Hello Guys
I have divided problems into posts according to difficulty. Hope u all don’t mind this. 
This is the part 2 of two posts of unofficial editorials for November Long Challenge.
For editorials of problems VILTRIBE, CLRL, PERPALIN and CHEFHPAL, click here.
This post has editorials for problems SEGPROD and CSUBQ.
Problem SEGPROD
Problem Difficulty : Medium
Prerequisites:Modular Multiplicative Inverse (euclid’s method), Simple maths, and patience.
Problem Explanation
Given an array and a number P (not necessarily prime), Answer range product queries modulo P. (Not as simple as it looks).
Solution
The first thing i want you all understood is the euclid method of finding modular multiplicative inverse, vital to my approach, about which you may read from wikipedia and find the algorithm here.
The naive brute force (not at all successful even for 20 points, due to integer overflow) would be to create a prefix product array (similar to prefix sum array) and for every query L to R, output prefixProd® / prefixProd(L-1) and output modulo P. That’s correct, but incapable of giving us 100 points.
So, we move to modular multiplicative inverse (MMI) for help!!. We will to the same thing. Instead of prefix product, we will make an array storing prefix modular product (mod P ofcourse). Now, to answer queries, we are going to answer queries just as prefixModProduct®/prefixModProduct(L-1).
This is same as (prefoxModProduct® * ModInverse(L-1))%P.
If you have followed the logic upto here and understood MMI from geeksforgeeks page, you have earned 20 points. Hurray.
The reason of only 20 points is that Modular multiplicative inverse of a number A exists only when A and P are co-prime (gcd(A,P)==1). For first subtask, it was given that P is prime and all numbers are smaller than P. From that, it’s obvious that gcd(A, P) is one for every element, so we get 20 points.
Now, for full solution, we will use the same technique, but with a slight change in plan. Now, we will first find prime factors of P (U’ll understand why) and handle numbers in a different way, which will be explained by an example.
Suppose P = 6 and given array is 1 2 3 4 5 6 7 8 9 10
Now, Prime factors of 6 are 2 and 3.
Now, we will handle powers of 2,3 separately and rest numbers separately.
Create an array factors which will have value {2,3} for this example. create a 2d array factorPowSum[NumberOfFactors][N]
factorPowSum array will look like below (explained how to make this below that)
factor 0 1 2 3 4 5 6 7 8 9 10 11
2… 0 0 1 1 3 3 4 4 7 7 8 8 (dots to adjust position, spaces are truncated automatically)
3… 0 0 0 1 1 1 2 2 2 4 4 4
And we will think(it’s necessary) we are given given array 1 1 1 1 5 1 7 1 1 5 11 (all numbers are divided by 2 and 3). Now, you will see that all numbers are co-prime to 6. Now we can use the algorithm to solve the queries for reduced array.
Let’s assign factor 2 to index 0 and factor 3 to index 1.
Creation of factPowSum array
Just assign factPowSum{factorIndex}{0} = 0 and for 0<i<=N
factPowSum{factorIndex}{i} = factPowSum{factorIndex}{i-1} + Power of factor divided from ith number(1-based indexing).
For example, from 8, 3rd pow of 2 was divided, factorPowSum{0}{3} = factorPowSum{0}{4} + 3.
Hope i made the array clear.
Now, consider that we are given only powers of factors of P.
Continuing same example, The same array we were working with now becomes (I know this is lengthy as hell, but I tried making it as easy to understand as possible)
1 2 3 4 1 6 1 8 9 2 1 (Only powers of 2 and 3 are considered from given array).
Now, if someone ask us range query on this array, we can answer each query within O(number of factors of P) time.
This is simple, just refer to factPowSum array and give it a try. Read further only after a try.
let ans = 1.
for every factor of P, ans = ans*pow(factor, factPowSum{R+1}-factPowSum{L}).
Now, combining above two sub-problems, we get answer of query as
let ans = (prefixModProd(R+1)*MMI(L) * product(pow(factor, factPowSum[R+1]-factPowSum[L])) )%P.
// The queries in question are 0-indexed
You are about to lose 100 points if you do this. Shocked!!. Be sure to take modulo after every multiplication so as to lose your AC to Integer overflow, use long long int/long only.
Now you all deserve 100 points in this problem, but you will get TLE because test cases were too tight. Calculating powers with fast modulo exp even takes O(logN) time, which is too much for 10^7 queries. So we calculate powers of factors of P before answering queries and store in an array. I am not going to explain this but it’s easily understandable from my code. In case you need help, feel free to ask.
Added Proof:
lemma: P can have at most 10 different prime factors.
Proof: Try multiplying first eleven prime numbers. The product will be greater than 10^9. but as P <= 10^9, it follows that maximum number of distinct prime factors P can have is 10.
Here’s a link to my
[4]. (Took nearly 30 submissions for 100 points :D)
### Problem [CSUBQ][5]
#
### Problem difficulty:Medium
### Prerequisites:Segment Tree, Too much patience unless u r a segment tree expert (msg me if u are, want a talk.)
#### Problem Explanation
An array of Size N and M queries, Tow Integers L and R (L<=R), handle following queries.
