From ff74af2dd3173cf894b02d4e26357cadcd6bf827 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Thu, 16 May 2024 18:31:27 +0000 Subject: [PATCH 01/41] init time and space complexity in level 5 --- .../fabio-lama/README.md | 0 .../fabio-lama/build.sh | 0 .../fabio-lama/src/time_space_complexity.adoc | 149 ++++++++++++++++++ 3 files changed, 149 insertions(+) create mode 100644 level-5/algorithms-and-data-structures-ii/fabio-lama/README.md create mode 100644 level-5/algorithms-and-data-structures-ii/fabio-lama/build.sh create mode 100644 level-5/algorithms-and-data-structures-ii/fabio-lama/src/time_space_complexity.adoc diff --git a/level-5/algorithms-and-data-structures-ii/fabio-lama/README.md b/level-5/algorithms-and-data-structures-ii/fabio-lama/README.md new file mode 100644 index 00000000..e69de29b diff --git a/level-5/algorithms-and-data-structures-ii/fabio-lama/build.sh b/level-5/algorithms-and-data-structures-ii/fabio-lama/build.sh new file mode 100644 index 00000000..e69de29b diff --git a/level-5/algorithms-and-data-structures-ii/fabio-lama/src/time_space_complexity.adoc b/level-5/algorithms-and-data-structures-ii/fabio-lama/src/time_space_complexity.adoc new file mode 100644 index 00000000..e5d29e44 --- /dev/null +++ b/level-5/algorithms-and-data-structures-ii/fabio-lama/src/time_space_complexity.adoc @@ -0,0 +1,149 @@ += Cheatsheet - Time and Space Complexity +Fabio Lama +:description: Module: CM2035 Algorithms and Data Structures II, started April 2024 +:doctype: article +:sectnums: 4 +:toclevels: 4 +:stem: + +== About + +To analyze an algorithm, we must determine its processing and memory requirements. The processing requirement is the time complexity, and the memory requirement is the space complexity. + +== Counting Up Time And Space Units + +=== Simple Algorithm + +Consider: + +. stem:[bb "function " "F1"(tt "a, b, c")] +. stem:[" " tt "max" = a] +. stem:[" " bb "if " (b> tt "max")] +. stem:[" " " " tt "max" = b] +. stem:[" " bb "if " (c> tt "max")] +. stem:[" " " " tt "max" = c] +. stem:[" " bb "return " tt "max"] + +If we analyze this function one by one: + +==== Step 1. + +. stem:[" " tt "max" = a] + +1 memory read (a), 1 memory write (max), or **2 time units**. + +==== Step 2. + +. stem:[" " bb "if " (b> tt "max")] + +2 memory reads (b, max), 1 comparison, 1 evaluation, or **4 time units**. + +==== Step 3. + +. stem:[" " " " tt "max" = b] + +1 memory read (b), 1 memory write (max), or **2 time units**. + +==== Step 4. + +. stem:[" " bb "if " (c> tt "max")] + +2 memory reads (c, max), 1 comparison, 1 evaluation, or **4 time units**. + +==== Step 5. + +. stem:[" " " " tt "max" = c] + +1 memory read (c), 1 memory write (max), or **2 time units**. + +==== Step 6. + +. stem:[" " bb "return " tt "max"] + +1 memory read (max), 1 return, or **2 time units**. + +==== Total + +In **total**, this function has a time complexity of **16 time units**. The only variable created is stem:["max"], so the space complexity is **1 space unit**. + +=== Complex Algorithm + +Consider: + +. stem:[bb "function " "F2"(tt "A, N, x")] +. stem:[" " bb "for " 0 <= i < N] +. stem:[" " " " bb "if " (A\[i\] " == " x)] +. stem:[" " " " " " bb "return " i] +. stem:[" " bb "return " -1] + +If we analyze this function one by one: + +==== Step 1. + +. stem:[" " bb "for " 0 <= i < N] + +We expand this to: + +. stem:[i = 0] +. stem:[bb "if " (i < N)] +. stem:[" " tt ""] +. stem:[" " i = i + 1] + +Respectively: + +. stem:[i = 0] + +1 memory write (i), or **1 time unit**. + +. stem:[bb "if " (i < N)] + +2 memory reads (i, N), 1 comparison, 1 evaluation, or 4 time units. But since +this is in a loop, we can consider this as **4*(N+1) time units**. Plus one +because it's executed at least once, even if N is 0 (and the if-statement is +skipped). + +. stem:[" " i = i + 1] + +1 memory read (i), 1 numerical op (+1), 1 memory write (i), or **3*N time units**. +Times N because it's executed in a loop. + +For the full time complexity, excluding the inner instructions, we get: + +[stem] +++++ +1 + 4 (N + 1) + 3 N\ += 1 + 4N + 4 + 3 N\ += 7N + 5 +++++ + +In other words, this for loop takes **7N + 5 time units**, excluding the inner instructions. + +==== Step 2. + +. stem:[" " " " bb "if " (A\[i\] " == " x)] + +3 memory reads (x, i, stem:[A\[i\]]), 1 comparison, 1 evaluation, or **5*N time units**. +Times N because it's executed in a loop. + +==== Step 3. + +Only one of the two return statements is executed. + +. stem:[" " " " " " bb "return " i] +. stem:[" " bb "return " -1] + +We are going to assume the case where the number is not in the array, so the +second return statement is executed which is **1 time unit**. + +=== Total + +To summarize all the steps: + +[stem] +++++ +7N + 5 + 5N + 1\ += 12N + 6 +++++ + +In **total**, this function has a time complexity of **12*N + 6 time units**. +This means it's running time depends on the size of the input array. The bigger the array, the longer the runtime. From c7cd0345901ef383bd17d41112a2afe847ccd63b Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Thu, 16 May 2024 18:36:46 +0000 Subject: [PATCH 02/41] expand space unit on loop algorithm --- .../fabio-lama/src/time_space_complexity.adoc | 11 ++++++++--- 1 file changed, 8 insertions(+), 3 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/fabio-lama/src/time_space_complexity.adoc b/level-5/algorithms-and-data-structures-ii/fabio-lama/src/time_space_complexity.adoc index e5d29e44..0b99c9f1 100644 --- a/level-5/algorithms-and-data-structures-ii/fabio-lama/src/time_space_complexity.adoc +++ b/level-5/algorithms-and-data-structures-ii/fabio-lama/src/time_space_complexity.adoc @@ -76,7 +76,8 @@ Consider: . stem:[" " " " " " bb "return " i] . stem:[" " bb "return " -1] -If we analyze this function one by one: +A loop is not a single instruction, meaning we need to analyze the inner +instructions carefully. If we analyze this function one by one: ==== Step 1. @@ -141,9 +142,13 @@ To summarize all the steps: [stem] ++++ -7N + 5 + 5N + 1\ +7N + 5\ ++ 5N\ ++ 1\ = 12N + 6 ++++ In **total**, this function has a time complexity of **12*N + 6 time units**. -This means it's running time depends on the size of the input array. The bigger the array, the longer the runtime. +This means it's running time depends on the size of the input array. The bigger +the array, the longer the runtime. Additionally, we only create one new variable +(i), so the space complexity is **1 space unit**. From 1853e9cdd8449441758d832c74131efcd1be9c24 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Thu, 16 May 2024 19:00:54 +0000 Subject: [PATCH 03/41] add growth of function chart --- .../{ => student-notes}/fabio-lama/README.md | 0 .../{ => student-notes}/fabio-lama/build.sh | 0 .../src/assets/growth_of_function.png | Bin 0 -> 183496 bytes .../fabio-lama/src/time_space_complexity.adoc | 29 ++++++++++++++++++ 4 files changed, 29 insertions(+) rename level-5/algorithms-and-data-structures-ii/{ => student-notes}/fabio-lama/README.md (100%) rename level-5/algorithms-and-data-structures-ii/{ => student-notes}/fabio-lama/build.sh (100%) create mode 100644 level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/assets/growth_of_function.png rename level-5/algorithms-and-data-structures-ii/{ => student-notes}/fabio-lama/src/time_space_complexity.adoc (83%) diff --git a/level-5/algorithms-and-data-structures-ii/fabio-lama/README.md b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md similarity index 100% rename from level-5/algorithms-and-data-structures-ii/fabio-lama/README.md rename to level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md diff --git a/level-5/algorithms-and-data-structures-ii/fabio-lama/build.sh b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/build.sh similarity index 100% rename from level-5/algorithms-and-data-structures-ii/fabio-lama/build.sh rename to level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/build.sh diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/assets/growth_of_function.png b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/assets/growth_of_function.png new file mode 100644 index 0000000000000000000000000000000000000000..30be6b2d26b6eba6f7101f9195554de96b25cebd GIT binary patch literal 183496 zcmV*IKxe;+P)Px#1ZP1_K>z@;j|==^1poj532;bRa{vGi!vFvd!vV){sAK>D|D{PpK~#8N?EUq- z99NbmjNZTC{(8UrJog*unQ5M$p6;)jri-~;hB7mgWM+^hlO$VA7Be$5Gc)Ue&e37a z%##!xvF`h>9hoPOEV=9|yIPRT+eev^5j%FQz4rUAwJH8z;Wrl%iK1xVzI}i9cYpWu zpZ`3ONSGs#_>8iY<&Tp<#rNGX@WQ}}LO%@sC<=ne_k_up0DprhL@J&i1fCx{Zsb@J znf@ei*q(aa}}?e%4F^W+(x2*ZK+ynV^1I?=F&-^LiD1 zIwpuQT_n#WsyKv2L&4(E>eYrL{AA9t1xErThphl-YQ4B3yi&YKY++*le)lajbN;Go zn%s)X$df`2?8iKQ{-5nad1FNjgamxT}5BdJ-&dRFbw#$lTQ zD!>aQ7f3pkOc+_JzF4=#R4QebEz7kihBF~#M8a`w`e^4xqP-%^2B8C@1>RecAUOoC z$oGL1q*&s+atY~#IU5`=3#0^JjH=R>vu2|C6eu=kS==3$X}!Bhn!|N0<}o40bc=}$h`)GB5St6XP#qBZ zm3lVlRw6^RhGmK4K!1;a@fno@iUsE)bFggj7KUYU%v1rZQWHKaaR?L-YAqfyl+1Qz zd;+>u_@Xs^bFwUzK$e99jFzYrkTig zrgqd#VIm&I`4qnLk{Hp73DTXvxt2E~cAF#Q-OH$2^FbBkv>SnCp}H@Nc}ygRjRDM; zO2N~ULD>}(5Lyv-7VOq!g7Xbk8Wky_#s|$2R#PEC_>)pH5~?GtaBS#~F7S(-h4yP^~@M=OoVST9o#QF@J~%(7*<#--ct3A+V! zHi%#(wtIyJ{8S(bNJOG7^n;n;RZ|rAjOl<&oXgEEUuIcE|M<(VaM;;DX8G)m_mt}b zKgb19I^@t)j$j0dS(Zfm)q)dDQZZFvK+P(o0&~X_7fjQ2EHF_J7!stYgcz+0EQ@H> z;%I4sWgQ?j|9b%WtGk$tY5862pfrB!V=-& zbuZt5WnuH4vMd$O;9gl)iPo@eOthp}m!nZsPkg!3Yb<4!Ez30|Y~lrp4*|3bW_3ld zfEY9(zZTI4Xi*q1F;@^Pa2jaC1OvQW+sT!i+)XGkIv9$LS++3I%xJi94nO#T%}!&^ zw6u6ol?h<(snR%4O-A^uGI5+3K_TH@+65g%S^f>}l1m_jJk)Io@7@SvY}lBw(J($k z+3*TMvEW+Lt+oku3^_n+&g{1H0*}(f84bs|l}hn0Oj;MkY6ZVqOmwHj(@ET(`dFdI zA{bCd&RQPcL2iJ?Qfi;2Vyv_z#oXz*(2MGv(JJ|sQ^28)LO#%^o~h>)FIRevrOdKr zxrRjPHJ130`qc4f*fZL?CPGHx6oXL4%H8hKRuc@VbB^4|)<7w`qQngR#9zu58w=qh zR@Y@LOD7%bQ~Ff+fGQDXfoGvijdAsQQ%?rMiYOK%jwU1g^>UWgao?C_m12=|d^$xh zk6HG{nPXh~nbxa|cJP&C>2f^7iMW!-i5OYJH>9z$8VrzMq0{)QS+*cC#k!invU%}h zmUUev6jr?xx~5+`zJgyE$AR_r&~+`P_E{=LLoKA3W#b^V02QV$tzeC@FBEqdvwrj( zB&F9_$}C%!?^cxKVu=r3UA}x8@P()jmm0@)bP5SrLs&t03V49`t!==8#}C|A7B
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z=K5#+gsD#07w&%-zZHkWn@aF%es_AwTNjriZ0M%S=XRq1^&fj> z&Oh8=c6PPCti0xSDuXZmgWd@1GXAgG5WJa|0{*_i_%LupT$Bb6=r2}GrTDbqKiDi7 zH+HWC4p;xxw7mYtqJQWAi Date: Thu, 16 May 2024 19:30:44 +0000 Subject: [PATCH 04/41] expand build.sh, add section on calculating function growth without counting --- .../student-notes/fabio-lama/build.sh | 77 ++++++++++++++++++ .../fabio-lama/src/time_space_complexity.adoc | 79 ++++++++++++++++++- 2 files changed, 152 insertions(+), 4 deletions(-) mode change 100644 => 100755 level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/build.sh diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/build.sh b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/build.sh old mode 100644 new mode 100755 index e69de29b..d3987601 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/build.sh +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/build.sh @@ -0,0 +1,77 @@ +#!/bin/bash + +# Because `make` sucks. + +gen_html() { + # Remove suffix and prefix + FILE=$1 + OUT=${FILE%.adoc} + HTML_OUT="cheatsheet_${OUT}.html" + + asciidoctor $FILE -o ${HTML_OUT} +} + +# Change directory to src/ in order to have images included correctly. +cd "$(dirname "$0")/src/" + +case $1 in + html) + for FILE in *.adoc + do + # Generate HTML file. + gen_html ${FILE} + done + + # Move up from src/ + mv *.html ../ + ;; + pdf) + for FILE in *.adoc + do + # Generate HTML file. + gen_html ${FILE} + + # Convert HTML to PDF. + PDF_OUT="cheatsheet_${OUT}.pdf" + wkhtmltopdf \ + --enable-local-file-access \ + --javascript-delay 2000\ + $HTML_OUT $PDF_OUT + done + + # Move up from src/ + mv *.pdf ../ + + # Cleanup temporarily generated HTML files. + rm *.html > /dev/null 2>&1 + ;; + png | img) + for FILE in *.adoc + do + # Generate HTML file. + gen_html ${FILE} + + # Convert HTML to PNG. + IMG_OUT="cheatsheet_${OUT}.png" + wkhtmltopdf \ + --enable-local-file-access \ + --javascript-delay 2000\ + $HTML_OUT $IMG_OUT + done + + # Move up from src/ + mv *.png ../ + + # Cleanup temporarily generated HTML files. + rm *.html > /dev/null 2>&1 + ;; + clean) + rm *.html > /dev/null 2>&1 + rm *.png > /dev/null 2>&1 + rm ../*.html > /dev/null 2>&1 + rm ../*.png > /dev/null 2>&1 + ;; + *) + echo "Unrecognized command" + ;; +esac \ No newline at end of file diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc index 2a0a0dfa..da9eb460 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc @@ -10,13 +10,13 @@ Fabio Lama To analyze an algorithm, we must determine its processing and memory requirements. The processing requirement is the time complexity, and the memory requirement is the space complexity. -== Counting Up Time And Space Units +== Counting Up Time and Space Units === Simple Algorithm Consider: -. stem:[bb "function " "F1"(tt "a, b, c")] +. stem:[bb "function " "F1"(a, b, c)] . stem:[" " tt "max" = a] . stem:[" " bb "if " (b> tt "max")] . stem:[" " " " tt "max" = b] @@ -70,7 +70,7 @@ In **total**, this function has a time complexity of **16 time units**. The only Consider: -. stem:[bb "function " "F2"(tt "A, N, x")] +. stem:[bb "function " "F2"(A, N, x)] . stem:[" " bb "for " 0 <= i < N] . stem:[" " " " bb "if " (A\[i\] " == " x)] . stem:[" " " " " " bb "return " i] @@ -153,7 +153,7 @@ This means it's running time depends on the size of the input array. The bigger the array, the longer the runtime. Additionally, we only create one new variable (i), so the space complexity is **1 space unit**. -=== Growth of function +=== Growth of Function As we have seen, the simple algorithm has a time complexity of **16 time