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47 lines
10 KiB
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Travelling Salesman Problem
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<div class="ques-view"><p>The "travelling salesman problem" asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science. (Quoted from "<a href="https://en.wikipedia.org/wiki/Travelling_salesman_problem">https://en.wikipedia.org/wiki/Travelling_salesman_problem</a>".)</p>
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<p>In this problem, you are supposed to find, from a given list of cycles, the one that is the closest to the solution of a travelling salesman problem.</p>
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<h3 id="input-specification-">Input Specification:</h3>
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<p>Each input file contains one test case. For each case, the first line contains 2 positive integers <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><mrow><mn>2</mn><mo><</mo><mi>N</mi><mo>≤</mo><mn>2</mn><mn>0</mn><mn>0</mn></mrow>2< N \le 200</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span><span class="mrel"><</span><span class="mord mathit" style="margin-right:0.10903em;">N</span><span class="mrel">≤</span><span class="mord mathrm">2</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span></span></span></span>), the number of cities, and <span class="katex"><span class="katex-mathml"><math><mrow><mi>M</mi></mrow>M</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">M</span></span></span></span>, the number of edges in an undirected graph. Then <span class="katex"><span class="katex-mathml"><math><mrow><mi>M</mi></mrow>M</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">M</span></span></span></span> lines follow, each describes an edge in the format <code>City1 City2 Dist</code>, where the cities are numbered from 1 to <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> and the distance <code>Dist</code> is positive and is no more than 100. The next line gives a positive integer <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">K</span></span></span></span> which is the number of paths, followed by <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.07153em;">K</span></span></span></span> lines of paths, each in the format:</p>
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<p><span class="katex"><span class="katex-mathml"><math><mrow><mi>n</mi></mrow>n</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span></span></span></span> <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>C</mi><mn>1</mn></msub></mrow>C_1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span></span></span></span> <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>C</mi><mn>2</mn></msub></mrow>C_2</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span></span></span></span> ... <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>C</mi><mi>n</mi></msub></mrow>C_n</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span></span></span></span></p>
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<p>where <span class="katex"><span class="katex-mathml"><math><mrow><mi>n</mi></mrow>n</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span></span></span></span> is the number of cities in the list, and <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow>C_i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.07153em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span></span></span></span>'s are the cities on a path.</p>
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<h3 id="output-specification-">Output Specification:</h3>
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<p>For each path, print in a line <code>Path X: TotalDist (Description)</code> where <code>X</code> is the index (starting from 1) of that path, <code>TotalDist</code> its total distance (if this distance does not exist, output <code>NA</code> instead), and <code>Description</code> is one of the following:</p>
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<ul>
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<li><code>TS simple cycle</code> if it is a simple cycle that visits every city;</li>
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<li><code>TS cycle</code> if it is a cycle that visits every city, but not a simple cycle;</li>
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<li><code>Not a TS cycle</code> if it is NOT a cycle that visits every city.</li>
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</ul>
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<p>Finally print in a line <code>Shortest Dist(X) = TotalDist</code> where <code>X</code> is the index of the cycle that is the closest to the solution of a travelling salesman problem, and <code>TotalDist</code> is its total distance. It is guaranteed that such a solution is unique.</p>
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<h3 id="sample-input-">Sample Input:</h3>
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<pre><code class="lang-in">6 10
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6 2 1
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3 4 1
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1 5 1
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2 5 1
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3 1 8
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4 1 6
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1 6 1
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6 3 1
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1 2 1
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4 5 1
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7
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7 5 1 4 3 6 2 5
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7 6 1 3 4 5 2 6
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6 5 1 4 3 6 2
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9 6 2 1 6 3 4 5 2 6
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4 1 2 5 1
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7 6 1 2 5 4 3 1
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7 6 3 2 5 4 1 6
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</code></pre>
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<h3 id="sample-output-">Sample Output:</h3>
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<pre><code class="lang-out">Path 1: 11 (TS simple cycle)
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Path 2: 13 (TS simple cycle)
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Path 3: 10 (Not a TS cycle)
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Path 4: 8 (TS cycle)
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Path 5: 3 (Not a TS cycle)
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Path 6: 13 (Not a TS cycle)
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Path 7: NA (Not a TS cycle)
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Shortest Dist(4) = 8
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</code></pre>
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</div> |