I currently have this function:
public double Max(double[] x, double[] y)
{
//Get min and max of x array as integer
int xMin = Convert.ToInt32(x.Min());
int xMax = Convert.ToInt32(x.Max());
// Generate a list of x values for input to Lagrange
double i = 2;
double xOld = Lagrange(xMin,x,y);
double xNew = xMax;
do
{
xOld = xNew;
xNew = Lagrange(i,x,y);
i = i + 0.01;
} while (xOld > xNew);
return i;
}
This will find the minimum value on a curve with decreasing slope...however, given this curve, I need to find three minima.
How can I find the three minima and output them as an array or individual variables? This curve is just an example--it could be inverted--regardless, I need to find multiple variables. So once the first min is found it needs to know how to get over the point of inflection and find the next... :/
*The Lagrange function can be found here.** For all practical purposes, the Lagrange function will give me f(x) when I input x...visually, it means the curve supplied by wolfram alpha.
*The math-side of this conundrum can be found here.**
Possible solution?
Generate an array of input, say x[1,1.1,1.2,1.3,1.4...], get an array back from the Lagrange function. Then find the three lowest values of this function? Then get the keys corresponding to the values? How would I do this?
It's been a while since I've taken a numerical methods class, so bear with me. In short there are a number of ways to search for the root(s) of a function, and depending on what your your function is (continuous? differentiable?), you need to choose one that is appropriate.
For your problem, I'd probably start by trying to use Newton's Method to find the roots of the second degree Lagrange polynomial for your function. I haven't tested out this library, but there is a C# based numerical methods package on CodePlex that implements Newton's Method that is open source. If you wanted to dig through the code you could.
The majority of root finding methods have cousins in the broader CS topic of 'search'. If you want a really quick and dirty approach, or you have a very large search space, consider something like Simulated Annealing. Finding all of your minima isn't guaranteed but it's fast and easy to code.
Assuming you're just trying to "brute force" calculate this to a certain level of prcision, you need your algorithm to basically find any value where both neighbors are greater than the current value of your loop.
To simplify this, let's just say you have an array of numbers, and you want to find the indices of the three local minima. Here's a simple algorithm to do it:
public void Test()
{
var ys = new[] { 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 4, 5, 4 };
var indices = GetMinIndices(ys);
}
public List<int> GetMinIndices(int[] ys)
{
var minIndices = new List<int>();
for (var index = 1; index < ys.Length; index++)
{
var currentY = ys[index];
var previousY = ys[index - 1];
if (index < ys.Length - 1)
{
var neytY = ys[index + 1];
if (previousY > currentY && neytY > currentY) // neighbors are greater
minIndices.Add(index); // add the index to the list
}
else // we're at the last index
{
if (previousY > currentY) // previous is greater
minIndices.Add(index);
}
}
return minIndices;
}
So, basically, you pass in your array of function results (ys) that you calculated for an array of inputs (xs) (not shown). What you get back from this function is the minimum indices. So, in this example, you get back 8, 14, and 17.
Related
I recently read an article explaining how to generate a weighted random number, and there's a piece of the code that I don't understand:
int r = ((int)(rand.Next() * (323567)) % prefix[n - 1]) + 1;
Why is rand.Next being multiplied by a constant 323567? Would the code work without this constant?
Below is the full code for reference, and you can find the full article here: https://www.geeksforgeeks.org/random-number-generator-in-arbitrary-probability-distribution-fashion/
Any help is appreciated, thank you!!
// C# program to generate random numbers
// according to given frequency distribution
using System;
class GFG{
// Utility function to find ceiling
// of r in arr[l..h]
static int findCeil(int[] arr, int r,
int l, int h)
{
int mid;
while (l < h)
{
// Same as mid = (l+h)/2
mid = l + ((h - l) >> 1);
if (r > arr[mid])
l = mid + 1;
else
h = mid;
}
return (arr[l] >= r) ? l : -1;
}
// The main function that returns a random number
// from arr[] according to distribution array
// defined by freq[]. n is size of arrays.
static int myRand(int[] arr, int[] freq, int n)
{
// Create and fill prefix array
int[] prefix = new int[n];
int i;
prefix[0] = freq[0];
for(i = 1; i < n; ++i)
prefix[i] = prefix[i - 1] + freq[i];
// prefix[n-1] is sum of all frequencies.