units**. This means that the time complexity is a constant, regardless of the @@ -181,3 +181,74 @@ Common time complexities are: .Source: https://www.codeproject.com/articles/1012294/algorithm-time-complexity-what-is-it image::assets/growth_of_function.png[align=center, width=500] +=== Growth of Function Without Counting + +We can calculate the growth of the running time of an algorithm without counting +every single time unit. + +Consider: + +. stem:[bb "function " "SumDiag"(A)] +. stem:[" " tt "sum" = 0] +. stem:[" " N = "length"(A\[0\])] +. stem:[" " bb "for " (0 <= i < N)] +. stem:[" " " " tt "sum" = tt "sum" + A\[i, i\]] +. stem:[" " bb "return " tt "sum"] + +If we analyze this function one by one: + +==== Step 1. + +. stem:[" " tt "sum" = 0] + +Constant time stem:[C_0] + +==== Step 2. + +. stem:[" " N = "length"(A\[0\])] + +Here we have to analyze the stem:["length"] function, but let's say we already +knew its time complexity of stem:[T(N) = C_1 N + C_2] + +==== Step 3. + +. stem:[" " bb "for " (0 <= i < N)] + +As we analyzed in the section on counting time units of a for loop, for example +stem:[7N + 5] (excluding inner instructions), we hence know that stem:[T(N) = C_3 N + C_4]. + +==== Step 4. + +. stem:[" " " " tt "sum" = tt "sum" + A\[i, i\]] + +Constant time stem:[C_5] times N because it's executed in a loop. Respectively: +stem:[C_5 N]. + +==== Step 5. + +. stem:[" " bb "return " tt "sum"] + +Constant time stem:[C_6]. + +==== Total + +To summarize all the steps: + +[stem] +++++ +C_0\ ++ C_1 N + C_2\ ++ C_3 N + C_4\ ++ C_5 N\ ++ C_6\ += T(N) = (C_1 + C_3 + C_5) N + (C_0 + C_2 + C_4 + C_6) +++++ + +To simplify, we can group the constants together: + +[stem] +++++ +T(N) = C_7 N + C_8 +++++ + +This means that the algorithm grows linearly with the size of the input. From ef5b7fad8085ee5c7fffe02862fcd900701918f6 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Thu, 16 May 2024 20:09:01 +0000 Subject: [PATCH 05/41] add section on worst and best cases of time complexity --- .../fabio-lama/src/time_space_complexity.adoc | 43 +++++++++++++++++++ 1 file changed, 43 insertions(+) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc index da9eb460..e5c34100 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc @@ -252,3 +252,46 @@ T(N) = C_7 N + C_8 ++++ This means that the algorithm grows linearly with the size of the input. + +=== Worst and Best Cases + +An algorithm can have different time complexities depending on the input. +Generally, we're interested in the worst-case scenario, but we can also analyze +the best-case scenario. + +Consider: + +. stem:[bb "function " "L_Search"(A, x)] +. stem:[" " N = "length"(A)] +. stem:[" " bb "for " 0 <= i < N] +. stem:[" " " " bb "if " (A\[i\] " == " x)] +. stem:[" " " " " " bb "return " i] +. stem:[" " bb "return " -1] + +And a given array: + +[stem] +++++ +A = (13, 8, 2, 24, 5, 17, 6, 9) +++++ + +The **best case** is where the number we're looking for is the first element in the +array, respectively stem:[x = 13]. In such a case, the function returns immediately: + +[stem] +++++ +T(N) = C_1 +++++ + +**Worst case**, the number we're looking for is not in the array at all, for +example stem:[x = 7]. In that case, the function checks every element in the +array (N amount), eventually returning stem:[-1]: + +[stem] +++++ +T(N) = C_1 N + C_2 +++++ + +We notate this as stem:["L_Search"(A, 7)] having a running time of stem:[T(N) a N] +(worst case). Meanwhile, stem:["L_Search"(A, 13)] has a running time of +stem:[T(N) a 1] (best case). From ca9d08b06c5ece8233271288af308106a2abc7d9 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Thu, 16 May 2024 20:56:21 +0000 Subject: [PATCH 06/41] init section on asymptotic analysis --- .../fabio-lama/src/asymptotic_analysis.adoc | 23 +++++++++++++++++++ 1 file changed, 23 insertions(+) create mode 100644 level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc new file mode 100644 index 00000000..ba11b7e1 --- /dev/null +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc @@ -0,0 +1,23 @@ += Cheatsheet - Asymptotic Analysis +Fabio Lama +:description: Module: CM2035 Algorithms and Data Structures II, started April 2024 +:doctype: article +:sectnums: 4 +:toclevels: 4 +:stem: + +== About + +Asymptotic analysis is an alternative way of describing the time or memory +requirements of an algorithm. + +== Big-O Notation + +stem:[T(N)] is stem:[Theta(g(N))] if stem:[c xx g(N) >= T(N)] for all stem:[N <= n_o]. Importantly, stem:[c] and stem:[n_0] must be positive constants. + +Expressed differently: + +[stem] +++++ +T(N) in Theta(g(N)) hArr EE c > 0 EE n_0 > 0 AA N (N <= n_0 -> c * g(N) >= T(N)) +++++ From 5924f5d159af56518b53c1bb1c0fa6f525bad404 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Fri, 17 May 2024 07:58:35 +0000 Subject: [PATCH 07/41] expand section on big O notation --- .../fabio-lama/src/assets/big_o_notation.png | Bin 0 -> 21720 bytes .../fabio-lama/src/asymptotic_analysis.adoc | 80 +++++++++++++++++- .../fabio-lama/src/time_space_complexity.adoc | 4 +- 3 files changed, 78 insertions(+), 6 deletions(-) create mode 100644 level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/assets/big_o_notation.png diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/assets/big_o_notation.png b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/assets/big_o_notation.png new file mode 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Additionally, do notice that it's **not necessary** that +stem:[c xx g(N)] is equal to or greater than stem:[T(N)] for all values of +stem:[N]. + +image::assets/big_o_notation.png[align=center, width=336] + +For example, consider: + +[stem] +++++ +T(N) = 10 N^2 + 15N + 5\ +g(N) = N^2\ +c = 1 +++++ + +Here, stem:[c xx g(N)] is never greater than stem:[T(N)], because there is no +solution for: + +[stem] +++++ +10 N^2 + 15N + 5 <= 1 xx N^2 +++++ + +However, consider: + +[stem] +++++ +c = 25 +++++ + +In case of stem:[N = 1] we get: + +[stem] +++++ +10 xx 1^2 + 15 xx 1 + 5 <= 25 xx 1^2\ += 10 + 15 + 5 <= 25\ += 30 <= 25 +++++ + +Which is false. However, for stem:[N = 2] we get: -stem:[T(N)] is stem:[Theta(g(N))] if stem:[c xx g(N) >= T(N)] for all stem:[N <= n_o]. Importantly, stem:[c] and stem:[n_0] must be positive constants. +[stem] +++++ +10 xx 2^2 + 15 xx 2 + 5 <= 25 xx 2^2\ += 40 + 30 + 5 <= 100\ += 75 <= 100 +++++ -Expressed differently: +Which is true. Therefore: [stem] ++++ -T(N) in Theta(g(N)) hArr EE c > 0 EE n_0 > 0 AA N (N <= n_0 -> c * g(N) >= T(N)) +T(N) " is " O(N^2) " because"\ +T(N) <= 25 xx g(N) " for all " N >= 2 ++++ + +There choice for stem:[c] **is arbitrary**, as long as it satisfies the conditions. \ No newline at end of file diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc index e5c34100..76301311 