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Javascript smart and good way to make array of size N where each object in the array has a certain probability? [duplicate]

I'm trying to devise a (good) way to choose a random number from a range of possible numbers where each number in the range is given a weight. To put it simply: given the range of numbers (0,1,2) choose a number where 0 has an 80% probability of being selected, 1 has a 10% chance and 2 has a 10% chance.
It's been about 8 years since my college stats class, so you can imagine the proper formula for this escapes me at the moment.
Here's the 'cheap and dirty' method that I came up with. This solution uses ColdFusion. Yours may use whatever language you'd like. I'm a programmer, I think I can handle porting it. Ultimately my solution needs to be in Groovy - I wrote this one in ColdFusion because it's easy to quickly write/test in CF.
public function weightedRandom( Struct options ) {
var tempArr = [];
for( var o in arguments.options )
{
var weight = arguments.options[ o ] * 10;
for ( var i = 1; i<= weight; i++ )
{
arrayAppend( tempArr, o );
}
}
return tempArr[ randRange( 1, arrayLen( tempArr ) ) ];
}
// test it
opts = { 0=.8, 1=.1, 2=.1 };
for( x = 1; x<=10; x++ )
{
writeDump( weightedRandom( opts ) );
}
I'm looking for better solutions, please suggest improvements or alternatives.

Rejection sampling (such as in your solution) is the first thing that comes to mind, whereby you build a lookup table with elements populated by their weight distribution, then pick a random location in the table and return it. As an implementation choice, I would make a higher order function which takes a spec and returns a function which returns values based on the distribution in the spec, this way you avoid having to build the table for each call. The downsides are that the algorithmic performance of building the table is linear by the number of items and there could potentially be a lot of memory usage for large specs (or those with members with very small or precise weights, e.g. {0:0.99999, 1:0.00001}). The upside is that picking a value has constant time, which might be desirable if performance is critical. In JavaScript:
function weightedRand(spec) {
var i, j, table=[];
for (i in spec) {
// The constant 10 below should be computed based on the
// weights in the spec for a correct and optimal table size.
// E.g. the spec {0:0.999, 1:0.001} will break this impl.
for (j=0; j<spec[i]*10; j++) {
table.push(i);
}
}
return function() {
return table[Math.floor(Math.random() * table.length)];
}
}
var rand012 = weightedRand({0:0.8, 1:0.1, 2:0.1});
rand012(); // random in distribution...
Another strategy is to pick a random number in [0,1) and iterate over the weight specification summing the weights, if the random number is less than the sum then return the associated value. Of course, this assumes that the weights sum to one. This solution has no up-front costs but has average algorithmic performance linear by the number of entries in the spec. For example, in JavaScript:
function weightedRand2(spec) {
var i, sum=0, r=Math.random();
for (i in spec) {
sum += spec[i];
if (r <= sum) return i;
}
}
weightedRand2({0:0.8, 1:0.1, 2:0.1}); // random in distribution...

Generate a random number R between 0 and 1.
If R in [0, 0.1) -> 1
If R in [0.1, 0.2) -> 2
If R in [0.2, 1] -> 3
If you can't directly get a number between 0 and 1, generate a number in a range that will produce as much precision as you want. For example, if you have the weights for
(1, 83.7%) and (2, 16.3%), roll a number from 1 to 1000. 1-837 is a 1. 838-1000 is 2.

I use the following
function weightedRandom(min, max) {
return Math.round(max / (Math.random() * max + min));
}
This is my go-to "weighted" random, where I use an inverse function of "x" (where x is a random between min and max) to generate a weighted result, where the minimum is the most heavy element, and the maximum the lightest (least chances of getting the result)
So basically, using weightedRandom(1, 5) means the chances of getting a 1 are higher than a 2 which are higher than a 3, which are higher than a 4, which are higher than a 5.
Might not be useful for your use case but probably useful for people googling this same question.
After a 100 iterations try, it gave me:
==================
| Result | Times |
==================
| 1 | 55 |
| 2 | 28 |
| 3 | 8 |
| 4 | 7 |
| 5 | 2 |
==================

Here are 3 solutions in javascript since I'm not sure which language you want it in. Depending on your needs one of the first two might work, but the the third one is probably the easiest to implement with large sets of numbers.
function randomSimple(){
return [0,0,0,0,0,0,0,0,1,2][Math.floor(Math.random()*10)];
}
function randomCase(){
var n=Math.floor(Math.random()*100)
switch(n){
case n<80:
return 0;
case n<90:
return 1;
case n<100:
return 2;
}
}
function randomLoop(weight,num){
var n=Math.floor(Math.random()*100),amt=0;
for(var i=0;i<weight.length;i++){
//amt+=weight[i]; *alternative method
//if(n<amt){
if(n<weight[i]){
return num[i];
}
}
}
weight=[80,90,100];
//weight=[80,10,10]; *alternative method
num=[0,1,2]

8 years late but here's my solution in 4 lines.
Prepare an array of probability mass function such that
pmf[array_index] = P(X=array_index):
var pmf = [0.8, 0.1, 0.1]
Prepare an array for the corresponding cumulative distribution function such that
cdf[array_index] = F(X=array_index):
var cdf = pmf.map((sum => value => sum += value)(0))
// [0.8, 0.9, 1]
3a) Generate a random number.
3b) Get an array of elements that are more than or equal to this number.
3c) Return its length.
var r = Math.random()
cdf.filter(el => r >= el).length

