Skip to content

Commit 2535885

Browse files
github-actions[bot]github-actions[bot]
committed
1 parent b2ea3c8 commit 2535885

Some content is hidden

Large Commits have some content hidden by default. Use the searchbox below for content that may be hidden.

50 files changed

+636
-378
lines changed

‎1369/algebra/bit-manipulation.html

Lines changed: 0 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -6660,12 +6660,6 @@
66606660

66616661

66626662

6663-
6664-
6665-
6666-
6667-
6668-
66696663

66706664

66716665

‎1369/algebra/euclid-algorithm.html

Lines changed: 6 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -6561,6 +6561,12 @@
65616561

65626562

65636563

6564+
6565+
6566+
6567+
6568+
6569+
65646570

65656571

65666572

‎1369/algebra/factorization.html

Lines changed: 15 additions & 3 deletions
Original file line numberDiff line numberDiff line change
@@ -6612,6 +6612,12 @@
66126612

66136613

66146614

6615+
6616+
6617+
6618+
6619+
6620+
66156621

66166622

66176623

@@ -6704,6 +6710,10 @@
67046710

67056711

67066712

6713+
6714+
6715+
6716+
67076717

67086718

67096719

@@ -6716,7 +6726,7 @@
67166726
<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
67176727

67186728
Last update:
6719-
<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="April 16, 2024 21:08:03">April 16, 2024</span>&emsp;
6729+
<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="April 15, 2025 02:32:08">April 15, 2025</span>&emsp;
67206730

67216731
<!-- Tags -->
67226732

@@ -6919,7 +6929,9 @@ <h2 id="pollards-rho-algorithm">Pollard's rho algorithm<a class="headerlink" hre
69196929
If <span class="arithmatex">$p$</span> is smaller than <span class="arithmatex">$\sqrt{n}$</span>, the repetition will likely start in <span class="arithmatex">$O(\sqrt[4]{n})$</span>.</p>
69206930
<p>Here is a visualization of such a sequence <span class="arithmatex">$\{x_i \bmod p\}$</span> with <span class="arithmatex">$n = 2206637$</span>, <span class="arithmatex">$p = 317$</span>, <span class="arithmatex">$x_0 = 2$</span> and <span class="arithmatex">$f(x) = x^2 + 1$</span>.
69216931
From the form of the sequence you can see very clearly why the algorithm is called Pollard's <span class="arithmatex">$\rho$</span> algorithm.</p>
6922-
<center>![Pollard's rho visualization](pollard_rho.png)</center>
6932+
<div style="text-align: center;">
6933+
<img src="pollard_rho.png" alt="Pollard's rho visualization">
6934+
</div>
69236935

69246936
<p>Yet, there is still an open question.
69256937
How can we exploit the properties of the sequence <span class="arithmatex">$\{x_i \bmod p\}$</span> to our advantage without even knowing the number <span class="arithmatex">$p$</span> itself?</p>
@@ -7072,7 +7084,7 @@ <h2 id="practice-problems">Practice Problems<a class="headerlink" href="#practic
70727084

70737085
<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
70747086
<span class="contributors-text">Contributors:</span>
7075-
<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (85.05%)</li><li><a href="https://github.com/chloeimb" title="chloeimb" data-bi-name="contributorprofile" target="_blank">chloeimb</a> (9.81%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (3.51%)</li><li><a href="https://github.com/0xGodspeed" title="0xGodspeed" data-bi-name="contributorprofile" target="_blank">0xGodspeed</a> (0.7%)</li><li><a href="https://github.com/pokorj54" title="pokorj54" data-bi-name="contributorprofile" target="_blank">pokorj54</a> (0.47%)</li><li><a href="https://github.com/jxu" title="jxu" data-bi-name="contributorprofile" target="_blank">jxu</a> (0.23%)</li><li><a href="https://github.com/kyomukyomupurin" title="kyomukyomupurin" data-bi-name="contributorprofile" target="_blank">kyomukyomupurin</a> (0.23%)</li></ul>
7087+
<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (84.42%)</li><li><a href="https://github.com/chloeimb" title="chloeimb" data-bi-name="contributorprofile" target="_blank">chloeimb</a> (9.77%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (3.49%)</li><li><a href="https://github.com/mhayter" title="mhayter" data-bi-name="contributorprofile" target="_blank">mhayter</a> (0.7%)</li><li><a href="https://github.com/0xGodspeed" title="0xGodspeed" data-bi-name="contributorprofile" target="_blank">0xGodspeed</a> (0.7%)</li><li><a href="https://github.com/pokorj54" title="pokorj54" data-bi-name="contributorprofile" target="_blank">pokorj54</a> (0.47%)</li><li><a href="https://github.com/jxu" title="jxu" data-bi-name="contributorprofile" target="_blank">jxu</a> (0.23%)</li><li><a href="https://github.com/kyomukyomupurin" title="kyomukyomupurin" data-bi-name="contributorprofile" target="_blank">kyomukyomupurin</a> (0.23%)</li></ul>
70767088
</ul>
70777089