// Generate a random number with
// value from 1 to this sum
Random rand = new Random();
int r = ((int)(rand.Next() * (323567)) % prefix[n - 1]) + 1; // <--- RNG * Constant
// Find index of ceiling of r in prefix array
int indexc = findCeil(prefix, r, 0, n - 1);
return arr[indexc];
}
// Driver Code
static void Main()
{
int[] arr = { 1, 2, 3, 4 };
int[] freq = { 10, 5, 20, 100 };
int i, n = arr.Length;
// Let us generate 10 random numbers
// according to given distribution
for(i = 0; i < 5; i++)
Console.WriteLine(myRand(arr, freq, n));
}
}
UPDATE:
I ran this code to check it:
int[] intArray = new int[] { 1, 2, 3, 4, 5 };
int[] weights = new int[] { 5, 20, 20, 40, 15 };
List<int> results = new List<int>();
for (int i = 0; i < 100000; i++)
{
results.Add(WeightedRNG.GetRand(intArray, weights, intArray.Length));
}
for (int i = 0; i < intArray.Length; i++)
{
int itemsFound = results.Where(x => x == intArray[i]).Count();
Console.WriteLine($"{intArray[i]} was returned {itemsFound} times.");
}
And here are the results with the constant:
1 was returned 5096 times.
2 was returned 19902 times.
3 was returned 20086 times.
4 was returned 40012 times.
5 was returned 14904 times.
And without the constant...
1 was returned 100000 times.
2 was returned 0 times.
3 was returned 0 times.
4 was returned 0 times.
5 was returned 0 times.
It completely breaks without it.
The constant does serve a purpose in some environments, but I don't believe this code is correct for C#.
To explain, let's look at the arguments to the function. The first sign something is off is passing n as an argument instead of inferring it from the arrays. The second sign is it's poor practice in C# to deal with paired arrays rather than something like a 2D array or sequence of single objects (such as a Tuple). But those are just indicators something is odd, and not evidence of any bugs.
So let's put that on hold for a moment and explain why a constant might matter by looking a small example.
Say you have three numbers (1, 2, and 3) with weights 3, 2, and 2. This function first builds up a prefix array, where each item includes the chances of finding the number for that index and all previous numbers.
We end up with a result like this: (3, 5, 7). Now we can use the last value and take a random number from 1 to 7. Values 1-3 result in 1, values 4 and 5 result in 2, and values 6 and 7 result in 3.
To find this random number the code now calls rand.Next(), and this is where I think the error comes in. In many platforms, the Next() function returns a double between 0 and 1. That's too small to use to lookup your weighted value, so you then multiply by a prime constant related the platform's epsilon value to ensure you have a reasonably large result that will cover the entire possible range of desired weights (in this case, 1-7) and then some. Now you take the remainder (%) vs your max weight (7), and map it via the prefix array to get the final result.
So the first error is, in C#, .Next() does not return a double. It is documented to return a non-negative random integer between 0 and int.MaxValue. Multiply that by 323567 and you're asking for integer overflow exceptions. Another sign of this mistake is the cast to int: the result of this function is already an int. And let's not even talk the meaningless extra parentheses around (323567).
There is also another, more subtle, error.
Let's the say the result of the (int)(rand.Next() * 323567) expression is 10. Now we take this value and get the remainder when dividing by our max value (%7). The problem here is we have two chances to roll a 1, 2, or 3 (the extra chance is if the original was 8, 9, or 10), and only once chance for the remaining weights (4-7). So we have introduced unintended bias into the system, completely throwing off the expected weights. (This is a simplification. The actual number space is not 1-10; it's 0 * 323567 - 0.999999 * 323567. But the potential for bias still exists as long that max value is not evenly divisible by the max weight value).
It's possible the constant was chosen because it has properties to minimize this effect, but again: it was based on a completely different kind of .Next() function.
Therefore the rand.Next() line should probably be changed to look like this:
int r = rand.Next(prefix[n - 1]) +1;
or this:
int r = ((int)(rand.NextDouble() * (323567 * prefix[n - 1])) % prefix[n - 1]) + 1;
But, given the other errors, I'd be wary of using this code at all.
For fun, here's an example running several different options:
https://dotnetfiddle.net/Y5qhRm
The original random method (using NextDouble() and a bare constant) doesn't fare as badly as I'd expect.
