;">，啥意思呢？就是你选择要走下一步之前，先确认要走的那个网格到终点的“代价“，优先选择“代价“较小的那个网格走，这个”代价“可以是这个网格到终点的直接距离（欧式距离），可以是最少的网格数（曼哈顿距离）等等，我们以欧式距离为例，如下图所示：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.80897583429229" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicKR6183yRbbcicJAoKv3UfOnpKA5LQwYib03IdMZ2fpU4TcHQTHnk6Yqg/640?wx_fmt=png" data-type="png" data-w="869" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">小明下一步该走哪个网格呢？答案是3号网格，因为3号网格离终点的欧式距离最短，代价最小，所以小明下一步是去3号网格，下一步时，也是按照这个思路继续，直到到达终点为止，最后得到的路线如下所示：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.8010403120936281" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicvUAv4kK6771WM7yEUXHZPiaObdl1kLHiaJ0JJXLm2jZVGGgGiaA4mLfFg/640?wx_fmt=png" data-type="png" data-w="769" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">这种每一次都选择当下最优的方法被称为<strong>贪婪算法</strong>，但是贪婪算法只关注<strong>局部最优</strong>，得到的结果可能不是全局最优。那这个路径到底是不是最优路径呢？答案是是的，这很容易理解，因为这样走的网格数肯定是最少的。但是在下面的情形中，这样做得到的路径就不是最优路径了。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.8110944527736131" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicbNBsDRnicgyNaHicky4tR31OEd5ZYL2RxYtVsgTyI11M0qHDJOZ4qbHQ/640?wx_fmt=png" data-type="png" data-w="667" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">绿色网格是障碍物，无法通行，我们再用贪心最佳优先试试：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.7873799725651578" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicUKwUeyhxhHV0MXMocA7WAEuaVjWsLHFic3V1jxiaGDQs5PPu5y0FB8vA/640?wx_fmt=png" data-type="png" data-w="729" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">这条路线肯定不是最优路径，你肯定能找到比它更短的路径。那问题出在哪？因为贪心最佳优先算法过分贪心地接近终点，这就好比拉力赛车比赛时，有弯道和直道，直道离终点更近，所以我们往往会优先选择直道，但是我们不能预计路上会发生什么，如果前面有一条河，那可能就只能目送走弯道的赛车手冲向终点了。</span></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.18333333333333332" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicibBkOZibCbPWPjOemGfStMMo5EBkEKC7OW2lz1XqL9OYsibSKic5KeYPAQ/640?wx_fmt=png" data-type="png" data-w="1080" style="box-sizing: border-box;width: 100%;height: auto;" data-backw="542" data-backh="99" /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">那有没有办法得到有障碍物的最优路径？有的，这里我们介绍另外一种古老的算法: </span><span style="color: rgb(255, 255, 255);font-size: 14px;background-color: rgb(0, 0, 0);"><strong><span style="color: rgb(255, 255, 255);font-size: 14px;background-image: initial;background-position: initial;background-size: initial;background-repeat: initial;background-attachment: initial;background-origin: initial;background-clip: initial;">广度优先搜索(Breadth First Search)</span></strong></span><span style="font-size: 14px;">。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">为了方便展示广度优先搜索的执行过程，我将每个网格都进行编号，例如左上角的网格，我叫它（1，1），那么小明家就叫（3，1）。