add cycle finding algo

content-org/all-posts.en.org

@@ -3944,6 +3944,40 @@ I use termux[fn:termux] to do these:
 #+CAPTION: enable tcpip on the phone
 [[file:../static/images/posts/connect-to-your-android-wirelessly/enable-tcpip-on-android.png]]
 
+* DONE Cycle Finding Algorithms :cycle_finding:@Data_Structure_and_Algorithm:
+CLOSED: [2023-12-19 Tue 00:27]
+:PROPERTIES:
+:EXPORT_HUGO_BUNDLE: cycle-finding-algorithms
+:EXPORT_FILE_NAME: index
+:EXPORT_DATE: [2023-12-17 Sun 23:24]
+:EXPORT_HUGO_CUSTOM_FRONT_MATTER: :image "/images/icons/TortoiseHare.jpg"
+:EXPORT_HUGO_CUSTOM_FRONT_MATTER+: :libraries '(mathjax)
+:EXPORT_HUGO_CUSTOM_FRONT_MATTER+: :description "this is a description"
+:END:
+:LOGBOOK:
+- State "DONE"       from "TODO"       [2023-12-19 Tue 00:27]
+:END:
+
+** Floyd's tortoise and hare
+
+
+#+CAPTION: tortoise and hare
+[[../static/images/posts/cycle-finding/tortoise-and-hare.svg]]
+
+hare and tortoise both starts from point *Start*, at speed of $2$ and $1$ respectively. They'll meet at some point in the cycle.
+
+ when they meet, hare traveled $S_2 - S_{1}$ more points than tortoise. We have: ($\lambda$ is the cycle length)
+
+$$
+S_{2} - S_{1} = 0  = S_{1} = x + m \pmod{\lambda}
+$$
+
+after they meet, one starts over from Start point, the other starts from Meet point, at speed of $1$, they'll meet again at *Cycle Start* point. because:
+
+   $$
+   x = x + m + x \pmod{\lambda}
+   $$
+
 * Footnotes
 
 [fn:org-download] [[https://github.com/abo-abo/org-download][org-download]] facilitates moving images from point A to B.

content/en/posts/cycle-finding-algorithms/index.md

@@ -0,0 +1,28 @@
++++
+title = "Cycle Finding Algorithms"
+date = 2023-12-19T00:27:00+08:00
+tags = ["cycle-finding"]
+categories = ["Data-Structure-and-Algorithm"]
+draft = false
+image = "/images/icons/TortoiseHare.jpg"
+libraries = ["mathjax"]
+description = "this is a description"
++++
+
+## Floyd's tortoise and hare {#floyd-s-tortoise-and-hare}
+
+{{< figure src="/images/posts/cycle-finding/tortoise-and-hare.svg" caption="<span class=\"figure-number\">Figure 1: </span>tortoise and hare" >}}
+
+hare and tortoise both starts from point **Start**, at speed of \\(2\\) and \\(1\\) respectively. They'll meet at some point in the cycle.
+
+when they meet, hare traveled \\(S\_2 - S\_{1}\\) more points than tortoise. We have: (\\(\lambda\\) is the cycle length)
+
+\\[
+S\_{2} - S\_{1} = 0  = S\_{1} = x + m \pmod{\lambda}
+\\]
+
+after they meet, one starts over from Start point, the other starts from Meet point, at speed of \\(1\\), they'll meet again at **Cycle Start** point. because:
+
+\\[
+   x = x + m + x \pmod{\lambda}
+   \\]