So we had an Interview Question at Codenation, This is the question Question Image. I understand that we can do the update node and sum of values through segment trees but what I can't understand is how to find the aith beautiful number, can someone help me out please! Also the constraints of X in the question are below 10^5

Auto comment: topic has been updated by kstheking (previous revision, new revision, compare).Hi!

A simple bfs should be enough.

Here's a code for creating series whose

sum of square of digits <= x.The remaining is just classical euler tour and update using segment tree.

I'll leave the complexity analysis to you.

Thanks!

Got it, Thanks for answering!

Dsxv, could you please clear this for me.

Won't taking mod here change the number itself. It'll yield the wrong number when used further, right?

For instance, for x=1, we get the first 15 numbers as this:

Output for x=11 10 100 1000 10000 100000 1000000 10000000 100000000 1755647 17556470 175564700 757402647 586315999 871938225

Hi!

The deciding parameter for the series is actually the second number i.e,

sum + i*i.As far as the numbers in series are concerned, notice that the question is asking us to take mod of sum of all the values

(fi)in the subtree.Where the values

(fi)in the subtree are nothing but the elements of our series.Now,

property of modulus addition:

`(A+B+C...) % mod = (A%mod + B%mod ...) % mod`

so, taking mod of A, B ... beforehand doesn't affect our answer.

Feel free to ask anything if you still have any doubt.

Thanks!

Yup, The deciding parameter for the series is actually the second number i.e, sum + i*i.

I get it now. Thanks.

Hi, I have a little doubt, since we are beginning with 1, wouldn't this code leave out numbers which begin with 2, 3 etc.

Yes, Updated. Thanks for pointing out!

Your logic won't work for x = 5, you push {1, 1} and then push {10*1 + i, 1 + i*i}, but the number {2, 4} also satisfies the constraints.....how will your algo detect that??

Updated. Thanks!

Hi bro can you pls suggest some beginner level problems involving euler tours or similar concepts? would be really helpful

Hi!

I suggest you to read LCA and learn the idea behind it as well as solve the questions mentioned at the bottom (First one is literally just template check).

Thanks!

why a segment tree is required?

It's required for finding the sum all nodes of subtree as well as updating the nodes.

Edit: please read my comment below for more infoHow to update using segment tree, it is not given in the question that the tree is binary tree , how exactly will the indexing of segment tree will work????????

Hi!

We can't apply segment tree directly onto the tree for solving this problem. First we have to flatten the tree using Euler tour. After this we know the

`in`

and`out`

time of every node.`in`

and`out`

of X) so we can just get sum of this subarray using segment tree and divide this by 2.`in`

and`out`

indexesI think flattening the tree into an array of size

`n`

is more "natural". There, you only increase t when you enter a new node, so you still have`t_in`

and`t_out`

but you only need to update once and sum doesn't over-count.I thought companies ask to reverse a linked list.

how did you solve 2nd question? question 2 laser tag

i thought of dp recurrence donno if its correct tho as hackerrank was very slow during the test(my test was completed before knowing the status of submission).

Spoilerdp[i] = 1 + max(dp[i/2] , dp[i/2 +1] ) (if i is odd) dp[i] = 1 + dp[i/2]. (if i is even)

dp[i] represents the number of games needed for i people. at last check if dp[i] <= 2*x

Just take ceil of log2(n). This will be your number of matches.

as one_last_chance mentioned, what we can do is first distribute them in teams of half, now consider one half and swap its half people to the other side,

now you have a quarter of people who have been together, swap half of them to the other side

By doing this log2(n) times you can generate configurations in which each person has fought with every other person

I implemented the same solution. It passed all tescases.

actually mine also should work! as it is same taking ceil(log2(i))

but donno about the status of the submission though

if(i%2==1){dp[i] = 1 + dp[i/2 +1];}

else{dp[i] = 1 + dp[i/2];}

Can u share the 3rd problem?

sure! link for 3rd problem

just sharing a similar question if wanna try

https://www.codechef.com/SEPT19A/problems/DOOFST

it is correct

Did anyone solve the xor one?

i observed binomial coefficients pattern (pascal triangle) and solved using it.

Spoilerlets say the numbers are arr = a b c d e f g h and every time arr is modified

k = 1 ;arr = 1a xor 1b , 1b xor 1c , 1c xor 1d ... so on

k=2 ; arr = 1a xor 2b xor 1c , 1b xor 2c xor 1d ... so on

k=3; arr = 1a xor 3b xor 3c xor 1d , .. so on

k=4; arr = 1a xor 4b xor 6c xor 4d xor 1e

after finding the binomial coefficients of level k = kgiven+1 (we can do that in O(K) time). initialise your ans as 0 ans start from idx = m (given) ans only xor array[idx] with ans if our current coefficients is odd. here array is initial array provided for us.

but we can't calculate ncr (binomial coefficients list) for k = 1e5 level (as it overflows the integer). Actually we don't need to (think for a bit you'll get it).