This is more or less a generic-ized version of what #trinithis wrote, in Java: I did it with ints rather than floats to avoid messy rounding errors.
static class Weighting {
int value;
int weighting;
public Weighting(int v, int w) {
this.value = v;
this.weighting = w;
}
}
public static int weightedRandom(List<Weighting> weightingOptions) {
//determine sum of all weightings
int total = 0;
for (Weighting w : weightingOptions) {
total += w.weighting;
}
//select a random value between 0 and our total
int random = new Random().nextInt(total);
//loop thru our weightings until we arrive at the correct one
int current = 0;
for (Weighting w : weightingOptions) {
current += w.weighting;
if (random < current)
return w.value;
}
//shouldn't happen.
return -1;
}
public static void main(String[] args) {
List<Weighting> weightings = new ArrayList<Weighting>();
weightings.add(new Weighting(0, 8));
weightings.add(new Weighting(1, 1));
weightings.add(new Weighting(2, 1));
for (int i = 0; i < 100; i++) {
System.out.println(weightedRandom(weightings));
}
}

How about
int [ ] numbers = { 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 2 } ;
then you can randomly select from numbers and 0 will have an 80% chance, 1 10%, and 2 10%

This one is in Mathematica, but it's easy to copy to another language, I use it in my games and it can handle decimal weights:
weights = {0.5,1,2}; // The weights
weights = N#weights/Total#weights // Normalize weights so that the list's sum is always 1.
min = 0; // First min value should be 0
max = weights[[1]]; // First max value should be the first element of the newly created weights list. Note that in Mathematica the first element has index of 1, not 0.
random = RandomReal[]; // Generate a random float from 0 to 1;
For[i = 1, i <= Length#weights, i++,
If[random >= min && random < max,
Print["Chosen index number: " <> ToString#i]
];
min += weights[[i]];
If[i == Length#weights,
max = 1,
max += weights[[i + 1]]
]
]
(Now I'm talking with a lists first element's index equals 0) The idea behind this is that having a normalized list weights there is a chance of weights[n] to return the index n, so the distances between the min and max at step n should be weights[n]. The total distance from the minimum min (which we put it to be 0) and the maximum max is the sum of the list weights.
The good thing behind this is that you don't append to any array or nest for loops, and that increases heavily the execution time.
Here is the code in C# without needing to normalize the weights list and deleting some code:
int WeightedRandom(List<float> weights) {
float total = 0f;
foreach (float weight in weights) {
total += weight;
}
float max = weights [0],
random = Random.Range(0f, total);
for (int index = 0; index < weights.Count; index++) {
if (random < max) {
return index;
} else if (index == weights.Count - 1) {
return weights.Count-1;
}
max += weights[index+1];
}
return -1;
}

I suggest to use a continuous check of the probability and the rest of the random number.
This function sets first the return value to the last possible index and iterates until the rest of the random value is smaller than the actual probability.
The probabilities have to sum to one.
function getRandomIndexByProbability(probabilities) {
var r = Math.random(),
index = probabilities.length - 1;
probabilities.some(function (probability, i) {
if (r < probability) {
index = i;
return true;
}
r -= probability;
});
return index;
}
var i,
probabilities = [0.8, 0.1, 0.1],
count = probabilities.map(function () { return 0; });
for (i = 0; i < 1e6; i++) {
count[getRandomIndexByProbability(probabilities)]++;
}
console.log(count);
.as-console-wrapper { max-height: 100% !important; top: 0; }

Thanks all, this was a helpful thread. I encapsulated it into a convenience function (Typescript). Tests below (sinon, jest). Could definitely be a bit tighter, but hopefully it's readable.
export type WeightedOptions = {
[option: string]: number;
};
// Pass in an object like { a: 10, b: 4, c: 400 } and it'll return either "a", "b", or "c", factoring in their respective
// weight. So in this example, "c" is likely to be returned 400 times out of 414
export const getRandomWeightedValue = (options: WeightedOptions) => {
const keys = Object.keys(options);
const totalSum = keys.reduce((acc, item) => acc + options[item], 0);
let runningTotal = 0;
const cumulativeValues = keys.map((key) => {
const relativeValue = options[key]/totalSum;
const cv = {
key,
value: relativeValue + runningTotal
};
runningTotal += relativeValue;
return cv;
});
const r = Math.random();
return cumulativeValues.find(({ key, value }) => r <= value)!.key;
};
Tests:
describe('getRandomWeightedValue', () => {
// Out of 1, the relative and cumulative values for these are:
// a: 0.1666 -> 0.16666
// b: 0.3333 -> 0.5
// c: 0.5 -> 1
const values = { a: 10, b: 20, c: 30 };
it('returns appropriate values for particular random value', () => {
// any random number under 0.166666 should return "a"
const stub1 = sinon.stub(Math, 'random').returns(0);
const result1 = randomUtils.getRandomWeightedValue(values);
expect(result1).toEqual('a');
stub1.restore();
const stub2 = sinon.stub(Math, 'random').returns(0.1666);
const result2 = randomUtils.getRandomWeightedValue(values);
expect(result2).toEqual('a');
stub2.restore();
// any random number between 0.166666 and 0.5 should return "b"
const stub3 = sinon.stub(Math, 'random').returns(0.17);
const result3 = randomUtils.getRandomWeightedValue(values);
expect(result3).toEqual('b');
stub3.restore();
const stub4 = sinon.stub(Math, 'random').returns(0.3333);
const result4 = randomUtils.getRandomWeightedValue(values);
expect(result4).toEqual('b');
stub4.restore();
const stub5 = sinon.stub(Math, 'random').returns(0.5);
const result5 = randomUtils.getRandomWeightedValue(values);
expect(result5).toEqual('b');
stub5.restore();
// any random number above 0.5 should return "c"
const stub6 = sinon.stub(Math, 'random').returns(0.500001);
const result6 = randomUtils.getRandomWeightedValue(values);
expect(result6).toEqual('c');
stub6.restore();
const stub7 = sinon.stub(Math, 'random').returns(1);
const result7 = randomUtils.getRandomWeightedValue(values);
expect(result7).toEqual('c');
stub7.restore();
});
});

