Segment tree
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The document discusses segment trees, which allow querying which stored intervals contain a given point. It presents the basic functions of segment trees - initialization, querying, and updating. Initialization builds the tree by dividing segments into halves and storing sums at each node. Querying returns the sum in a range by checking if a segment is fully within, outside of, or partially within the range. Updating modifies values by adding to all relevant nodes. Segment trees enable efficient range minimum/maximum queries and have numerous applications.
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Segment tree
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in matlab.
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Task4output.txt 2 5 9 13 15 10 1 0 3 7 11 14 1.docx
Task4/output.txt
2 5 9 13
15 10 1 0
3 7 11 14
12 8 6 4
2 5 1 0
3 7 6 4
12 8 9 13
15 10 11 14
0 1 2 5
7 6 4 3
8 9 12 13
15 14 11 10
0 1 2 3
7 6 4 5
8 9 11 10
15 14 12 13
0 1 2 3
7 6 5 4
8 9 10 11
15 14 13 12
Task4/task4.cpp
#include <cstdio>
#include <iostream>
using namespace std;
#define MAXROW 4
#define MAXCOL 4
void bidirection_bbsort(int a[], int n, bool ascending)
{
int i,j, temp;
//This is the original bubble sort
// Make SMALL change such that it handles descending sorting too
for (i = n-1; i >= 1; i--) {
for (j = 1; j <= i; j++) {
if (a[j-1] > a[j]) {
temp = a[j-1] ;
a[j-1] = a[j] ;
a[j] = temp ;
}
}
}
}
void column_sort(int m[][MAXCOL], int col)
{
//Use the bidirection_bbsort() once it is implemented
// No need to invent / write a new sorting logic
}
void print_matrix(int m[][MAXCOL], int nRow, int nCol)
{
int i, j;
for (i = 0; i < nRow; i++){
for (j = 0; j < nCol; j++){
printf("%2d ", m[i][j]); //use printf for easier formatting
//cout << setw(2) << m[i][j] << " "; //equivalent c++ output formatting
}
cout << endl;
}
cout << endl;
}
void phase_A(int mat[][MAXCOL], int nRow, int nCol)
{
int i;
bool order = true;
for (i = 0; i < nRow; i++, order = !order){
bidirection_bbsort(mat[i], nCol, order);
}
}
void phase_B(int mat[][MAXCOL], int nRow, int nCol)
{
int i;
for (i = 0; i < nCol; i++){
column_sort(mat, i);
}
}
void magic_sort(int mat[][MAXCOL], int nRow, int nCol)
{
int i;
for (i = 1; i < nRow; i *= 2){
phase_A(mat, nRow, nCol);
print_matrix(mat, nRow, nCol);
phase_B(mat, nRow, nCol);
print_matrix(mat, nRow, nCol);
}
phase_A(mat, nRow, nCol);
print_matrix(mat, nRow, nCol);
}
int main()
{
int arr[] = { 5, 12, 6, 7, 9, 6, 1};
int i;
int mat[MAXROW][MAXCOL] = {
{5, 2, 13, 9},
{10, 1, 15, 0},
{7, 3, 11, 14},
{12, 8, 4, 6}
};
magic_sort(mat, 4, 4);
return 0;
}
Task3/task3.cpp
#include <cstdio>
#include <iostream>
using namespace std;
struct Point {
int X, Y;
};
void sort(Point a[], int n)
{
//You can adapt any of the insertion / selection / bubble sort
}
int main()
{
int i;
Point ptArray[5] = { {11, 34}, {5, 73}, {11, 19}, {13, 5}, {11,
68}};
sort(ptArray, 5);
//The output should match the order given in question
for (i = 0; i < 5; i++){
printf("(%d, %d)\n", ptArray[i].X, ptArray[i].Y);
}
return 0;
}
Task2/sample_length_3.txt
***
%%%
%##
%$$
##%
$$%
Task2/sample_length_5.txt
***%%
***##
***$$
%***%
%%***
%%%%%
%%%##
%%%$$
%%##%
%%$$%
%##%%
%####
%##$$
%$$%%
%$$##
%$$$$
##***
##%%%
##%##
##%$$
####%
##$$%
$$***
$$%%%
$$%.
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Segment tree
- 2. Presented BY: Shohanur Rahman ID# 1321368042 CSE 225 (Data Structure) North South University Email: shohan.nsu.cse@gmail.com 2
- 3. Segment Tree Definition: In computer science, a segment tree is a tree data structure for storing intervals, or segments. It allows querying which of the stored segments contain a given point. It can be implemented as a dynamic structure. 3
- 5. Sample Problem: You are given n (1<=n<=100000) integers & 100000 query. Each query you have to change a value of an element Or you have to given the minimum value of a range. 5
- 6. Function Of segment Tree 6
- 7. Initial Functions: We start with a segment arr[1 . . . n]. and every time we divide the current segment into two halves(if it has not yet become a segment of length 1), and then call the same procedure on both halves, and for each such segment we store the sum in corresponding node. 7
- 9. Initial Functions( in tree look): 9
- 10. Initial Functions(Algorithm): #define mx 100001 int arr[mx]; int tree[mx*3]; void init(int node,int b,int e) { if(b==e) { tree[node]=arr[b]; return; } int Left=node*2; int Right=node*2+1; int mid=(b+e)/2; init(Left,b,mid); init(Right,mid+1,e); tree[node]=tree[Left]+tree[Right]; } 10
- 12. Query Functions: Once the tree is initialized, how to get the sum using the initialized segment tree.Now,we use Query function.Here 3 types of cases can be happening. 12
- 14. Query Functions: case A: if(b>=i&&e<=j);then the segment is :Releavent segment. case B: if (i > e || j < b);means current segment is outside of i-j,then its valuless. case C: if case A and case B are not true and some of the parts is included in i-j , then we need to break the segment and take some of the parts. 14
- 15. Query Functions(Algorithm): int query(int node,int b,int e,int i,int j) { if (i > e || j < b) return 0; if (b >= i && e <= j) return tree[node]; int Left=node*2; int Right=node*2+1; int mid=(b+e)/2; int p1=query(Left,b,mid,i,j); int p2=query(Right,mid+1,e,i,j); return p1+p2; } 15
- 16. Update Function: Like tree initialization and query operations, update can also be done recursively. We are given an index which needs to update. Let new value be the value to be added. We start from root of the segment tree, and add new value to all nodes which have given index in their range. If a node doesn’t have given index in its range, we don’t make any changes to that node. 16
- 18. Update Function(Algorithm): void update(int node,int b,int e,int i,int newvalue) { if (i > e || i < b) return; if (b >= i && e <= i) { tree[node]=newvalue; return; } int Left=node*2; int Right=node*2+1; int mid=(b+e)/2; update(Left,b,mid,i,newvalue); update(Right,mid+1,e,i,newvalue); tree[node]=tree[Left]+tree[Right]; } 18
- 19. Time Complexity: Function Name: Time: 1.Initial Function O(n log n) 2.Query Function O(log n) 3.Update Function O(log n) 19
- 20. Applications: Using segment trees we get an <O(N), O(log N)> algorithm. Segment trees are very powerful, not only because they can be used for RMQ(Range Minimum Query). They are a very flexible data structure, can solve even the dynamic version of RMQ problem, and have numerous applications in range searching problems. 20
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- 22. Time to run code…….. 22
- 23. Thank You ! 23