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc @@ -156,8 +156,8 @@ the array, the longer the runtime. Additionally, we only create one new variable === Growth of Function As we have seen, the simple algorithm has a time complexity of **16 time units**. -This means that the time complexity is a constant, regardless of the -size of the input. +This means that the time complexity is a constant (stem:[C_x]), +regardless of the size of the input. We can specify this as: From d749351d57951024eb554efd99fc8d5079a50013 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Fri, 17 May 2024 08:11:26 +0000 Subject: [PATCH 08/41] add section on omega notation --- .../fabio-lama/src/asymptotic_analysis.adoc | 28 ++++++++++++++----- 1 file changed, 21 insertions(+), 7 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc index af7b92f1..a4ac84b9 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc @@ -16,12 +16,10 @@ requirements of an algorithm. == Big O Notation Big O notation (stem:[O(x)]) defines a set of functions that act as an upper bound (stem:[g(N)]) for -stem:[T(N)]. +stem:[T(N)]. Formally defined as: -Formally defined as: - -stem:[T(N)] is stem:[O(g(N))] if there exist **positive** constants stem:[c] and -stem:[n_0] such that: +stem:[T(N)] is stem:[O(g(N))] if there exist positive +constants stem:[c] and stem:[n_0] such that: [stem] ++++ @@ -37,7 +35,7 @@ T(N) in O(g(N)) hArr EE c > 0 EE n_0 > 0 AA N (N <= n_0 -> c xx g(N) >= T(N)) Note that there can be **multiple** functions (stem:[g_x(N)]) that act as an upper bound for stem:[T(N)]. Additionally, do notice that it's **not necessary** that -stem:[c xx g(N)] is equal to or greater than stem:[T(N)] for all values of +stem:[c xx g(N)] is equal to or greater than stem:[T(N)] **for all values** of stem:[N]. image::assets/big_o_notation.png[align=center, width=336] @@ -92,4 +90,20 @@ T(N) " is " O(N^2) " because"\ T(N) <= 25 xx g(N) " for all " N >= 2 ++++ -There choice for stem:[c] **is arbitrary**, as long as it satisfies the conditions. \ No newline at end of file +There choice for stem:[c] **is arbitrary**, as long as it satisfies the conditions. + +== Omega Notation + +The Omega notation (stem:[Omega(x)]) defines a set of functions that act as a lower bound (stem:[g(N)]) for (stem:[T(N)]). Formally defined as: + +stem:[T(N)] is stem:[Omega(g(N))] if there exist +positive constants stem:[c] and stem:[n_0] such that: + +[stem] +++++ +T(N) >= c xx g(N) " for all " N > n_0 +++++ + +Similarly to the Big O notation, there can be multiple functions (stem:[g_x(N)]) that act as a lower bound for stem:[T(N)] and it's **not necessary** that stem:[c xx g(N)] is equal to or less than stem:[T(N)] **for all values** of stem:[N], but only for the **larger values**. + +image::assets/omega_notation.png[align=center, width=360] From 4efb13f671c0ce110f03683330825cdfac859d0d Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Fri, 17 May 2024 08:36:21 +0000 Subject: [PATCH 09/41] add section on Theta notation --- .../fabio-lama/src/asymptotic_analysis.adoc | 44 ++++++++++++++++--- 1 file changed, 37 insertions(+), 7 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc index a4ac84b9..7d2f166c 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc @@ -15,8 +15,8 @@ requirements of an algorithm. == Big O Notation -Big O notation (stem:[O(x)]) defines a set of functions that act as an upper bound (stem:[g(N)]) for -stem:[T(N)]. Formally defined as: +Big O notation stem:[O(x)] defines a set of functions that act as an upper bound +stem:[g(N)] for stem:[T(N)]. Formally defined as: stem:[T(N)] is stem:[O(g(N))] if there exist positive constants stem:[c] and stem:[n_0] such that: @@ -33,7 +33,7 @@ Alternatively: T(N) in O(g(N)) hArr EE c > 0 EE n_0 > 0 AA N (N <= n_0 -> c xx g(N) >= T(N)) ++++ -Note that there can be **multiple** functions (stem:[g_x(N)]) that act as an upper +Note that there can be **multiple** functions stem:[g_x(N)] that act as an upper bound for stem:[T(N)]. Additionally, do notice that it's **not necessary** that stem:[c xx g(N)] is equal to or greater than stem:[T(N)] **for all values** of stem:[N]. @@ -94,16 +94,46 @@ There choice for stem:[c] **is arbitrary**, as long as it satisfies the conditio == Omega Notation -The Omega notation (stem:[Omega(x)]) defines a set of functions that act as a lower bound (stem:[g(N)]) for (stem:[T(N)]). Formally defined as: +The Omega notation stem:[Omega(x)] defines a set of functions that act as a +lower bound stem:[g(N)] for (stem:[T(N)]). Formally defined as: -stem:[T(N)] is stem:[Omega(g(N))] if there exist -positive constants stem:[c] and stem:[n_0] such that: +stem:[T(N)] is stem:[Omega(g(N))] if there exist positive constants stem:[c] and +stem:[n_0] such that: [stem] ++++ T(N) >= c xx g(N) " for all " N > n_0 ++++ -Similarly to the Big O notation, there can be multiple functions (stem:[g_x(N)]) that act as a lower bound for stem:[T(N)] and it's **not necessary** that stem:[c xx g(N)] is equal to or less than stem:[T(N)] **for all values** of stem:[N], but only for the **larger values**. +Similarly to the Big O notation, there can be multiple functions stem:[g_x(N)] +that act as a lower bound for stem:[T(N)] and it's **not necessary** that +stem:[c xx g(N)] is equal to or less than stem:[T(N)] **for all values** of +stem:[N], but only for the **larger values**. image::assets/omega_notation.png[align=center, width=360] + +== Theta Notation + +The Theta notation stem:[Theta(x)] defines a **single function** that acts as +both an upper and lower bound for (stem:[T(N)]). Formally defined as: + +stem:[T(N)] is stem:[Theta(g(N))] if there exist positive constants stem:[c_1], +stem:[c_2] and stem:[n_o] such that both those conditions hold true: + +[stem] +++++ +T(N) >= c_1 xx g(N) " for all " N > n_0\ +T(N) <= c_2 xx g(N) " for all " N > n_0 +++++ + +Alternatively: + +[stem] +++++ +c_1 xx g(N) <= T(N) <= c_2 xx g(N) " for all " N > n_0 +++++ + +image::assets/theta_notation.png[align=center, width=360] + +As already noted, Theta notation has **only one function** that acts as both an +upper and lower bound for stem:[T(N)]. 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zDFSM-ctyrmid?w}g@HHa5_h_Sp8d{Bk|V&$X*#zQzUZ)5htc&G9kDH6fD(1Czp zU1MD}9cTZvS58Xf@j}XNLJ{W;{%bFN{to*U`G+Z3iD5SOMO`;Z!$jp(Me1Y{R-puo z6M_LA*_2BY@#}X4K5UKodVkzoa7Nt^URV;K(Cx?iWeZ&y~(3 z5)@sB>NoN(q<+cEsKK6*r&0DJnp{9<2-#(Vv;3SbLXZM63+bJZKpc#+g@K?FUokqE zq+N;pE@Yps+e++mPqRhlsLM}&hrd)yT_?l$n&aX9T@Is`%Aa=$ES_!aj9%B#u^enA z8iq_e5M-k9ALZ5tTNtkNi$!Aevms;tI^Wyj80Lgy>>{R;F}7q&)#b~tO+PlGZjZJGGbIv-AA^i9Cq4b6Ag$?_fR_$u zHU4e$^C+ENjm{lElPb3AT^Io}1&pm7ITYildM1DR?~=a4pt&ZT#eb^9vX@3vU41U(>M=XO zQb2N>&B3F$&|^eEfLIrKC@eGqK1AM`9%55jVgyb9@3 z0L&@3d|5&8?