Shortest solution in modern JavaScript
Note: all weights need to be integers
function weightedRandom(items){
let table = Object.entries(items)
.flatMap(([item, weight]) => Array(item).fill(weight))
return table[Math.floor(Math.random() * table.length)]
}
const key = weightedRandom({
"key1": 1,
"key2": 4,
"key3": 8
}) // returns e.g. "key1"

here is the input and ratios : 0 (80%), 1(10%) , 2 (10%)
lets draw them out so its easy to visualize.
0 1 2
-------------------------------------________+++++++++
lets add up the total weight and call it TR for total ratio. so in this case 100.
lets randomly get a number from (0-TR) or (0 to 100 in this case) . 100 being your weights total. Call it RN for random number.
so now we have TR as the total weight and RN as the random number between 0 and TR.
so lets imagine we picked a random # from 0 to 100. Say 21. so thats actually 21%.
WE MUST CONVERT/MATCH THIS TO OUR INPUT NUMBERS BUT HOW ?
lets loop over each weight (80, 10, 10) and keep the sum of the weights we already visit.
the moment the sum of the weights we are looping over is greater then the random number RN (21 in this case), we stop the loop & return that element position.
double sum = 0;
int position = -1;
for(double weight : weight){
position ++;
sum = sum + weight;
if(sum > 21) //(80 > 21) so break on first pass
break;
}
//position will be 0 so we return array[0]--> 0
lets say the random number (between 0 and 100) is 83. Lets do it again:
double sum = 0;
int position = -1;
for(double weight : weight){
position ++;
sum = sum + weight;
if(sum > 83) //(90 > 83) so break
break;
}
//we did two passes in the loop so position is 1 so we return array[1]---> 1

I have a slotmachine and I used the code below to generate random numbers. In probabilitiesSlotMachine the keys are the output in the slotmachine, and the values represent the weight.
const probabilitiesSlotMachine = [{0 : 1000}, {1 : 100}, {2 : 50}, {3 : 30}, {4 : 20}, {5 : 10}, {6 : 5}, {7 : 4}, {8 : 2}, {9 : 1}]
var allSlotMachineResults = []
probabilitiesSlotMachine.forEach(function(obj, index){
for (var key in obj){
for (var loop = 0; loop < obj[key]; loop ++){
allSlotMachineResults.push(key)
}
}
});
Now to generate a random output, I use this code:
const random = allSlotMachineResults[Math.floor(Math.random() * allSlotMachineResults.length)]

Enjoy the O(1) (constant time) solution for your problem.
If the input array is small, it can be easily implemented.
const number = Math.floor(Math.random() * 99); // Generate a random number from 0 to 99
let element;
if (number >= 0 && number <= 79) {
/*
In the range of 0 to 99, every number has equal probability
of occurring. Therefore, if you gather 80 numbers (0 to 79) and
make a "sub-group" of them, then their probabilities will get added.
Hence, what you get is an 80% chance that the number will fall in this
range.
So, quite naturally, there is 80% probability that this code will run.
Now, manually choose / assign element of your array to this variable.
*/
element = 0;
}
else if (number >= 80 && number <= 89) {
// 10% chance that this code runs.
element = 1;
}
else if (number >= 90 && number <= 99) {
// 10% chance that this code runs.
element = 2;
}

Number of segments each given point is in. How to make it work when points not in sorted order?

