distance | Distance PHP class can calculate the distance | Map library

This wouldn't be easy to implement but could be sublinear for many cases, and at most linear. Consider representing the perimeter of each tree as four corners (they mark a square rotated 45 degrees). For each tree compute it's perimeter intersection with the current intersection. The difficulty comes with managing the corners of the intersection, which could include more than one point because of the diagonal alignments. Run inside the final intersection to count how many empty spots are within it.

Unfortunately I don't know if any way to use that nice glue syntax with anything that's not on the left side of a :=. That's there the magic happens. You can get something to work if you take care of the explicity conversion to sumbol your self and do the string building manually. It's not pretty, but this works

First of All, Yes you can do this with high accuracy if the GPS coordinates are accurate.

Second, the main problem is rotation if the field are straight with lat lng lines this would be easy and straightforward (no bun intended).

The easy way is to convert coordinate to rotated image similar to the real field then rotated every X,Y point to the new straight image. (see the image below)

Here is how to rotate x,y knowing the angel:

1. Despite the test, this implementation of Dijkstra may put Ω(E) items in the priority queue. This will cost Ω(E log E) with every comparison-based priority queue.

2. Why not E log V? Well, assuming a connected, simple, nontrivial graph, we have Θ(E log V) = Θ(E log E) since log (V−1) ≤ log E < log V² = 2 log V.

3. The O(E + V log V)-time implementations of Dijkstra's algorithm depend on a(n amortized) constant-time DecreaseKey operation, avoiding multiple entries for an individual vertex. The implementation in this question will likely be faster in practice on sparse graphs, however.

I tried applying a Genetic Algorithm to the problem above. I made an initial guess that two additional nodes would connect all three disconnected components.

Just quick speedups:

1. You can omit math.sqrt()
2. Create dictionary of colors instead of a list (that way you don't have to search for the index each iteration)
3. use min() instead of sorted()

Some interesting articles/posts:

How to track coordinates on the quadraticCurve

https://coderedirect.com/questions/385964/nearest-point-on-a-quadratic-bezier-curve

And if it doesn't work maybe you can take a look at this library: https://pomax.github.io/bezierjs/

As suggested by Pomax in the comments the thing you're looking for is in the library and it looks like there is a proper explanation.

There is a live demo if you want to try it: https://pomax.github.io/bezierinfo/#projections
The source code of it is here: https://pomax.github.io/bezierinfo/chapters/projections/project.js

To use it install it using the steps from GitHub: https://github.com/Pomax/bezierjs

Of course credit to Pomax for suggesting his library

You can see the cost of instructions on most mainstream architecture here and there. Based on that and assuming you use for example an Intel Skylake processor, you can see that one 32-bit imul instruction can be computed per cycle but with a latency of 3 cycles. In the optimized code, 2 lea instructions (which are very cheap) can be executed per cycle with a 1 cycle latency. The same thing apply for the sal instruction (2 per cycle and 1 cycle of latency).

This means that the optimized version can be executed with only 2 cycle of latency while the first one takes 3 cycle of latency (not taking into account load/store instructions that are the same). Moreover, the second version can be better pipelined since the two instructions can be executed for two different input data in parallel thanks to a superscalar out-of-order execution. Note that two loads can be executed in parallel too although only one store can be executed in parallel per cycle. This means that the execution is bounded by the throughput of store instructions. Overall, only 1 value can only computed per cycle. AFAIK, recent Intel Icelake processors can do two stores in parallel like new AMD Ryzen processors. The second one is expected to be as fast or possibly faster on the chosen use-case (Intel Skylake processors). It should be significantly faster on very recent x86-64 processors.

Note that the lea instruction is very fast because the multiply-add is done on a dedicated CPU unit (hard-wired shifters) and it only supports some specific constant for the multiplication (supported factors are 1, 2, 4 and 8, which mean that lea can be used to multiply an integer by the constants 2, 3, 4, 5, 8 and 9). This is why lea is faster than imul/mul.

I can reproduce the slower execution with -O2 using GCC 11.2 (on Linux with a i5-9600KF processor).

The main source of source of slowdown comes from the higher number of micro-operations (uops) to be executed in the -O2 version certainly combined with the saturation of some execution ports certainly due to a bad micro-operation scheduling.

Here is the assembly of the loop with -Os:

I have such a solution. And I'm surprised myself that I used two loops for!! Incredibly, I did it. First things first.

My proposal is based on a simplification. However, the mistake you will make at short distances will be relatively small. But the time gain is huge!

Well, I propose to count the distance in Cartesian coordinates, not spherical.

So we're going to need a simple function that computes the Cartesian coordinates based on the two arguments latitude and longitude. Here is our LatLong2Cart feature.