So I use a randomly generated dataset and I need to find the moving average of the sample size the user inputted. For example, the dataset is a list with {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and the sample size the user inputted is 2. The program must calculate first the mean of:
1 and 2 = 1.5, 2 and 3 = 2.5, 3 and 4 = 3.5,
and so on. How do I do this? Thanks! 😊
You can keep track of the sum and queue up all the values so you know what to subtract from the sum once you get to the sample size.
public static IEnumerable<decimal> MovingAverages(IEnumerable<decimal> values, int sample)
{
var queue = new Queue<decimal>(sample);
decimal sum = 0;
foreach(var x in values)
{
sum += x;
queue.Enqueue(x);
if(queue.Count == sample)
{
yield return sum / sample;
sum -= queue.Dequeue();
}
}
}
Basically this will queue up the first n values (equal to sample) and the sum. Once it gets the desired number of values to average it yields the sum divided by the sample size and then removes the oldest value from the queue and subtracts it from the sum. Note that if the sample size is larger than the number of values this will return an empty enumerable.
This can be done with a simple loop.
for (int i = 0; i <= array.Length - samplesize; i++)
Console.WriteLine(array.Skip(i).Take(samplesize).Average());
The Skip(i).Take(samplesize) portion selects only the elements you are interested in at the moment.
C# why does binarysearch have to be made on sorted arrays and lists?
Is there any other method that does not require me to sort the list?
It kinda messes with my program in a way that I cannot sort the list for it to work as I want to.
A binary search works by dividing the list of candidates in half using equality. Imagine the following set:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
We can also represent this as a binary tree, to make it easier to visualise:
Source
Now, say we want to find the number 3. We can do it like so:
Is 3 smaller than 8? Yes. OK, now we're looking at everything between 1 and 7.
Is 3 smaller than 4? Yes. OK, now we're looking at everything between 1 and 3.
Is 3 smaller than 2? No. OK, now we're looking at 3.
We found it!
Now, if your list isn't sorted, how will we divide the list in half? The simple answer is: we can't. If we swap 3 and 15 in the example above, it would work like this:
Is 3 smaller than 8? Yes. OK, now we're looking at everything between 1 and 7.
Is 3 smaller than 4? Yes. OK, now we're looking at everything between 1 and 3 (except we swapped it with 15).
Is 3 smaller than 2? No. OK, now we're looking at 15.
Huh? There's no more items to check but we didn't find it. I guess it's not in the list.
The solution is to use an appropriate data type instead. For fast lookups of key/value pairs, I'll use a Dictionary. For fast checks if something already exists, I'll use a HashSet. For general storage I'll use a List or an array.
Dictionary example:
var values = new Dictionary<int, string>();
values[1] = "hello";
values[2] = "goodbye";
var value2 = values[2]; // this lookup will be fast because Dictionaries are internally optimised inside and partition keys' hash codes into buckets.
HashSet example:
var mySet = new HashSet<int>();
mySet.Add(1);
mySet.Add(2);
if (mySet.Contains(2)) // this lookup is fast for the same reason as a dictionary.
{
// do something
}
List exmaple:
var list = new List<int>();
list.Add(1);
list.Add(2);
if (list.Contains(2)) // this isn't fast because it has to visit each item in the list, but it works OK for small sets or places where performance isn't so important
{
}
var idx2 = list.IndexOf(2);
If you have multiple values with the same key, you could store a list in a Dictionary like this:
var values = new Dictionary<int, List<string>>();
if (!values.ContainsKey(key))
{
values[key] = new List<string>();
}
values[key].Add("value1");
values[key].Add("value2");
There is no way you use binary search on unordered collections. Sorting collection is the main concept of the binary search. The key is that on every move u take the middle index between l and r. On first step they are 0 and size - 1, after every step one of them becomes middle index between them. If x > arr[m] then l becomes m + 1, otherwise r becomes m - 1. Basically, on every step you take half of the array you had and, of course, it remains sorted. This code is recursive, if you don't know what recursion is(which is very important in programming), you can review and learn here.