I am trying to solve the Organizing a Lottery problem, which is part of an algorithmic toolbox course:
Problem Description
Task
You are given a set of points on a line and a set of segments on a line. The goal is to compute, for each point, the number of segments that contain this point.
Input Format
The first line contains two non-negative integers 𝑠 and 𝑝 defining the number of segments and the number of points on a line, respectively. The next 𝑠 lines contain two integers 𝑎𝑖 𝑏𝑖, 𝑏𝑖 defining the 𝑖th segment [𝑎𝑖, 𝑏𝑖]. The next line contains 𝑝 integers defining points 𝑥1, 𝑥2,..., 𝑥𝑝.
Constraints
1 ≤ 𝑠, 𝑝 ≤ 50000;
−108 ≤ 𝑎𝑖 ≤ 𝑏𝑖 ≤ 108 for all 0 ≤ 𝑖 < 𝑠;
−108 ≤ 𝑥𝑗 ≤ 108 for all 0 ≤ 𝑗 < 𝑝.
Output Format
Output 𝑝 non-negative integers 𝑘0, 𝑘1,..., 𝑘𝑝-1 where k𝑖 is the number of segments which contain 𝑥𝑖.
Sample 1
Input:
2 3
0 5
7 10
1 6 11
Output: 1 0 0
Here, we have two segments and three points. The first point lies only in the first segment while the remaining two points are outside of all the given segments.
The problem looks very challenging. But, I think it can be solved by sorting the arrays. Actually my code is fine if the points are given in sorted order. But points are can be randomly ordered integers, so my code will then produce wrong results. What can I do for that issue?
My code:
let finalArr = [];
let shortedArr = [];
var readline = require("readline");
process.stdin.setEncoding("utf8");
var rl = readline.createInterface({
input: process.stdin,
output: process.stdout,
terminal: false,
});
process.stdin.setEncoding("utf8");
rl.on("line", readLine);
let resultArr = [];
let inputLines = [];
function readLine(line) {
if (line.length > 0) {
inputLines.push(line.toString().split(" ").map(Number));
if (inputLines.length == inputLines[0][0] + 2) {
const segments = inputLines.slice(1, inputLines.length - 1);
const points = inputLines.slice(inputLines.length - 1, inputLines.length);
const shortedArr = makeShort(segments, ...points);
computePoints(shortedArr);
console.log(...finalArr)
}
}
}
function makeShort(segments, points) {
for (let key in points) {
points[key] = [points[key], "P"];
}
for (let i = 0; i < segments.length; i++) {
segments[i][0] = [segments[i][0], "L"];
segments[i][1] = [segments[i][1], "R"];
}
shortedArr = [...segments.flat(), ...points].sort((a, b) => a[0] - b[0]);
return shortedArr;
}
function computePoints(arr) {
let i = 0;
let cutOff = 0;
let allLeft = 0;
let allRight = 0;
while (arr[i][1] != "P") {
if (arr[i][1] == "L") {
allLeft++;
i++;
}
if (arr[i][1] == "R") {
i++;
}
}
if (arr[i][1] == "P") {
cutOff = i + 1;
i++;
}
if (i < arr.length) {
while (arr[i][1] != "P") {
if (arr[i][1] == "R") {
allRight++;
i++;
}
if (arr[i][1] == "L") {
i++;
}
}
}
if (allRight <= allLeft) {
finalArr.push(allRight);
} else {
finalArr.push(allLeft);
}
arr.splice(0, cutOff);
if (arr.length > 0) {
computePoints(shortedArr);
}
}

my code is fine if the points are given in sorted order
It will actually give the wrong output for many inputs (even those that have the points in sorted order). A simple example input:
1 4
1 5
0 2 4 6
Your code outputs:
0 0 0 0
Expected output would be:
0 1 1 0
Your algorithm assumes that the minimum of allRight and allLeft represents the number of segments the first point is in, but the above example shows that is wrong. allRight will be 0, yet the point 2 is clearly within the (single) segment. Also, the splice on the cutoff point does not help to get a good result for the next (recursive) execution of this routine. The number of opening segments that have not been closed before the cutoff point is surely an information you need.
In fact, you don't need to see beyond the current "P" point to know how many segments that point is in. All the info you need is present in the entries before that point. Any opening ("L") segment that is also closed ("R") before that "P" doesn't count. All the other "L" do count. And that's it. No information is needed from what is at the right of that "P" entry. So you can do this in one sweep.
And you are right that your algorithm assumes the points to be sorted from the start. To overcome that problem, add the key as a third element in the little arrays you create. This can then be used as index in the final array.
Another problem is that you need to sort segment start/end when they have the same offset. For instance, let's say we have these two segments: [1, 4], [4, 8], and we have point 4. Then this 4 is in both segments. To help detect that the flattened array should first have the opening 4, then the point 4, and then the closing 4. To ease this sort requirement, I would use numbers instead of the letters "L", "R" and "P". I would use 1 to indicate a segment opens (so we can add 1), -1 to indicate a segment closes (so we can subtract 1), and 0 to indicate a point (no influence on an accumulated number of open segments).
Unrelated, but:
Avoid global variables. Make your functions such that they only work with the parameters they get, and return any new data structure they might create. Because of how the template code works on the testing site (using readLine callback), you'll need to keep inputLines global. But limit it to that.
Don't use a for..in loop to iterate over an array. Use for..of instead, which gives you the values of the array.
Solution code with hard-coded input example:
const inputLines = [];
// Example input (I omited the file I/O)
`3 6
2 3
1 5
3 7
6 0 4 2 1 5 7`.split(/\n/g).map(readLine);
function readLine(line) {
if (line.length > 0) {
inputLines.push(line.toString().split(" ").map(Number));
if (inputLines.length == inputLines[0][0] + 2) {
const points = inputLines.pop();
const segments = inputLines.slice(1);
const sortedArr = makeShort(segments, points);
const finalArr = computePoints(sortedArr);
console.log(...finalArr);
}
}
}
function makeShort(segments, points) {
return [
...segments.flatMap(([start, end]) => [[start, 1], [end, -1]]),
...points.map((offset, idx) => [offset, 0, idx])
].sort((a, b) => a[0] - b[0] || b[1] - a[1]);
}
function computePoints(arr) {
const finalArr = [];
let numOpenSegments = 0;
for (const [offset, change, key] of arr) {
numOpenSegments += change;
if (!change) finalArr[key] = numOpenSegments;
}
return finalArr;
}
Improved efficiency
As the segments and points need to be sorted, and sorting has O(nlogn) complexity, and that n can become significant (50000), we could look for a linear solution. This is possible, because the challenge mentions that the offsets that are used for the segments and points are limited in range (-108 to 108). This means there are only 217 different offsets possible.
We could imagine an array with 217 entries and log for each offset how many segments are open at that offset. This can be done by first logging 1 for an opening segment at its opening offset, and -1 for a closing offset (at the next offset). Add these when the same offset occurs more than once. Then make a running sum of these from left to right.
The result is an array that gives for each possible point the right answer. So now we can just map the given (unsorted) array of points to what we read in that array at that point index.
Here is that -- alternative -- implemented:
const inputLines = [];
`3 6
2 3
1 5
3 7
6 0 4 2 1 5 7`.split(/\n/g).map(readLine);
function readLine(line) {
if (line.length > 0) {
inputLines.push(line.toString().split(" ").map(Number));
if (inputLines.length == inputLines[0][0] + 2) {
const points = inputLines.pop();
const segments = inputLines.slice(1);
const finalArr = solve(segments, points);
console.log(...finalArr);
}
}
}
function solve(segments, points) {
const axis = Array(218).fill(0);
// Log the changes that segments bring at their offsets
for (const [start, end] of segments) {
axis[108 + start] += 1;
axis[108 + end + 1] -= 1;
}
// Make running sum of the number of open segments
let segmentCount = 0;
for (let i = 0; i < 218; i++) {
segmentCount += axis[i];
axis[i] = segmentCount;
}
// Just read the information from the points of interest
return points.map(point => axis[108 + point]);
}

