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Draft:Klee's algorithm - Revision history
2025-06-23T03:08:20Z
Revision history for this page on the wiki
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TakuyaMurata: #redirect Klee's measure problem; the algorithm is unreferenced so I don't think it can be merged into the existing article
2025-06-15T06:41:57Z
<p>#redirect <a href="/wiki/Klee%27s_measure_problem" title="Klee's measure problem">Klee's measure problem</a>; the algorithm is unreferenced so I don't think it can be merged into the existing article</p>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>As an additional challenge, Klee invited the readers to consider the problem in multiple dimensions and how efficiently someone could solve the complexity of finding a disjoint set of d-dimensional intervals, reducing that set to the minimum number of intervals possible, and computing the measure of their union. This last question was eventually labeled [[Klee's measure problem]].</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=== Logical error in the original ===</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The original pseudocode had the following lines inside of the loop through the endpoints:</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td colspan="2" class="diff-empty diff-side-added"></td>
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<tr>
<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <math>EXCESS = EXCESS + t_i</math></div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> if <math>EXCESS = 1</math> then <math>m = m + 1</math> and <math>c_m = e_i</math></div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> if <math>EXCESS = 0</math> then <math>d_m = e_i</math></div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td colspan="2" class="diff-empty diff-side-added"></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>EXCESS</math> is the interval nesting count, <math>i</math> is the current endpoint index, <math>t_i</math> = +1 for left endpoints and -1 for right endpoints, <math>e_i</math> is the current endpoint value, <math>m</math> is the index of the current interval for the new set, and </div></td>
<td colspan="2" class="diff-empty diff-side-added"></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>c_m</math> and <math>d_m</math> are the start and end of the current interval for the new set, respectively. This code incorrectly jumps ahead <math>m</math> at the right endpoint of the second outermost interval and uses that endpoint as the left endpoint of the next recorded interval.</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td colspan="2" class="diff-empty diff-side-added"></td>
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<tr>
<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>To fix this problem, the pseudocode could be rewritten as follows:</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td colspan="2" class="diff-empty diff-side-added"></td>
</tr>
<tr>
<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> if <math>EXCESS = 0</math> then <math>m = m + 1</math> and <math>c_m = e_i</math></div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <math>EXCESS = EXCESS + t_i</math></div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> if <math>EXCESS = 0</math> then <math>d_m = e_i</math></div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
<td colspan="2" class="diff-empty diff-side-added"></td>
</tr>
<tr>
<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This way, the first line only triggers before the first endpoint of a nested set of endpoints and the second line only triggers after the end of the last endpoint of such a set.</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=== Javascript implementation ===</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><syntaxhighlight lang="javascript" line></div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>function measureOfUnion(intervals = []) {</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> // the input is expected to be an array of intervals,</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> // with each interval being a 2-element array</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> const endpoints = [];</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> for (let interval of intervals) {</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> endpoints.push({v: interval[0], nestingChange: +1});</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> endpoints.push({v: interval[1], nestingChange: -1});</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> }</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> endpoints.sort(endpointSort);</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> const disjointIntervals = [];</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> let nestingLevel = 0;</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> let currentInterval;</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> for (let endpoint of endpoints) {</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> if (nestingLevel === 0) {</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> currentInterval = [endpoint.v];</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> disjointIntervals.push(currentInterval);</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> }</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> nestingLevel += endpoint.nestingChange;</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> if (nestingLevel === 0)</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> currentInterval.push(endpoint.v);</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> }</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> let measure = 0;</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> for (let interval of disjointIntervals) {</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> measure += interval[1] - interval[0];</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> }</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> return measure;</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>}</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>function endpointSort(a, b) {</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> // normal expression for ascending order</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> if (a.v != b.v) return a.v - b.v;</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> // swap endpoints if the first one ends an interval</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> if (a.nestingChange == -1) return +1;</div></td>
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<td class="diff-marker" data-marker="−"></td>
<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> // otherwise, stay in this order</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> return -1;</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>}</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div></syntaxhighlight></div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><br /></td>
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TakuyaMurata
https://en.wikipedia.org/w/index.php?title=Draft:Klee%27s_algorithm&diff=1288861645&oldid=prev
David of Earth: Refreshed my inquiry
2025-05-05T03:34:43Z
<p>Refreshed my inquiry</p>
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David of Earth
https://en.wikipedia.org/w/index.php?title=Draft:Klee%27s_algorithm&diff=1263730042&oldid=prev
David of Earth: Where can I put an original correction?
