Finiteness: Algorithm must complete after a finite number of instructions have been executed
absence of Ambiguity: Each step must be clearly defines, having only one interpretation
definition of Sequence: Each step must have a unique defined preceding and succeeding step. The first step (start step) and last step
(halt step) must be clearly noted.
input/Output: Number and types of required inputs and results must be specified.
Feasibility: It must be possible to perform each instruction
logn, n, nlogn, n^2, n^3, 2^n
Big-O: Suppose that f(n) and g(n) are nonnegative functions of n. Then we say that f(n) is O(g(n)) provided that there are constants C
> 0 and N > 0 such that for all n > N, f(n) <= Cg(n).
-expresses and upper bound
-Theorems: 1.) K is O(1)
2.) A polynomial is O(the term containing the highest power of n)
3.) K*f(n) is O(f(n)) (i.e. you can drop constants)
4.) If f(n) is O(g(n)) and g(n) is O(h(n)) then f(n) is O(h(n)) [transitivity]
5.) K, logb(n), n, nlogb(n), n^2, n^higher power, 2^n, 3^n, larger constants^n, n!, n^n
6.) f(n) is O(the dominant term of f(n))
7.) for any base b, logb(n) is O(log(n))
Big-Omega: Suppose that f(n) and g(n) are nonnegative functions of n. Then we say that f(n) is
๎ถ๎g๎n๎๎
provided that there are
constants C > 0 and N > 0 such that for all n > N, f(n) >= Cg(n).
-expresses a lower bound
Big-Theta: Suppose that f(n) and g(n) are nonnegative functions of n. Then we say that f(n) is
๎ฎ๎ g๎n๎๎
provided that f(n) is
O(g(n)) and also that f(n) is
๎ถ๎g๎n๎๎
.
-Theroems:
8.) f(n) is
๎ฎ๎ g๎n๎๎
if
lim
n๎โ
f๎n๎
g๎n๎=c , where 0๎c๎ โ
1. corollary: if the limit above is 0 then f(n) is O(g(n)), and if the limit is infinity then f(n) is
๎ถ๎g๎n๎๎
.
9.) if f(n) is
๎ฎ๎ g๎n๎๎
then
lim
n๎โ
f๎n๎
g๎n๎=c , where 0๎c๎โ
provided that the limit exists
10.) If K > 0 is a constant, then K is
๎ฎ๎1๎
.
11.) A polynomial is
๎ฎ๎ the highest power of n๎
12.) K * f(n) is
๎ฎ๎ f๎n๎๎
[i.e. constants coefficients can be dropped]
13.) In general, f(n) is
๎ฎ
of the dominant term of f(n), where โdominantโ may usually be determined from theorem 5
14.) for any base b, logb(n) is
๎ฎ๎ log ๎n๎ ๎
)
15.) If f(n) is
๎ฎ๎ g๎n๎๎
and g(n) is
๎ฎ๎ h๎n๎๎
then f(n) is
๎ฎ๎ h๎n๎๎
16.) If f(n) is
๎ฎ๎ g๎n๎๎
then g(n) is
๎ฎ๎ f๎n๎๎
17.) If f(n) is
๎ฎ๎ f๎n๎๎
-by theorems 15-17,
๎ฎ
is an equivalence relation on the set of positive-valued functions
AVL Tree: a binary search tree in which the heights of the left and right subtrees of the root differ by at most 1, and in which the left
and right subtrees are themselves AVL trees.
Each node has a balance factor indicating the relative heights of left/right subtrees