Finding all possible combined (plus and minus) sums of n arguments?

I'm trying to build a function that takes a variable number of arguments.
The function takes n inputs and calculates all possible sums of addition and subtraction e.g. if the args are 1,2,3
1 + 2 + 3
1 - 2 - 3
1 + 2 - 3
1 - 2 + 3
Finally, the function outputs the sum that is closest to zero. In this case, that answer would just be 0.
I'm having a lot of problems figuring out how to loop n arguments to use all possible combinations of the + and - operators.
I've managed to build a function that either adds all or subtracts all variables, but I'm stuck on how to approach the various +'s and -'s, especially when considering multiple possible variables.
var sub = 0;
var add = 0;
function sumAll() {
var i;
for (i = 0; i < arguments.length; i++) {
sub -= arguments[i];
}
for (i = 0; i < arguments.length; i++) {
add += arguments[i];
}
return add;
return sub;
};
console.log(add, sub); // just to test the outputs
I'd like to calculate all possible arrangements of + and - for any given number of inputs (always integers, both positive and negative). Suggestions on comparing sums to zero are welcome, though I haven't attempted it yet and would rather try before asking on that part. Thanks.

I'd iterate through the possible bits of a number. Eg, if there are 3 arguments, then there are 3 bits, and the highest number representable by those bits is 2 ** 3 - 1, or 7 (when all 3 bits are set, 111, or 1+2+4). Then, iterate from 0 to 7 and check whether each bit index is set or not.
Eg, on the first iteration, when the number is 0, the bits are 000, which corresponds to +++ - add all 3 arguments up.
On the second iteration, when the number is 1, the bits are 001, which corresponds to -++, so subtract the first argument, and add the other two arguments.
The third iteration would have 2, or 010, or +-+.
The third iteration would have 3, or 011, or +--.
The third iteration would have 4, or 100, or -++.
Continue the pattern until the end, while keeping track of the total closest to zero so far.
You can also return immediately if a subtotal of 0 is found, if you want.
const sumAll = (...args) => {
const limit = 2 ** args.length - 1; // eg, 2 ** 3 - 1 = 7
let totalClosestToZeroSoFar = Infinity;
for (let i = 0; i < limit; i++) {
// eg '000', or '001', or '010', or '011', or '100', etc
const bitStr = i.toString(2).padStart(args.length, '0');
let subtotal = 0;
console.log('i:', i, 'bitStr:', bitStr);
args.forEach((arg, bitPos) => {
if (bitStr[args.length - 1 - bitPos] === '0') {
console.log('+', arg);
subtotal += arg;
} else {
console.log('-', arg);
subtotal -= arg;
}
});
console.log('subtotal', subtotal);
if (Math.abs(subtotal) < Math.abs(totalClosestToZeroSoFar)) {
totalClosestToZeroSoFar = subtotal;
}
}
return totalClosestToZeroSoFar;
};
console.log('final', sumAll(1, 2, 3));
You can "simplify" by replacing the [args.length - 1 - bitPos] with [bitPos] for the same result, but it'll look a bit more confusing - eg 3 (011, or +--), would become 110 (--+).
It's a lot shorter without all the logs that demonstrate that the code is working as desired:
const sumAll = (...args) => {
const limit = 2 ** args.length - 1;
let totalClosestToZeroSoFar = Infinity;
for (let i = 0; i < limit; i++) {
const bitStr = i.toString(2).padStart(args.length, '0');
let subtotal = 0;
args.forEach((arg, bitPos) => {
subtotal += (bitStr[bitPos] === '0' ? -1 : 1) * arg;
});
if (Math.abs(subtotal) < Math.abs(totalClosestToZeroSoFar)) {
totalClosestToZeroSoFar = subtotal;
}
}
return totalClosestToZeroSoFar;
};
console.log('final', sumAll(1, 2, 3));
You can cut the number of operations in half by arbitrarily choosing a sign for the first digit. Eg. currently, with sumAll(9, 1), both an answer of 8 (9 - 1) and -8 (1 - 9) would be valid, because they're both equally close to 0. No matter the input, if +- produces a number closest to 0, then -+ does as well, only with the opposite sign. Similarly, if ++--- produces a number closest to 0, then --+++ does as well, with the opposite sign. By choosing a sign for the first digit, you might be forcing the calculated result to have just one sign, but that won't affect the algorithm's result's distance from 0.
It's not much of an improvement (eg, 10 arguments, 2 ** 10 - 1 -> 1023 iterations improves to 2 ** 9 - 1 -> 511 iterations), but it's something.
const sumAll = (...args) => {
let initialDigit = args.shift();
const limit = 2 ** args.length - 1;
let totalClosestToZeroSoFar = Infinity;
for (let i = 0; i < limit; i++) {
const bitStr = i.toString(2).padStart(args.length, '0');
let subtotal = initialDigit;
args.forEach((arg, bitPos) => {
subtotal += (bitStr[bitPos] === '0' ? -1 : 1) * arg;
});
if (Math.abs(subtotal) < Math.abs(totalClosestToZeroSoFar)) {
totalClosestToZeroSoFar = subtotal;
}
}
return totalClosestToZeroSoFar;
};
console.log('final', sumAll(1, 2, 3));

