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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/algorithms/graphs/bfs-tree/"><meta data-rh="true" property="og:locale" content="en"><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-algorithms-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-algorithms-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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<p>As we have talked on the seminar, if we construct from some vertex <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> 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="data:image/svg+xml;base64,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
<img loading="lazy" src="data:image/svg+xml;base64,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
<p>We run BFS from the vertex <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> and obtain the following BFS tree:</p>
<p><img loading="lazy" src="data:image/svg+xml;base64,PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiIHN0YW5kYWxvbmU9Im5vIj8+CjwhRE9DVFlQRSBzdmcgUFVCTElDICItLy9XM0MvL0RURCBTVkcgMS4xLy9FTiIKICJodHRwOi8vd3d3LnczLm9yZy9HcmFwaGljcy9TVkcvMS4xL0RURC9zdmcxMS5kdGQiPgo8IS0tIEdlbmVyYXRlZCBieSBncmFwaHZpeiB2ZXJzaW9uIDIuNDMuMCAoMCkKIC0tPgo8IS0tIFRpdGxlOiAlMyBQYWdlczogMSAtLT4KPHN2ZyB3aWR0aD0iMjA2cHQiIGhlaWdodD0iMjYwcHQiCiB2aWV3Qm94PSIwLjAwIDAuMDAgMjA2LjAwIDI2MC4wMCIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIiB4bWxuczp4bGluaz0iaHR0cDovL3d3dy53My5vcmcvMTk5OS94bGluayI+CjxnIGlkPSJncmFwaDAiIGNsYXNzPSJncmFwaCIgdHJhbnNmb3JtPSJzY2FsZSgxIDEpIHJvdGF0ZSgwKSB0cmFuc2xhdGUoNCAyNTYpIj4KPHRpdGxlPiUzPC90aXRsZT4KPHBvbHlnb24gZmlsbD0id2hpdGUiIHN0cm9rZT0idHJhbnNwYXJlbnQiIHBvaW50cz0iLTQsNCAtNCwtMjU2IDIwMiwtMjU2IDIwMiw0IC00LDQiLz4KPCEtLSBhIC0tPgo8ZyBpZD0ibm9kZTEiIGNsYXNzPSJub2RlIj4KPHRpdGxlPmE8L3RpdGxlPgo8ZWxsaXBzZSBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBjeD0iMTM1IiBjeT0iLTIzNCIgcng9IjI3IiByeT0iMTgiLz4KPHRleHQgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iMTM1IiB5PSItMjMwLjMiIGZvbnQtZmFtaWx5PSInQ2FzY2FkaWEgQ29kZSBQTCcsICdKZXRCcmFpbnMgTW9ubycsICdJb3NldmthJywgJ0ZpcmEgQ29kZScsICdIYWNrJywgbW9ub3NwYWNlIiBmb250LXNpemU9IjE0LjAwIj5hPC90ZXh0Pgo8L2c+CjwhLS0gYyAtLT4KPGcgaWQ9Im5vZGUyIiBjbGFzcz0ibm9kZSI+Cjx0aXRsZT5jPC90aXRsZT4KPGVsbGlwc2UgZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIgY3g9Ijk5IiBjeT0iLTE2MiIgcng9IjI3IiByeT0iMTgiLz4KPHRleHQgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iOTkiIHk9Ii0xNTguMyIgZm9udC1mYW1pbHk9IidDYXNjYWRpYSBDb2RlIFBMJywgJ0pldEJyYWlucyBNb25vJywgJ0lvc2V2a2EnLCAnRmlyYSBDb2RlJywgJ0hhY2snLCBtb25vc3BhY2UiIGZvbnQtc2l6ZT0iMTQuMDAiPmM8L3RleHQ+CjwvZz4KPCEtLSBhJiM0NTsmZ3Q7YyAtLT4KPGcgaWQ9ImVkZ2UxIiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5hJiM0NTsmZ3Q7YzwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik0xMjYuNjUsLTIxNi43NkMxMjIuMjksLTIwOC4yOCAxMTYuODUsLTE5Ny43MSAxMTEuOTYsLTE4OC4yIi8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjExNC45OSwtMTg2LjQ0IDEwNy4zLC0xNzkuMTUgMTA4Ljc3LC0xODkuNjQgMTE0Ljk5LC0xODYuNDQiLz4KPC9nPgo8IS0tIGUgLS0+CjxnIGlkPSJub2RlMyIgY2xhc3M9Im5vZGUiPgo8dGl0bGU+ZTwvdGl0bGU+CjxlbGxpcHNlIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGN4PSIxNzEiIGN5PSItMTYyIiByeD0iMjciIHJ5PSIxOCIvPgo8dGV4dCB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSIxNzEiIHk9Ii0xNTguMyIgZm9udC1mYW1pbHk9IidDYXNjYWRpYSBDb2RlIFBMJywgJ0pldEJyYWlucyBNb25vJywgJ0lvc2V2a2EnLCAnRmlyYSBDb2RlJywgJ0hhY2snLCBtb25vc3BhY2UiIGZvbnQtc2l6ZT0iMTQuMDAiPmU8L3RleHQ+CjwvZz4KPCEtLSBhJiM0NTsmZ3Q7ZSAtLT4KPGcgaWQ9ImVkZ2UyIiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5hJiM0NTsmZ3Q7ZTwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik0xNDMuMzUsLTIxNi43NkMxNDcuNzEsLTIwOC4yOCAxNTMuMTUsLTE5Ny43MSAxNTguMDQsLTE4OC4yIi8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjE2MS4yMywtMTg5LjY0IDE2Mi43LC0xNzkuMTUgMTU1LjAxLC0xODYuNDQgMTYxLjIzLC0xODkuNjQiLz4KPC9nPgo8IS0tIGIgLS0+CjxnIGlkPSJub2RlNCIgY2xhc3M9Im5vZGUiPgo8dGl0bGU+YjwvdGl0bGU+CjxlbGxpcHNlIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGN4PSIyNyIgY3k9Ii05MCIgcng9IjI3IiByeT0iMTgiLz4KPHRleHQgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iMjciIHk9Ii04Ni4zIiBmb250LWZhbWlseT0iJ0Nhc2NhZGlhIENvZGUgUEwnLCAnSmV0QnJhaW5zIE1vbm8nLCAnSW9zZXZrYScsICdGaXJhIENvZGUnLCAnSGFjaycsIG1vbm9zcGFjZSIgZm9udC1zaXplPSIxNC4wMCI+YjwvdGV4dD4KPC9nPgo8IS0tIGMmIzQ1OyZndDtiIC0tPgo8ZyBpZD0iZWRnZTMiIGNsYXNzPSJlZGdlIj4KPHRpdGxlPmMmIzQ1OyZndDtiPC90aXRsZT4KPHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIgZD0iTTg0LjQzLC0xNDYuODNDNzQuMjUsLTEzNi45NCA2MC40OCwtMTIzLjU1IDQ4Ljk3LC0xMTIuMzYiLz4KPHBvbHlnb24gZmlsbD0iYmxhY2siIHN0cm9rZT0iYmxhY2siIHBvaW50cz0iNTEuNDEsLTEwOS44NSA0MS44LC0xMDUuMzggNDYuNTMsLTExNC44NyA1MS40MSwtMTA5Ljg1Ii8+CjwvZz4KPCEtLSBpIC0tPgo8ZyBpZD0ibm9kZTUiIGNsYXNzPSJub2RlIj4KPHRpdGxlPmk8L3RpdGxlPgo8ZWxsaXBzZSBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBjeD0iOTkiIGN5PSItOTAiIHJ4PSIyNyIgcnk9IjE4Ii8+Cjx0ZXh0IHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9Ijk5IiB5PSItODYuMyIgZm9udC1mYW1pbHk9IidDYXNjYWRpYSBDb2RlIFBMJywgJ0pldEJyYWlucyBNb25vJywgJ0lvc2V2a2EnLCAnRmlyYSBDb2RlJywgJ0hhY2snLCBtb25vc3BhY2UiIGZvbnQtc2l6ZT0iMTQuMDAiPmk8L3RleHQ+CjwvZz4KPCEtLSBjJiM0NTsmZ3Q7aSAtLT4KPGcgaWQ9ImVkZ2U0IiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5jJiM0NTsmZ3Q7aTwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik05O
<img loading="lazy" src="data:image/svg+xml;base64,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
<p>Let&#x27;s consider pair of vertices <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> and <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>. For them we can safely lay, from the BFS tree, following properties:</p>
<ul>
<li>lower bound: <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></li>
<li>upper bound: <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></li>
</ul>
<p>By having a look at the graph we started from, we can see that we have a path ‹<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>› that has a length 2. Apart from that we can also notice there is another path from <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> to <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> and that is ‹<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>›. And that path has a length of <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>. Doesn&#x27;t this break our statements at the beginning? (<em>I&#x27;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&#x27;s keep the same graph, but break the lower bound, i.e. I have gotten a lower bound <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>, 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&#x27;s no shorter path than the one with length <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>. So what can we do about it? We&#x27;ll add an edge to have a shorter path. Now we have gotten a lower bound of <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>, which means the only shorter path we can construct has <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> edge and that is ‹<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>› (no intermediary vertices). Let&#x27;s do this!</p>
