Similar Arrays Solution Codechef - You are given two arrays of positive integers, AA and BB, both of length NN. You must choose a sequence of NN non-increasing real numbers R1,R2,…RNR1,R2,…RN

Similar Arrays Solution Codechef -  

SOLUTION

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You are given two arrays of positive integers, A$A$ and B$B$, both of length N$N$. You must choose a sequence of N$N$ non-increasing real numbers R1,R2,…RN$R_1,R_2,\dots R_N$ to multiply with B1,B2,…BN$B_1,B_2,\dots B_N$ respectively, such that the sum of the squares of their differences with corresponding terms A1,A2,…AN$A_1,A_2,\dots A_N$ is minimized.

Formally, find the minimum possible value of ∑i=1N(Ai−RiBi)2${\displaystyle \sum _{i=1}^N(A_i-R_i{B}_i{)}^2}$, over all possible N$N$-tuples of real numbers R=(R1,R2,…RN)$R=(R_1,R_2,\dots R_N)$ satisfying R1≥R2≥⋯≥RN$R_1\ge R_2\ge \cdots \ge R_N$

Input

• The first line contains an integer T$T$, the number of test cases. Then the test cases follow.
• The first line of each test case contains an integer N$N$, the length of arrays A$A$ and B$B$.
• The second line contains N$N$ space-separated integers A1,A2,…AN$A_1,A_2,\dots A_N$.
• The third line contains N$N$ space-separated integers B1,B2,…BN$B_1,B_2,\dots B_N$.

Output

For each test case, print the minimum possible value of the objective as described in the statement.

Your answer is considered correct if its absolute or relative error does not exceed 10−6${10}^{-6}$.

Constraints

• 2≤N≤5⋅105$2\le N\le 5\cdot {10}^5$
• The sum of N$N$ over all test cases doesn't exceed 5⋅105$5\cdot {10}^5$
• 1≤Ai,Bi≤1000$1\le A_i,B_i\le 1000$ for each valid i$i$.

Subtasks

• Subtask 1 (5 points): T=1,N=2$T=1,N=2$.
• Subtask 2 (11 points): The sum of N$N$ over all test cases doesn't exceed 20$20$.
• Subtask 3 (13 points): The sum of N$N$ over all test cases doesn't exceed 300$300$.
• Subtask 4 (14 points): The sum of N$N$ over all test cases doesn't exceed 2000$2000$.
• Subtask 5 (20 points): The sum of N$N$ over all test cases doesn't exceed 7000$7000$.
• Subtask 6 (37 points): No additional constraints

Sample Input

3
2
2 5
1 8
5
7 9 1 4 3
9 8 6 13 1
10
66 23 51 81 60 7 26 127 66 8
9 88 77 12 2 38 7 63 90 111

Sample Output

0.000000000000000
12.247238031469687
17698.696831405897683

Explanation

In the first test case, we choose R1=2$R_1=2$ and R2=0.625$R_2=0.625$

Date Added:27-05-2021

Time Limit:1 secs

Source Limit:50000 Bytes

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SOLUTION

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