# 1, Algorithm

++n

n + + differences

int n=1,n1,n2;

n1=++n;

System.out.println(n1);

System.out.println(n);

n2=n++;

System.out.println(n2);

System.out.println(n);

2

2

2

3

## 1. Concept

Philosophy of writing algorithms:

From simple to complex: verify step by step and print more intermediate results

Part first and then whole: subdivide when there is no idea

Rough first and then fine: variable renaming, statement merging and boundary processing (length-1 and length-2) are all tried. You can write a non tangled and refined method first

Philosophy of de bug ging:

Step 1: read through the program to make sure you understand the logic of the whole program and you can't read the bug

Step 2: if there is still a bug, output the intermediate value

Step 3: if there are still bug s, reduce the functions and leave the core

In the worst case, you can't do it just by looking at it

algorithm: Algorithms for different solutions to the same problem are often specific to specific data structures

Data Structure: different ways of storing data in Data Structure: a tube of apples (there is no gap in the middle of the array, data cannot be inserted into the linked list, and data can be inserted) a basket of apples

The search is much faster than the linked list for the array. The linked list should look down the chain one by one

Big O: academic time representation to determine the quality of the algorithm

Time measurement: the shorter the time, the better. The time is calculated by the time difference (System.currentTimeMillis() prints out the start time and end time respectively). However, for some not very complex algorithms, the time difference may be rounded to 1s. This time can put the algorithm in the cycle and increase the time, that is, the amplitude is not enough to round up

Space calculation: counting the apples in the basket needs the help of another basket, which takes up other space. The less space, the better the algorithm

Time complexity: the complexity of time is the change law of time with the expansion of the problem scale

O (ignoring the higher-order term of the coefficient)

The sample is large enough, and those are not very important, so they can be ignored

Represents the worst time complexity

There are average and best, these interviews are not tested

Spatial complexity: with the expansion of the scale of the problem, the change law of space is to see the additional space rather than the allocation of spatial variables necessary for the problem. Those local variables that must be allocated are not included in the spatial complexity

Operation not to be considered: program initialization with initial value

Ignore: O(2n) and O(n) are almost the same. Ignoring 2 O(n^2+2n) is also directly ignoring 2n

eg1. For an array with a size of 10 and an array with a size of 10000, there is no difference in accessing the last element of the array

That is, with the expansion of the scale of the problem, the time basically does not change, expressed as O(1), a constant

eg2. For accessing a certain location in the linked list (if it is all location 1, it must be O(1), but we generally say that the time complexity is the worst case), so it is assumed that the value scale of the last location in the linked list is 10 1s and the scale is 100 10s O(n) linear change

The most common are O(1) and O(n). If the last position of the access linked list is cycled twice, it is O(n2), and the third cycle is O(n3)

eg2: time complexity of averaging an array

O(n) because you want to add up all the numbers of the array

## 2. sort

Sorting is to arrange a group of numbers without order from small to large or from large to small

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The four most important types are: inserting, stacking, merging and fast sorting

Selective bubble insertion

Quick return to the Greek statistical base

EN Fang, en Lao, en 13

Yes, Enga k, Enga times k

Unstable, unstable, unstable

Unstable unstable stable

### 2.1 select sort

The so-called selective sorting is to continuously select the smallest (largest)

The simplest and most useless (time complexity is too high O(n^2) and unstable) sorting algorithm has a certain optimization space

By comparing the subscript that continuously determines the minimum value, this subscript is constantly changing. Traverse the subscript that determines the minimum value and exchange the value at the first position. The second traverse starts from the second number, then exchanges with the second value, and so on.