<p><img loading="lazy" src="data:image/svg+xml;base64,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
<img loading="lazy" src="data:image/svg+xml;base64,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
<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 admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><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_BuS1"><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="data:image/svg+xml;base64,PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiIHN0YW5kYWxvbmU9Im5vIj8+CjwhRE9DVFlQRSBzdmcgUFVCTElDICItLy9XM0MvL0RURCBTVkcgMS4xLy9FTiIKICJodHRwOi8vd3d3LnczLm9yZy9HcmFwaGljcy9TVkcvMS4xL0RURC9zdmcxMS5kdGQiPgo8IS0tIEdlbmVyYXRlZCBieSBncmFwaHZpeiB2ZXJzaW9uIDIuNDMuMCAoMCkKIC0tPgo8IS0tIFRpdGxlOiAlMyBQYWdlczogMSAtLT4KPHN2ZyB3aWR0aD0iMjc4cHQiIGhlaWdodD0iMjYwcHQiCiB2aWV3Qm94PSIwLjAwIDAuMDAgMjc4LjAwIDI2MC4wMCIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIiB4bWxuczp4bGluaz0iaHR0cDovL3d3dy53My5vcmcvMTk5OS94bGluayI+CjxnIGlkPSJncmFwaDAiIGNsYXNzPSJncmFwaCIgdHJhbnNmb3JtPSJzY2FsZSgxIDEpIHJvdGF0ZSgwKSB0cmFuc2xhdGUoNCAyNTYpIj4KPHRpdGxlPiUzPC90aXRsZT4KPHBvbHlnb24gZmlsbD0id2hpdGUiIHN0cm9rZT0idHJhbnNwYXJlbnQiIHBvaW50cz0iLTQsNCAtNCwtMjU2IDI3NCwtMjU2IDI3NCw0IC00LDQiLz4KPCEtLSBhIC0tPgo8ZyBpZD0ibm9kZTEiIGNsYXNzPSJub2RlIj4KPHRpdGxlPmE8L3RpdGxlPgo8ZWxsaXBzZSBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBjeD0iMTM1IiBjeT0iLTIzNCIgcng9IjI3IiByeT0iMTgiLz4KPHRleHQgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iMTM1IiB5PSItMjMwLjMiIGZvbnQtZmFtaWx5PSInQ2FzY2FkaWEgQ29kZSBQTCcsICdKZXRCcmFpbnMgTW9ubycsICdJb3NldmthJywgJ0ZpcmEgQ29kZScsICdIYWNrJywgbW9ub3NwYWNlIiBmb250LXNpemU9IjE0LjAwIj5hPC90ZXh0Pgo8L2c+CjwhLS0gYyAtLT4KPGcgaWQ9Im5vZGUyIiBjbGFzcz0ibm9kZSI+Cjx0aXRsZT5jPC90aXRsZT4KPGVsbGlwc2UgZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIgY3g9Ijk5IiBjeT0iLTE2MiIgcng9IjI3IiByeT0iMTgiLz4KPHRleHQgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iOTkiIHk9Ii0xNTguMyIgZm9udC1mYW1pbHk9IidDYXNjYWRpYSBDb2RlIFBMJywgJ0pldEJyYWlucyBNb25vJywgJ0lvc2V2a2EnLCAnRmlyYSBDb2RlJywgJ0hhY2snLCBtb25vc3BhY2UiIGZvbnQtc2l6ZT0iMTQuMDAiPmM8L3RleHQ+CjwvZz4KPCEtLSBhJiM0NTsmZ3Q7YyAtLT4KPGcgaWQ9ImVkZ2UxIiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5hJiM0NTsmZ3Q7YzwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik0xMjYuNjUsLTIxNi43NkMxMjIuMjksLTIwOC4yOCAxMTYuODUsLTE5Ny43MSAxMTEuOTYsLTE4OC4yIi8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjExNC45OSwtMTg2LjQ0IDEwNy4zLC0xNzkuMTUgMTA4Ljc3LC0xODkuNjQgMTE0Ljk5LC0xODYuNDQiLz4KPC9nPgo8IS0tIGUgLS0+CjxnIGlkPSJub2RlMyIgY2xhc3M9Im5vZGUiPgo8dGl0bGU+ZTwvdGl0bGU+CjxlbGxpcHNlIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGN4PSIxNzEiIGN5PSItMTYyIiByeD0iMjciIHJ5PSIxOCIvPgo8dGV4dCB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSIxNzEiIHk9Ii0xNTguMyIgZm9udC1mYW1pbHk9IidDYXNjYWRpYSBDb2RlIFBMJywgJ0pldEJyYWlucyBNb25vJywgJ0lvc2V2a2EnLCAnRmlyYSBDb2RlJywgJ0hhY2snLCBtb25vc3BhY2UiIGZvbnQtc2l6ZT0iMTQuMDAiPmU8L3RleHQ+CjwvZz4KPCEtLSBhJiM0NTsmZ3Q7ZSAtLT4KPGcgaWQ9ImVkZ2UyIiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5hJiM0NTsmZ3Q7ZTwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik0xNDMuMzUsLTIxNi43NkMxNDcuNzEsLTIwOC4yOCAxNTMuMTUsLTE5Ny43MSAxNTguMDQsLTE4OC4yIi8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjE2MS4yMywtMTg5LjY0IDE2Mi43LC0xNzkuMTUgMTU1LjAxLC0xODYuNDQgMTYxLjIzLC0xODkuNjQiLz4KPC9nPgo8IS0tIGIgLS0+CjxnIGlkPSJub2RlNCIgY2xhc3M9Im5vZGUiPgo8dGl0bGU+YjwvdGl0bGU+CjxlbGxpcHNlIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGN4PSIyNyIgY3k9Ii05MCIgcng9IjI3IiByeT0iMTgiLz4KPHRleHQgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