I didn't know about Pascal Triangle, couldn't make sense of the pattern, thanks!

the nth level is same as

nC1 , nC2 , nC3 , nC4 ... .. nCn

ncr is n!/((n-r)! * r!)

how do u solve tournament question.i solved in o(n2) using dp stilll some cases are not passing.

I knew about Pascal Triangle, still couldn't make sense of the pattern, thanks!

Lemme describe the question first for those who don't know.

QuestionGiven an array

`A`

of length`n`

, apply the following operation`k`

times :Create a new array

`B`

of length`n - 1`

such that`B_0 = A_0 XOR A_1`

,`B_1 = A_1 XOR A_2`

,......,`B_{n - 2} = A_{n - 2} XOR A_{n - 1}`

.Replace A with B i.e.

`A = B`

Now you have to tell

`m-th`

element of final array`A`

Constraints`1 <= n <= 1e5`

`1 <= k <= n - 1`

`0 <= m < n - k`

`1 <= A_i <= 1e9`

HintTry to find how

`A`

will look if`k = power of 2`

.SpoilerLet

`n = 10`

and`k = 4`

`A`

will look like (let final array is denoted by`B`

)`B_0`

=`A_0 XOR A_1 XOR A_2 XOR A_3`

`B_1`

=`A_1 XOR A_2 XOR A_3 XOR A_4`

and

`so on`

.Code`P.S. :`

I was not able to code it during the contest and just verified with bruteforce soln after the contest, so there might be some mistakes. I would appreciate if anyone points out any issue.How did u do tournamnet questions??i used dp for that like:dp[2]=1,dp[1]=1;thenfor(i=3;i<=n;i++){int t=INT_MAX;for(j=1;j<i;j++){t=min(t,dp[j]+dp[i-j]+1);}}if(x>=(dp[n]+1)/2))"1";else "0";what is wrong in this

Could you please use good formatting?

88179101 plz see at this dp[1]=0

N was the total numbers of player . Assume N is a perfect power of 2. Divide them in teams of size N/2 and let them fight. Team A = N/2 Team B = N/2 players Now each player in Team A will fight against every player in Team B . Only the players of Team A & B haven't fight among themselves . So again divide them in two halves Team A ( N/2 ) -> Team C (N/4) Team D (N/4)

Team B ( N/2 ) -> Team E (N/4) Team F (N/4)

Team C & D & E & F will be of size N/4.

Now if you let fight between C & D and E & F then +2 will get added in answer . But is this the optimal ?

No , Because you can club Team C and E , Team D and F then let them fight . doing this +1 will get to the answer.

So basically what is the terminating condition ? when size is equal to 1

N -> N/2 -> N/4 — > N/8 -------------> 1 solve it to get the answer.

So ans is log(N) base 2 if it is a perfect power of 2. if not then log(N) base 2 + 1 ( for the remaining )

thnx.

Just brute force with help of bitsets.

For Question 3

Simple/well known fact is that for any given number $$$x$$$ we know $$$xor(x,x)=0$$$,

so if we take xor of $$$x$$$ with itself an even number of times then $$$xor(x,x,x....)=0$$$,

but if we take xor of $$$x$$$ with itself odd times the result is $$$x$$$.

Notation Used- $$$xor(x,x,x...x)[k-times]=x^k$$$

Consider the simple array $$$a=1,2,3,4,5$$$

For observing the pattern assume $$$m=0$$$

So we will look only at the value $$$a[0]$$$

Performing the operation 1 time we get —

$$$a[0]=[1\cdot2]$$$

here $$$1\cdot2=xor(1,2)$$$ same notation is followed below

2nd time —

$$$a[0]=[1\cdot2^2\cdot3]$$$

3rd time —

$$$a[0]=[1\cdot2^3\cdot3^3\cdot4^1]$$$

4th time —

$$$a[0]=[1\cdot2^4\cdot3^6\cdot4^4\cdot5^1]$$$

5th time —

$$$a[0]=[1\cdot2^5\cdot 3^{10} \cdot4^{10}\cdot5^5\cdot6^1]$$$

Now observe the powers of $$$1,2,3,4,5,6$$$ which are respectively $$$1,5,10,10,5,1$$$ which helps us notice the pattern as we can now see that powers obtained after 5th time are nothing but

But how to find $$$4^{10}$$$ or $$$2^5$$$(xor-value) ? For finding the value we just need to know for any $$$k$$$, $$$i$$$ when $$${k \choose i}$$$ is even and when it is odd. or we just need to find $$${k \choose i} \% \space 2$$$ for doing this many methods are given on gfg you just need to select the appropriate method keeping in mind the time-complexity as here $$$k$$$ is the of the order of $$$10^5$$$. Some methods are given here, and here.