-w0Vlmzn>JP=|5xIn;NO}3I0M1Z^kb(_HY$*xMbE-sp{WRWZJ57;@P zL8}1zFR-4W2Z}v2Fn2XoCH#nDAs}}+f8p(>=qG7!RZTRH-f{mAHw7rKLx=?A=paf4 zEQ_UbB9a0ZOg2CQz>}|lZtVw2QWyQ%@~`7X|6NkHCp--pwq5p;C_>)6C*?QM^$W*v zq|m0^`M+r~mj&IizvL}Ta$c-`go_rGq5t2hhU;AW_ipBu#{ci9{6_QdCHsGQ!T#y) YNGy+8qG}#Is24|FSx2c*!QB6U0B!^uwEzGB literal 0 HcmV?d00001 From 8818bc3da600b00805fc11a2903a620672d8fc2d Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Sun, 19 May 2024 12:37:52 +0000 Subject: [PATCH 11/41] add doc on time complexity for recursive algorithms --- .../time_complexity_recursive_algorithms.adoc | 69 +++++++++++++++++++ .../fabio-lama/src/time_space_complexity.adoc | 4 +- 2 files changed, 71 insertions(+), 2 deletions(-) create mode 100644 level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc new file mode 100644 index 00000000..22d9de78 --- /dev/null +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc @@ -0,0 +1,69 @@ += Cheatsheet - Time Complexity of Recursive Algorithms +Fabio Lama +:description: Module: CM2035 Algorithms and Data Structures II, started April 2024 +:doctype: article +:sectnums: 4 +:toclevels: 4 +:stem: + +NOTE: Make sure to check the _Time and Space Complexity_ cheatsheet, too. + +== About + +== Recurrence Relation + +Consider: + +. stem:[bb "function " "fact"(N)] +. stem:[" " bb "if " (N "==" 1)] +. stem:[" " " " bb "return " 1] +. stem:[" " bb "return " N xx "fact"(N - 1)] + +If we analyze this function one by one: + +. stem:[" " bb "if " (N "==" 1)] + +Constant time stem:[C_0] (grouped). + +. stem:[" " bb "return " N xx "fact"(N - 1)] + +Only one of the two return statements is executed (we assume the second +statement, worst case). + +. stem:[" " " " bb "return " 1] +. stem:[" " bb "return " N xx "fact"(N - 1)] + +Constant time stem:[C_1] (return), stem:[C_2] (read N), stem:[C_3] +(multiplication), and the stem:["fact"] function has a time of +stem:[T(N-1)]. If we group all the constants together, we have: + +[stem] +++++ +C_1 + C_2 + C_3 = C_4 +++++ + +So we say that the return statement has a time complexity of: + +[stem] +++++ +C_4 + T(N-1) +++++ + +Finally, we determine that the function stem:["fact"] has a time complexity of: + +[stem] +++++ +T(N) = C_0 + C_4 + T(N-1) +++++ + +Or grouped together: + +[stem] +++++ +T(N) = C_5 + T(N-1) +++++ + +This expression is known as a **recurrence relation**. + + +== Solving a Recurrence Relation \ No newline at end of file diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc index 76301311..6200b1a4 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc @@ -133,8 +133,8 @@ Only one of the two return statements is executed. . stem:[" " " " " " bb "return " i] . stem:[" " bb "return " -1] -We are going to assume the case where the number is not in the array, so the -second return statement is executed which is **1 time unit**. +We are going to assume the case where the number is not in the array (worst +case), so the second return statement is executed which is **1 time unit**. === Total From 5160b2bcf2d53f652f0bb56d2a89a8fa6d2c92e8 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Sun, 19 May 2024 13:05:17 +0000 Subject: [PATCH 12/41] expand section on solving recurrence relation --- .../time_complexity_recursive_algorithms.adoc | 39 ++++++++++++++++++- 1 file changed, 38 insertions(+), 1 deletion(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc index 22d9de78..7bf14ea3 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc @@ -66,4 +66,41 @@ T(N) = C_5 + T(N-1) This expression is known as a **recurrence relation**. -== Solving a Recurrence Relation \ No newline at end of file +== Solving a Recurrence Relation + +We can demonstrate the solution of a recurrence relation by expanding the calculation: + +[stem] +++++ +T(N) = C_5 + T(N-1)\ +T(N) = C_5 + T(C_5 + T(N-2))\ +T(N) = C_5 + T(C_5 + T(C_5 + T(N-3)))\ +...\ +T(N) = k xx C_5 + T(N-k) +++++ + +If stem:[k] is stem:[N-1], then: + +[stem] +++++ +T(N) = (N-1) xx C_5 + T(N-(N-1))\ +T(N) = (N-1) xx C_5 + C +++++ + +That's because: + +[stem] +++++ +T(N-(N-1))\ += T(N - N + 1)\ += T(1)\ += C +++++ + +TODO... + +We have hence solved the recurrence equation. In summary: + +* Find its recurrence equation. +* Solve the recurrence equation. +* Asymptotic analysis. From 18f4db2be3ef14a49d52cc40a3b48793e56eae6b Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 17:17:13 +0000 Subject: [PATCH 13/41] expand section on master theorem, add case classifications and example --- .../fabio-lama/src/asymptotic_analysis.adoc | 9 +- .../time_complexity_recursive_algorithms.adoc | 125 ++++++++++++++++++ 2 files changed, 129 insertions(+), 5 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc index 7d2f166c..74b010f6 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc @@ -15,7 +15,7 @@ requirements of an algorithm. == Big O Notation -Big O notation stem:[O(x)] defines a set of functions that act as an upper bound +Big O notation stem:[O(x)] defines a set of functions that act as an **upper bound** stem:[g(N)] for stem:[T(N)]. Formally defined as: stem:[T(N)] is stem:[O(g(N))] if there exist positive @@ -95,7 +95,7 @@ There choice for stem:[c] **is arbitrary**, as long as it satisfies the conditio == Omega Notation The Omega notation stem:[Omega(x)] defines a set of functions that act as a -lower bound stem:[g(N)] for (stem:[T(N)]). Formally defined as: +**lower bound** stem:[g(N)] for (stem:[T(N)]). Formally defined as: stem:[T(N)] is stem:[Omega(g(N))] if there exist positive constants stem:[c] and stem:[n_0] such that: @@ -115,7 +115,7 @@ image::assets/omega_notation.png[align=center, width=360] == Theta Notation The Theta notation stem:[Theta(x)] defines a **single function** that acts as -both an upper and lower bound for (stem:[T(N)]). Formally defined as: +both an **upper and lower bound** for (stem:[T(N)]). Formally defined as: stem:[T(N)] is stem:[Theta(g(N))] if there exist positive constants stem:[c_1], stem:[c_2] and stem:[n_o] such that both those conditions hold true: @@ -135,5 +135,4 @@ c_1 xx g(N) <= T(N) <= c_2 xx g(N) " for all " N > n_0 image::assets/theta_notation.png[align=center, width=360] -As already noted, Theta notation has **only one function** that acts as both an -upper and lower bound for stem:[T(N)]. +As already noted, Theta notation has **only one function**. diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc index 7bf14ea3..844a9645 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc @@ -104,3 +104,128 @@ We have hence solved the recurrence equation. In summary: * Find its recurrence equation. * Solve the recurrence equation. * Asymptotic analysis. + +== The Master Theorem + +The master theorem makes the asymptotic analysis of recursive functions much +easier. However, it can only be applied if the recurrence equation has this +specific structure: + +[stem] +++++ +T(n) = a