From e8ffef5c2f2eba2ff30232cd7554835187e2a52c Mon Sep 17 00:00:00 2001 From: Andrew Date: Thu, 13 Feb 2025 20:55:42 +0000 Subject: [PATCH] [CS4423]: Add Wk05-2 lecture notes --- year4/semester2/CS4423/notes/CS4423.pdf | Bin 1733958 -> 1737748 bytes year4/semester2/CS4423/notes/CS4423.tex | 21 +++++++++++++++++++++ 2 files changed, 21 insertions(+) diff --git a/year4/semester2/CS4423/notes/CS4423.pdf b/year4/semester2/CS4423/notes/CS4423.pdf index 2d1c6abca86e7286048da476faf7bd307441e6be..01b8866da76abaa1e12ffa08a610a1bd7f3d45cd 100644 GIT binary patch delta 44724 zcmY(KWl&s8w6=l4-5K27-6gnN2<`-TcLsO2!7aEu0fM^*cXvY2U_n35Ik)at_s6R0 zy?aSlPwm;ynzv{8mvw5SO??CcfP;-4JmUbQh2mlZ2XWxzf=WKOD+Ack&mS0H0LZSZ zW(ro+nDCgyrh<|%!Qt2%FCzPt5!pX6U^1ES z*zKAKZf^SS&7KR8$&T&Tq&b9G1na*%Vmg#%J927GxOF69YWFpoAWzH(AdMiG;wmD_ zqE7H^8~DPR3!pQDiWb+g18#vqwh0i*!!Cf=EAEt~axwIsvcO4{ww z6dzu{uzLVMilHXJr`4-&Hj>`Kb)NOCUfaSK=~MTFxOt1CFFEa#!LXZO9#@lRae9?4 zB0`@Y(y?Nzw;^J&0RA>~?nk!5C0ucLs<`r?bH{aL<1qPD!VpO#^5e4->?qO8D&`RJ 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b/year4/semester2/CS4423/notes/CS4423.tex index 30ee76f5..5f674cdb 100644 --- a/year4/semester2/CS4423/notes/CS4423.tex +++ b/year4/semester2/CS4423/notes/CS4423.tex @@ -729,6 +729,27 @@ Given a rooted tree $T$ with root $x$, to visit all nodes in the tree: Many questions on networks regarding distance \& connectivity can be answered by a versatile strategy involving a subgraph which is a tree and then searching that; such a tree is called \textbf{spanning tree} of the underlying graph. +\subsubsection{Graph Diameter} +A natural problem arising in many practical applications is the following: given a pair of nodes $x,y$, find one or all the paths from $x$ to $y$ with the fewest number of edges possible. +This is a somewhat complex measure on a network (compared to, say, statistics on node degrees) and we will therefore need a more complex procedure, that is, an algorithm, in order to solve such problems systematically. +\\\\ +\textbf{Definition:} let $G=(X,E)$ be a simple graph and let $x,y \in X$. +Let $P(x,y)$ be the set of all paths from $x$ to $y$. +Then: +\begin{itemize} + \item The \textbf{distance} $d(x,y)$ from $x$ to $y$ is + \begin{align*} + d(x,y) = \text{min}\{ l(p) : p \in P(x,y) \}, + \end{align*} + the shortest possible length of a path from $x$ to $y$, and a \textbf{shortest path} from $x$ to $y$ is a path $p \in P(x,y)$ of length $l(p) = d(x,y)$. + + \item The \textbf{diameter} $\text{diam}(G)$ of the network $G$ is the length of the longest shortest path between any two nodes: + \begin{align*} + \text{diam}(G) = \text{max}\{ l(p) : p \in P(x,y) \} + \end{align*} +\end{itemize} + + \end{document}