# BROCLK - Editorial

#1

Author: Trung Nguyen
Tester: Hanlin Ren
Editorialist: Hanlin Ren

MEDIUM

### PROBLEM:

A clock has a minute hand which goes x angle per second. The minute hand has length l and the center of clock has coordinate (0,0). Initially, the other endpoint is at coordinate (0,l), and after 1 second its y-coordinate becomes d. Given l,d,t(x is unknown), calculate the y-coordinate of this endpoint after t seconds.

### QUICK EXPLANATION:

By the problem statement we have \cos(x)=d/l, and the final answer is \cos(tx)\cdot l. So we only need to compute \cos(tx). There is a formula computing \cos(nx):

\cos(nx)=2\cos(x)\cos((n-1)x)-\cos((n-2)x),

and we can use fast matrix exponentiation to compute \cos(tx) in O(\log t) time.

### EXPLANATION:

##A note to modular arithmetic

Click to view

During the contest, a lot of people asked problems on the output section, in particular related to “modular inverse”. So let’s answer them here.

Let m,a,b be integers. a is the modular inverse of b in modulo m if (a\cdot b)\equiv 1\pmod m. For example, 8 is the modular inverse of 7 in modulo 11 since

8 imes 7=56\equiv 1\pmod {11}.

We can prove that, in modulo m, a has an inverse if and only if \gcd(m,a)=1. (\gcd(7,11)=1 in above case.) To find the modulo inverse of a in modulo m, one solves the following equation

ax+my=1

by the Extended Euclidean Algorithm. Note that x,y are variables to solve; x is the modulo inverse of a. The modular inverse, if exists, is unique in modulo m. (You may say 19 is also a modular inverse of 7 in modulo 11, but 19 and 8 are equivalent.)

If m is a prime(in our problem, m=10^9+7 is a prime), then every integer 1\le a < m has a modular inverse, and let’s denote it as a^{-1}. The inverse can be computed in another way:
by Fermat’s little theorem, it’s simply (a^{m-2}\bmod m).

For a rational number \frac{p}{q}, if \gcd(q,m)=1, then it’s equivalent to p\cdot q^{-1} in modulo m. For example, \frac{2}{3}\equiv 2\cdot 3^{-1}\equiv 2\cdot 333333336\equiv 666666672\pmod {10^9+7}.

Now, in modulo m where m is a prime, we can manipulate every rational number as if it was an integer. For example, let’s compute \cos(x)=d/l and \sin^2(x)=1-\cos^2(x). The corresponding Python code is:

mod = 1000000007
cos_x = (d * pow(l, mod - 2, mod)) % mod
sqr_sin_x = (1 - cos_x * cos_x) % mod


and no manipulation of fractions is involved. Everything is just elements in \mathbb{Z}/(10^9+7)\mathbb{Z}.

(Note: \sin(x) can’t be computed even if \sin^2(x) can. This is because \sin(x) might not be rational. In this case, \sin^2(x) is a quadratic non-residue modulo 10^9+7.)

(Note 2: if you use languages such as C++, beware of overflow and underflow. For example, multiplying a and b should be written as (1ll*a*b)%mod rather than (a*b)%mod; the correct way to subtract b from a is (a-b+mod)%mod rather than (a-b)%mod. Think about the reasons yourself.)

##Setter’s Solution
First notice that x is given by \cos(x)=d/l. To see this, consider the following figure which illustrates the problem statement: OA and OB are the position of minute hand initially and after 1 seconds, respectively.

And the answer after t seconds is \cos(tx)\cdot l. Therefore, our actual task is: given \cos(x) and integer t, compute \cos(tx).

Notice the following formula:

\cos(nx)=2\cos(x)\cos((n-1)x)-\cos((n-2)x)\ \ \ (*)

that means we can write \cos(tx) in a matrix-multiplication form:

\begin{pmatrix}\cos(tx)\\\cos((t-1)x)\end{pmatrix}=\begin{pmatrix}2\cos(x)&-1\\1&0\end{pmatrix}\begin{pmatrix}\cos((t-1)x)\\\cos((t-2)x)\end{pmatrix},

and therefore

\begin{pmatrix}\cos(tx)\\\cos((t-1)x)\end{pmatrix}=\begin{pmatrix}2\cos(x)&-1\\1&0\end{pmatrix}^{t-1}\begin{pmatrix}\cos(x)\\1\end{pmatrix}.

Thus we can compute \cos(tx) by fast matrix exponentiation in O(\log t) time.

##Tester’s Solution
Again, we’re given \cos(x) and integer t and we want to compute \cos(tx). But what if we don’t know formula (*)? We can use the following (better-known) formula:

\begin{align*} \cos(a+b)=&\cos(a)\cos(b)-\sin(a)\sin(b)\ \ \ (1)\\ \sin(a+b)=&\sin(a)\cos(b)+\cos(a)\sin(b)\ \ \ (2) \end{align*}

All numbers we come across will be in the form a+b\sin(x). Although we don’t know \sin(x), we can represent such a number as (a,b). Therefore we can define +,-, imes on these numbers:

\begin{align*} (a,b)\pm(c,d)=&(a\pm c,b\pm d)\\ (a,b)\cdot(c,d)=&(a+b\sin(x))(c+d\sin(x))\\ =&ac+bd(1-\cos^2(x))+(ad+bc)\sin(x)\\ =&(ac+bd(1-\cos^2(x)),ad+bc). \end{align*}

We can write a recursive procedure \mathrm{Solve}(t) for computing the pair (\cos(tx),\sin(tx)).