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.8102409638554217" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicACFhgx40FICrL0mXZYnMzGquMgEK6G4QwcialdDRG2a9umJTsvq47Aw/640?wx_fmt=png" data-type="png" data-w="664" style="box-sizing:border-box;width:&quot;auto&quot;;" /><span style="font-size: 14px;">&nbsp;</span></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">首先，小明第一步能选择的网格为（4，1）、（3，2）和（2，1），我们将这三个网格放到一个队列里面：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.45445134575569357" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicIILDe1kQaZicEFiajtLF4P0DEGhbXgK2weVGfOPiaQBbzRHlGasjiaofaA/640?wx_fmt=png" data-type="png" data-w="966" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">（4，1）不是终点，所以我们可以将它拿掉，并将（4，1）的下一步所能达到的网格即（4，2）加入到这个队列后面；与此同时创造一个树结构，用于记录拿掉的网格和这个网格的上一步，如下图所示：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.40015015015015015" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODiclcFzwanicEEXicWfibZvObZJyibXk2jmxanxJhFK7YiccgzjaUs5FQgibzEA/640?wx_fmt=png" data-type="png" data-w="1332" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">&nbsp;&nbsp;对后续的网格进行同样的操作，操作流程如下：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.7862232779097387" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicJH3XdR4ex1BiaCkz5zG6M1GicghVLyK5ibiaBRJC2ibxfHcicgXe48sIW2tA/640?wx_fmt=png" data-type="png" data-w="842" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">到达终点（1，4）时，结构树如下：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="1.045045045045045" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODiczibT9slwvHTI2MicnRkoQNGVugVv8Ht34Bb2iasReAIh42y33Itea6jdA/640?wx_fmt=png" data-type="png" data-w="666" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">从商场开始，逐个记录相应的上一步网格，如红圈所示，将这个路径展示如下：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.795134443021767" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicbvmbMTXj24QEDCicZl7x4tuBMibrDn9IicgLN3icOSkFnxYx19cf7f3rPA/640?wx_fmt=png" data-type="png" data-w="781" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">这条路径就是最优路径，因为广度优先是以地毯式地逐层搜索，一旦它发现了终点，就结束了。换句话说，它没结束之前，之前的每一层都不可能到达终点，这个到达终点的“层数“肯定是最少的，即走的步数最少。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">到目前为止，我们已经基本解决了“简单“的问题，下一步就需要朝着复杂化的问题出发，正如你之前所惑：二维网格中等长的路径，在现实生活中可能不等长啊，有些是平路，有些山路，还有红绿灯等等……，这些指标如何量化？有一种办法就是给每个网格分配一个权重值，比如这个网格是山路，就给它赋值6，是平路就给它赋值3，如下所示，小明走到（2，1），开销就是3，走到（3，2）开销就是2，（你肯定好奇我为什么把表格越缩越小了，后面我会告诉你原因）：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.8045602605863192" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicjQW9pYEQauF9GF7m616iau3P94UjOLFd2Gk3efDp2TZePN2nJAFiabUQ/640?wx_fmt=png" data-type="png" data-w="614" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.18333333333333332" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODic3iaiaNlKic53j0Z9LGoX1Ew6bIwyU2DUxask55H9Qp5icomXySfwia5W6EA/640?wx_fmt=png" data-type="png" data-w="1080" style="box-sizing: border-box;width: 100%;height: auto;" data-backw="542" data-backh="99" /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;"><br /></span></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">因为障碍物不可穿透，所以我们把它赋值为正无穷，这时候我们用广度优先搜索，会得到最优路径吗？答案是不能，不信你可以自己试试，广度优先只能解决每条路权重都相等的情况，对于这种权重变化的复杂问题，广度优先不起作用，那到底该怎么办呢？