1. update ith value to X.
2. Find number of subarrays betweeen L and R, whose maximum values lie between L and R. (both inclusive)
#### Solution
**One basic formula. (very basic)**
let function c denote c*(c+1)/2 (The number of all the possible contigious subarrays from an array of length c.)
1+2+3+4+5 ... (N-1) + N = N(N+1)/2
Now, take L = 1, R = 10 and given array 2 0 11 3 0 (example given in problem)
(I'm directly taking an array instead of a series of updates to explain it).
Required subarrays between 1 to 5 are {1, 1}, {1,2},{4,4} and {4,5}
An interesting thing to observe is that ans = c(2)-c(1)+c(2)-c(1) (I'll explain how i came up with this.)
**Two Facts:**
1. Any element > R is never included in any subarray (like element 3).
2. Any number of elements smaller than L can be included in subarray as long as there is atleast one single element between L and R inclusive.
So, we are going to get 25 points first using this fact.
See [this][6] solution for details. in this solution, inc means consecutive elements till now which are smaller or equal to R, and exc means all consecutive elements < L till current position. Whenever we reach an element >= L, subtract excluded subarray count ( c(exc) ) from count and set exc = 0 and inc++ and whenever u get an element > R, add c(inc), subtract c(exc) and set both to 0. otherwise inc++, exc++.
Ans is count + c(inc)-c(exc). Got 25 points. This solution runs in O(QN) time.
I hope this logic is clear. Because if it isn't, be sure to read again once, twice, 10 times, 10^2 times, 2^ 10 times (I like this quote :P).
Now, Getting 52 points isn't difficult if You have heard about **square root decomposition**. I have submitted a [solution][7] using this technique too. :P. (I submitted too many solutions for this) I used in this solution apart from the counting technique we discussed above and one thing mentioned below
Cheers, now we got 52 points with this ease, but no longer. Problem setters aren't going to gift 52 points with this ease.
Now, think here, what information about a segment L to I and a segment from I+1 to R we need to find out all required information.
Refer to my [solution][8] (One more solution :P) alongwith to understand what's going on.
Let me help u. You need 8 variables (too much u feel i guess), as follows:
1. count => The count of subarrays already included in range, except those at borders.
2. incL => Number of elements from left to leftmost element which is > R. (0 if there's no element > R in range)
3. incR => Number of elements from right to rightmost element which is > R. (0 if there's no element > R in range)
4. excL => Number of elements from left to leftmost element which is >= L. (0 if there's no element >= L in range)
5. excR => Number of elements from right to rightmost element which is >= L. (0 if there's no element >= L in range)
6. size => size of range
7. found1 => boolean, which tells whether range contains atleast one element > R.
8. found2 => boolean, which tells whether range contains atleast one element >= L.
Any ideas how we are going to merge ranges?, it's simple enough.
Consider array 2 0 1 11 3 0 5
Answer of this Problems is c(3)-c(1)+c(3)-c(1) = 6-1+6-1 = 10
Think of this problem in two parts. for elements > R and for elements >= L.
For first part, consider any element > R as dividing element. for above example, 11 is dividing the total ranges into two sub-ranges of length 3 each.
For second part, consider any element >= L as dividing element. For above example, 2,1,11,3 and 5 are dividing elements, resulting in 2 ranges of length 1. ( {1,1} and {5,5} ).
If you understood this part, You can divide the given problem into two parts, the one dealing with elements to be included, and the one dealing with elements to be excluded.
Here comes the Main part of solution,**The Segment tree**. Now, as you may see, i have used an iterative segment tree,(I messed up recursive one and also, iterative is slightly faster if u take the trouble to understand it) about which you may [read here][9]. I first of all, created a class S (struct equivalent in java) which hold all this info about each segment of tree.
Now, the most important thing is to create a merge function, which will merge two segments into a larger one correctly.
You may see from my code, i have dealt with inclusion and exclusion separately, making my code simpler.
For first part, i check if both sides have element > R (our dividing element) (using boolean found1). if both has, include Left of large range will be outL of left segment and include right will be includeRight of right segment. count will be increase by c(left.incR + right.incL) because they are no longer boundary elements of segment.
In case only one segment has dividing element, incL and incR of output range is assigned accordingly.
Same way for large. And Here we go. We have solved the problem. Bet you didn't realize that.
All that is left to implement it. And Now, nothing is going to stop you from 100 points except a TLE in one file. (Not Sure, because i had use the following tip to save time.)
Link to My
10.
PS: This was my first editorial using segment trees. So, i am sorry if any error might have crept in (without my knowledge, of course) Do give your review for this editorial.
As always, i wholeheartedly invite your suggestions and thank for your response to my previous editorials.
PS:Sorry for delay, was held up in something important. Delay gift will be posted as soon as I learn the technique of problem Polynomial. 
Hope you don’t mind delay in delay gift.