• If t=1, we return \cos(tx)=(\cos(x),0) and \sin(tx)=(0,1);
• If t is odd, then we find (\cos((t-1)x),\sin((t-1)x)) by calling \mathrm{Solve}(t-1), and (\cos(tx),\sin(tx)) can be computed by formula (1) and (2);
• If t is even, then we call \mathrm{Solve}(\frac{t}{2}) to find (\cos(\frac{t}{2}x),\sin(\frac{t}{2}x)), and (\cos(tx),\sin(tx)) can be computed similarly.

This procedure runs in O(\log t) time since every two steps decrease t by half.

Suppose we’ve computed \cos(tx)=(a,b). b must be equal to 0 since it’s possible to represent \cos(tx) by only \cos(x) terms. Therefore \cos(tx)=a and we’re done.

## On formula (*)

How to prove formula (*)? In fact I don’t know, but one might look into this page for answer. The Chebyshev polynomial of the first kind is defined as T_0(x)=1, T_1(x)=x and T_{n+1}(x)=2xT_n(x)-T_{n-1}(x). It can be proved that T_n(\cos(x))=\cos(nx), which directly implies formula (*).

### ALTERNATIVE SOLUTION:

Please feel free to share your approach

### AUTHOR’S AND TESTER’S SOLUTIONS:

Author’s solution can be found here.
Tester’s solution can be found here.

### RELATED PROBLEMS:

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#2

For formula proof:

#3

Wow, I found (another) simple proof for the formula in @taran_1407’s editorial! Link.

#4

I followed the Setter’s approach, but I am still getting a TLE. Can anybody help me why this solution fails?

#5

I solved it using a function similar to modular exponentiation.
\cos tx = 2\cos^2(t/2) - 1 for even t
\cos tx = 2\cos(\lfloor t/2 \rfloor) \cos(\lceil t/2 \rceil) - \cos x for odd t

Simple recursive function with memoization passes. Though I couldn’t derive an upper bound on number distinct recursive calls. I guess its around 2 * \log t.

#6

You seriously need better editorialists.

#7

Really good problem… never used matrix exponentiation like fibonacci’s anywhere else before

#8

Can someone help me understand the formula manipulation to get from the first matrix representation of the formula to the second one with the second 2x2 matrix raised to the (t-1) power?

#9

I calculated cosnx by breaking n into a sum of powers of 2 and then calculated each power recursively using the formula cos2x=2cos^2-1. I used memoization to avoid repeating the same calculations. This required no more than log(n) operations.

To combine the powers of 2, I used the formula cos(a+b)=2cos(a)cos(b)-cos(a-b). I also used memoisation in this approach by mapping the previously calculated values of cosx otherwise it would result in TLE.

Link to my solution

#10

@r_64 halin ren how to read ur page in english

#11

need some help!
first i found out the angle with respect to positive y axis here

double angle=acos((double)D/I)

then i found the angle with respect to y axis after t seconds as this

angle=angle*T; // the final angle

ll rem=angle/(2*pi);
angle-=rem*(2*pi); // here we got the final angle


now I found the quadrant where the final angle will lie and calculate the value of d(the final y coordinate)

the i converted d into fraction using the function

and just find the inverse and print it where i am wrong

my solution my solution for BROCLK

#12

hey everybody,
I found the x correctly. using cos(x) = d/l;
now why does doesn’t we get the final length by cos(x*t) this can +ve or -ve depending on the quadrent.
I believe cos will take care of everything as it changes the quadrent it switched the sign of value i.e from -ve to +ve or vice versa.
Please confirm… my broken code is https://www.codechef.com/viewsolution/17355814

#13

After finding x, and also given t, why can’t we straight away find cos(tx)?

#14

There is no need to use matrices or trigonometry at all. In my solution, all you need to know is how to deal with complex numbers.

First, swap X and Y coords for convenience. Then, we have complex number (d/l, sqrt(1-(d/l)^2)). Note, that real part of its Nth power times l is the answer, according to properties of complex number multiplication. But its imaginary part is irrational, how do we tackle that?

Let’s denote that nasty root sqrt(1-(d/l)^2) as R. Take a look at what happens when we multiplying complex numbers that looks like (a, b*R):

(a+b*R*i) * (c+d*R*i) = (a*c-b*d*R*R) + (ad+bc*R)*i = (a*c-b*d*(1-d/l)) + (ad+bc)*R*i


As we can see, field of numbers of that kind is closed under multiplication. So, all we need is to store complex number (a, b*R) as a pair of integers (a, b), write our own multiplication function, and use binpow.


[1]

[1]: https://www.codechef.com/viewsolution/17300590

#15

When I use the normal iterative to compute the chebyshev and fermat’s little theorem for modular inverse,
I get wrong answer on the first subtask.

The given test cases are computed right tho, and yes, I need something better for when t is large
but could someone tell me what’s wrong in this code, at least for t <= 3?

https://www.codechef.com/viewsolution/17436393

#16

There is other way to solve the question. Below is my approach:

We know e^ix = cos(x) + isin(x)

=> e^inx = cos(nx) + isin(nx)

We know cos(nx) + isin(nx) = (cos(x) + isin(x))^n

After computing we would have cos(nx) = real((cos(x) + isin(x))^n)

The multiplication can can be done in logn.

Here is the code: https://www.codechef.com/viewsolution/17348197

#17

@manjuransari at one place u used the line

t_x = (xn_x)%mod - ((((y_xn_y_x)%mod)*diff)%mod)%mod;

cant we write it as t_x = [(xn_x)- ((((y_xn_y_x))*diff))]%mod;

i.e after multiplying all the values at last we are calculating mod

why using mod everytime as question clearly mentions you have to give FINAL result as ans%mod

why using mod every time will not affect our answer

#18

Can someone please tell me what is wrong with this code??
solution

#19

@albert it is possible that multiplying the two values may result in integer overflow. Hence after multiplying two values take mod, and then multiply with third value.

#20

@manjuransari ok… thanks !!