70787090
</article>

‎1369/algebra/linear-diophantine-equation.html

Lines changed: 6 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -6570,6 +6570,12 @@
65706570

65716571

65726572

6573+
6574+
6575+
6576+
6577+
6578+
65736579

65746580

65756581

‎1369/algebra/phi-function.html

Lines changed: 0 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -6606,12 +6606,6 @@
66066606

66076607

66086608

6609-
6610-
6611-
6612-
6613-
6614-
66156609

66166610

66176611

‎1369/algebra/sieve-of-eratosthenes.html

Lines changed: 15 additions & 3 deletions
Original file line numberDiff line numberDiff line change
@@ -6597,6 +6597,12 @@
65976597

65986598

65996599

6600+
6601+
6602+
6603+
6604+
6605+
66006606

66016607

66026608

@@ -6767,6 +6773,10 @@
67676773

67686774

67696775

6776+
6777+
6778+
6779+
67706780

67716781

67726782

@@ -6779,7 +6789,7 @@
67796789
<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
67806790

67816791
Last update:
6782-
<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="October 11, 2024 10:03:16">October 11, 2024</span>&emsp;
6792+
<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="April 15, 2025 02:32:08">April 15, 2025</span>&emsp;
67836793

67846794
<!-- Tags -->
67856795

@@ -6816,7 +6826,9 @@ <h1 id="sieve-of-eratosthenes">Sieve of Eratosthenes<a class="headerlink" href="
68166826
The next unmarked number is 5, which is the next prime number, and we mark all proper multiples of it.
68176827
And we continue this procedure until we have processed all numbers in the row.</p>
68186828
<p>In the following image you can see a visualization of the algorithm for computing all prime numbers in the range <span class="arithmatex">$[1; 16]$</span>. It can be seen, that quite often we mark numbers as composite multiple times.</p>
6819-
<center>![Sieve of Eratosthenes](sieve_eratosthenes.png)</center>
6829+
<div style="text-align: center;">
6830+
<img src="sieve_eratosthenes.png" alt="Sieve of Eratosthenes">
6831+
</div>
68206832

68216833
<p>The idea behind is this:
68226834
A number is prime, if none of the smaller prime numbers divides it.
@@ -7016,7 +7028,7 @@ <h2 id="practice-problems">Practice Problems<a class="headerlink" href="#practic
70167028

70177029
<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
70187030
<span class="contributors-text">Contributors:</span>
7019-
<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (42.7%)</li><li><a href="https://github.com/ellen-interpret" title="ellen-interpret" data-bi-name="contributorprofile" target="_blank">ellen-interpret</a> (23.36%)</li><li><a href="https://github.com/hieplpvip" title="hieplpvip" data-bi-name="contributorprofile" target="_blank">hieplpvip</a> (16.79%)</li><li><a href="https://github.com/Morass" title="Morass" data-bi-name="contributorprofile" target="_blank">Morass</a> (5.47%)</li><li><a href="https://github.com/tcNickolas" title="tcNickolas" data-bi-name="contributorprofile" target="_blank">tcNickolas</a> (3.65%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (2.19%)</li><li><a href="https://github.com/boxlesscat" title="boxlesscat" data-bi-name="contributorprofile" target="_blank">boxlesscat</a> (2.19%)</li><li><a href="https://github.com/roll-no-1" title="roll-no-1" data-bi-name="contributorprofile" target="_blank">roll-no-1</a> (1.09%)</li><li><a href="https://github.com/TrietMinh799" title="TrietMinh799" data-bi-name="contributorprofile" target="_blank">TrietMinh799</a> (0.73%)</li><li><a href="https://github.com/jxu" title="jxu" data-bi-name="contributorprofile" target="_blank">jxu</a> (0.73%)</li><li><a href="https://github.com/zdr256" title="zdr256" data-bi-name="contributorprofile" target="_blank">zdr256</a> (0.36%)</li><li><a href="https://github.com/infiniteasp8" title="infiniteasp8" data-bi-name="contributorprofile" target="_blank">infiniteasp8</a> (0.36%)</li><li><a href="https://github.com/joaquingx" title="joaquingx" data-bi-name="contributorprofile" target="_blank">joaquingx</a> (0.36%)</li></ul>
7031+
<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (42.03%)</li><li><a href="https://github.com/ellen-interpret" title="ellen-interpret" data-bi-name="contributorprofile" target="_blank">ellen-interpret</a> (23.19%)</li><li><a href="https://github.com/hieplpvip" title="hieplpvip" data-bi-name="contributorprofile" target="_blank">hieplpvip</a> (16.67%)</li><li><a href="https://github.com/Morass" title="Morass" data-bi-name="contributorprofile" target="_blank">Morass</a> (5.43%)</li><li><a href="https://github.com/tcNickolas" title="tcNickolas" data-bi-name="contributorprofile" target="_blank">tcNickolas</a> (3.62%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (2.17%)</li><li><a href="https://github.com/boxlesscat" title="boxlesscat" data-bi-name="contributorprofile" target="_blank">boxlesscat</a> (2.17%)</li><li><a href="https://github.com/mhayter" title="mhayter" data-bi-name="contributorprofile" target="_blank">mhayter</a> (1.09%)</li><li><a href="https://github.com/roll-no-1" title="roll-no-1" data-bi-name="contributorprofile" target="_blank">roll-no-1</a> (1.09%)</li><li><a href="https://github.com/TrietMinh799" title="TrietMinh799" data-bi-name="contributorprofile" target="_blank">TrietMinh799</a> (0.72%)</li><li><a href="https://github.com/jxu" title="jxu" data-bi-name="contributorprofile" target="_blank">jxu</a> (0.72%)</li><li><a href="https://github.com/zdr256" title="zdr256" data-bi-name="contributorprofile" target="_blank">zdr256</a> (0.36%)</li><li><a href="https://github.com/infiniteasp8" title="infiniteasp8" data-bi-name="contributorprofile" target="_blank">infiniteasp8</a> (0.36%)</li><li><a href="https://github.com/joaquingx" title="joaquingx" data-bi-name="contributorprofile" target="_blank">joaquingx</a> (0.36%)</li></ul>
70207032
</ul>
70217033