1. Treatment process

package com.liu.algorithm.sort; public class SelectionSort { public static void main(String[] args) { int[] arr={4,3,5,1,9,6,7,2,8}; //Optimization 1: handle the boundary value, change arr.length to arr.length-1, and the outer cycle will be less than once for (int i=0;i<arr.length-1;i++){ int minPostion=i; for (int j=i+1;j<arr.length;j++){ //Optimization 2: statement merging // if (arr[j]<arr[minPostion]){ // minPostion=j; // } minPostion=arr[j]<arr[minPostion]?j:minPostion; } // System.out.println("the subscript of the minimum value is:" + minpostion "); //Optimization 5: two position exchange refining method swap(arr,i,minPostion); //Optimization 3: print the sorting results after each cycle System.out.println("This is the second"+i+"After one loop, the contents of the array"); print(arr); //Optimization 4: repeated code writing method // for (int k =0;k<arr.length;k++){ // System.out.print(arr[k]+" "); // } System.out.println(); } System.out.println("The final result is:"); print(arr); // for (int i=0;i<arr.length;i++){ // System.out.print(arr[i]+" "); // } // for (int arrays:arr){ // System.out.print(arrays+" "); // } } static void swap(int[] arr,int i,int j){ int temp=arr[i]; arr[i]=arr[j]; arr[j]=temp; } static void print(int[] arr){ for (int i=0;i<arr.length;i++){ System.out.print(arr[i]+" "); } } } /* Step 1: define an array (array initialization) Step 2: print out the array Step 3: print out the subscript of the smallest group of elements (compare the first element with the remaining elements one by one, and constantly refresh the defined subscript value) Step 4: exchange the position between the smallest element and the first element. At this time, the first element is arranged Step 5: repeat the above process and compare the remaining elements, that is, put a for loop on the outermost layer Step 6: Optimization: the code for exchanging and printing arrays is abstracted into methods, and the outermost boundary value of ternary operators prints out the results of each cycle Step 7: explore deep optimization, that is, you can find the minimum and maximum values at the same time, put them at both ends, or compare the first element with the second and third elements at the same time */

2. Complete code

package com.liu.algorithm.sort; public class SelectionSort_1 { public static void main(String[] args) { int[] arr = {31, 33, 38, 22, 45, 16, 90, 35, 90}; for (int i = 0; i < arr.length - 1; i++) { int minPostion = i; for (int j = i + 1; j < arr.length; j++) { minPostion = arr[j] < arr[minPostion] ? j : minPostion; } swap(arr, i, minPostion); System.out.println("This is the second" + (i + 1) + "Results after cycle"); print(arr); System.out.println(); } System.out.println("======This is the final result======"); print(arr); } static void swap(int[] arr, int i, int j) { int temp = arr[j]; arr[j] = arr[i]; arr[i] = temp; } static void print(int[] arr) { for (int i = 0; i < arr.length; i++) { System.out.print(arr[i] + " "); } } }

### (1) bigO analysis

Find the one that consumes the most time for analysis. You don't need to analyze all of them. If there are two for loops, it depends on the execution time of the statement in the middle of the two for loops

The best and worst are O(n^2). The so-called best is that it has been sorted, but it still needs to traverse all the numbers in the array

Space complexity O(1) indicates that no additional space is occupied

Instability is that two equal numbers are sorted, and their relative order may change after sorting

How to prove that the selection sorting is unstable:

5 6 5 2 1 the order of the second 5 will be in front of the first 5

Consequences of instability: if Xiao Wang and Lao Wang, who are also bank customers, are of the same age, but the bank's deposits and membership levels are different, if the order is reversed, some business logic of the bank may be disrupted. For example, in the case of the same age, people with high membership levels can handle well, resulting in logical contradictions.

### (2) Verification algorithm - logarithm

Logarithm: compare what you write with what the system writes (Arrays.sort(arr);) and the output result is a logarithm

How to verify the correctness of the algorithm?

