The time complexity and spatial complexity of the sorting algorithm commonly used in "Computer Foundation"

Time complexity and spatial complexity of commonly used sorting algorithms

Sorting method	Worst-time analysis	Average Time complexity	Degree of stability	Complexity of space
Bubble sort	O (N2)	O (N2)	Stability	O (1)
Quick Sort	O (N2)	O (N*LOG2N)	Not stable	O (log2n) ~o (n)
Select sort	O (N2)	O (N2)	Stability	O (1)
Binary Tree Sorting	O (N2)	O (N*LOG2N)	Top	O (N)
Insert Sort	O (N2)	O (N2)	Stability	O (1)
Heap Sort	O (N*LOG2N)	O (N*LOG2N)	Not stable	O (1)
Hill sort	O	O	Not stable	O (1)

1, time complexity  
(1) Time frequency an algorithm takes time to execute, theoretically it can not be calculated, it must be run on the computer test to know. But we can not and do not need to test each algorithm, just know which algorithm spends more time, which algorithm spends less time on it. And the time that an algorithm spends is proportional to the number of executions of the statement in the algorithm, which algorithm takes more time than the number of statements executed. The number of times a statement is executed in an algorithm is called a statement frequency or time frequency. Note as T (N).  
(2) time complexity in the time frequency mentioned just now, N is called the scale of the problem, and when N is constantly changing, the time frequency t (n) will change constantly. But sometimes we want to know what the pattern is when it changes. To do this, we introduce the concept of time complexity. Under normal circumstances, the number of iterations of the basic operation of the algorithm is a function of the problem size n, denoted by T (n), if there is an auxiliary function f (n), so that when n approaches infinity, the limit value of T (n)/f (n) is not equal to zero constant, then f (n) is the same order of magnitude function of t As T (n) =o (f (n)), called O (f (n)) is the progressive time complexity of the algorithm, which is referred to as the complexity of time.  
in a variety of different algorithms, if the In addition, the time frequency is different, the time complexity may be the same, such as T (N) =n2 +3n+4 and T (n) =4n2+2n+1 have different frequency, but the time complexity is the same, all O (N2). In order of magnitude increment, the common time complexity is: Constant order O (1), Logarithmic order O (log2n), linear order O (n), linear logarithmic order O (nlog2n), square order O (n2), Cubic O (n3),..., K-order O (NK), exponent-order (2n). With the increasing of the problem scale N, the complexity of the time is increasing and the efficiency of the algorithm is less. 2, spatial complexity is similar to the time complexity, the spatial complexity is the measurement of the storage space required when the algorithm executes in the computer. Note: S (n) =o (f (n)) We are generally talking about the size of the secondary storage unit in addition to the normal memory overhead. The discussion method is similar to the complexity of time, and is not discussed.  
(3) The time performance of progressive time complexity evaluation algorithm is mainly used to evaluate the time performance of an algorithm based on the order of magnitude of time complexity (i.e., the asymptotic time complexity of the algorithm).

2. Similar to the discussion of time complexity, an algorithm's spatial complexity (space complexity) S (n) is defined as the storage space consumed by the algorithm, and it is also a function of the problem size n. Asymptotic spatial complexity is also often referred to as spatial complexity.
Spatial complexity (space complexity) is a measure of the amount of storage space that is temporarily occupied by an algorithm while it is running. The storage space occupied by an algorithm in the computer memory, including the storage space occupied by the storage algorithm itself, the storage space occupied by the input and output data of the algorithm and the storage space occupied by the algorithm in the running process three aspects. The storage space occupied by the input and output data of the algorithm is determined by the problem to be solved, which is passed by the calling function by the parameter table, and it does not change with the algorithm. Storage algorithm itself occupies the storage space and the length of the algorithm written in proportion, to compress the storage space, you must write a shorter algorithm. Algorithm in the running process of temporary occupied storage space varies with the algorithm, some algorithms only need to occupy a small amount of temporary work units, and does not vary with the size of the problem, we call this algorithm "in-place/", is to save the memory of the algorithm, as described in this section of the algorithm is so , some algorithms need to occupy the number of temporary work and solve the problem of the size of N, it increases with the increase of N, when n is large, will occupy more storage units, such as in the Nineth chapter described in the Quick Sort and merge sorting algorithm is the case.

If the spatial complexity of an algorithm is a constant, that is, it does not change with the size of the processed data n, it can be represented as O (1); when the spatial complexity of an algorithm is proportional to the logarithm of the 2 base n, it can be represented as 0 (10g2n) When an algorithm's empty I-division complexity is linearly proportional to n, it can be represented as 0 (n). If the parameter is an array, it is only necessary to allocate a space for it to store an address pointer transmitted by the argument, that is, a machine word space, and if the parameter is a reference, you only need to assign it a space to store one It is used to store the address of the corresponding argument variable so that the argument variable is automatically referenced by the system.

The time complexity and spatial complexity of the sorting algorithm commonly used in "Computer Foundation"