2024-12-18T08:19:25Z
<p>Where can I put an original correction?</p>
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David of Earth
https://en.wikipedia.org/w/index.php?title=Draft:Klee%27s_algorithm&diff=1232452813&oldid=prev
TakuyaMurata: TakuyaMurata moved page Draft:Klee's Algorithm to Draft:Klee's algorithm: Lowercase
2024-07-03T20:20:07Z
<p>TakuyaMurata moved page <a href="/wiki/Draft:Klee%27s_Algorithm" class="mw-redirect" title="Draft:Klee's Algorithm">Draft:Klee's Algorithm</a> to <a href="/wiki/Draft:Klee%27s_algorithm" class="mw-redirect" title="Draft:Klee's algorithm">Draft:Klee's algorithm</a>: Lowercase</p>
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TakuyaMurata
https://en.wikipedia.org/w/index.php?title=Draft:Klee%27s_algorithm&diff=1231089246&oldid=prev
Felix QW: Declining submission: nn - Submission is about a topic not yet shown to meet general notability guidelines (be more specific if possible) and v - Submission is improperly sourced (AFCH)
2024-06-26T11:50:55Z
<p>Declining submission: nn - Submission is about a topic not yet shown to meet general notability guidelines (be more specific if possible) and v - Submission is improperly sourced (<a href="/wiki/Wikipedia:AFCH" class="mw-redirect" title="Wikipedia:AFCH">AFCH</a>)</p>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{AFC comment|1=The sections on "Logical error in the original" and the Javascript implementation are unsourced. Apart from the bullet-listed description of the algorithm, the page [[Klee's measure problem]] already contains the verifiable, encyclopedic content of this draft. As the latest reference provided here is from 1978, it is unclear currently that this topic is notable separately from [[Klee's measure problem]], which puts it in relevant context. [[User:Felix QW|Felix QW]] ([[User talk:Felix QW|talk]]) 11:50, 26 June 2024 (UTC)}}</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{Short description|Algorithm for the length of a union of intervals}}</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Framed as an open research problem published in 1977 in ''The American Mathematical Monthly'',<ref>{{cite journal</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Framed as an open research problem published in 1977 in ''The American Mathematical Monthly'',<ref>{{cite journal</div></td>
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Felix QW
https://en.wikipedia.org/w/index.php?title=Draft:Klee%27s_algorithm&diff=1218021434&oldid=prev
David of Earth: Submitting using AfC-submit-wizard
2024-04-09T07:55:16Z
<p>Submitting using <a href="/wiki/Wikipedia:AFCSW" class="mw-redirect" title="Wikipedia:AFCSW">AfC-submit-wizard</a></p>
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David of Earth
https://en.wikipedia.org/w/index.php?title=Draft:Klee%27s_algorithm&diff=1218020917&oldid=prev
David of Earth: Initial page draft creation
2024-04-09T07:50:20Z
<p>Initial page draft creation</p>
<p><b>New page</b></p><div>{{AfC submission|t||ts=20240409074923|u=David of Earth|ns=118|demo=}}<!-- Important, do not remove this line before article has been created. --><br />
<br />
Framed as an open research problem published in 1977 in ''The American Mathematical Monthly'',<ref>{{cite journal<br />
|first1=Victor<br />
|last1=Klee<br />
|author-link1=Victor Klee<br />
|year=1977<br />
|title=Can the measure of <math>\cup[a_i, b_i]</math> be computed in less than <math>O(n \log n)</math> steps?<br />
|journal=[[American Mathematical Monthly]]<br />
|volume=84<br />
|issue=4<br />
|pages=284–285<br />
|mr=0436661<br />
|doi=10.2307/2318871<br />
|jstor=2318871<br />
}}</ref> Victor Klee presented a challenge to the journal's readers to find an algorithm to compute the measure of the union of a set of intervals that would be more efficient than sorting the interval endpoints and sequentially adding the resulting lengths together. Klee then provided an algorithm in pseudocode that performs the sorting and summation as a baseline solution for this problem with a complexity of <math>O(n \log n)</math>. This complexity was later proven to be optimal in 1978 by Michael Freman and Bruce Weide.<ref>*{{cite journal<br />
|first1=Michael L.<br />
|last1=Fredman<br />
|author-link1=Michael Fredman<br />
|first2=Bruce<br />
|last2=Weide<br />
|year=1978<br />