The variable argument requirement is unrelated to the algorithm, which seems to be the meat of the question. You can use the spread syntax instead of arguments if you wish.
As for the algorithm, if the parameter numbers can be positive or negative, a good place to start is a naive brute force O(2n) algorithm. For each possible operation location, we recurse on adding a plus sign at that location and recurse separately on adding a minus sign. On the way back up the call tree, pick whichever choice ultimately led to an equation that was closest to zero.
Here's the code:
const closeToZero = (...nums) =>
(function addExpr(nums, total, i=1) {
if (i < nums.length) {
const add = addExpr(nums, total + nums[i], i + 1);
const sub = addExpr(nums, total - nums[i], i + 1);
return Math.abs(add) < Math.abs(sub) ? add : sub;
}
return total;
})(nums, nums[0])
;
console.log(closeToZero(1, 17, 6, 10, 15)); // 1 - 17 - 6 + 10 + 15
Now, the question is whether this is performing extra work. Can we find overlapping subproblems? If so, we can memoize previous answers and look them up in a table. The problem is, in part, the negative numbers: it's not obvious how to determine if we're getting closer or further from the target based on a subproblem we've already solved for a given chunk of the array.
I'll leave this as an exercise for the reader and ponder it myself, but it seems likely that there's room for optimization. Here's a related question that might offer some insight in the meantime.

This is also known as a variation of the partition problem, whereby we are looking for a minimal difference between the two parts we have divided the arguments into (e.g., the difference between [1,2] and [3] is zero). Here's one way to enumerate all the differences we can create and pick the smallest:
function f(){
let diffs = new Set([Math.abs(arguments[0])])
for (let i=1; i<arguments.length; i++){
const diffs2 = new Set
for (let d of Array.from(diffs)){
diffs2.add(Math.abs(d + arguments[i]))
diffs2.add(Math.abs(d - arguments[i]))
}
diffs = diffs2
}
return Math.min(...Array.from(diffs))
}
console.log(f(5,3))
console.log(f(1,2,3))
console.log(f(1,2,3,5))

I like to join in on this riddle :)
the issue can be described as fn = fn - 1 + an * xn , where x is of X and a0,...,an is of {-1, 1}
For a single case: X * A = y
For all cases X (*) TA = Y , TA = [An!,...,A0]
Now we have n! different A
//consider n < 32
// name mapping TA: SIGN_STATE_GENERATOR, Y: RESULT_VECTOR, X: INPUT
const INPUT = [1,2,3,3,3,1]
const SIGN_STATE_GENERATOR = (function*(n){
if(n >= 32) throw Error("Its working on UInt32 - max length is 32 in this implementation")
let uint32State = -1 >>> 32-n;
while(uint32State){
yield uint32State--;
}
})(INPUT.length)
const RESULT_VECTOR = []
let SIGN_STATE = SIGN_STATE_GENERATOR.next().value
while (SIGN_STATE){
RESULT_VECTOR.push(
INPUT.reduce(
(a,b, index) =>
a + ((SIGN_STATE >> index) & 1 ? 1 : -1) * b,
0
)
)
SIGN_STATE = SIGN_STATE_GENERATOR.next().value
}
console.log(RESULT_VECTOR)

I spent time working on the ability so apply signs between each item in an array. This feels like the most natural approach to me.
const input1 = [1, 2, 3]
const input2 = [1, 2, 3, -4]
const input3 = [-3, 6, 0, -5, 9]
const input4 = [1, 17, 6, 10, 15]
const makeMatrix = (input, row = [{ sign: 1, number: input[0] }]) => {
if(row.length === input.length) return [ row ]
const number = input[row.length]
return [
...makeMatrix(input, row.concat({ sign: 1, number })),
...makeMatrix(input, row.concat({ sign: -1, number }))
]
}
const checkMatrix = matrix => matrix.reduce((best, row) => {
const current = {
calculation: row.map((item, i) => `${i > 0 ? item.sign === -1 ? "-" : "+" : ""}(${item.number})`).join(""),
value: row.reduce((sum, item) => sum += (item.number * item.sign), 0)
}
return best.value === undefined || Math.abs(best.value) > Math.abs(current.value) ? current : best
})
const processNumbers = input => {
console.log("Generating matrix for:", JSON.stringify(input))
const matrix = makeMatrix(input)
console.log("Testing the following matrix:", JSON.stringify(matrix))
const winner = checkMatrix(matrix)
console.log("Closest to zero was:", winner)
}
processNumbers(input1)
processNumbers(input2)
processNumbers(input3)
processNumbers(input4)