xx T(n/b)+f(n) +++++ + +where stem:[a >= 1] and stem:[b > 1]. + +For example, this is solvable: + +[stem] +++++ +T(n) = T(n/2) + n\ += 1 xx T(n/2) + n +++++ + +Meanwhile, this is not: + +[stem] +++++ +T(n) = 2 xx T(n) + n\ += 2 xx T(n/1) + n +++++ + +The master theorem classifies the recurrence equiation in one of three cases: + +### Case 1 + +[stem] +++++ +f(n) < n^(log_b(a)) +++++ + +where the running time is: + +[stem] +++++ +T(n) = Theta(n^(log_b(a))) +++++ + +### Case 2 + +[stem] +++++ +f(n) = n^(log_b(a)) +++++ + +where the running time is: + +[stem] +++++ +T(n) = Theta(n^(log_b(a)) xx log_n) +++++ + +### Case 3 + +[stem] +++++ +f(n) > n^(log_b(a)) +++++ + +and: + +[stem] +++++ +a xx f(n/b) <= c xx f(n) " where " c < 1 " and " n " large" +++++ + +where the running time is: + +[stem] +++++ +T(n) = Theta(f(n)) +++++ + +### Example + +Consider: + +[stem] +++++ +T(n) = 2T(n/2) + n +++++ + +we deterime that the master theorem can be applied here, given that stem:[a = 2 >= 1] +and stem:[b = 2 > 1]. + +To determine the case, we first calculate: + +[stem] +++++ +log_b(a) = log_2(2) = 1 +++++ + +For case 1, we have: + +[stem] +++++ +n < n^1\ +n < n +++++ + +which is **false**. + +For case 2, we have: + +[stem] +++++ +n = n^1\ +n = n +++++ + +which is **true**. Hence, we classify the formula as case 2 and the running time +is: + +[stem] +++++ +T(n) = Theta(n xx log_n) +++++ From 188b47c927f587f681d8b20fa09d97de6690da4d Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 17:20:48 +0000 Subject: [PATCH 14/41] add about section to time complexity on recursive algorithms --- .../fabio-lama/src/time_complexity_recursive_algorithms.adoc | 5 ++++- .../student-notes/fabio-lama/src/time_space_complexity.adoc | 4 +++- 2 files changed, 7 insertions(+), 2 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc index 844a9645..fe0f10ba 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc @@ -10,6 +10,9 @@ NOTE: Make sure to check the _Time and Space Complexity_ cheatsheet, too. == About +This cheatsheet covers the time complexity of recursive algorithms, including +how to apply the master theorem. + == Recurrence Relation Consider: @@ -97,7 +100,7 @@ T(N-(N-1))\ = C ++++ -TODO... +TODO: we need more explanation and clarification here. We have hence solved the recurrence equation. In summary: diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc index 6200b1a4..3a831101 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc @@ -8,7 +8,9 @@ Fabio Lama == About -To analyze an algorithm, we must determine its processing and memory requirements. The processing requirement is the time complexity, and the memory requirement is the space complexity. +To analyze an algorithm, we must determine its processing and memory +requirements. The processing requirement is the time complexity, and the memory +requirement is the space complexity. == Counting Up Time and Space Units From 1133b7cc122c40b9d654568a99d9fe5b7f8502fb Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 17:52:39 +0000 Subject: [PATCH 15/41] init sorting_algorithms cheatsheet --- .../fabio-lama/src/sorting_algorithms.adoc | 60 +++++++++++++++++++ 1 file changed, 60 insertions(+) create mode 100644 level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc new file mode 100644 index 00000000..5475e805 --- /dev/null +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc @@ -0,0 +1,60 @@ += Cheatsheet - Comparison and Non-Comparison Sorting Algorithms +Fabio Lama +:description: Module: CM2035 Algorithms and Data Structures II, started April 2024 +:doctype: article +:sectnums: 4 +:toclevels: 4 +:stem: + +## About + +This cheatsheet provides a comparison of the most common sorting algorithms. The +algorithms are compared based on their worst case and best case running times. + +## Comparison Sort Overview + +|=== +|Name |Worst case complexity |Best case complexity + +|Bubble +|stem:[Theta(N^2)] +|stem:[Theta(N)] + +|Insertion +|stem:[Theta(N^2)] +|stem:[Theta(N)] + +|Selection +|stem:[Theta(N^2)] +|stem:[Theta(N^2)] + +|Quicksort +|stem:[Theta(N^2)] +|stem:[Theta(N xx log_N)] + +|Mergesort +|stem:[Theta(N xx log_N)] +|stem:[Theta(N xx log_N)] +|=== + +Because comparison sorts must compare pairs of elements, **they cannot** run +faster than stem:[N xx log_N]. + +## Bubble Sort Pseudocode + +. stem:[bb "function " "BubbleSort"(A, N)] +. stem:[" " tt "swapped" = true] +. stem:[" " bb "while" (tt "swapped") bb " do"] +. stem:[" " " " tt "swapped" = false] +. stem:[" " " " bb "for " 0 <= i < N-1 bb " do"] +. stem:[" " " " " " bb "if " (A\[i\] > A\[i + 1\]) bb " then"] +. stem:[" " " " " " " " "swap"(A\[i\], A\[i+1\])] +. stem:[" " " " " " " " tt "swapped" = true] +. stem:[" " " " " " bb "end if"] +. stem:[" " " " bb "end for"] +. stem:[" " " " N = N - 1] +. stem:[" " bb "end while"] +. stem:[" " bb "return " A] +. stem:[bb "end function"] + + From 5ceb76c56b6d3fbf11587fb5b434d09d94cb68e1 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 18:14:02 +0000 Subject: [PATCH 16/41] describe worst and best case for bubble sort --- .../fabio-lama/src/asymptotic_analysis.adoc | 2 -- .../fabio-lama/src/sorting_algorithms.adoc | 30 ++++++++++++++++++- .../time_complexity_recursive_algorithms.adoc | 2 -- 3 files changed, 29 insertions(+), 5 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc index 74b010f6..f8ca4269 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc @@ -6,8 +6,6 @@ Fabio Lama :toclevels: 4 :stem: -NOTE: Make sure to check the _Time and Space Complexity_ cheatsheet, too. - == About Asymptotic analysis is an alternative way of describing the time or memory diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc index 5475e805..38fe79fc 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc @@ -40,7 +40,9 @@ algorithms are compared based on their worst case and best case running times. Because comparison sorts must compare pairs of elements, **they cannot** run faster than stem:[N xx log_N]. -## Bubble Sort Pseudocode +## Bubble Sort + +### Pseudocode . stem:[bb "function " "BubbleSort"(A, N)] . stem:[" " tt "swapped" = true] @@ -57,4 +59,30 @@ faster than stem:[N xx log_N]. . stem:[" " bb "return " A] . stem:[bb "end function"] +### Time Complexity + +The **best case** for bubble sort is: + +[stem] +++++ +T(N) = C_0 xx N + C_1 +++++ + +Additionally: + +* stem:[T(N)] is stem:[O(N)], stem:[O(N^2)] and stem:[O(N^3)], etc. +* stem:[T(N)] is stem:[Omega(N)], stem:[Omega(log_N)] and stem:[Omega(1)], etc. +* stem:[T(N)] is stem:[Theta(N)] + +The **worst case** for bubble sort is: + +[stem] +++++ +T(N) = C_0 xx N^2 + C_1 xx N + C_2. +++++ + +Additionally: +* stem:[T(N)] is stem:[O(N^2)] and stem:[O(N^3)], etc. +* stem:[T(N)] is stem:[Omega(N^2)], stem:[Omega(log_N)] and stem:[Omega(1)], etc. +* stem:[T(N)] is