以 |xa| 形式查询多个方程在 x 处的最小值 + b

Question

Kind of unrelated to what I was doing, but I stumbled on this interesting data structure. I can't figure out a few details.

Suppose I have like N<10^5 equations in the form y = |xa| + b, with 0 < x, a, b < 10^5. After O(N) preprocessing, for any x, I want to find the minimum value out of all these equations in constant time.

Here's what I have so far: add the functions one by one in increasing values of a, somehow find the intersection between the new function and the min existing function , then store intersections. The intersections can tell us what equation to use for the min. Then we can use binary search for log time queries, but since x<10^5, we can just go through all values and precompute the correct min value for each x

Visualization, where the black is the min values

Answer 1

The basic property of modulus is

1. |x| = x if x>=0
2. |x| = -x if x<0

Using this, the equation y = |xa|+b can be written as

1. y = x + (b-a) if x >= a
2. y = (b+a) - x if x<a

Since b and a are constants, we can use pre-computation to get answers per query in constant time.
For a given x , there are two cases:

1. The minima lies to the left of x (ie a < x ). If we know the minimum value of ba for all lines with a < x , we can get the answer in O(1) time.
2. Similarly, when the minima lies on the right of x (ie a >= x ), if we know the minimum value of b+a for all lines with a >= x , we can get the answer in O(1) time.

To summarise, we need to know the minimum value of ba for all a on the left of x and the minimum value of b+a for all a on the right of x . This can easily be done using a simple prefix and/or suffix array implementation.