kristoferssolo/Theory-of-Algorithms-Cheatsheet
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Kristofers Solo 2025-06-15 11:06:08 +03:00
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@ -552,11 +552,11 @@ QED.
Notācija, kas tiek izmantota, lai raksturotu *funkciju* sarežģītību
asimptotiski.
=== Lielais-O (formālā definīcija)
=== Lielais-$O$ (formālā definīcija)
$f(n) in O(g(n))$, ja:
$exists C > 0, exists n_0 > 0:$ $(forall n >= n_0: f(n) <= c * g(n))$
$exists C > 0, exists n_0 > 0:$ $(forall n >= n_0: f(n) <= c dot g(n))$
Tas nozīmē, ka funkcija $f(n)$ asimptotiski nepārsniedz konstanti $c$ reizinātu
$g(n)$.
@ -767,7 +767,7 @@ Laiks $O(f(N)) ->$ atmiņa $O(f(N))$.
=== Asimptotiskas augšanas hierarhija
Sekojošas funkcijas pieaugums pie $x -> infinity$:
Sekojošas funkcijas pieaugums pie $x -> oo$:
$log(x) << x << x dot log(x) << x^k << a^x << x! << x^x$
@ -1072,13 +1072,14 @@ Ir spēkā sakarība $"INDSET"(G, k) = "CLIQUE"(G', k)$.
log_8(3x-4)=log_8(5x+2) \
"so," 3x-4=5x+2
$,
[Pow. to log],
$
a^(log_a (x)) = x
$,
$
2^(log_2 (x)) = x
$
$,
))
]
@ -1120,8 +1121,8 @@ Ir spēkā sakarība $"INDSET"(G, k) = "CLIQUE"(G', k)$.
[Reizinājums], [$ f(x) * g(x) $],
[
$
f'(x) * g(x) + \
f(x) * g'(x)
f'(x) * g(x) + \
f(x) * g'(x)
$
],
@ -1151,10 +1152,10 @@ Ir spēkā sakarība $"INDSET"(G, k) = "CLIQUE"(G', k)$.
columns: 2,
[*Rule Name*], [*Formula*],
[Reizinājums], [$ a^m * a^n = a^(m+n) $],
[Reizinājums], [$ a^m dot a^n = a^(m+n) $],
[Dalījums], [$ a^m / a^n = a^(m-n) $],
[Pakāpes pakāpe], [$ (a^m)^n = a^(m*n) $],
[Reizinājuma pakāpe], [$ (a*b)^m = a^m * b^m $],
[Pakāpes pakāpe], [$ (a^m)^n = a^(m dot n) $],
[Reizinājuma pakāpe], [$ (a dot b)^m = a^m dot b^m $],
[Dalījuma pakāpe], [$ (a/b)^m = a^m / b^m $],
[0-pakāpe], [$ a^0 = 1 $],
[Negatīva pakāpe], [$ a^(-m) = 1 / a^m $],
@ -1171,8 +1172,8 @@ $
sum_(i=1)^(n) i^2 = (n(n+1)(2n+1))/(6)\
sum_(i=1)^(n) i^3 = ( (n(n+1))/(2))^2 \
// Geometric series (ratio r \neq 1)
r > 1: sum_(i=0)^(n) a*r^i = a * (r^(n+1)-1)/(r-1) quad \
r < 1: sum_(i=0)^(infinity) a*r^i = (a)/(1-r) \
r > 1: sum_(i=0)^(n) a dot r^i = a dot (r^(n+1)-1)/(r-1) quad \
r < 1: sum_(i=0)^(oo) a dot r^i = (a)/(1-r) \
// Logarithmic sum
sum_(i=1)^(n) log i = log(n!) approx n log n - n + O(log n) \
// Exponential sum (appears in brute-force algorithms)