|title=The complexity of computing the measure of <math>\cup[a_i, b_i]</math><br />
|journal=[[Communications of the ACM]]<br />
|volume=21 |issue=7<br />
|pages=540–544<br />
|mr=0495193<br />
|doi=10.1145/359545.359553<br />
|s2cid=16493364<br />
|doi-access=free<br />
}}</ref><br />
<br />
As an additional challenge, Klee invited the readers to consider the problem in multiple dimensions and how efficiently someone could solve the complexity of finding a disjoint set of d-dimensional intervals, reducing that set to the minimum number of intervals possible, and computing the measure of their union. This last question was eventually labeled [[Klee's measure problem]].<br />
<br />
== Algorithm ==<br />
<br />
# Separate the endpoint pairs for each interval and tag if each endpoint was on the left or the right.<br />
# Sort the individual endpoints in ascending order of value, with the additional condition that left endpoints are before right endpoints when two endpoint values are the same. (This changes any partially overlapping intervals into a series of nested ones.)<br />
# Iterate through the sorted endpoints, keeping track of the interval nesting count and recording just the outermost interval of each nesting.<br />
# Add up the measures of each of the resulting set of disjointed intervals.<br />
<br />
=== Logical error in the original ===<br />
<br />
The original pseudocode had the following lines inside of the loop through the endpoints:<br />
<br />
<math>EXCESS = EXCESS + t_i</math><br />
if <math>EXCESS = 1</math> then <math>m = m + 1</math> and <math>c_m = e_i</math><br />
if <math>EXCESS = 0</math> then <math>d_m = e_i</math><br />
<br />
where <math>EXCESS</math> is the interval nesting count, <math>i</math> is the current endpoint index, <math>t_i</math> = +1 for left endpoints and -1 for right endpoints, <math>e_i</math> is the current endpoint value, <math>m</math> is the index of the current interval for the new set, and <br />
<math>c_m</math> and <math>d_m</math> are the start and end of the current interval for the new set, respectively. This code incorrectly jumps ahead <math>m</math> at the right endpoint of the second outermost interval and uses that endpoint as the left endpoint of the next recorded interval.<br />
<br />
To fix this problem, the pseudocode could be rewritten as follows:<br />
<br />
if <math>EXCESS = 0</math> then <math>m = m + 1</math> and <math>c_m = e_i</math><br />
<math>EXCESS = EXCESS + t_i</math><br />
if <math>EXCESS = 0</math> then <math>d_m = e_i</math><br />
<br />
This way, the first line only triggers before the first endpoint of a nested set of endpoints and the second line only triggers after the end of the last endpoint of such a set.<br />
<br />
=== Javascript implementation ===<br />
<br />
<syntaxhighlight lang="javascript" line><br />
function measureOfUnion(intervals = []) {<br />
// the input is expected to be an array of intervals,<br />
// with each interval being a 2-element array<br />
const endpoints = [];<br />
for (let interval of intervals) {<br />
endpoints.push({v: interval[0], nestingChange: +1});<br />
endpoints.push({v: interval[1], nestingChange: -1});<br />
}<br />
endpoints.sort(endpointSort);<br />
const disjointIntervals = [];<br />
let nestingLevel = 0;<br />
let currentInterval;<br />
for (let endpoint of endpoints) {<br />
if (nestingLevel === 0) {<br />
currentInterval = [endpoint.v];<br />
disjointIntervals.push(currentInterval);<br />
}<br />
nestingLevel += endpoint.nestingChange;<br />
if (nestingLevel === 0)<br />
currentInterval.push(endpoint.v);<br />
}<br />
let measure = 0;<br />
for (let interval of disjointIntervals) {<br />
measure += interval[1] - interval[0];<br />
}<br />
return measure;<br />
}<br />
<br />
function endpointSort(a, b) {<br />
// normal expression for ascending order<br />
if (a.v != b.v) return a.v - b.v;<br />
// swap endpoints if the first one ends an interval<br />
if (a.nestingChange == -1) return +1;<br />
// otherwise, stay in this order<br />
return -1;<br />
}<br />
</syntaxhighlight><br />
<br />
== References ==<br />
<!-- Inline citations added to your article will automatically display here. See en.wikipedia.org/wiki/WP:REFB for instructions on how to add citations. --><br />
{{reflist}}</div>
David of Earth