70227034
</article>

‎1369/combinatorics/burnside.html

Lines changed: 0 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -6624,12 +6624,6 @@
66246624

66256625

66266626

6627-
6628-
6629-
6630-
6631-
6632-
66336627

66346628

66356629

‎1369/data_structures/fenwick.html

Lines changed: 15 additions & 3 deletions
Original file line numberDiff line numberDiff line change
@@ -6654,6 +6654,12 @@
66546654

66556655

66566656

6657+
6658+
6659+
6660+
6661+
6662+
66576663

66586664

66596665

@@ -6898,6 +6904,10 @@
68986904

68996905

69006906

6907+
6908+
6909+
6910+
69016911

69026912

69036913

@@ -6910,7 +6920,7 @@
69106920
<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
69116921

69126922
Last update:
6913-
<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="July 31, 2024 14:14:13">July 31, 2024</span>&emsp;
6923+
<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="April 15, 2025 02:32:08">April 15, 2025</span>&emsp;
69146924

69156925
<!-- Tags -->
69166926

@@ -7029,7 +7039,9 @@ <h3 id="definition-of-gi">Definition of <span class="arithmatex">$g(i)$</span><a
70297039
<p>where <span class="arithmatex">$|$</span> is the bitwise OR operator.</p>
70307040
<p>The following image shows a possible interpretation of the Fenwick tree as tree.
70317041
The nodes of the tree show the ranges they cover.</p>
7032-
<center>![Binary Indexed Tree](binary_indexed_tree.png)</center>
7042+
<div style="text-align: center;">
7043+
<img src="binary_indexed_tree.png" alt="Binary Indexed Tree">
7044+
</div>
70337045

70347046
<h2 id="implementation">Implementation<a class="headerlink" href="#implementation" title="Permanent link">&para;</a></h2>
70357047
<h3 id="finding-sum-in-one-dimensional-array">Finding sum in one-dimensional array<a class="headerlink" href="#finding-sum-in-one-dimensional-array" title="Permanent link">&para;</a></h3>
@@ -7345,7 +7357,7 @@ <h2 id="other-sources">Other sources<a class="headerlink" href="#other-sources"
73457357

