## Segment Trees

A segment tree is a heap-like data structure that can be used for making update/query operations upon array intervals in logarithmical time. We define the segment tree for the interval [i, j] in the following recursive manner:

- the first node will hold the information for the interval [i, j]
- if i<j the left and right son will hold the information for the intervals
**[i, (i+j)/2]**and**[(i+j)/2+1, j]**

See the picture below to understand more :

We can use segment trees to solve Range Minimum/Maximum Query Problems (RMQ). The time complexity is **T(N, log N)** where O(N) is the time required to build the tree and each query takes O(log N) time. Here’s a C++ template implementation :

#include<iostream> using namespace std; #include<math.h> template<class T> class SegmentTree { int *A,size; public: SegmentTree(int N) { int x = (int)(ceil(log2(N)))+1; size = 2*(int)pow(2,x); A = new int[size]; memset(A,-1,sizeof(A)); } void initialize(int node, int start, int end, T *array) { if (start==end) A[node] = start; else { int mid = (start+end)/2; initialize(2*node,start,mid,array); initialize(2*node+1,mid+1,end,array); if (array[A[2*node]]<= array[A[2*node+1]]) A[node] = A[2 * node]; else A[node] = A[2 * node + 1]; } } int query(int node, int start, int end, int i, int j, T *array) { int id1,id2; if (i>end || j<start) return -1; if (start>=i && end<=j) return A[node]; int mid = (start+end)/2; id1 = query(2*node,start,mid,i,j,array); id2 = query(2*node+1,mid+1,end,i,j,array); if (id1==-1) return id2; if (id2==-1) return id1; if (array[id1]<=array[id2]) return id1; else return id2; } }; int main() { int i,j,N; int A[1000]; scanf("%d",&N); for (i=0;i<N;i++) scanf("%d",&A[i]); SegmentTree<int> s(N); s.initialize(1,0,N-1,A); while (scanf("%d%d",&i,&j)!=EOF) printf("%d\n",A[s.query(1,0,N-1,i-1,j-1,A)]); }

Resources:

NJOY!

-fR0D

Hey, nice tut. If u add a tutorial for updating the values in segment tree, it will be great!

BorisSeptember 18, 2010 at 12:34 PM

when i five n= 800 in input then this code crashes.

please help

AnujOctober 13, 2010 at 8:53 AM

please add the nessary discription about n,A[i]. and also add the necassary printf statements . please reply immediately

berinNovember 12, 2010 at 7:22 PM

Do you think this guy’s your slave or what?

Andrés MejíaNovember 14, 2010 at 8:55 PM

could u plzzz add a good tutorial on binary indexed tree as they r easy to code????

aayush kumarJune 9, 2011 at 4:49 PM

There already is a post on that https://comeoncodeon.wordpress.com/2009/09/17/binary-indexed-tree-bit/

fR0DDYJune 9, 2011 at 8:30 PM

Hi I Don’t Understand Whats the application of segment tree ? i mean if we can search the element in given range of sorted array in O(logn) then why we need such complex DS or m i missing sum-thing so do u mean we can find the elements in unsorted array in O(logn) is it so .?? as Heap can unsorted array as 5 4 3 1 2 isn’t it .?? also please explain the in detail the initialize & query part & also write update part as you have mentioned..i am really interest in algorithms & so i wants to know what we can do with segment tree once you will reply my question i will really look & analyze it…i mean really really interested & appreciate ur attempt.

i mean when i m giving input for i & j 0 ,5 or i=0 & j=1 to 9 for N=10 array then i am getting output of query is 0 m not getting what exactly query function is doing ?? whats the purpose of it does it s giving element in range or its searching particular element & returning that element.

Reply ASAP.

AlgoseekarJune 12, 2011 at 10:08 PM

It can be used to find maximum/minimum element in a range of an unsorted array. Also between the queries also, you can update any element.

fR0DDYJune 15, 2011 at 10:27 PM

Bugs in the code above.

-> log is logarithm to the base 10, whereas log to the base 2 should be used.

-> only the position should be returned in line numbers 50,52,55,57.

To verify the bug run the code with the following input

N = 8

array = { 1, 2, 0, -1, 5, 5, 5, 5 }

first i,j -> 1 2 ( gives correct output of 0 )

second i,j -> 3 4 (gives incorrect output of 0)

mozzakOctober 1, 2011 at 12:43 PM

This is the output for this case- remember output is minimum element🙂

8

1 2 0 -1 5 5 5 5

1 2

1

3 4

-1

shashank jainAugust 2, 2012 at 7:25 PM

It would be better if in the tree initialization N = 2^X, you would get faster solution. Your current solution would get TL on some test cases… I don’t remeber testcases, but I promise you that N should be equal 2^x🙂

vilvlerFebruary 27, 2012 at 3:02 PM

Why would it be faster if you are adding more nodes to the tree?

Andrés MejíaFebruary 28, 2012 at 5:38 AM

Nice tutorial, I also wanted to know about updation of segment trees. How is it achieved? Can you explain a little more?

amanJune 18, 2012 at 8:54 AM

hey how to implement if i need all intervals with 0<=i<j<n

like for eg in the above tree i need to fint the max from the interval [1,8]

aichemzeeDecember 9, 2012 at 2:55 AM

I beleive the time complexity of construction the segment tree is $O(n\log n)$.

MacropodusFebruary 23, 2013 at 9:51 PM

There’s a small bug, not so important yet proves pain in neck if tested under certain input ranges. For very large range say 1 to 100000. The Query function goes so deep in recursion that it exceeds the recursion depth & hence will result as “Segmentation Fault”. I’ve tried it locally on my machine and on online competition too to verify.

Ravi OjhaMay 12, 2013 at 1:59 PM

[…] wcipeg.com/wiki/Segment_tree comeoncodeon.wordpress.com/2009/09/15/segment-trees/ letuskode.blogspot.com/2013/01/segtrees.html […]

Segment Trees | Sport CoderSport CoderDecember 5, 2013 at 1:25 AM

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Segment Trees - Lazy UpdatesDecember 10, 2013 at 10:03 PM

hey it fails to be in ti,e limit if no of queries ranges to 10^8…… some more better approach required …..

AnonymousApril 13, 2015 at 1:45 AM