Dears, I need your help on that CP-SAT implementation. Here a simplified version of what I'm doing:
assume that I have 3 boolean decision variables: x1, x2, x3 and boolean coefficents a1 a2 a3
model.Add(num == a1*x1+a2*x2+a3*x3)
model.Add(denom == x1+x2+x3)
model.Add(denom > 0)
model.AddDivisionEquality(division, num, denom)
model.Minimize(division)
When I solve it I receive MODEL_INVALID
The divisor cannot span across zero in constraint: int_div { target { vars: 1004 coeffs: 1 } exprs { vars: 1002 coeffs: 1 } exprs { vars: 1003 coeffs: 1 } }
Seems the denominator is getting zero value temporary during the search. Is there a way to avoid this?
Thanks!
Related
I have the follow function:
import copy
rn = 0
def check_vals(x, y):
global rn
if (y != None) & (int(x)+1) == int(y):
return rn + 1
else:
# Using copy to deepcopy and not forming a shallow one.
res = copy.copy(rn)
# Increment so that the next value with start form +1
rn += 1
# Return the same value as we want to group using this
return res + 1
return 0
#pandas_udf(IntegerType(), functionType=PandasUDFType.GROUPED_AGG)
def check_final(x, y):
return lambda x, y: check_vals(x, y)
I need apply this function in a follow df:
index initial_range final_range
1 1 299
1 300 499
1 500 699
1 800 1000
2 10 99
2 100 199
So I need that follow output:
index min_val max_val
1 1 699
1 800 1000
2 10 199
See, that the grouping field there are a news abrangencies, that are the values min(initial) and max(final), until the sequence is broken, applying the groupBy.
I tried:
w = Window.partitionBy('index').orderBy(sf.col('initial_range'))
df = (df.withColumn('nextRange', sf.lead('initial_range').over(w))
.fillna(0,subset=['nextRange'])
.groupBy('index')
.agg(check_final("final_range", "nextRange").alias('check_1'))
.withColumn('min_val', sf.min("initial_range").over(Window.partitionBy("check_1")))
.withColumn('max_val', sf.max("final_range").over(Window.partitionBy("check_1")))
)
But, don't worked.
Anyone can help me?
I think pure Spark SQL API can solve your question and it doesn't need to use any UDF, which might be an impact of your Spark performance. Also, I think two window function is enough to solve this question:
df.withColumn(
'next_row_initial_diff', func.col('initial_range')-func.lag('final_range', 1).over(Window.partitionBy('index').orderBy('initial_range'))
).withColumn(
'group', func.sum(
func.when(func.col('next_row_initial_diff').isNull()|(func.col('next_row_initial_diff')==1), func.lit(0))
.otherwise(func.lit(1))
).over(
Window.partitionBy('index').orderBy('initial_range')
)
).groupBy(
'group', 'index'
).agg(
func.min('initial_range').alias('min_val'),
func.max('final_range').alias('max_val')
).drop(
'group'
).show(100, False)
Column next_row_initial_diff: Just like the lead you use to shift/lag the row and check if it's in sequence.
Column group: To group the sequence in index partition.
I'm looking to create a function that returns a solve for x math equation that can be preformed in ones head (Clearly thats a bit subjective but I'm not sure how else to phrase it).
Example problem: (x - 15)/10 = 6
Note: Only 1 x in the equation
I want to use the operations +, -, *, /, sqrt (Only applied to X -> sqrt(x))
I know that let mathExpression = NSExpression(format: question) converts strings into math equations but when solving for x I'm not sure how to go about doing this.
I previously asked Generating random doable math problems swift for non solving for x problems but I'm not sure how to convert that answer into solving for x
Edit: Goal is to generate an equation and have the user solve for the variable.