SpoilerThat's an interesting pattern. Why does this happen?

(most of what is written below is based on this amazing video by kartik8800 I really recommend watching it link)

Really Long comment ahead —

We start off by looking at the really cool animation of pascal triangle on wikipedia, which offers us some satisfaction because something similar is happening there.

Note how in the above animation the value of one row is calculated on the basis of previous row, which itself is based on the recurrence relation

So for this particular question what we are essentially doing is sort of like pascal triangle.

ImageIn the above example the value of a node is simply xor of the two nodes below it(like xor(1,2)=3). This makes sure that the size of the array is reduced by 1 after every step(as in our original question).

Indeed another way of stating the above visual operation can be - given an array at each step $$$a[i]=xor(a[i],a[i+1])$$$ is done for a particular $$$k$$$ number of times.

Let's now jump to a seemingly unrelated problem the famous dp problem.

This has a very short dp solution based on the recurrence - $$$f(m,n)=f(m,n-1)+f(m-1,n)$$$ where $$$f(m,0)=1$$$ and $$$f(0,n)=1$$$.

However the number of such paths can be directly proven to be $$${m+n \choose m}$$$.

A very nice combinatorial/intutive proof is given at brilliant and stackoverflow.

Jumping back to our question assume we have an array of size $$$k+1$$$ , since performing the operation each time reduces the size of our array by 1, so after $$$k$$$ such operations only one element will be left and we want to find that value.

For finding the value all we need to find is for each index $$$i$$$ in the original array ( that is $$$i=0,1,2...,k-1,k$$$) what is its contribution in the final array.

ImageNote that the number on top of each node is just it's index.

The image also shows at which positons the index=3 is contributing.

Now the real insight for linking the two problems is that

the contribution of the index $$$i$$$ is nothing but the number of ways to reach the top element if we are only allowed to either move up-left or up-right(as shown in image)The reason for that is based on the idea that each time the element goes up-left or up-right it contributes it's current value to that index, this is even more clear by looking at the recurrence relation.

Recurrence relationLets denote the contribution of the element $$$i$$$ in the original array to a position $$$j$$$ after performing $$$x$$$ operations be $$$f_i(j,x)$$$.Then

(base case for above recurrence is left for the reader if any :P)

So all that is left to calculate is to find the number of ways to reach the top-most element for the element at index $$$i$$$.

It is easy to observe now that it takes exactly $$$i$$$ up-left moves and exactly $$$k-i$$$ up-right moves for it to reach the top(for example the element at index 3 requires 3 up-left move and 4-3=1 up-right move).

So by the formula for number of such paths= $$${m+n \choose m}$$$. For our question the number of such paths is given by $$$m=i,n=k-i$$$.

Which explains the formula I got in my previous comment.

Additional questionCan you observe something similar if instead of $$$xor$$$ , the operation $$$sum$$$ is used ? what about multiplication ?

BonusYou may have come across a solution which uses bit-manipulation however that is also based on this pattern and uses lucas-theorem(without even knowing about it), it is essentially based on the fact that all we need is $$${k \choose i} mod \space 2$$$, can you come up with that solution after seeing this pattern ?

That is a very neat and clever observation.

Here is my O(N) solution to ZaurXOR: https://youtu.be/lccWXAd7enI

I think the solution is very easy to comprehend and does not use any pattern observation etc. I have used number of paths in a graph to evaluate the answer.

Basically I have developed a DP solution and optimized it using some very basic combinatorics!

Hey... How do you get to know about these tests?? I missed Facebook HackerCup and now this. Can anyone please tell me how can I keep track of these interview tests or competitions? Plz, it would be a great help!!!

use https://clist.by/ to know about ongoing contest. btw codenation test was only for college student for hiring and internship. Their college's training and placement cell float the test link.

The last Question zaurxor had weak test cases! I wrote a O(n) solution but some of my friends passed n.k solution which is not fair

The n.k brute force indeed passed majority of the test cases but I think it failed about 4 of them

Actually implemented O(n*k) soln and it failed just 1 testcase.

Can u share your code and approach?