PAT/source/html/a1053.html

Path of Equal Weight
<div class="ques-view"><p>Given a non-empty tree with root <span class="katex"><span class="katex-mathml"><math><mrow><mi>R</mi></mrow>R</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:0.00773em;">R</span></span></span></span>, and with weight <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>W</mi><mi>i</mi></msub></mrow>W_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:0.13889em;">W</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.13889em;"><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> assigned to each tree node <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>T</mi><mi>i</mi></msub></mrow>T_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:0.13889em;">T</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.13889em;"><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>.  The <strong>weight of a path from <span class="katex"><span class="katex-mathml"><math><mrow><mi>R</mi></mrow>R</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:0.00773em;">R</span></span></span></span> to <span class="katex"><span class="katex-mathml"><math><mrow><mi>L</mi></mrow>L</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">L</span></span></span></span></strong> is defined to be the sum of the weights of all the nodes along the path from <span class="katex"><span class="katex-mathml"><math><mrow><mi>R</mi></mrow>R</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:0.00773em;">R</span></span></span></span> to any leaf node <span class="katex"><span class="katex-mathml"><math><mrow><mi>L</mi></mrow>L</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">L</span></span></span></span>.</p>
<p>Now given any weighted tree, you are supposed to find all the paths with their weights equal to a given number.  For example, let's consider the tree showed in the following figure: for each node, the upper number is the node ID which is a two-digit number, and the lower number is the weight of that node.  Suppose that the given number is 24, then there exists 4 different paths which have the same given weight: {10 5 2 7}, {10 4 10}, {10 3 3 6 2} and {10 3 3 6 2}, which correspond to the red edges in the figure.</p>
<p><img alt="" src="https://images.ptausercontent.com/212"/></p>
<h3 id="input-specification-">Input Specification:</h3>
<p>Each input file contains one test case.  Each case starts with a line containing <span class="katex"><span class="katex-mathml"><math><mrow><mn>0</mn><mo>&lt;</mo><mi>N</mi><mo>≤</mo><mn>1</mn><mn>0</mn><mn>0</mn></mrow>0 &lt; N \le 100</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">0</span><span class="mrel">&lt;</span><span class="mord mathit" style="margin-right:0.10903em;">N</span><span class="mrel">≤</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span></span></span></span>, the number of nodes in a tree, <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> (<span class="katex"><span class="katex-mathml"><math><mrow><mo>&lt;</mo><mi>N</mi></mrow>&lt; N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mrel">&lt;</span><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span>), the number of non-leaf nodes, and <span class="katex"><span class="katex-mathml"><math><mrow><mn>0</mn><mo>&lt;</mo><mi>S</mi><mo>&lt;</mo><msup><mn>2</mn><mrow><mn>3</mn><mn>0</mn></mrow></msup></mrow>0 &lt; S &lt; 2^{30}</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.853208em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">0</span><span class="mrel">&lt;</span><span class="mord mathit" style="margin-right:0.05764em;">S</span><span class="mrel">&lt;</span><span class="mord"><span class="mord mathrm">2</span><span class="msupsub"><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord scriptstyle uncramped mtight"><span class="mord mathrm mtight">3</span><span class="mord mathrm mtight">0</span></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>, the given weight number. The next line contains <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> positive numbers where <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>W</mi><mi>i</mi></msub></mrow>W_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:0.13889em;">W</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.13889em;"><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> (<span class="katex"><span class="katex-mathml"><math><mrow><mo>&lt;</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn></mrow>&lt;1000</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.68354em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mrel">&lt;</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span></span></span></span>) corresponds to the tree node <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>T</mi><mi>i</mi></msub></mrow>T_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:0.13889em;">T</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.13889em;"><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>.  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 in the format:</p>
<pre><code>ID K ID[1] ID[2] ... ID[K]
</code></pre><p>where <code>ID</code> is a two-digit number representing a given non-leaf node, <code>K</code> is the number of its children, followed by a sequence of two-digit <code>ID</code>'s of its children. For the sake of simplicity, let us fix the root ID to be <code>00</code>.</p>
<h3 id="output-specification-">Output Specification:</h3>
<p>For each test case, print all the paths with weight S in <strong>non-increasing</strong> order.  Each path occupies a line with printed weights from the root to the leaf in order.  All the numbers must be separated by a space with no extra space at the end of the line.</p>
<p>Note: sequence <span class="katex"><span class="katex-mathml"><math><mrow><mo>{</mo><msub><mi>A</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>A</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub><mo>}</mo></mrow>\{A_1, A_2, \cdots , A_n\}</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">{</span><span class="mord"><span class="mord mathit">A</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><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 class="mpunct">,</span><span class="mord"><span class="mord mathit">A</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><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 class="mpunct">,</span><span class="minner">⋯</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">A</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><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 class="mclose">}</span></span></span></span> is said to be <strong>greater than</strong> sequence <span class="katex"><span class="katex-mathml"><math><mrow><mo>{</mo><msub><mi>B</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>B</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mo separator="true">,</mo><msub><mi>B</mi><mi>m</mi></msub><mo>}</mo></mrow>\{B_1, B_2, \cdots , B_m\}</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">{</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.05017em;"><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 class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.05017em;"><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 class="mpunct">,</span><span class="minner">⋯</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.05017em;"><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">m</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 class="mclose">}</span></span></span></span> if there exists <span class="katex"><span class="katex-mathml"><math><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>&lt;</mo><mi>m</mi><mi>i</mi><mi>n</mi><mo>{</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo>}</mo></mrow>1 \le k &lt; min\{n, m\}</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span><span class="mrel">≤</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mrel">&lt;</span><span class="mord mathit">m</span><span class="mord mathit">i</span><span class="mord mathit">n</span><span class="mopen">{</span><span class="mord mathit">n</span><span class="mpunct">,</span><span class="mord mathit">m</span><span class="mclose">}</span></span></span></span> such that <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>A</mi><mi>i</mi></msub><mo>=</mo><msub><mi>B</mi><mi>i</mi></msub></mrow>A_i = B_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">A</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><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 class="mrel">=</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.05017em;"><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> for <span class="katex"><span class="katex-mathml"><math><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>⋯</mo><mo separator="true">,</mo><mi>k</mi></mrow>i=1, \cdots , k</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="minner">⋯</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span>, and <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>A</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>&gt;</mo><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow>A_{k+1} &gt; B_{k+1}</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">A</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">1</span></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 class="mrel">&gt;</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.05017em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">1</span></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>
<h3 id="sample-input-">Sample Input:</h3>
<pre><code class="lang-in">20 9 24
10 2 4 3 5 10 2 18 9 7 2 2 1 3 12 1 8 6 2 2
00 4 01 02 03 04
02 1 05
04 2 06 07
03 3 11 12 13
06 1 09
07 2 08 10
16 1 15
13 3 14 16 17
17 2 18 19
</code></pre>
<h3 id="sample-output-">Sample Output:</h3>
<pre><code class="lang-out">10 5 2 7
10 4 10
10 3 3 6 2
10 3 3 6 2
</code></pre>
<h3 id="special-thanks-to-zhang-yuan-and-yang-han-for-their-contribution-to-the-judge-s-data-">Special thanks to Zhang Yuan and Yang Han for their contribution to the judge's data.</h3>
</div>