kristoferssolo/Theory-of-Algorithms-Cheatsheet
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@ -802,9 +802,9 @@ saprātīgie deterministiskie skaitļošanas modeļi ir polinomiāli ekvivalenti
- Dots grafs $G$ un divas virsotnes $u$, $v$. - Dots grafs $G$ un divas virsotnes $u$, $v$.
- Jautājums: vai eksistē ceļš no $u$ uz $v$? - Jautājums: vai eksistē ceļš no $u$ uz $v$?
- Rupjais-spēks: pārbaudīt visus ceļus — eksponenciāls laiks. - Rupjais-spēks: pārbaudīt visus ceļus -- eksponenciāls laiks.
- Efektīvs algoritms: meklēšana plašumā (breadth-first search); laika - Efektīvs algoritms: meklēšana plašumā (breadth-first search); laika
sarežģītība: $O(|V| + |E|)$. sarežģītība: $O(abs(V) + abs(E))$.
== Piemērs ($"RELPRIME"$) == Piemērs ($"RELPRIME"$)
@ -820,7 +820,7 @@ saprātīgie deterministiskie skaitļošanas modeļi ir polinomiāli ekvivalenti
#NP (nederminēti-polinomiālas) problēmas #NP (nederminēti-polinomiālas) problēmas
ir problēmas (2 ekvivalentas definīcijas): ir problēmas (2 ekvivalentas definīcijas):
+ $L in NP$, ja eksistē pārbaudes algoritms - $O(n^c)$ laika Tjūringa mašīna $M$: + $L in NP$, ja eksistē pārbaudes algoritms -- $O(n^c)$ laika Tjūringa mašīna $M$:
+ Ja $L(x) = 1$, tad eksistē y: $M(x, y) = 1$. + Ja $L(x) = 1$, tad eksistē y: $M(x, y) = 1$.
+ Ja $L(x) = 0$, tad visiem y: $M(x, y) = 0$. + Ja $L(x) = 0$, tad visiem y: $M(x, y) = 0$.
+ _Informācija $y$ var saturēt brīvi definētu informāciju._ + _Informācija $y$ var saturēt brīvi definētu informāciju._
@ -1006,39 +1006,40 @@ Vārdiski. Jauns grafs $G$, kurā ir visas virsotnes no $V$, bet
visas šķautnes, kas ir $G$ nav $G'$ un pretēji -- visas šķautnes visas šķautnes, kas ir $G$ nav $G'$ un pretēji -- visas šķautnes
kā nav $G$ ir $G'$. kā nav $G$ ir $G'$.
#figure(diagram( #figure(
cell-size: 1mm, diagram(
node-stroke: 0pt, cell-size: 1mm,
spacing: 1em, node-stroke: 0pt,
node-shape: circle, spacing: 1em,
// phantom location nodes node-shape: circle,
dot-node((0, 0), <b1>), // phantom location nodes
dot-node((-2, 1), <a1>), dot-node((0, 0), <b1>),
dot-node((0, 2), <c1>), dot-node((-2, 1), <a1>),
dot-node((0, 2), <c1>),
dot-node((0, 4), <b2>), dot-node((0, 4), <b2>),
dot-node((-2, 5), <a2>), dot-node((-2, 5), <a2>),
dot-node((0, 6), <c2>), dot-node((0, 6), <c2>),
// label nodes // label nodes
node((rel: (0.7em, 0.7em), to: <b1>), $B$), node((rel: (0.7em, 0.7em), to: <b1>), $B$),
node((rel: (-0.7em, 0em), to: <a1>), $A$), node((rel: (-0.7em, 0em), to: <a1>), $A$),
node((rel: (0.7em, 0.7em), to: <c1>), $C$), node((rel: (0.7em, 0.7em), to: <c1>), $C$),
node((1, 1), text(green, $G$)), node((1, 1), text(green, $G$)),
node((rel: (0.7em, 0.7em), to: <b2>), $B$), node((rel: (0.7em, 0.7em), to: <b2>), $B$),
node((rel: (-0.7em, 0em), to: <a2>), $A$), node((rel: (-0.7em, 0em), to: <a2>), $A$),
node((rel: (0.7em, 0.7em), to: <c2>), $C$), node((rel: (0.7em, 0.7em), to: <c2>), $C$),
node((1.6, 5), text(green, $G "papildinājums"$)), node((1.6, 5), text(green, $G "papildinājums"$)),
edge(<a1>, <b1>), edge(<a1>, <b1>),
edge(<c1>, <b1>, "--", stroke: yellow), edge(<c1>, <b1>, "--", stroke: yellow),
edge(<c1>, <a1>), edge(<c1>, <a1>),
edge(<a2>, <b2>, "--", stroke: yellow), edge(<a2>, <b2>, "--", stroke: yellow),
edge(<c2>, <b2>), edge(<c2>, <b2>),
edge(<c2>, <a2>, "--", stroke: yellow), edge(<c2>, <a2>, "--", stroke: yellow),
)), ),
caption: "Papildgrafa piemērs", caption: "Papildgrafa piemērs",
) )
Ir spēkā sakarība $"INDSET"(G, k) = "CLIQUE"(G', k)$. Ir spēkā sakarība $"INDSET"(G, k) = "CLIQUE"(G', k)$.