- Visual observation (the test program can be used for real engineering calculation, and it is impossible to cite all cases)
- Generate enough random samples
- Calculate the sample results with the correct algorithm
- Verify with your own algorithm and compare the verification results

### 2.2 bubble sorting

Bubbling is to compare the smallest or largest number to the last

Spatial complexity: O(1) does not involve any additional space

Time complexity: O(n^2)

Stability: the stable pairwise comparison is not the jumping comparison like the selective sorting, but may jump to the back

Optimize bubble sorting: the best time complexity is O(1)

### 2.3 insert sort (*)

It is better to use insert sort for basically ordered arrays, which is twice as fast as bubble sort and faster than selection sort

stable

Time complexity:

O(n^2) (6 5 4 3 2 1) compare n+n-1+n-2 + 1 arithmetic sequence

It is better to compare O(n) and find that there is no need to exchange without moving (1 2 3 4 5)

Compare once

Space complexity: O(1)

Insertion sorting is to compare the second element to the last element, and insert and move slowly

It's kind of like the bubble sort ahead

### 2.4 summary selection bubble insertion

Play the game called Fighting the Landlord

The space complexity of bubble selection and insertion is O(1) and the time complexity is O(n^2). The last two are O(n)

Bubbling: basically no pairwise comparison, pairwise exchange is too slow

Choice: basically not unstable

Insertion: the efficiency is high when the sample is small and basically orderly

### 2.5 Hill sorting

I don't use too much, and I don't use too much for the interview

Improved insertion Sort Hill sort is a special insertion sort with interval > = 1. The interval of ordinary insertion sort is 1

When the interval is large, the number of moves is relatively small, the moving distance is relatively long, and the jumping row is unstable

Take out a number every other interval to form a new array

The optimization of hill sorting is mainly reflected in the interval sequence

At first, it takes half the length of the array as the interval sequence, and then half, half and so on

Knuth sequence

H = 3 * H + 1 h = 1, 1, 4, 7, 10, 13, H max. < = 1 / 3 of the entire array length

Time complexity: best O(n^1.3)

Space complexity: O(1)

Stability:

### 2.6 merge sort (*)

### 2.6.1 recursion

A method calls itself in its internal execution process. Dogs bite their own tail. Strictly speaking, they are not the same method, because the parameters are different.

Recursion requires a stop condition, that is, Base case

package com.liu.algorithm.sort; public class Recursion { public static void main(String[] args) { System.out.println(f(10)); } static long f(int n){ //Without these two conditions, stack overflow will occur if (n<1){ return -1; } if (n==1){ return 1; } return n+f(n-1); } } Operation result: 5

merge sort

Special for Java object sorting (TIM SORT improved merge sorting)

Continue to divide into half, divide until it can't be divided again (that is, there are only two numbers or one number in the array), and then merge (the premise of merging is that the two arrays have been arranged in order)

Time complexity: because it is constantly divided into two, logarithmic function

Space complexity:

Stability:

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java object sorting

- Object sorting generally requires more stable object attributes

### 2.7 quick sort (*)

Single shaft fast exhaust

Double axis fast exhaust

Now there is an array with the last number as the reference. Those less than it are placed on the left and those greater than it are placed on the right. Of course, both the left and right are to the left of the last number

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java internal Array double axis fast sorting, which is an improved algorithm for fast sorting

Arrays.sort(arr);

### 2.8 counting and sorting

Non comparative sorting is suitable for some special cases

A kind of bucket thought

Algorithm idea: large quantity and small range

- The age of tens of thousands of employees in a large company is 18-80
- College entrance examination score 0-750

The number of occurrences of each number is taken as the element value of the new array, and the size of the value is taken as the subscript value of the new array, and then these numbers are re formed into a new array from small to large

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Result array: 001222223334455566667777899

Space complexity: n+k(k buckets)

Time complexity: n+k (n)

### 2.9 cardinality sorting

Non comparison sort

A kind of bucket thought

Multi keyword sorting

The first place is ten and the first place is hundred

Space complexity: O(n)

Time complexity: O(n)