Since all you want is a string representing an equation and a value for x, you don't need to do any solving. Just start with x and transform it until you have a nice equation. Here's a sample: (copy and paste it into a Playground to try it out)
import UIKit
enum Operation: String {
case addition = "+"
case subtraction = "-"
case multiplication = "*"
case division = "/"
static func all() -> [Operation] {
return [.addition, .subtraction, .multiplication, .division]
}
static func random() -> Operation {
let all = Operation.all()
let selection = Int(arc4random_uniform(UInt32(all.count)))
return all[selection]
}
}
func addNewTerm(formula: String, result: Int) -> (formula: String, result: Int) {
// choose a random number and operation
let operation = Operation.random()
let number = chooseRandomNumberFor(operation: operation, on: result)
// apply to the left side
let newFormula = applyTermTo(formula: formula, number: number, operation: operation)
// apply to the right side
let newResult = applyTermTo(result: result, number: number, operation: operation)
return (newFormula, newResult)
}
func applyTermTo(formula: String, number:Int, operation:Operation) -> String {
return "\(formula) \(operation.rawValue) \(number)"
}
func applyTermTo(result: Int, number:Int, operation:Operation) -> Int {
switch(operation) {
case .addition: return result + number
case .subtraction: return result - number
case .multiplication: return result * number
case .division: return result / number
}
}
func chooseRandomNumberFor(operation: Operation, on number: Int) -> Int {
switch(operation) {
case .addition, .subtraction, .multiplication:
return Int(arc4random_uniform(10) + 1)
case .division:
// add code here to find integer factors
return 1
}
}
func generateFormula(_ numTerms:Int = 1) -> (String, Int) {
let x = Int(arc4random_uniform(10))
var leftSide = "x"
var result = x
for i in 1...numTerms {
(leftSide, result) = addNewTerm(formula: leftSide, result: result)
if i < numTerms {
leftSide = "(" + leftSide + ")"
}
}
let formula = "\(leftSide) = \(result)"
return (formula, x)
}
func printFormula(_ numTerms:Int = 1) {
let (formula, x) = generateFormula(numTerms)
print(formula, " x = ", x)
}
for i in 1...30 {
printFormula(Int(arc4random_uniform(3)) + 1)
}
There are some things missing. The sqrt() function will have to be implemented separately. And for division to be useful, you'll have to add in a system to find factors (since you presumably want the results to be integers). Depending on what sort of output you want, there's a lot more work to do, but this should get you started.
Here's sample output:
(x + 10) - 5 = 11 x = 6
((x + 6) + 6) - 1 = 20 x = 9
x - 2 = 5 x = 7
((x + 3) * 5) - 6 = 39 x = 6
(x / 1) + 6 = 11 x = 5
(x * 6) * 3 = 54 x = 3
x * 9 = 54 x = 6
((x / 1) - 6) + 8 = 11 x = 9
Okay, let’s assume from you saying “Note: Only 1 x in the equation” that what you want is a linear equation of the form y = 0 = β1*x + β0, where β0 and β1 are the slope and intercept coefficients, respectively.
The inverse of (or solution to) any linear equation is given by x = -β0/β1. So what you really need to do is generate random integers β0 and β1 to create your equation. But since it should be “solvable” in someone’s head, you probably want β0 to be divisible by β1, and furthermore, for β1 and β0/β1 to be less than or equal to 12, since this is the upper limit of the commonly known multiplication tables. In this case, just generate a random integer β1 ≤ 12, and β0 equal to β1 times some integer n, 0 ≤ n ≤ 12.
If you want to allow simple fractional solutions like 2/3, just multiply the denominator and the numerator into β0 and β1, respectively, taking care to prevent the numerator or denominator from getting too large (12 is again a good limit).
Since you probably want to make y non-zero, just generate a third random integer y between -12 and 12, and change your output equation to y = β1*x + β0 + y.
Since you mentioned √ could occur over the x variable only, that is pretty easy to add; the solution (to 0 = β1*sqrt(x) + β0) is just x = (β0/β1)**2.
Here is some very simple (and very problematic) code for generating random integers to get you started:
import func Glibc.srand
import func Glibc.rand
import func Glibc.time
srand(UInt32(time(nil)))
print(rand() % 12)
There are a great many answers on this website that deal with better ways to generate random integers.
To find how many ways we have of making change for the amount 4 given the coins [1,2,3], we can create a DP algorithm that produces the following table:
table[amount][coins.count]
0 1 2 3 4
-----------
(0) 1 | 1 1 1 1 1
(1) 2 | 1 1 2 2 3
(2) 3 | 1 1 2 3 4
The last position being our answer. The answer is 4 because we have the following combinations: [1,1,1,1],[2,1],[2,2],[3,1].
My question is, is it possible to retrieve these combinations from the table I just generated? How?
For completeness, here's my algorithm
func coinChange(coins: [Int], amount: Int) -> Int {
// int[amount+1][coins]
var table = Array<Array<Int>>(repeating: Array<Int>(repeating: 0, count: coins.count), count: amount + 1)
for i in 0..<coins.count {
table[0][i] = 1
}
for i in 1...amount {
for j in 0..<coins.count {
//solutions that include coins[j]
let x = i - coins[j] >= 0 ? table[i - coins[j]][j] : 0
//solutions that don't include coins[j]
let y = j >= 1 ? table[i][j-1] : 0
table[i][j] = x + y
}
}
return table[amount][coins.count - 1];
}
Thanks!
--
Solution
Here's an ugly function that retrieves the combinations, based on #Sayakiss 's explanation:
func getSolution(_ i: Int, _ j: Int) -> [[Int]] {
if j < 0 || i < 0 {
//not a solution
return []
}
if i == 0 && j == 0 {
//valid solution. return an empty array where the coins will be appended
return [[]]
}
return getSolution(i - coins[j], j).map{var a = $0; a.append(coins[j]);return a} + getSolution(i, j - 1)
}
getSolution(amount, coins.count-1)
Output:
[[1, 3], [2, 2], [1, 1, 2], [1, 1, 1, 1]]
Sure you can. We define a new function get_solution(i,j) which means all solution for your table[i][j].