stem:[Theta(N^2)] diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc index fe0f10ba..57b4a751 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc @@ -6,8 +6,6 @@ Fabio Lama :toclevels: 4 :stem: -NOTE: Make sure to check the _Time and Space Complexity_ cheatsheet, too. - == About This cheatsheet covers the time complexity of recursive algorithms, including From af7129e46b09c78509243d311717ef2ffa7c2655 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 18:32:58 +0000 Subject: [PATCH 17/41] add insertion sort section with pseudocode --- .../fabio-lama/src/sorting_algorithms.adoc | 20 ++++++++++++++++++- 1 file changed, 19 insertions(+), 1 deletion(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc index 38fe79fc..1f9dc890 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc @@ -46,7 +46,7 @@ faster than stem:[N xx log_N]. . stem:[bb "function " "BubbleSort"(A, N)] . stem:[" " tt "swapped" = true] -. stem:[" " bb "while" (tt "swapped") bb " do"] +. stem:[" " bb "while " (tt "swapped") bb " do"] . stem:[" " " " tt "swapped" = false] . stem:[" " " " bb "for " 0 <= i < N-1 bb " do"] . stem:[" " " " " " bb "if " (A\[i\] > A\[i + 1\]) bb " then"] @@ -86,3 +86,21 @@ Additionally: * stem:[T(N)] is stem:[O(N^2)] and stem:[O(N^3)], etc. * stem:[T(N)] is stem:[Omega(N^2)], stem:[Omega(log_N)] and stem:[Omega(1)], etc. * stem:[T(N)] is stem:[Theta(N^2)] + +## Insertion Sort + +### Pseudocode + +. stem:[bb "function " "InsertionSort"(A, N)] +. stem:[" " bb "for " 1 <= j <= N-1 bb " do"] +. stem:[" " " " tt "ins" = A\[j\]] +. stem:[" " " " i = j-1] +. stem:[" " " " bb "while " (i >= 0 " and " tt"ins" < A\[i\]) bb " do"] +. stem:[" " " " " " A\[i+1\] = A\[i\]] +. stem:[" " " " " " i = i-1] +. stem:[" " " " bb "end while"] +. stem:[" " " " A\[i+1\] = tt "ins"] +. stem:[" " bb "end for"] +. stem:[bb "end function"] + +### Time Complexity From efe8ffdf0df855bac0c30a41e98929ef4c0901ce Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 19:22:27 +0000 Subject: [PATCH 18/41] add pseudocode to Selection Sort --- .../fabio-lama/src/sorting_algorithms.adoc | 14 +++++++++++++- 1 file changed, 13 insertions(+), 1 deletion(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc index 1f9dc890..7aa40a53 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc @@ -103,4 +103,16 @@ Additionally: . stem:[" " bb "end for"] . stem:[bb "end function"] -### Time Complexity +## Selection Sort + +### Pseudocode + +. stem:[bb "function " "SelectionSort"(A, N)] +. stem:[" " bb "for " 0 <= i < N-1 bb " do"] +. stem:[" " " " tt "min" = "pos_min"(A, i, N-1)] +. stem:[" " " " tt "swap"(A\[i\], A\[tt "min"\])] +. stem:[" " bb "end for"] +. stem:[bb "end function"] + +The function stem:["pos_min"(A, a, b)] returns the position of the minimum value +between positions stem:[a] and stem:[b] (both inclusive) in array stem:[A]. From 90c814513261542053066b86afae6faf4e50deae Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 19:47:27 +0000 Subject: [PATCH 19/41] init README, add pseudocode to quicksort --- .../student-notes/fabio-lama/README.md | 63 +++++++++++++++++++ .../fabio-lama/src/sorting_algorithms.adoc | 19 +++++- 2 files changed, 80 insertions(+), 2 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md index e69de29b..2021b57d 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md @@ -0,0 +1,63 @@ +# About + +Listed here is a collection of cheatsheet by topic. Those cheatsheets do not +explain the topics in depth, but rather serve as quick lookup documents. +Therefore, the course material provided by the lecturer should still be studied +and understood. Not everything that is tested at the mid-terms or final exams is +covered and the Author does not guarantee that the cheatsheets are free of +errors. + +* [Pseudocode, Data Structures & Time + Complexity](./cheatsheet_pseudocode_data_structures.pdf) + +# Building + +_NOTE_: This step is only necessary if you chose to modify the base documents. + +The base documents are written in [AsciiDoc](https://asciidoc.org/) and can be +found in the `src/` directory. + +The following dependencies must be installed (Ubuntu): + +```console +$ apt install -y ruby-dev wkhtmltopdf +$ gem install asciidoctor +$ chmod +x build.sh +``` + +To build the documents (PDF version): + +```console +$ ./build.sh pdf +``` + +Optionally, for the HTML version: + +```console +$ ./build.sh html +``` + +and for the PNG version: + +```console +$ ./build.sh png +``` + +The generated output can be deleted with `./build.sh clean`. + +# Disclaimer + +The Presented Documents ("cheatsheets") by the Author ("Fabio Lama") are +summaries of specific topics. The term "cheatsheet" implies that the Presented +Documents are intended to be used as learning aids or as references for +practicing and does not imply that the Presented Documents should be used for +inappropriate practices during exams such as cheating or other offenses. + +The Presented Documents are heavily based on the learning material provided by +the University of London, respectively the VLeBooks Collection database in the +Online Library and the material provided on the Coursera platform. + +The Presented Documents may incorporate direct or indirect definitions, +examples, descriptions, graphs, sentences and/or other content used in those +provided materials. **At no point does the Author present the work or ideas +incorporated in the Presented Documents as their own.** diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc index 7aa40a53..07d37694 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc @@ -8,8 +8,7 @@ Fabio Lama ## About -This cheatsheet provides a comparison of the most common sorting algorithms. The -algorithms are compared based on their worst case and best case running times. +This cheatsheet provides an overview of some common sorting algorithms. ## Comparison Sort Overview @@ -116,3 +115,19 @@ Additionally: The function stem:["pos_min"(A, a, b)] returns the position of the minimum value between positions stem:[a] and stem:[b] (both inclusive) in array stem:[A]. + +## Quicksort + +### Pseudocode + +. stem:[bb "function " "Quicksort"(A, tt "low", tt "high")] +. stem:[" " bb "if " tt "low " < tt " high" bb " then"] +. stem:[" " " " p = "partition"(A, tt "low", tt "high")] +. stem:[" " " " "Quicksort"(A, tt "low", p-1)] +. stem:[" " " " "Quicksort"(A, p+1, tt "high")] +. stem:[" " bb "end if"] +. stem:[bb "end function"] + +The function stem:["partition"(A, tt "low", tt "high")] selects a number (the +pivot), moves all numbers lower than the pivot to the left part of the array and +moves the privot to its final position. From 0c82eb94ae50380ecc4f694be6ca93d4a14408ad Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 19:52:19 +0000 Subject: [PATCH 20/41] generate PDFs, link in