// C# implementation of recursive Binary Search
using System;
class GFG {
// Returns index of x if it is present in
// arr[l..r], else return -1
static int binarySearch(int[] arr, int l,
int r, int x)
{
if (r >= l) {
int mid = l + (r - l) / 2;
// If the element is present at the
// middle itself
if (arr[mid] == x)
return mid;
// If element is smaller than mid, then
// it can only be present in left subarray
if (arr[mid] > x)
return binarySearch(arr, l, mid - 1, x);
// Else the element can only be present
// in right subarray
return binarySearch(arr, mid + 1, r, x);
}
// We reach here when element is not present
// in array
return -1;
}
// Driver method to test above
public static void Main()
{
int[] arr = { 2, 3, 4, 10, 40 };
int n = arr.Length;
int x = 10;
int result = binarySearch(arr, 0, n - 1, x);
if (result == -1)
Console.WriteLine("Element not present");
else
Console.WriteLine("Element found at index "
+ result);
}
}
Output:
Element is present at index 3
Sure there is.
var list = new List<int>();
list.Add(42);
list.Add(1);
list.Add(54);
var index = list.IndexOf(1); //TADA!!!!
EDIT: Ok, I hoped the irony was obvious. But strictly speaking, if your array is not sorted, you are pretty much stuck with the linear search, readily available by means of IndexOf() or IEnumerable.First().
Hi i have an array of size N. The array values will always have either 1, 2, 3 integer values only. Now i need to find the lowest number between a given range of array indices. So for e.g. array = 2 1 3 1 2 3 1 3 3 2. the lowest value for ranges like [2-4] = 1, [4-5] = 2, [7-8] = 3, etc.
Below is my code :
static void Main(String[] args) {
string[] width_temp = Console.ReadLine().Split(' ');
int[] width = Array.ConvertAll(width_temp,Int32.Parse); // Main Array
string[] tokens_i = Console.ReadLine().Split(' ');
int i = Convert.ToInt32(tokens_i[0]);
int j = Convert.ToInt32(tokens_i[1]);
int vehicle = width[i];
for (int beg = i+1; beg <= j; beg++) {
if (vehicle > width[beg]) {
vehicle = width[beg];
}
}
Console.WriteLine("{0}", vehicle);
}
The above code works fine. But my concern is about efficiency. In above I am just taking one set of array range, but in actual there will be n number of ranges and I would have to return the lowest for each range. Now the problem is if there is a range like [0-N], N is array size, then I would end up comparing all the items for lowest. So I was wondering if there is a way around to optimize the code for efficiency???
I think it is a RMQ (Range Minimum Query) and there is several implementation which may fit your scenario.
Here is a nice TopCoder Tutorial cover a lot of them, I recommend two of them:
Using the notation in the tutorial, define <P, T> as <Preprocess Complexity, Query Complexity>, there is two famous and common implementation / data structure which can handle RMQ: Square Rooting Array & Segment Tree.
Segment Tree is famous yet hard to implement, it can solve RMQ in <O(n), O(lg n)> though, which has better complexity than Square Rooting Array (<O(n), O(sqrt(n))>)
Square Rooting Array (<O(n), O(sqrt(n))>)
Note That It is not a official name of the technique nor any data structure, indeed I do not know if there is any official naming of this technique since I learnt it...but here we go
For query time, it is definitely not the best you can got to solve RMQ, but it has an advantage: Easy Implementation! (Compared to Segment Tree...)
Here is the high level concept of how it works:
Let N be the length of the array, we split the array into sqrt(N) groups, each contain sqrt(N) elements.
Now we use O(N) time to find the minimum value of each groups, store them into another array call M
So using the above array, M[0] = min(A[0..2]), M[1] = min(A[3..5]), M[2] = min(A[6..8]), M[3] = min(A[9..9])
(The image from TopCoder Tutorial is storing the index of the minimum element)
Now let's see how to query:
For any range [p..q], we can always split this range into 3 parts at most.
Two parts for the left boundaries which is some left over elements that cannot be form a whole group.
One part is the elements in between, which forms some groups.
Using the same example, RMQ(2,7) can be split into 3 parts:
Left Boundary (left over elements): A[2]
Right Boundary (left over elements): A[6], A[7]
In between elements (elements across whole group): A[3],A[4],A[5]
Notice that for those in between elements, we have already preprocessed their minimum using M, so we do not need to look at each element, we can look and compare M instead, there is at most O(sqrt(N)) of them (it is the length of M afterall)
For boundary parts, as they cannot form a whole group by definition, means there is at most O(sqrt(N)) of them (it is the length of one whole group afterall)
So combining two boundary parts, with one part of in between elements, we only need to compare O(3*sqrt(N)) = O(sqrt(N)) elements
You can refer to the tutorial for more details (even for some pseudo codes).