iMjciIHk9Ii04Ni4zIiBmb250LWZhbWlseT0iJ0Nhc2NhZGlhIENvZGUgUEwnLCAnSmV0QnJhaW5zIE1vbm8nLCAnSW9zZXZrYScsICdGaXJhIENvZGUnLCAnSGFjaycsIG1vbm9zcGFjZSIgZm9udC1zaXplPSIxNC4wMCI+YjwvdGV4dD4KPC9nPgo8IS0tIGMmIzQ1OyZndDtiIC0tPgo8ZyBpZD0iZWRnZTMiIGNsYXNzPSJlZGdlIj4KPHRpdGxlPmMmIzQ1OyZndDtiPC90aXRsZT4KPHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIgZD0iTTg0LjQzLC0xNDYuODNDNzQuMjUsLTEzNi45NCA2MC40OCwtMTIzLjU1IDQ4Ljk3LC0xMTIuMzYiLz4KPHBvbHlnb24gZmlsbD0iYmxhY2siIHN0cm9rZT0iYmxhY2siIHBvaW50cz0iNTEuNDEsLTEwOS44NSA0MS44LC0xMDUuMzggNDYuNTMsLTExNC44NyA1MS40MSwtMTA5Ljg1Ii8+CjwvZz4KPCEtLSBpIC0tPgo8ZyBpZD0ibm9kZTUiIGNsYXNzPSJub2RlIj4KPHRpdGxlPmk8L3RpdGxlPgo8ZWxsaXBzZSBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBjeD0iOTkiIGN5PSItOTAiIHJ4PSIyNyIgcnk9IjE4Ii8+Cjx0ZXh0IHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9Ijk5IiB5PSItODYuMyIgZm9udC1mYW1pbHk9IidDYXNjYWRpYSBDb2RlIFBMJywgJ0pldEJyYWlucyBNb25vJywgJ0lvc2V2a2EnLCAnRmlyYSBDb2RlJywgJ0hhY2snLCBtb25vc3BhY2UiIGZvbnQtc2l6ZT0iMTQuMDAiPmk8L3RleHQ+CjwvZz4KPCEtLSBjJiM0NTsmZ3Q7aSAtLT4KPGcgaWQ9ImVkZ2U0IiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5jJiM0NTsmZ3Q7aTwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik05O
<img loading="lazy" src="data:image/svg+xml;base64,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
<p>Oops, we have gotten a new BFS tree, that has a height difference of 1.</p>
<div class="theme-admonition theme-admonition-tip admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><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_BuS1"><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="/algorithms/tags/graphs/">graphs</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/bfs/">bfs</a></li></ul></div></div><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://github.com/mfocko/blog/tree/main/algorithms/10-graphs/2022-04-30-bfs-tree.md" target="_blank" rel="noopener noreferrer" 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 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="col lastUpdated_vwxv"><span class="theme-last-updated">Last updated<!-- --> on <b><time datetime="2022-04-30T00:00:00.000Z">Apr 30, 2022</time></b></span></div></div></footer></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Docs pages"><a class="pagination-nav__link pagination-nav__link--prev" href="/algorithms/graphs/iterative-and-iterators/"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Iterative algorithms via iterators</div></a></nav></div></div><div class="col col--3"><div class="tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#introduction" class="table-of-contents__link toc-highlight">Introduction</a></li><li><a href="#lower-bound" class="table-of-contents__link toc-highlight">Lower bound</a></li><li><a href="#proof-by-contradiction" class="table-of-contents__link toc-highlight">Proof by contradiction</a></li></ul></div></div></div></div></main></div></div></div><footer class="footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="col footer__col"><div class="footer__title">Git</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/mfocko" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitHub<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://gitlab.com/mfocko" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitLab<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://git.mfocko.xyz/mfocko" target="_blank" rel="noopener noreferrer" class="footer__link-item">Gitea (self-hosted)<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLin
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