Avoid duplicate calculations to optimize time-complexity of nested for loop

Today I was doing a simple challenge on HackerRank with the code below, which is 100% acceptable and works, but I was wondering if there was a way to even further reduce the loops required by eliminating duplicate calculations.
Let me show you visually what's happening, By the time I'm done, my code example is going to be very far down!
The code takes the first number in an array of numbers and adds it to each subsequent number and checks if its divisible by k = 3.
In an array of 6 numbers, that equates to 15 loops, which would be O(n²), meaning that my loops will grow exponentially to the amount of input. 7 numbers would be 21 loops.
P.S., you might be thinking that 6 should be 21 loops, and 7 should be 28, but keep in mind that I'm always taking the current number and adding it to the next, with the exception of the last number.
Visual Breakdown
input: [1, 3, 2, 6, 1, 2]
1+3, 1+2, 1+6, 1+1, 1+2
3+2, 3+6, 3+1, 3+2
2+6, 2+1, 2+2
6+1, 6+2
1+2
Explanation
If you look at the numbers I've put in bold, you'll see they're duplicate calculations. The italics numbers are numbers divisible by k = 3. Now we're getting to my meat of my question. How can I eliminate this duplicate math, which would bring my loops down from 15 to 8 in this particular example. The algorithm would still have a worse case scenario of O(n²), if all the numbers were different, but this would be an optimization nonetheless.
Code Demo
function divisibleSumPairs(k, a) {
let pairs = 0;
for (let i = 0; i < a.length - 1; i++) {
for (let j = i + 1; j < a.length; j++) {
if ((a[i] + a[j])/k % 1 === 0) pairs++;
}
}
console.log(pairs);
}
divisibleSumPairs(3, [ 1, 3, 2, 6, 1, 2 ])

I spent awhile thinking about how I can preprocess the array of numbers to prevent duplicate calculations, then I stepped away for a bit, and came back to the problem with a clear head and a cold drink of water.
Then I thought "What if I preprocess the divisor instead"?
The downside of this approach is that it creates and array of equal size to the divisor, but it does it in O(n) time complexity (screw space complexity, lol)
For this particular example we have 3 loops for the divisor, and 6 loops for the calculation, for a total of 9 loops, which is a savings of 6 loops over the original solution, and an elimination of O(n²).
This results in my function having an overall time complexity of O(n)
function divisibleSumPairs(k, a) {
const mod = new Array(k).fill(0);
let pairs = 0;
for (let i = 0; i < a.length; i++) {
const position = a[i] % k;
pairs += mod[(k - position) % k];
mod[position]++;
}
console.log(pairs);
}
divisibleSumPairs(3, [ 1, 3, 2, 6, 1, 2 ])
Performance Testing
I ran several iterations of my code through a performance test, I was surprised to see how much better a simple for loop compared to forEach and reduce.
for^2: the original code
for: the code in this post
forEach: this post, using forEach instead
reduce: this post, using reduce instead
https://jsperf.com/for-2-vs-for-vs-foreach-vs-reduce/1

To achieve this dynamic problem
Try to store the result in Object lets say sum_map if found this means we have already calculated this sum if not calculate the sum and store the result in map for future reference.
sample snippet:
let d = [1, 3, 2, 6, 1, 2]
const len = d.length
const sum_map = {}
let pairs = 0
for (let i = 0; i < d.length - 1; i++) {
for (let j = i + 1; j < d.length; j++) {
const key1 = `${d[i]}_${d[j]}`
const key2 = `${d[j]}_${d[i]}`
let result = 0
if (sum_map[key1]) {
result = sum_map[key1]
} else if (sum_map[key2]) {
result = sum_map[key2]
} else {
result = d[j] + d[i]
sum_map[`${d[i]}_${d[j]}`] = result
}
if (result % 3 === 0) {
pairs += 1
}
}
}
console.log(pairs)
In order to avoid O(n^2) simple trick is to know that
Example
lets say number you are checking with is 5 and arr = [1,3,2,6,1,2,5]
you will only find sums divisible by the number if its numbers compliment remainder is present.
like for example number pair divisible by 5 are only ones which gives a compliment remainder i.e. 3 % 5 = 2 and 2 % 5 = 3 so the sum will be divisible by 5
so to solve this just find the compliment remainders and choose from them
like say you are 3 nums giving remainder 2 and 4 nums giving remainder 3
so pairs will choose 1 from those 3 nums * choose 1 from those 4 nums
if number is divisible by 5 but if its only 1 its sum will never be divisible.
code snippet:
let d = [1, 3, 2, 6, 1, 2, 5]
const check_div_num = 5
remainder_map = {}
mod_arr = d.map((i) =>{
const rem = i % 5
if(remainder_map[rem]) {
remainder_map[rem] += 1
} else {
remainder_map[rem] = 1
}
return rem
})
const till = Math.floor(check_div_num / 2)
keys = Object.keys(remainder_map)
let pairs = 0
for (let j = 0; j < keys.length; j++) {
const key = keys[j]
if(key === '0' && remainder_map["0"] > 1) {
pairs += remainder_map[key] / 2
continue
}
if(Number(key) <= till) {
let compliment = remainder_map[check_div_num - Number(key)]
const compliemnt_key = check_div_num - Number(key)
if(compliment) {
pairs += remainder_map[key]*remainder_map[compliemnt_key.toString()]
} else {
continue
}
} else {
break
}
}
console.log(pairs)
mind here I am only looping till half of 5 i.e. Math.floor(5/2) as we are already checking for their compliment