Stable sorting

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### 2.10 bucket sorting

Maximum value - minimum value = difference is divided into buckets according to the difference. One bucket represents a range, and each bucket is sorted separately

unimportance

## 3. Binary sorting

Find whether a number exists in an ordered array

Keep cutting in half

How many times

Look at 2 ^ (several times) = array length

Several times = log2 array length

The complexity is (log2 array length) = log2n

This means that the log is based on 2

logn is based on 2 by default in the computer

Disordered dichotomy

### 4. XOR

exclusive OR is abbreviated as xor or eor

The difference is 1. The symbol is ^ no carry addition

0^N== N

N^N===0

XOR satisfies commutative law and associative law (the result of the same number XOR must be the same)

The same position only depends on the even number or odd number of 1, which has nothing to do with the order

Use of XOR

Title 1: int a=x;

int b=y；

Let the value of ab exchange

a=a^b

b=a^b

a=a^b

That's it

This is true if the values are the same, but not if the memory is the same

package com.liu.algorithm.erfen; /** * @author : Liu Ke * @date : 2021-12-01 23:23 **/ public class Demo { public static void main(String[] args) { int[] arrs={1,2,3}; swap(arrs,0,0); System.out.println(arrs[0]); System.out.println(arrs[1]); } public static void swap(int[] arrs,int i,int j){ arrs[i]=arrs[i]^arrs[j]; System.out.println("arr[0]="+arrs[0]); arrs[j]=arrs[i]^arrs[j]; System.out.println("arr[0]="+arrs[0]); arrs[i]=arrs[i]^arrs[j]; } } Topic 2: In an array, only one number is odd and the others are even Find this number Take all the numbers in the array out of XOR, and the most result is this number The XOR result between even numbers is 0 The odd number is the last one int xor=0； int xor=xor^a[0] int xor=xor^a[1] . . int xor =xor^a[n] Topic 3: Binary number N，Take its rightmost 1 N&(Reverse N+1) Topic 4: Two numbers in a group appear odd times, and the other numbers appear even times. Find these two numbers int xor=a^b!=0 therefore xor The first t Bit must be 1, so either a The first t Bit is 1, or b The first t Bit 1.(Find the confirmation with 1 on the far right t In fact, you don't have to find the position. The rightmost is 1. Any 1 is OK) Divide the whole group into two groups. One group is the second group t Bit is 1, and one group is the second t Bit 0. One of the XOR groups gets a or b，xor',Then another number is xor^xor' ![Insert picture description here](https://img-blog.csdnimg.cn/1a8234ff5f384570937478b1ea5476e0.png?x-oss-process=image/watermark,type_d3F5LXplbmhlaQ,shadow_50,text_Q1NETiBA5Y-v5Y-v6KW_6YeM55qE6aOe6KGM,size_19,color_FFFFFF,t_70,g_se,x_16) expand: Find a number, how many 1 ![Insert picture description here](https://img-blog.csdnimg.cn/c19d5306af5d49eaacedc8a5e90fb2f5.png?x-oss-process=image/watermark,type_d3F5LXplbmhlaQ,shadow_50,text_Q1NETiBA5Y-v5Y-v6KW_6YeM55qE6aOe6KGM,size_14,color_FFFFFF,t_70,g_se,x_16) # 2, Data structure Non comparative sorting is suitable for some special cases A kind of bucket thought Algorithm idea: large quantity and small range - Age ranking of tens of thousands of employees in a large company 18-80 - College entrance examination score 0-750 The number of occurrences of each number is taken as the element value of the new array, and the size of the value is taken as the subscript value of the new array, and then these numbers are re formed into a new array from small to large [External chain picture transfer...(img-yCLBv4OP-1638342991085)] Result array: 001222223334455566667777899 Space complexity: n+k(k Barrels) Time complexity: n+k(n) ## 2.9 cardinality sorting Non comparison sort A kind of bucket thought Multi keyword sorting The first place is ten and the first place is hundred Space complexity: O(n) Time complexity: O(n) Stable sorting [External chain picture transfer...(img-gDocPa6Q-1638342991088)] ## 2.10 bucket sorting Maximum-minimum value=The difference is divided into buckets according to the difference. One bucket represents a range, and each bucket is sorted separately unimportance