I. problem elicitation
Internet products should be designed with a ranking list, which is based on scores. The score range is [11000], and each score can have more than one person.
1. Xiao Ming's score is 300. Find out its ranking.
2. After a period of time, the scores of some users have changed, and Xiaoming's scores have also changed. Ask for his ranking.
Binary tree array
1. Concept
All positive integers can be represented in binary.
For example:
34 = 2^1 + 2 ^7
12 = 2^2 + 2^3
The binary representation of 34 ， and 12 ， is as follows
For an integer ， i, lowbit(i) represents the integer value represented by the last binary position 1 ， of ， i.
lowbit(12)= 2^2 = 4
lowbit(34) = 2^1 = 2
2. Data structure diagram
Tree array is a kind of tree relational structure established on the array.
3 description
The black array represents the original array (represented by A[i]), and the red structure represents the tree array (represented by C[i]).
C[i] = A[ i - lowbit(i) + 1 ] + A[ i - lowbit(i) + 2 ] + ... + A[i];
The mathematical meaning of the above formula is: the value of the tree element with subscript () I) = the sum (including) A[i]) lowbit(i) elements in the element array.
Based on this formula, the value of each tree element can be calculated:
C[1] = A[1]; (lowbit(1) = 1)
C[2] = A[1] + A[2]; (lowbit(2) = 2)
C[3] = A[3]; (lowbit(3) = 1)
C[4] = A[1] + A[2] + A[3] + A[4]; (lowbit(4) = 4)
C[5] = A[5]; (lowbit(5) = 1)
C[6] = A[5] + A[6]; (lowbit(6) = 1)
C[7] = A[7]; (lowbit(7) = 1)
C[8] = A[1] + A[2] + A[3] + A[4] + A[5] + A[6] + A[7] + A[8]; (lowbit(8) = 8)
III. application scenarios
With the tree array C[i], we can find the prefix sum of the original array A[i] and quickly update the array C[i] when an element in A[i] is modified.
1 ， find the prefix sum of the original array ， A ， and
Assuming that the sum of the first # I # elements of the original array # A # is # Sum[i], the integer interval [1,i] can be expressed as [1,i-lowbit[i]] and [i-lowbit[i]+1,i].
Assuming j = i-lowbit[i], J < = I "and Sum[0] = 0, Sum[i] = Sum[j] + C[i] (Sum[j] represents the sum of the first interval and C[i] represents the sum of the second interval)
Then calculate the prefix and Sum[i] step by step until Sum[0] = 0.
2. When an element in the original array # A[i] is modified, the tree array # C is modified synchronously
When an element A[i] is modified, all tree array elements C[j] containing A[i] must be modified accordingly.
According to the tree structure above, we can get
C[k+1] = C[k] + lowbit(k)
IV. realization
1. Demand
a get the ranking of people with 10 points.
b) after a period of competition, get a ranking of 10 ￠ points.