@ -1118,26 +1119,26 @@ Ir spēkā sakarība $"INDSET"(G, k) = "CLIQUE"(G', k)$.
[Summa], [$ f(x) + g(x) $], [$ f'(x) + g'(x) $], [Summa], [$ f(x) + g(x) $], [$ f'(x) + g'(x) $],
[Starpība], [$ f(x) - g(x) $], [$ f'(x) - g'(x) $], [Starpība], [$ f(x) - g(x) $], [$ f'(x) - g'(x) $],
[Reizinājums], [$ f(x) * g(x) $], [Reizinājums], [$ f(x) dot g(x) $],
[ [
$ $
f'(x) * g(x) + \ f'(x) dot g(x) + \
f(x) * g'(x) f(x) dot g'(x)
$ $
], ],
/* /*
[Quotient Rule], [$ (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2 $], [$ (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2 $], [Quotient Rule], [$ (f'(x) dot g(x) - f(x) * g'(x)) / (g(x))^2 $], [$ (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2 $],
[Chain Rule], [$ f(g(x)) $], [$ f'(g(x)) * g'(x) $], [Chain Rule], [$ f(g(x)) $], [$ f'(g(x)) dot g'(x) $],
[Euler’s Number Exponent Rule], [$ e^x $], [$ e^x $], [Euler’s Number Exponent Rule], [$ e^x $], [$ e^x $],
[Constant Exponent Rule], [$ a^x $], [$ a^x * ln(a) $], [Constant Exponent Rule], [$ a^x $], [$ a^x dot ln(a) $],
[Natural Log Rule], [$ ln(x) $], [$ 1 / x $], [Natural Log Rule], [$ ln(x) $], [$ 1 / x $],
[Logarithm Rule], [$ log_a(x) $], [$ 1 / (x * ln(a)) $], [Logarithm Rule], [$ log_a(x) $], [$ 1 / (x dot ln(a)) $],
[Sine Rule], [$ sin(x) $], [$ cos(x) $], [Sine Rule], [$ sin(x) $], [$ cos(x) $],
[Cosine Rule], [$ cos(x) $], [$ -sin(x) $], [Cosine Rule], [$ cos(x) $], [$ -sin(x) $],
[Tangent Rule], [$ tan(x) $], [$ sec^2(x) $], [Tangent Rule], [$ tan(x) $], [$ sec^2(x) $],
[Cosecant Rule], [$ csc(x) $], [$ -csc(x) * cot(x) $], [Cosecant Rule], [$ csc(x) $], [$ -csc(x) dot cot(x) $],
[Secant Rule], [$ sec(x) $], [$ sec(x) * tan(x) $], [Secant Rule], [$ sec(x) $], [$ sec(x) dot tan(x) $],
[Cotangent Rule], [$ cot(x) $], [$ -csc^2(x) $], [Cotangent Rule], [$ cot(x) $], [$ -csc^2(x) $],
*/ */
) )