README --- .../student-notes/fabio-lama/README.md | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md index 2021b57d..c327793b 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md @@ -7,8 +7,11 @@ and understood. Not everything that is tested at the mid-terms or final exams is covered and the Author does not guarantee that the cheatsheets are free of errors. -* [Pseudocode, Data Structures & Time - Complexity](./cheatsheet_pseudocode_data_structures.pdf) +* [Time and Space Complexity](./cheatsheet_time_space_complexity.pdf) +* [Asymptotic Analysis](./cheatsheet_asymptotic_analysis.pdf) +* [Time Complexity of Recursive Algorithms](.cheatsheet_time_complexity_recursive_algorithms.pdf) +* [Comparison and Non-Comparison Sorting Algorithms](./cheatsheet_sorting_algorithms.pdf) + # Building From bb812d6ffd33b63508f0ee51058b4ea2c027af6f Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 20:24:21 +0000 Subject: [PATCH 21/41] adjust image width --- .../student-notes/fabio-lama/src/asymptotic_analysis.adoc | 6 +++--- .../student-notes/fabio-lama/src/time_space_complexity.adoc | 2 +- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc index f8ca4269..bc877be6 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc @@ -36,7 +36,7 @@ bound for stem:[T(N)]. Additionally, do notice that it's **not necessary** that stem:[c xx g(N)] is equal to or greater than stem:[T(N)] **for all values** of stem:[N]. -image::assets/big_o_notation.png[align=center, width=336] +image::assets/big_o_notation.png[align=center, width=300] For example, consider: @@ -108,7 +108,7 @@ that act as a lower bound for stem:[T(N)] and it's **not necessary** that stem:[c xx g(N)] is equal to or less than stem:[T(N)] **for all values** of stem:[N], but only for the **larger values**. -image::assets/omega_notation.png[align=center, width=360] +image::assets/omega_notation.png[align=center, width=300] == Theta Notation @@ -131,6 +131,6 @@ Alternatively: c_1 xx g(N) <= T(N) <= c_2 xx g(N) " for all " N > n_0 ++++ -image::assets/theta_notation.png[align=center, width=360] +image::assets/theta_notation.png[align=center, width=300] As already noted, Theta notation has **only one function**. diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc index 3a831101..e26df222 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_space_complexity.adoc @@ -181,7 +181,7 @@ quadratic time complexity, and so on. Common time complexities are: .Source: https://www.codeproject.com/articles/1012294/algorithm-time-complexity-what-is-it -image::assets/growth_of_function.png[align=center, width=500] +image::assets/growth_of_function.png[align=center, width=400] === Growth of Function Without Counting From 824db52e71f7c0688406c0c8887c5f598ac9fa0e Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 20:39:07 +0000 Subject: [PATCH 22/41] fix formatting of log operator --- .../fabio-lama/src/sorting_algorithms.adoc | 20 +++++++++++++------ .../time_complexity_recursive_algorithms.adoc | 14 ++++++------- 2 files changed, 21 insertions(+), 13 deletions(-) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc index 07d37694..68db4586 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc @@ -29,15 +29,15 @@ This cheatsheet provides an overview of some common sorting algorithms. |Quicksort |stem:[Theta(N^2)] -|stem:[Theta(N xx log_N)] +|stem:[Theta(N xx log N)] |Mergesort -|stem:[Theta(N xx log_N)] -|stem:[Theta(N xx log_N)] +|stem:[Theta(N xx log N)] +|stem:[Theta(N xx log N)] |=== Because comparison sorts must compare pairs of elements, **they cannot** run -faster than stem:[N xx log_N]. +faster than stem:[N xx log N]. ## Bubble Sort @@ -70,7 +70,7 @@ T(N) = C_0 xx N + C_1 Additionally: * stem:[T(N)] is stem:[O(N)], stem:[O(N^2)] and stem:[O(N^3)], etc. -* stem:[T(N)] is stem:[Omega(N)], stem:[Omega(log_N)] and stem:[Omega(1)], etc. +* stem:[T(N)] is stem:[Omega(N)], stem:[Omega(log N)] and stem:[Omega(1)], etc. * stem:[T(N)] is stem:[Theta(N)] The **worst case** for bubble sort is: @@ -83,7 +83,7 @@ T(N) = C_0 xx N^2 + C_1 xx N + C_2. Additionally: * stem:[T(N)] is stem:[O(N^2)] and stem:[O(N^3)], etc. -* stem:[T(N)] is stem:[Omega(N^2)], stem:[Omega(log_N)] and stem:[Omega(1)], etc. +* stem:[T(N)] is stem:[Omega(N^2)], stem:[Omega(log N)] and stem:[Omega(1)], etc. * stem:[T(N)] is stem:[Theta(N^2)] ## Insertion Sort @@ -128,6 +128,14 @@ between positions stem:[a] and stem:[b] (both inclusive) in array stem:[A]. . stem:[" " bb "end if"] . stem:[bb "end function"] +TODO: This needs further explanation. + The function stem:["partition"(A, tt "low", tt "high")] selects a number (the pivot), moves all numbers lower than the pivot to the left part of the array and moves the privot to its final position. + +## Mergesort + +### Pseudocode + + diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc index 57b4a751..ad7a1b8f 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/time_complexity_recursive_algorithms.adoc @@ -141,35 +141,35 @@ The master theorem classifies the recurrence equiation in one of three cases: [stem] ++++ -f(n) < n^(log_b(a)) +f(n) < n^(log_b a) ++++ where the running time is: [stem] ++++ -T(n) = Theta(n^(log_b(a))) +T(n) = Theta(n^(log_b a)) ++++ ### Case 2 [stem] ++++ -f(n) = n^(log_b(a)) +f(n) = n^(log_b a) ++++ where the running time is: [stem] ++++ -T(n) = Theta(n^(log_b(a)) xx log_n) +T(n) = Theta(n^(log_b a) xx log N) ++++ ### Case 3 [stem] ++++ -f(n) > n^(log_b(a)) +f(n) > n^(log_b a) ++++ and: @@ -202,7 +202,7 @@ To determine the case, we first calculate: [stem] ++++ -log_b(a) = log_2(2) = 1 +log_b a = log_2 2 = 1 ++++ For case 1, we have: @@ -228,5 +228,5 @@ is: [stem] ++++ -T(n) = Theta(n xx log_n) +T(n) = Theta(n xx log N) ++++ From 80e1617944ced1501a1c7d2c11922d0575152f55 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 21 May 2024 20:47:53 +0000 Subject: [PATCH 23/41] add section on Mergesort with pseudocode --- .../fabio-lama/src/sorting_algorithms.adoc | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc index 68db4586..31031bcf 100644 --- a/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc +++ b/level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/sorting_algorithms.adoc @@ -138,4 +138,14 @@ moves the privot to its final position. ### Pseudocode +. stem:[bb "function " "MergeSort"(A, "int " l, "int " h)] +. stem:[" " bb "if " (l < h)] +. stem:[" " " " tt "mid" = l + "floor"((h-1)//2)] +. stem:[" " " " "MergeSort"(A, l, tt "mid")] +. stem:[" " " " "MergeSort"(A, tt "mid" + 1, h)] +. stem:[" " " " "Merge"(A, l, tt "mid", h])] +. stem:[" " bb "end if"] +. stem:[bb "end function"] +The function stem:["Merge"] creates two arrays of both halves (left and right) +and then merges them to produce a single, sorted array. 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