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<title data-rh="true">Distance boundaries from BFS tree on undirected graphs | mf</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://blog.mfocko.xyz/ib002/graphs/bfs-tree"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-ib002-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-ib002-current"><meta data-rh="true" property="og:title" content="Distance boundaries from BFS tree on undirected graphs | mf"><meta data-rh="true" name="description" content="Short explanation of distance boundaries deduced from a BFS tree.
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1-1v-7h1.7c.46 0 .68-.57.33-.87L12.67 3.6c-.38-.34-.96-.34-1.34 0l-8.36 7.53c-.34.3-.13.87.33.87H5v7c0 .55.45 1 1 1h3c.55 0 1-.45 1-1z" fill="currentColor"></path></svg></a></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item"><a class="breadcrumbs__link" itemprop="item" href="/ib002/category/graphs"><span itemprop="name">Graphs</span></a><meta itemprop="position" content="1"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link" itemprop="name">Distance boundaries from BFS tree on undirected graphs</span><meta itemprop="position" content="2"></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><header><h1>Distance boundaries from BFS tree on undirected graphs</h1></header><h2 class="anchor anchorWithStickyNavbar_LWe7" id="introduction">Introduction<a href="#introduction" class="hash-link" aria-label="Direct link to Introduction" title="Direct link to Introduction"></a></h2><p>As we have talked on the seminar, if we construct from some vertex <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">u</span></span></span></span></span> BFS tree on an undirected graph, we can obtain:</p><ul><li>lower bound of length of the shortest path between 2 vertices from the <em>height difference</em></li><li>upper bound of length of the shortest path between 2 vertices from the <em>path through the root</em></li></ul><h2 class="anchor anchorWithStickyNavbar_LWe7" id="lower-bound">Lower bound<a href="#lower-bound" class="hash-link" aria-label="Direct link to Lower bound" title="Direct link to Lower bound"></a></h2><p>Consider the following graph:</p><p><img loading="lazy" src="/assets/images/bfs_graph_light-6e21a942bccd92bcce6840da7c3cb056.png#gh-light-mode-only" width="252" height="539" class="img_ev3q">
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<img loading="lazy" src="/assets/images/bfs_graph_dark-f7a3a78eaf9de049469b4c64e0712867.png#gh-dark-mode-only" width="252" height="539" class="img_ev3q"></p><p>We run BFS from the vertex <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">a</span></span></span></span></span> and obtain the following BFS tree:</p><p><img loading="lazy" src="/assets/images/bfs_tree_light-61d6723c3c587d565b6280b8b8eca211.png#gh-light-mode-only" width="275" height="347" class="img_ev3q">
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<img loading="lazy" src="/assets/images/bfs_tree_dark-34f32262c6e4ffc14983b3ebf9a2f5a9.png#gh-dark-mode-only" width="275" height="347" class="img_ev3q"></p><p>Let's consider pair of vertices <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">e</span></span></span></span></span> and <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">h</span></span></span></span></span>. For them we can safely lay, from the BFS tree, following properties:</p><ul><li>lower bound: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span></span></span></span></span></li><li>upper bound: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">4</span></span></span></span></span></li></ul><p>By having a look at the graph we started from, we can see that we have a path ‹<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo separator="true">,</mo><mi>j</mi><mo separator="true">,</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">e, j, h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.05724em">j</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">h</span></span></span></span></span>› that has a length 2. Apart from that we can also notice there is another path from <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">e</span></span></span></span></span> to <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">h</span></span></span></span></span> and that is ‹<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo separator="true">,</mo><mi>a</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>i</mi><mo separator="true">,</mo><mi>d</mi><mo separator="true">,</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">e, a, c, i, d, h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">h</span></span></span></span></span>›. And that path has a length of <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">5</span></span></span></span></span>. Doesn't this break our statements at the beginning? (<em>I'm leaving that as an exercise ;)</em>)</p><h2 class="anchor anchorWithStickyNavbar_LWe7" id="proof-by-contradiction">Proof by contradiction<a href="#proof-by-contradiction" class="hash-link" aria-label="Direct link to Proof by contradiction" title="Direct link to Proof by contradiction"></a></h2><p>Let's keep the same graph, but break the lower bound, i.e. I have gotten a lower bound <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span></span></span></span></span>, but “there must be a shorter path”! ;)</p><p>Now the more important question, is there a shorter path in that graph? The answer is no, there's no shorter path than the one with length <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span></span></span></span></span>. So what can we do about it? We'll add an edge to have a shorter path. Now we have gotten a lower bound of <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">2</span></span></span></span></span>, which means the only shorter path we can construct has <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span></span> edge and that is ‹<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mo separator="true">,</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">e, h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">h</span></span></span></span></span>› (no intermediary vertices). Let's do this!</p><p><img loading="lazy" src="/assets/images/bfs_graph_with_additional_edge_light-799673ba333298d16327abe67c90507e.png#gh-light-mode-only" width="252" height="539" class="img_ev3q">
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<img loading="lazy" src="/assets/images/bfs_graph_with_additional_edge_dark-c2da6c052b067785e877b4654a13f328.png#gh-dark-mode-only" width="252" height="539" class="img_ev3q"></p><p>Okay, so we have a graph that breaks the rule we have laid. However, we need to run BFS to obtain the new BFS tree, since we have changed the graph.</p><div class="theme-admonition theme-admonition-tip alert alert--success admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_S0QG"><p>Do we need to run BFS after <strong>every</strong> change?</p><p>I am leaving that as an exercise ;)</p></div></div><p><img loading="lazy" src="/assets/images/bfs_tree_with_additional_edge_light-93df97e1f9f1883467248532809374d0.png#gh-light-mode-only" width="371" height="347" class="img_ev3q">
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<img loading="lazy" src="/assets/images/bfs_tree_with_additional_edge_dark-127aa5b00988d4569669a92f5d841dbf.png#gh-dark-mode-only" width="371" height="347" class="img_ev3q"></p><p>Oops, we have gotten a new BFS tree, that has a height difference of 1.</p><div class="theme-admonition theme-admonition-tip alert alert--success admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_S0QG"><p>Try to think about a way this can be generalized for shortening of minimal length 3 to minimal length 2 ;)</p></div></div></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-tags-row row margin-bottom--sm"><div class="col"><b>Tags:</b><ul class="tags_jXut padding--none margin-left--sm"><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/graphs">graphs</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/ib002/tags/bfs">bfs</a></li></ul></div></div><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://gitlab.com/mfocko/blog/tree/main/ib002/10-graphs/2022-04-30-bfs-tree.md" target="_blank" rel="noreferrer noopener" class="theme-edit-this-page"><svg fill="currentColor" height="20" width="20" viewBox="0 0 40 40" class="iconEdit_Z9Sw" aria-hidden="true"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 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