Reading Notes: Introduction to Algorithms in Chapter 1 of "Algorithm Diagrams"

Binary search #

Binary search is half search, and it is effective when the queue list is ordered.

For a list of n elements, binary search requires at most$lO{g}_{2}n$ steps, a simple sequential search requires at most n steps.

log #

Logarithm: The logarithmic operation is the inverse of the exponentiation operation

$N=a^x(a>0,a\ne 1)$, $x$that is$a$base$N$logarithm, denoted as$x={\log }_{a}N$,in:

• $a$ : end
• $N$ : Antilogarithm
• $x$ : by$a$base$N$logarithm of

power:

log refers to both ${\log }_2$

$\log 8$ = ${\log }_{2}8$ = 3 ($2^3=8$)

1. The logarithm to the base 10 is called the common logarithm, denoted as$\lg$
2. with irrational numbers$e$($e=2.71828\dots$) base logarithm is called natural logarithm, denoted as$\ln$
3. Zero has no logarithm
4. In the range of real numbers, negative numbers have no logarithm; in the range of complex numbers, negative numbers have logarithms

Time Complexity #

Practical complexity of simple sequential search $O\left(n\right)$

time complexity of binary search $O\left(\log n\right)$

Time complexity represents the worst-case running time

Common time complexity #

• $O\left(\log n\right)$ logarithmic time
• $O\left(n\right)$ linear time
• $O(n\times \log n)$
• $O\left(n^2\right)$
• $O(n!)$ factorial of n

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