USACO 2021 January Contest, Platinum

Problem 2. Minimum Cost Paths


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Farmer John's pasture can be regarded as an $N\times M$ ($2\le N\le 10^9$, $2\le M\le 2\cdot 10^5$) 2D grid of square "cells" (picture a huge chessboard). The cell at the $x$-th row from the top and $y$-th column from the right is denoted by $(x,y)$ for each $x\in [1,N], y\in [1,M]$. Furthermore, for each $y\in [1,M]$, the $y$-th column is associated with the cost $c_y$ ($1\le c_y\le 10^9$).

Bessie starts at the cell $(1,1)$. If she is currently located at the cell $(x,y)$, then she may perform one of the following actions:

  • If $y<M$, Bessie may move to the next column (increasing $y$ by one) for a cost of $x^2$.
  • If $x<N$, Bessie may move to the next row (increasing $x$ by one) for a cost of $c_y$.

Given $Q$ ($1\le Q\le 2\cdot 10^5$) independent queries each of the form $(x_i,y_i)$ ($x_i\in [1,N], y_i\in [1,M]$), compute the minimum possible total cost for Bessie to move from $(1,1)$ to $(x_i,y_i)$.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $N$ and $M$.

The second line contains $M$ space-separated integers $c_1,c_2,\ldots,c_M$.

The third line contains $Q$.

The last $Q$ lines each contain two space-separated integers $x_i$ and $y_i$.

OUTPUT FORMAT (print output to the terminal / stdout):

$Q$ lines, containing the answers for each query.

Note that the large size of integers involved in this problem may require the use of 64-bit integer data types (e.g., a "long long" in C/C++).

SAMPLE INPUT:

5 4
1 100 100 20
20
1 1
2 1
3 1
4 1
5 1
1 2
2 2
3 2
4 2
5 2
1 3
2 3
3 3
4 3
5 3
1 4
2 4
3 4
4 4
5 4

SAMPLE OUTPUT:

0
1
2
3
4
1
5
11
19
29
2
9
20
35
54
3
13
29
49
69

The output in grid format: 

    1  2  3  4
  *--*--*--*--*
1 | 0| 1| 2| 3|
  *--*--*--*--*
2 | 1| 5| 9|13|
  *--*--*--*--*
3 | 2|11|20|29|
  *--*--*--*--*
4 | 3|19|35|49|
  *--*--*--*--*
5 | 4|29|54|69|
  *--*--*--*--*

SCORING:

  • Test cases 1-3 satisfy $N,M\le 2000$.
  • Test cases 4-8 satisfy $c_2>c_3>\cdots>c_M$.
  • Test cases 9-15 satisfy $N\le 2\cdot 10^5$.
  • Test cases 16-20 satisfy no additional constraints.

Problem credits: Benjamin Qi

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