是的，你没猜错，我又要介绍另一种算法：</span><span style="color: rgb(255, 255, 255);font-size: 14px;background-color: rgb(0, 0, 0);"><strong><span style="color: rgb(255, 255, 255);font-size: 14px;background-image: initial;background-position: initial;background-size: initial;background-repeat: initial;background-attachment: initial;background-origin: initial;background-clip: initial;">Dijkstra算法</span></strong></span><span style="font-size: 14px;">。该算法是以计算机科学家Dijkstra的名字命名的，废话不多说，我们来看看他是如何解决这个问题的。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">首先，我们需要一个表格记录小明家到每个网格所需的开销，因为一开始我们不知道除了到达（2，1）和（3，2）的点所需要多少开销，所以可以暂时不做记录，并同样用结构树记录下每次走的网格和它的上一步。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.26111458985597996" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicQMQPF0YPQSYGrxJLOOkIIQwHpicbxhTC4pQS20aPwOlxjMhjEA79JKQ/640?wx_fmt=png" data-type="png" data-w="1597" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">下一步选择所需开销最小的网格，即（3，2），找到（3，2）的能到达的网格，即（2，2）和（3，3）,如果此时小明家到它们的开销可以更小（这里肯定可以，因为之前都不能到达这两点，开销相当于无穷大），就在表格里更新小明家到它们的开销，并在结构树中更新该更小的开销的上一步，如下图所示：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.3327935222672065" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODickicRQ5NZjiaepE73K6hmPutxsbILicPzrF8IBgqZTX7SusvlTGWoD2eew/640?wx_fmt=png" data-type="png" data-w="1235" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">接下来，选择剩下的所能到达的网格中开销最小的，重复上述过程。直到到达终点，并确认小明家到终点的开销是这个表格中剩下最小的，流程图如下所示：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.6132404181184669" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicTJMRuL9IvCIljMzEpOgEOjib8wKz73L3dm9lH6bmxWmRjEMVdEupMWQ/640?wx_fmt=png" data-type="png" data-w="861" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">流程执行的结果如下所示：</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:center;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><img data-ratio="0.4980430528375734" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODic2PRfInLeKIFwbkNyDKwBB7eHEjnHbQyaiaxkUiaO66E2AEeJCSyP7DGg/640?wx_fmt=png" data-type="png" data-w="1022" style="box-sizing:border-box;width:&quot;auto&quot;;" /></p><p style="text-align:center;text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">和广度优先一样，从商场向上回溯到达小明家，就是最短路径。我知道此时你的脑子里肯定有一万个问号：为什么这样做得到的路径就是最短的？数学上对此算法有严格的证明，但是如果仅仅以此来说服你，那也不符合本专栏力求通俗易懂的初衷，要想理解这个算法的意义，首先你需要认可一组归纳条件：</span></p><p style="text-indent: 0em;"><br /></p><section data-tools="135编辑器" data-id="93717" style="max-width: 100%;box-sizing: border-box;overflow-wrap: break-word !important;"><section style="max-width: 100%;box-sizing: border-box;background-image: url(&quot;https://mmbiz.qpic.cn/mmbiz_png/7QRTvkK2qC6tIGvc3ozCHcBonywI7fDV4Ksmoy5hWQHibq7TQpAQ5tricOQ7dnLg81iapxGeoUkic5Ob8vHOSI24xQ/640?wx_fmt=png&quot;);background-size: 100%;overflow-wrap: break-word !important;"><section style="padding: 1em;max-width: 100%;box-sizing: border-box;line-height: 1.75em;letter-spacing: 1px;overflow-wrap: break-word !important;"><section style="max-width: 100%;box-sizing: border-box;overflow-wrap: break-word !important;"><section style="margin-right: 8px;margin-left: 8px;max-width: 100%;text-align: left;box-sizing: border-box !important;overflow-wrap: break-word !important;"><section style="margin: 5px 8px;max-width: 100%;text-indent: 0em;letter-spacing: 2px;line-height: 1.75em;box-sizing: border-box !important;overflow-wrap: break-word !important;"><p style="text-align:justify;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="max-width: 100%;color: rgb(102, 102, 102);letter-spacing: 2px;text-indent: 28px;widows: 1;caret-color: rgb(102, 102, 102);font-size: 14px;font-family: 微软雅黑, sans-serif;box-sizing: border-box !important;overflow-wrap: break-word !important;">1. &nbsp;&nbsp; 从小明家出发，只走一个网格，如果这个网格是几个方向中开销最小的，那可以说，你就找到了小明家到该网格的最优路径，这就是图中的（3，2）网格。</span></p></section><section style="margin: 5px 8px;max-width: 100%;text-indent: 0em;letter-spacing: 2px;line-height: 1.75em;box-sizing: border-box !important;overflow-wrap: break-word !important;"><p style="text-align:justify;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="max-width: 100%;color: rgb(102, 102, 102);letter-spacing: 2px;text-indent: 28px;widows: 1;caret-color: rgb(102, 102, 102);font-size: 14px;font-family: 微软雅黑, sans-serif;box-sizing: border-box !important;overflow-wrap: break-word !important;">2. &nbsp;&nbsp;&nbsp;&nbsp;如果你已经找到了小明家通往某个网格的最优路径，并对这个网格更新其下一步所能到达的网格的开销，那它下一步所能到达的网格的开销不一定是最优的，但一定是经过这个网格中路径最优的。就如同图中，你通过（3，2）到达了（2，2），此时更新（2，2）的开销为3，这个开销不一定是到达（2，2）的最小开销，但一定是经过（3，2）后能到达（2，2）中开销最小的。</span></p></section></section></section></section></section></section><section style="margin-right: 8px;margin-left: 8px;max-width: 100%;text-align: center;line-height: 1.75em;box-sizing: border-box !important;overflow-wrap: break-word !important;"><br style="max-width: 100%;letter-spacing: 0.544px;background-color: rgb(255, 255, 255);font-family: -apple-system, BlinkMacSystemFont, &quot;Helvetica Neue&quot;, &quot;PingFang SC&quot;, &quot;Hiragino Sans GB&quot;, &quot;Microsoft YaHei UI&quot;, &quot;Microsoft YaHei&quot;, Arial, sans-serif;box-sizing: border-box !important;overflow-wrap: break-word !important;" /></section><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="text-indent: 0em;font-size: 14px;">针对第2个条件，你已经判断出了开销3是经过（3，2）后能到达（2，2）中开销最小的，那怎么证明它是全局最小的呢？你可别忘了，每次你都要判断未被判断的网格中当下小明家所到达的哪个是最小的，此时（2，2）是最小的，换句话说，你不可能再通过其他路径到达（2，2）后开销比3还小，所以开销3就是小明家到达（2，2）的全局最小开销。可以通过归纳法层层递归，当未被判断的网格中，最小开销对应的网格是商场时，这个开销就是小明家到商场的全局最小开销，得到的路径就是到达商场的最优路径（但愿你能懂我在说什么）。</span></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">所以虽然Dijkstra算法表面上看是一种贪婪算法，但是它执行的每一步都考虑到了全局最优的情况，所以必定能得到最优路径。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">在分析这些算法的过程中，尤其是到了广度优先和Dijkstra时，你可能已经注意到了我故意减少了网格数量。其实这是小编有意而为之，原因就是广度优先和Dijkstra在运行的过程中，会向周围一直发散，如果网格数量太多，这个过程会变得更加繁琐，需要更大的篇幅来描述。其实这对于这两个算法实际运行的过程也一样，如果我们需要考虑的网格越来越多，运行效率会越来越低，而且比如在Dijkstra中，像（1，1）这样开销很大的点最后都没有考虑。那如何对这些基本上不会被考虑的点进行剪枝呢，聪明的计算机科学家也把折衷的办法想到了：最佳优先算法运行效率很高，但是无法解决障碍物和有权重的情况，Dijkstra与之相反。