73467358
<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
73477359
<span class="contributors-text">Contributors:</span>
7348-
<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (37.33%)</li><li><a href="https://github.com/madhur4127" title="madhur4127" data-bi-name="contributorprofile" target="_blank">madhur4127</a> (22.75%)</li><li><a href="https://github.com/ngthanhtrung23" title="ngthanhtrung23" data-bi-name="contributorprofile" target="_blank">ngthanhtrung23</a> (20.96%)</li><li><a href="https://github.com/JJCUBER" title="JJCUBER" data-bi-name="contributorprofile" target="_blank">JJCUBER</a> (7.39%)</li><li><a href="https://github.com/Morass" title="Morass" data-bi-name="contributorprofile" target="_blank">Morass</a> (4.39%)</li><li><a href="https://github.com/tanmay-sinha" title="tanmay-sinha" data-bi-name="contributorprofile" target="_blank">tanmay-sinha</a> (1.6%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (1.2%)</li><li><a href="https://github.com/vatsalsharma376" title="vatsalsharma376" data-bi-name="contributorprofile" target="_blank">vatsalsharma376</a> (1.2%)</li><li><a href="https://github.com/CLown1331" title="CLown1331" data-bi-name="contributorprofile" target="_blank">CLown1331</a> (1.0%)</li><li><a href="https://github.com/tcNickolas" title="tcNickolas" data-bi-name="contributorprofile" target="_blank">tcNickolas</a> (0.6%)</li><li><a href="https://github.com/Zyad-Ayad" title="Zyad-Ayad" data-bi-name="contributorprofile" target="_blank">Zyad-Ayad</a> (0.4%)</li><li><a href="https://github.com/joaquingx" title="joaquingx" data-bi-name="contributorprofile" target="_blank">joaquingx</a> (0.4%)</li><li><a href="https://github.com/3centroids" title="3centroids" data-bi-name="contributorprofile" target="_blank">3centroids</a> (0.2%)</li><li><a href="https://github.com/harshit-jain52" title="harshit-jain52" data-bi-name="contributorprofile" target="_blank">harshit-jain52</a> (0.2%)</li><li><a href="https://github.com/invincible0" title="invincible0" data-bi-name="contributorprofile" target="_blank">invincible0</a> (0.2%)</li><li><a href="https://github.com/turfaa" title="turfaa" data-bi-name="contributorprofile" target="_blank">turfaa</a> (0.2%)</li></ul>
7360+
<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (36.98%)</li><li><a href="https://github.com/madhur4127" title="madhur4127" data-bi-name="contributorprofile" target="_blank">madhur4127</a> (22.66%)</li><li><a href="https://github.com/ngthanhtrung23" title="ngthanhtrung23" data-bi-name="contributorprofile" target="_blank">ngthanhtrung23</a> (20.87%)</li><li><a href="https://github.com/JJCUBER" title="JJCUBER" data-bi-name="contributorprofile" target="_blank">JJCUBER</a> (7.36%)</li><li><a href="https://github.com/Morass" title="Morass" data-bi-name="contributorprofile" target="_blank">Morass</a> (4.37%)</li><li><a href="https://github.com/tanmay-sinha" title="tanmay-sinha" data-bi-name="contributorprofile" target="_blank">tanmay-sinha</a> (1.59%)</li><li><a href="https://github.com/adamant-pwn" title="adamant-pwn" data-bi-name="contributorprofile" target="_blank">adamant-pwn</a> (1.19%)</li><li><a href="https://github.com/vatsalsharma376" title="vatsalsharma376" data-bi-name="contributorprofile" target="_blank">vatsalsharma376</a> (1.19%)</li><li><a href="https://github.com/CLown1331" title="CLown1331" data-bi-name="contributorprofile" target="_blank">CLown1331</a> (0.99%)</li><li><a href="https://github.com/mhayter" title="mhayter" data-bi-name="contributorprofile" target="_blank">mhayter</a> (0.6%)</li><li><a href="https://github.com/tcNickolas" title="tcNickolas" data-bi-name="contributorprofile" target="_blank">tcNickolas</a> (0.6%)</li><li><a href="https://github.com/Zyad-Ayad" title="Zyad-Ayad" data-bi-name="contributorprofile" target="_blank">Zyad-Ayad</a> (0.4%)</li><li><a href="https://github.com/joaquingx" title="joaquingx" data-bi-name="contributorprofile" target="_blank">joaquingx</a> (0.4%)</li><li><a href="https://github.com/3centroids" title="3centroids" data-bi-name="contributorprofile" target="_blank">3centroids</a> (0.2%)</li><li><a href="https://github.com/harshit-jain52" title="harshit-jain52" data-bi-name="contributorprofile" target="_blank">harshit-jain52</a> (0.2%)</li><li><a href="https://github.com/invincible0" title="invincible0" data-bi-name="contributorprofile" target="_blank">invincible0</a> (0.2%)</li><li><a href="https://github.com/turfaa" title="turfaa" data-bi-name="contributorprofile" target="_blank">turfaa</a> (0.2%)</li></ul>
73497361
</ul>
73507362

73517363
</article>

‎1369/data_structures/sqrt_decomposition.html

Lines changed: 0 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -6606,12 +6606,6 @@
66066606

66076607

66086608

6609-
6610-
6611-
6612-
6613-
6614-
66156609

66166610

66176611

‎1369/dynamic_programming/divide-and-conquer-dp.html

Lines changed: 0 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -6564,12 +6564,6 @@
65646564

65656565

65666566

6567-
6568-
6569-
6570-
6571-
6572-
65736567

65746568

65756569

0 commit comments

Comments
 (0)
pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy