Ciel and Karaoke
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You need to find the distance from the given piecewise-constant function to the set of all functions, satisfying Lipschitz condition with the constant K, in the space L1(−∞, +∞). If you understand nothing here then read the problem statement below :P
Chef Ciel and her friends go to karaoke after the New Year's party. In karaoke, they sing songs, and moreover, recent karaoke machines can score their performance. To get a high score, one of the most important things is to sing with correct pitches.
The correct pitch at time t for some song is represented as f(t), where f(t) is a piecewise-constant function. In this problem, let us consider the infinitely long time −∞ < t < +∞ for simplicity. Let N−1 integers A1 < A2 < ... < AN−1 denote the time moments when the correct pitch is changed. That is,
- f(t) = B0, for −∞ < t < A1,
- f(t) = Bk, for Ak ≤ t < Ak+1 (1 ≤ k ≤ N−2),
- f(t) = BN−1, for AN−1 ≤ t < +∞.
On the other hand, Ciel cannot change her pitch instantly. So Ciel's pitch g(t) must satisfy the condition |g(x) − g(y)| ≤ K |x − y| for all real x and y. In particular g(t) is a continuous function and |g'(t)| ≤ K for all t where g'(t) exists (g'(t) denotes the derivative of g(t)).
The score assigned to Ciel by the karaoke machine is the value of the integral of |f(t) − g(t)| over the interval (−∞, +∞). Your task is to calculate the minimum value of this integral over all possible functions g(t) that satisfy the above condition.
The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows.
Each test case consists of exactly 3 lines. The first line of each test case contains two space separated integers N and K. The second line contains N−1 space separated integers A1, A2, ..., AN−1. Finally, the third line contains N space separated integers B0, B1, ..., BN−1.
For each test case, output a single line containing the minimum value of the integral mentioned in the problem statement. The output must have an absolute or relative error at most 0.000001 (10−6). Please, note that your output should not have more than 1000 digits after the decimal point, otherwise you may (or may not) get wrong answer or runtime error (SIGXFSZ).
- 1 ≤ T ≤ 250
- 2 ≤ N ≤ 8
- 1 ≤ K ≤ 4
- 0 ≤ A1 < A2 < ... < AN−1 ≤ 20130120
- 0 ≤ B0, B1, ..., BN−1 ≤ 20130120
- It can occur that Bi−1 = Bi for some i = 1, 2, ..., N−1
- The sum of N3 in one test file does not exceed 2013
Input: 3 2 1 1000 900 1000 2 2 1000 900 1000 3 1 500 1200 20 60 30 Output: 2500.0000000000 1250.0000000000 625.0000000000
The following figure corresponds to the example case 1. Here the bold red line denotes the correct pitch f(t) and the blue line denotes the optimal Ciel's pitch g(t). The answer is the area of the green region.
|Tags||cook30, laycurse, math, medium-hard|
|Time Limit:||2.013 sec|
|Source Limit:||50000 Bytes|
|Languages:||C, CPP14, JAVA, PYTH, PYTH 3.6, CS2, PAS fpc, PAS gpc, RUBY, PHP, GO, NODEJS, HASK, SCALA, D, PERL, FORT, WSPC, ADA, CAML, ICK, BF, ASM, CLPS, PRLG, ICON, SCM qobi, PIKE, ST, NICE, LUA, BASH, NEM, LISP sbcl, LISP clisp, SCM guile, JS, ERL, TCL, PERL6, TEXT, PYP3, CLOJ, FS|
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If you are still having problems, see a sample solution here.