You can think it returns an array of array, for example, the output of get_solution(4,3) is [[1,1,1,1],[2,1],[2,2],[3,1]]. Then:
Case 1. Any solution from get_solution(i - coins[j], j) plus coins[j] is a solution for table[i][j].
Case 2. Any solution from get_solution(i, j - 1) is a solution for table[i][j].
You can prove Case 1 + Case 2 is all possible solution for table[i][j](note you get table[i][j] by this way).
The only problem remains is to implement get_solution(i,j) and I think it's good for you to do it by yourself.
If you still got any question, please don't hesitate to leave a comment here.
any of you knows how can I check if the division remainder is integer or zero?
if ( integer ( 3/2))
You should use the modulo operator like this
// a,b are ints
if ( a % b == 0) {
// remainder 0
} else
{
// b does not divide a evenly
}
It sounds like what you are looking for is the modulo operator %, which will give you the remainder of an operation.
3 % 2 // yields 1
3 % 1 // yields 0
3 % 4 // yields 1
However, if you want to actually perform the division first, you may need something a bit more complex, such as the following:
//Perform the division, then take the remainder modulo 1, which will
//yield any decimal values, which then you can compare to 0 to determine if it is
//an integer
if((a / b) % 1 > 0))
{
//All non-integer values go here
}
else
{
//All integer values go here
}
Walkthrough
(3 / 2) // yields 1.5
1.5 % 1 // yields 0.5
0.5 > 0 // true
swift 3:
if a.truncatingRemainder(dividingBy: b) == 0 {
//All integer values go here
}else{
//All non-integer values go here
}
You can use the below code to know which type of instance it is.
var val = 3/2
var integerType = Mirror(reflecting: val)
if integerType.subjectType == Int.self {
print("Yes, the value is an integer")
}else{
print("No, the value is not an integer")
}
let me know if the above was useful.
Swift 5
if numberOne.isMultiple(of: numberTwo) { ... }
Swift 4 or less
if numberOne % numberTwo == 0 { ... }
Swift 2.0
print(Int(Float(9) % Float(4))) // result 1
I often see code like
int hashCode(){
return a^b;
}
Why XOR?
Of all bit-operations XOR has the best bit shuffling properties.
This truth-table explains why:
A B AND
0 0 0
0 1 0
1 0 0
1 1 1
A B OR
0 0 0
0 1 1
1 0 1
1 1 1
A B XOR
0 0 0
0 1 1
1 0 1
1 1 0
As you can see for AND and OR do a poor job at mixing bits.
OR will on average produce 3/4 one-bits. AND on the other hand will produce on average 3/4 null-bits. Only XOR has an even one-bit vs. null-bit distribution. That makes it so valuable for hash-code generation.
Remember that for a hash-code you want to use as much information of the key as possible and get a good distribution of hash-values. If you use AND or OR you'll get numbers that are biased towards either numbers with lots of zeros or numbers with lots of ones.
XOR has the following advantages:
It does not depend on order of computation i.e. a^b = b^a
It does not "waste" bits. If you change even one bit in one of the components, the final value will change.
It is quick, a single cycle on even the most primitive computer.
It preserves uniform distribution. If the two pieces you combine are uniformly distributed so will the combination be. In other words, it does not tend to collapse the range of the digest into a narrower band.
More info here.
XOR operator is reversible, i.e. suppose I have a bit string as 0 0 1 and I XOR it with another bit string 1 1 1, the the output is
0 xor 1 = 1
0 1 = 1
1 1 = 0
Now I can again xor the 1st string with the result to get the 2nd string. i.e.
0 1 = 1
0 1 = 1
1 0 = 1
So, that makes the 2nd string a key. This behavior is not found with other bit operator
Please see this for more info --> Why is XOR used on Cryptography?
There is another use case: objects in which (some) fields must be compared without regarding their order. For example, if you want a pair (a, b) be always equal to the pair (b, a).
XOR has the property that a ^ b = b ^ a, so it can be used in hash function in such cases.
Examples: (full code here)
definition:
final class Connection {
public final int A;
public final int B;
// some code omitted
#Override
public boolean equals(Object o) {
if (this == o) return true;
if (o == null || getClass() != o.getClass()) return false;
Connection that = (Connection) o;
return (A == that.A && B == that.B || A == that.B && B == that.A);
}
#Override
public int hashCode() {
return A ^ B;
}
// some code omitted
}
usage:
HashSet<Connection> s = new HashSet<>();
s.add(new Connection(1, 3));
s.add(new Connection(2, 3));
s.add(new Connection(3, 2));
s.add(new Connection(1, 3));
s.add(new Connection(2, 1));
s.remove(new Connection(1, 2));
for (Connection x : s) {
System.out.println(x);
}
// output:
// Connection{A=2, B=3}
// Connection{A=1, B=3}