Master Theorem

$T(n)\; =\; a\; T\backslash left(\backslash frac\{n\}\{b\}\backslash right)\; +\; f(n)\backslash ,$

$a\; \backslash ge\; 1\backslash ,$

$b\; >\; 1\backslash ,$

$\backslash frac\{n\}\{b\}\backslash ,$

$\backslash left\backslash lfloor\; \backslash frac\{n\}\{b\}\; \backslash right\backslash rfloor\backslash ,$ (the floor function)

$\backslash left\backslash lceil\; \backslash frac\{n\}\{b\}\; \backslash right\backslash rceil\backslash ,$ (the ceiling function)

$\backslash frac\{n\}\{b\}.$

$T(n)\backslash ,$

$f(n)\; \backslash in\; O\backslash left(\; n^\{\backslash log\_b\; a\; -\; \backslash varepsilon\}\; \backslash right)$ (the big-O notation)

$\backslash varepsilon\; >\; 0\backslash ,$

$T(n)\; \backslash in\; \backslash Theta\backslash left(\; n^\{\backslash log\_b\; a\}\; \backslash right).\backslash ,$

$f(n)\; \backslash in\; \backslash Theta\backslash left(\; n^\{\backslash log\_b\; a\}\; \backslash right)\backslash ,$

$T(n)\; \backslash in\; \backslash Theta\backslash left(\; n^\{\backslash log\_b\; a\}\; \backslash log(n)\backslash right).\backslash ,$

$f(n)\; \backslash in\; \backslash Omega\backslash left(\; n^\{\backslash log\_b\; a\; +\; \backslash varepsilon\}\; \backslash right)$

$\backslash varepsilon\; >\; 0$

$a\; f\backslash left(\; \backslash frac\{n\}\{b\}\; \backslash right)\; \backslash le\; c\; f(n)$

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