2. Realization
package treearray;
import java.util.Random;
/**
* @className: TreeArray
* @description: Tree array
* @date: 2022/1/30
* @author: cakin
*/
public class TreeArray {
static int lengh = 15;
// Corresponding array element
static int a[] = new int[lengh + 1];
// Corresponding tree array
static int c[] = new int[lengh + 1];
/**
* Function Description: calculate the lowbit of x
*
* @param x subscript
* @return x Corresponding lowbit
* @author Bei Yi
* @date 2022/1/30
* @description:
*/
static int lowbit(int x) {
return x & (-x);
}
/**
* Function Description: sum the first i items of array A
*
* @param i A Subscript of array
* @return Sum the first i items of array A
* @author Bei Yi
* @date 2022/1/30
*/
static int sum(int i) {
int res = 0;
while (i > 0) {
res += c[i];
i -= lowbit(i);
}
return res;
}
/**
* Function Description: modify the element of element A[i] and add val value
*
* @param i A Subscript of
* @param val A Added value
* @author Bei Yi
* @date 2022/1/30
*/
static void update(int i, int val) {
while (i <= lengh) {
c[i] += val;
i += lowbit(i);
}
return;
}
public static void main(String[] args) {
// initialization
for (int i = 1; i <= lengh; i++) {
a[i] = new Random().nextInt(10) + 1;
update(i, a[i]);
}
for (int i : a) {
System.out.print(i + " ");
}
System.out.println();
for (int i : c) {
System.out.print(i + " ");
}
System.out.println();
// Ranking of people with 10 points
System.out.println("Ranking of people with 10 points:");
System.out.println(sum(lengh) - sum(10) + 1);
System.out.println("=============After a period of hard work=======================");
// After a period of hard work, the ranking has changed
for (int i = 1; i <= lengh; i++) {
int count = new Random().nextInt(3);
a[i] += count;
update(i, count);
}
for (int i : a) {
System.out.print(i + " ");
}
System.out.println();
for (int i : c) {
System.out.print(i + " ");
}
System.out.println();
// Ranking of people with 10 points
System.out.println("Ranking of people with 10 points:");
System.out.println(sum(lengh) - sum(10) + 1);
}
}3. Test
0 2 9 9 10 3 9 8 1 8 5 6 3 5 9 10 0 2 11 9 30 3 12 8 51 8 13 6 22 5 14 10 Ranking of people with 10 points: 34 =============After a period of hard work======================= 0 4 9 11 10 4 11 9 2 8 6 6 4 6 9 11 0 4 13 11 34 4 15 9 60 8 14 6 24 6 15 11 Ranking of people with 10 points: 37
The tide washes away the sand. If you don't advance, you will fall back.
Change lengh# to 100.
0 1 10 3 10 2 4 9 7 10 7 2 9 5 7 1 4 5 2 9 3 7 5 5 3 10 8 10 7 8 2 9 4 3 2 9 5 3 3 2 7 8 8 2 2 7 1 5 4 4 4 1 3 8 5 1 10 9 3 2 5 4 5 2 7 8 10 9 7 1 5 10 7 10 9 6 9 6 6 5 10 7 5 10 1 9 3 7 2 7 7 6 3 8 7 6 2 3 2 6 6 0 1 11 3 24 2 6 9 46 10 17 2 28 5 12 1 91 5 7 9 19 7 12 5 39 10 18 10 35 8 10 9 188 3 5 9 19 3 6 2 34 8 16 2 20 7 8 5 71 4 8 1 12 8 13 1 36 9 12 2 19 4 9 2 332 8 18 9 34 1 6 10 57 10 19 6 34 6 12 5 118 7 12 10 23 9 12 7 44 7 14 6 23 8 15 6 208 3 5 6 17 Ranking of people with 10 points: 495 =============After a period of hard work======================= 0 3 11 3 10 4 5 11 7 10 9 2 11 7 7 2 5 6 4 11 5 9 5 5 4 11 10 12 7 9 2 11 6 5 4 11 6 4 3 3 9 8 8 4 4 8 2 5 5 6 5 3 4 10 5 1 11 11 3 2 7 6 5 3 7 8 10 10 8 1 6 10 8 12 11 6 9 8 8 7 12 7 6 11 1 11 4 7 4 8 8 7 3 9 8 6 3 4 3 8 8 0 3 14 3 27 4 9 11 54 10 19 2 32 7 14 2 107 6 10 11 26 9 14 5 49 11 21 12 40 9 11 11 224 5 9 11 26 4 7 3 45 8 16 4 24 8 10 5 89 6 11 3 18 10 15 1 45 11 14 2 23 6 11 3 402 8 18 10 36 1 7 10 61 12 23 6 38 8 16 7 134 7 13 11 25 11 15 7 51 8 16 7 26 9 17 6 237 4 7 8 23 Ranking of people with 10 points: 590