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zt3?4E@c+O8gH$JB0nfPr3{dG4z`$uY4+g<@|Ep3Y3e1-zh;*9|e7K3DwQ>&nZP=^; zK<*8n^>du^=oSoi%A-hdCMv+$J!V%);PLH-EDGE{ zwg(FqI$2pl?c`p2YwLxB+Yf&#i$isNfY-GaC20rO1__A2Rni?8mVPWcF78!MxN#Om(;w>fqf@jc`gZNQvZ2X=OEW+Q2}*S}v=YO2iL7 zb7_p01b*}6m@KC-duqtvQg!*PTc8{`9aIi4H|T3kuJt?6=>fW92Y|MQ;8Sg$fC^c> zE;Jg{kQg;2(E&-&M58ppkQkGTlL*cP#t6+j;vbpDB!V)5-q5s$f?hFov2dwl1Q4v= zbVeYBrArA84xo9$XMh?2cx|YC+(%F<2w^rTgagnTiVqJdDj+rFgBgAZWgpE0)UdMB zG@^U5AyYIr(sf$nWQU_zGn~wPEKTX5tjIOCgrG1T9ausBq);1x1nd`B&@lI7{Fkg+mOgU1XdY1Uz%Vm%5zXeMnSP*buNJhY7 zz<}wNQc89q4QY2^6*G3V1wY7;c!?ISXuI= z!{NZ)n2?MBEvg05kcs&^3dq16ypRkF9p4C~11%Affl7pA;Os|8MiR!4h!VyMTo()FC1R06I)mm4 z=|>y~7a~zuKj2=3>L8(nY!`5Zkc8R`wjshY97YHO5? 0$ and $n_0 > 0$ such that $forall n >= n_0: 0 <= f(n) <= c g(n)$ + - The intuition is that for a large enough $n$, there is a #link()[function] $g$ and constant $c$, such that $f(n)$ is always lesser than $g$. +=== $Omega-$notation +- _Lower Bound_, that is function grows at least as fast as $c g(n)$ +- $f in O(g)$ if there is $c > 0$ and $n_0 > 0$ such that $forall n >= n_0: 0 <= c g(n) <= f(n)$ +=== $Theta-$notation +- Both upper and lower bounded by $c g(n)$ +- $f in O(g)$ if there is $c_1,c_2 > 0$ and $n_0 > 0$ such that $forall n >= n_0: 0 <= c_1 g(n) <= f(n) <= c_2 g(n)$ + +=== $o$-notation +- Strict upper bound, $0 <= f(n) < c g(n)$ +=== $omega$-notation +- Strict lower bound, $0 <= c g(n) < f(n)$ + +#[ +#set par(justify: true, leading: 0.5em) +=== Orders of Common Functions +- $O(1)$ +- $O(log log n)$ +- $O(log n)$ +- $O((log n)^c)$ +- $O(n^c), 0 < c < 1$ +- $O(n)$ +- $O(n log^* n)$ +- $O(n log n) = O(log n!)$ +- $O(n^2)$ +- $O(n^c)$ +- $O(c^n)$ +- $O(n!)$ +] + +=== Limits +- $limits(lim)_(n arrow.r infinity)(f(n)/g(n)) = 0 arrow.r.double f(n) = o(g(n))$ + - By defn of limits, $limits(lim)_(n arrow.r infinity)(f(n)/g(n)) = 0$ means: + - $forall epsilon > 0, exists n_0 > 0$, s.t. $forall n >= n_0, f(n)/g(n) < epsilon$ + - Hence, $f(n) < c * g(n)$ +- $limits(lim)_(n arrow.r infinity)(f(n)/g(n)) < infinity arrow.r.double f(n) = O(g(n))$ +- 0 < $limits(lim)_(n arrow.r infinity)(f(n)/g(n)) < infinity arrow.r.double f(n) = Theta(g(n))$ +- $limits(lim)_(n arrow.r infinity)(f(n)/g(n)) > 0 arrow.r.double f(n) = Omega(g(n))$ +- $limits(lim)_(n arrow.r infinity)(f(n)/g(n)) = infinity arrow.r.double f(n) = omega(g(n))$ + += Lecture 2 +== Merge Sort +- MERGE-SORT $A[1..n]$ + 1. If $n = 1$, done + 2. Recursively sort $A[1..ceil(n/2)]$ and $A[ceil(n/2)+1..n]$ + 3. Merge the 2 sorted lists +- $T(n) = $ + - $Theta(1)$ if $n = 1$ + - $2T(n/2) + Theta-(n)$ if n > 1 + +== Solving Recurrances +=== Telescoping Method +- For a sequence $sum^(n-1)_(k=0) (a_k - a_k+1) = vec(delim: #none,a_0-a_1,a_1-a_2,a_(n-1)-a_n) = a_0 - a_n$ +- E.g. $T(n) = 2T(n/2)+n$ + - $T(n)/n = T(n/2)/(n/2) + 1$ + - $T(n)/n=T(1)/1+log n$ + - $T(n) = n log n$ +- General Solution + - $T(n) = a T(n/b)+f(n)$ + - $T(n)/g(n) = T(n/b)/g(n/b) + h(n)$ + - And then sum up occurances of $h(n)$. +=== Recursion Tree +- Draw the tree, where each node is the $f(n)$ value. +- Figure out the height of the tree and number of leaves +=== Master Theorem +- Put recurrance in the form $ T(n) = a T(n/b) + f(n) $ +- $a >= 1, b > 1, f$ is asymptotically positive +- Compare $f(n)$ and $n^(log_b a)$ + +Cases ++ $f(n) = O(n^(log_b a - epsilon)), epsilon > 0$ (If $epsilon = 0$, case 2) + - $f(n)$ grows polynomically slower than $n^(log_b a)$ + - $therefore T(n) = Theta(n^(log_b a))$ ++ $f(n) = Theta(n^(log_b a)log^(k)n), k >= 0$ + - $f(n)$ and $n^(log_b a)$ grow similar rates + - $therefore T(n) = Theta(n^(log_b a) log^(k+1)n)$ ++ $f(n) = Omega(n^(log_b a + epsilon)), epsilon > 0$ + - $f(n)$ satisfies $a f(n/b) <= c f(n), c < 1$ + - The regularity condition satisifes that sum of subproblems $< f(n)$ + - $f(n)$ grows polynomically faster than $n^(log_b a)$ + - $therefore T(n) = Theta(f(n))$ + +=== Substitution +- For Upper bound: $T(n) <= c f(n)$ +- For Tight bound: $T(n) <= c_2n^2 - c_1n$ +Solve $T(n) = 4T(n/2) + n$ +- Guess $T(n) = O(n^3)$ + - Constant $c$ s.t. $T(n) <= c n^3, n >= n_0$ +- By induction + - $c = max{2, q}, n_0=1$ + - Base Case ($n = 1$): $T(1) = q <= c (1)^3$ + - Recursive Case ($n > 1$): + - Strong induction, assume $T(k) <= c k^3, n > k >= 1$ + - $T(n) = 4T(n/2) + n <= 4c(n/2)^3 + n = (c/2)n^3 + n <= c n^3$ += Tutorial 2 +== Telescoping +- $T(n) = 4T(n/4) + n/(log n)$ +- $T(n)/n - T(n/4)/(n/4) = 1/(log n)$ +- Let $a_i = T(4^i)/4^i, i = log_4 n, a_i - a_(i-1) = 1/(2i)$ +- $sum^(i-1)_(m=0)(a_(m+1)-a_m) = 1/(2i) +1/(2(i-1)) + ... + 1/2 = 1/2(H_i)$ ($H_i$ is harmonic) +- $a_i - a_0 = O(log i)$ +- $T(4^i) = O(4^i log i), T(n) = O(n log log n)$ += Lecture 3 +== Correctness of Iterative Algos using Invariants +- Initialization: Invariant is true before first iteration of loop +- Maintenance: Invariant is true at start of loop iteration, it is true at start of the _next iteration_ (Induction) +- Termination: Algo at the end gives correct answer +=== Insertion Sort +- Invariant 1: $A[1..i-1]$ are sorted values of $B[1..i-1]$ +- Invariant 2: +== Recursive Algorithms +- Usually use mathematical induction on _size of problem_ +=== Binary Search +- Induction on $"length" n = "ub" - "lb" + 1$ +- Base Case + - ```python if n <= 0: return False``` +- Induction Step: if $n > 0$, `ub >= lb` + - Assume algo works for all values `ub-lb+1 < n` + - `x == A["mid"]: return True`, answer returned correctly + - `x > A[mid]`, then x is in the array iff it is in `A[mid + 1..ub]`, as $A$ is sorted. Thus by induction answer must be `BinarySearch(A, mid+1, ub, x)` + - `x < A[mid]`, then x is in the array iff it is in `A[lb..mid-1]`, as $A$ is sorted. Thus by induction answer must be `BinarySearch(A, lb, mid-1, x)` + - Thus, answer returned is correct + +== Divide and Conquer ++ Divide problem into smaller subproblems ++ Solve subproblems recursively (conquer) ++ Combine / Use subproblem to get solution to full problem + +- $T(n) = a T(n/b) + f(n)$ + - $a$ subproblems + - Each subproblem is size of atmost $n/b$ + - $f(n)$ is time needed to _divide_ problem into subproblems + time to get solution from subproblems _(combine)_ +=== Merge Sort ++ `MergeSort(A[lb..ub])` ++ `If ub = lb` return ++ `If ub > lb` $"mid" = ("ub"+"lb")/2$ ($O(1)$) ++ `MergeSort(A[lb,mid]), MergeSort(A[mid+1,ub])` (2 Subproblems, each $ceil(n/2)$) ++ Merge 2 sorted list (Combining: $Theta(n)$) + +$T(n) = 2T(n/2) + O(n)$ +Complexity: $Theta(n * log n)$ + +=== Powering +- $F(a, n) = F(a, floor(n/2))^2$ if $n$ is even +- $F(a, n) = F(a, floor(n/2))^2 * F(a, 1)$ if $n$ is odd + +$T(n) = T(n/2) + O(1)$ + +=== Fibonacci + + += Tutorial 3 += Lecture 4 += Tutorial 4 += Lecture 5 += Tutorial 5 += Lecture 6 += Tutorial 6 diff --git a/cs3230/template.pdf b/cs3230/template.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bb0f6dafb01ce0cb9fa2df60e933e96e40b1743b GIT binary patch literal 1991 zcmah~O>W~h6rM#_DtZS(g8<3GrhY8{2(ku=HBN&@Gp_BXL4yFLC0f=*5(SEm6|ah( zpa+-(6um+hJxCAHEA&yaV|k{TawGBad%t}A-pRxpxAb;TIr#;aq8La+Z1F*H90egf zf66Js6B1Fmn-ESjfpmaO<=&@ZOfFc7?lcft1FLQHx)$np^mc!sYu3F4jaVi?t3c*n z?L5mE?CnuXw$q)PeGoNKQx z?nj7_#qSY`S3m#p+sCh8;lIE7Ra)P{_I_mW2S11WiaOS{%uoJ)7l#m{tj4rTV@eCQ z<^e5GRhcTWgz;HU0tzCC2f@N)$hNV)eZTlXgQjFc1SFF+FmFq7ITMiDUIw-rh3Omx zJc67Bm}tHT9n=jL*Usj~gznBbU3o z7<7BmOxx^L8CYus{j1?mx|(WsQzgAnxBv8 zD*krM>1_LM9&8_iL8!iQozgDTJO%5hOp~mz%b`l3L8uM?QjAqpLBzvQ1!QZ(V<@NU zIy_w3Q*XRegWXVF36a|vZ#J9u#%wblVSO+dV9mgW(E>!Pc*+DRTUqgXXTYOHz~fw& z5lMq&!Pa7^g3l_LFnBhauX$3L3FO5;5<5L6@pa&<;UI+W6@cd+sPv~{CUK0AF{{ugK`79Q=S4{#j*`Y z2`LIU=Zrx*wc$_B9LUrL62GCm00kGg+>zNgWp$p7?%rh%4+j38-<*4Rv*3TIa(I#d zL809vdZ-Cu1Z0q~Nb*5-Plq9K6Vnal6Hb>Drbm-Ny_f?x%D0BE4LMLM5{0Q!a|@C* zG;LC02}TqR^%U)Sp{`|_7FvF@mjUUa>J(+qGEIk&