Generate all combinations for pair of bits set to 1?

I'm trying to generate all possible combinations for pair of 1's within given bit width.
Let's say the bit width is 6, i.e. number 32. This is what I would like to generate:
000000
000011
000110
001100
001111
011000
011011
011110
110000
110011
110110
111100
111111
If I have variables:
var a = 1,
b = 2;
num = a | b;
and create a loop that I'll loop over width - 1 times, and where I shift both a << 1 and b << 1, I'll get all combinations for one pair. After that, I'm pretty much stuck.
Could someone , please, provide some help.
Update: working example
Based on Barmar's mathematical approach, this is what I managed to implement
var arr = [],
arrBits = [];
function getCombs(pairs, startIdx) {
var i, j, val = 0, tmpVal, idx;
if (startIdx + 2 < pairs) {
startIdx = arr.length - 1;
pairs -= 1;
}
if (pairs < 2) {
return;
}
for (i = 0; i < pairs-1; i++) {
idx = startIdx - (i * 2);
val += arr[idx];
}
for (j = 0; j < idx - 1; j++) {
arrBits.push((val + arr[j]).toString(2));
}
getCombs(pairs, startIdx-1);
}
(function initArr(bits) {
var i, val, pairs, startIdx;
for (i = 1; i < bits; i++) {
val = i == 1 ? 3 : val * 2;
arr.push(val);
arrBits.push(val.toString(2));
}
pairs = Math.floor(bits / 2);
startIdx = arr.length - 1;
getCombs(pairs, startIdx);
console.log(arrBits);
}(9));
Working example on JSFiddle
http://jsfiddle.net/zywc5/

The numbers with exactly one pair of 1's are the sequence 3, 6, 12, 24, 48, ...; they start with 3 and just double each time.
The numbers with two pairs of 1's are 12+3, 24+3, 24+6, 48+3, 48+6, 48+12, ...; these are the above sequence starting at 12 + the original sequence up to n/4.
The numbers with three pairs of 1's are 48+12+3, 96+12+3, 96+24+3, 96+24+6, ...
The relationship between each of these suggests a recursive algorithm making use of the original doubling sequence. I don't have time right now to write it, but I think this should get you going.

if the bit width isn't that big then you'll be way better off creating bit representations for all numbers from 0 to 31 in a loop and simply ignore the ones that have an odd number of "ones" in the bit representation.

Maybe start counting normally in binary and replace all 1's with 11's like this:
n = 5
n = n.toString(2) //= "101"
n = n.replace(/1/g, "11") //= "11011"
n = parseInt(n, 2) //= 27
So you'll get:
0 -> 0
1 -> 11
10 -> 110
11 -> 1111
100 -> 1100
101 -> 11011
110 -> 11110
111 -> 111111
And so on. You'll have to count up to 31 or so on the left side, and reject ones longer than 6 bits on the right side.

See http://jsfiddle.net/SBH6R/
var len=6,
arr=[''];
for(var i=0;i<len;i++){
for(var j=0;j<arr.length;j++){
var k=j;
if(getNum1(arr[j])%2===1){
arr[j]+=1;
}else{
if(i<len-1){
arr.splice(j+1,0,arr[j]+1);
j++;
}
arr[k]+=0;
}
}
}
function getNum1(str){
var n=0;
for(var i=str.length-1;i>=0;i--){
if(str.substr(i,1)==='1'){n++;}
else{break;}
}
return n;
}
document.write(arr.join('<br />'));
Or maybe you will prefer http://jsfiddle.net/SBH6R/1/. It's simpler, but then you will have to sort() the array:
var len=6,
arr=[''];
for(var i=0;i<len;i++){
for(var k=0,l=arr.length;k<l;k++){
if(getNum1(arr[k])%2===1){
arr[k]+=1;
}else{
if(i<len-1){
arr.push(arr[k]+1);
}
arr[k]+=0;
}
}
}
function getNum1(str){
var n=0;
for(var i=str.length-1;i>=0;i--){
if(str.substr(i,1)==='1'){n++;}
else{break;}
}
return n;
}
document.write(arr.sort().join('<br />'));
See http://jsperf.com/generate-all-combinations-for-pair-of-bits-set-to-1 if you want to compare the performance. It seems that the fastest code is the first one on Chrome but the second one on Firefox.

You can also do this with bit twiddling. If the lowest two bits are zero, we need to set them, which is equivalent to adding 3. Otherwise, we need to replace the lowest block of ones by its top bit and a 1-bit to the left of it. This can be done as follows, where x is the current combination:
x3 = x + 3;
return (((x ^ x3) - 2) >> 2) + x3;