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-indent: 0em;"><br /></p><section style="text-indent: 0em;margin-left: 0px;margin-right: 0px;"><img data-ratio="0.18333333333333332" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxAWJqTbP38UBOFianV3qODicbhuiaWFrbUrE5H3T3zx9QiauCZ2RHdJQKbSdlh8PMAfkFrlHUjwMejBQ/640?wx_fmt=png" data-type="png" data-w="1080" style="box-sizing: border-box;width: 100%;height: auto;" data-backw="562" data-backh="103" /></section><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">那我们可不可以将两者结合起来呢？对的，本篇将再简单介绍一下能将这两种优势融合的算法：</span><span style="color: rgb(255, 255, 255);font-size: 14px;background-color: rgb(0, 0, 0);"><strong><span style="color: rgb(255, 255, 255);font-size: 14px;background-image: initial;background-position: initial;background-size: initial;background-repeat: initial;background-attachment: initial;background-origin: initial;background-clip: initial;">A^*算法</span></strong></span><span style="font-size: 14px;">。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">A^*算法执行的过程与Dijkstra相似，但是在每一次找出未被判断的网格中开销最小的时候，这里的“开销”不是单指小明家到该网格的开销，还需加上该网格到商场的“估计代价”（我们前面在最佳优先里面提到的“代价“）,这个“开销”是“启发式”的，“启发式“可以简单理解为试图在合理的运行时间得到不错的结果，但这个结果有可能是次优的，这一点和最佳优先搜索类似。</span></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">举一个简单的例子即可理解：假设你在荒郊野外迷路了，此时你受Dijkstra算法的启发，想找到当下最优的路径走，但是这个方向实际与城邦的方向相反。此时你身上装着一个信号接收器，能接收到城邦的信号塔发射的信号，它告诉你：你正在离城邦越来越远。此时你肯定会再权衡一下，决定是否舍弃掉这条当下的最优路径。这个决定不一定绝对正确，有可能反方向再迂回的路是一马平川，正方向前面是一座大山，但是大部分情况下可能是不错的选择。</span></p><p style="text-indent: 0em;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">那能不能通过A^*算法得到全局最优解呢？计算机科学家们认为当“估计代价”小于“真实代价“的时候，A^*算法可以取得全局最优解，但运行效率不见得比Dijkstra高，要想理解这个过程比较复杂，在这里不多赘述。A^*算法的运用领域很广，很多地图软件都是根据此算法计算最优路径；智能机器人也会根据此算法绕开障碍物；当你在英雄联盟的峡谷探索的时候，点击小地图去往某个地方，也许此时A^*算法正在帮你计算最优路径。</span></p><section style="margin: 5px 8px;max-width: 100%;box-sizing: border-box;text-indent: 0em;text-align: center;line-height: 2em;overflow-wrap: break-word !important;"><p style="text-align:center;max-width: 100%;min-height: 1em;line-height: 25.5px;text-indent: 0em;box-sizing: border-box !important;overflow-wrap: break-word !important;"><br /></p></section><p style="text-align: center;"><img class="rich_pages" data-backh="103" data-backw="562" data-ratio="0.18333333333333332" data-s="300,640" data-src="https://mmbiz.qpic.cn/mmbiz_png/AJkYaTAujIxBKiaQNH3yAajSVKNDghYCqllMQ2LAm1s80CdZkCxos35cr0e3xRrWysHmPvWeohSYeaaZGRXfPvA/640?wx_fmt=png" data-type="png" data-w="1080" style="width: 100%;height: auto;" /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><br /></p><p style="text-align:justify;text-indent: 0em;margin: 5px 10px;line-height: 1.5em;letter-spacing: 1px;"><span style="font-size: 14px;">关于最佳路径的搜索算法就算介绍结束了，<strong>禾略城市地图的交通通达性、生活配套资源丰富度等等落到空间判断的底层都不乏它们的身影存在</strong>。<strong>身处于这样一个数据信息爆炸的时代，禾略总是站在巨人的肩膀上，旨在为客户，为社会创造价值。</strong></span><br /></p></section><section style="margin: 5px 8px;text-align: center;max-width: 100%;min-height: 1em;line-height: 1.5em;box-sizing: border-box !important;overflow-wrap: break-word !important;"><br /></section><section style="margin: 5px 8px;text-align: center;max-width: 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