You could do this using Linq extension methods.
List<int> numbers = new List<int> {2, 1, 3, 1, 2, 3, 1, 3, 3, 2};
int minindex =1, maxindex =3, minimum=-1;
if(minindex <= maxindex && maxindex>=0 && maxindex >=0 && maxindex < numbers.Count())
{
minimum = Enumerable.Range(minindex, maxindex-minindex+1) // max inclusive, remove +1 if you want to exclude
.Select(x=> numbers[x]) // Get the elements between given indices
.Min(); // Get the minimum among.
}
Check this Demo
This seems a fun little problem. My first point would be that scanning a fixed array tends to be pretty fast (millions per second), so you'd need a vast amount of data to warrant a more complex solution.
The obvious first thing, is to break from the loop when you have found a 1, as you've found your lowest value then.
If you want something more advanced.
Create a new array of int. Create a pre load function that populates each item of this array with the next index where it gets lower.
Create a loop that uses the new array to skip.
Here is what I mean. Take the following arrays.
int[] intialArray = new int[] { 3, 3, 3, 3, 2, 2, 2, 1 };
int[] searchArray = new int[] { 4, 4, 4, 4, 7, 7, 7, 7 };
So the idea is to find the lowest between positions 0-7.
Start at initialArray[0] and get value 3.
Read searchArray[0] and get the value 4. The 4 is the next index where the number is lower.
Read initialArray[4] and get the value 2.
etc.
So basically you'd need to put some effort to build the searcharray, but onces it's complete you would scan each range much faster.
Form your looping like the following:
int[] inputArray = { 2, 1, 3, 1, 2, 3, 1, 3, 3, 2 };
int minIndex = 2;
int maxIndex = 5;
int minVal = 3;
for (int i = minIndex; i <= maxIndex; i++)
{
if (inputArray[i] <= minVal)
minVal = inputArray[i];
}
Console.WriteLine("Minimum value in the Given range is ={0}", minVal);
I'm still learning the ropes with C# programming. I am trying to write an application to solve Knight's Tour but have run into some grief when trying to get possible future positions for the knight.
For positions I am using a very simple struct:
public struct Position
{
public enum BoardRank { A = 1, B, C, D, E, F, G, H };
public BoardRank X;
public int Y;
}
In the knight class, my PossibleMoves() method begins like this:
public List<Position> PossibleMoves(Position position)
{
List<Position> positions = new List<Position>();
int[] multiply = new int[]{1, -1};
foreach (int times in multiply)
{
try{
Position possible = new Position();
possible.X = position.X + (Behaviour.X * times);
possible.Y = position.Y + (Behaviour.Y * times);
positions.Add(possible);
}
...
For position = A1 and times = -1, you can see how Behaviour.X could quickly fall out of bounds, but I assumed this would have been picked up by the try block.
I tried adding a {get; set;} on the enum declaration but that threw some useless syntax errors in VS2010.
Is there anything else I can try here to stop my enum from going out of bounds?
I assumed this would have been picked up by the try block.
Nope. Enums in C# are "named numbers" effectively. They're not a complete set of values for the type.
Is there anything else I can try here to stop my enum from going out of bounds?
You can use Enum.IsDefined to check whether a value exists in the original enum. I would personally stop using public fields, and instead make Position immutable - then validate the value in the constructor. You could also have methods such as WithX which returned a new value based on the current value with just X changing. While you've got public fields, you're never going to be able to trust that any particular value is valid though.
It may be useful to use modulo to keep the values within a specific range:
possible.X = (position.X + (Behaviour.X * times)) % ((int)BoardRank.H + 1);
This way I am not sure whether an enum is the best solution here, as we're working with integers anyway. The numbers must be a sequence with no gaps and you have to make sure you take the highest defined enum value plus one. Thus, if you add a I to your enum, you need to change the modul.
Here I have a very simple program to illustrate how it works:
enum Foo { A, B, C }
static void Main(string[] args)
{
for (int i = 0; i < 10; i++)
{
Console.WriteLine(i % ((int)Foo.C + 1));
}
}
As you see we take i modulo C + 1 which makes C's integer value the actual range maximum. This is the output:
0, 1, 2, 0, 1, 2, 0, 1, 2, 0