Weighted random pick

Question

I have a set of items. I need to randomly pick one. The problem is that they each have a weight of 1-10. A weight of 2 means that the item is twice as likely to be picked than a weight of 1. A weight of 3 is three times as likely.

I currently fill an array with each item. If the weight is 3, I put three copies of the item in the array. Then, I pick a random item.

My method is fast, but uses a lot of memory. I am trying to think of a faster method, but nothing comes to mind. Anyone have a trick for this problem?

EDIT: My Code...

Apparently, I wasn't clear. I do not want to use (or improve) my code. This is what I did.

//Given an array $a where $a[0] is an item name and $a[1] is the weight from 1 to 100.
$b = array();
foreach($a as $t)
    $b = array_merge($b, array_fill(0,$t[1],$t));
$item = $b[array_rand($b)];

This required me to check every item in $a and uses max_weight/2*size of $a memory for the array. I wanted a COMPLETELY DIFFERENT algorithm.

Further, I asked this question in the middle of the night using a phone. Typing code on a phone is nearly impossible because those silly virtual keyboards simply suck. It auto-corrects everything, ruining any code I type.

An yet further, I woke up this morning with an entirely new algorithm that uses virtual no extra memory at all and does not require checking every item in the array. I posted it as an answer below.

Answer 1

I'll use this input array for my explanation:

$values_and_weights=array(
    "one"=>1,
    "two"=>8,
    "three"=>10,
    "four"=>4,
    "five"=>3,
    "six"=>10
);

The simple version isn't going to work for you because your array is so large. It requires no array modification but may need to iterate the entire array, and that's a deal breaker.

/*$pick=mt_rand(1,array_sum($values_and_weights));
$x=0;
foreach($values_and_weights as $val=>$wgt){
    if(($x+=$wgt)>=$pick){
        echo "$val";
        break;
    }
}*/

For your case, re-structuring the array will offer great benefits. The cost in memory for generating a new array will be increasingly justified as:

1. array size increases and
2. number of selections increases.

The new array requires the replacement of "weight" with a "limit" for each value by adding the previous element's weight to the current element's weight.

Then flip the array so that the limits are the array keys and the values are the array values.

The selection logic is: the selected value will have the lowest limit that is >= $pick.

// Declare new array using array_walk one-liner:
array_walk($values_and_weights,function($v,$k)use(&$limits_and_values,&$x){$limits_and_values[$x+=$v]=$k;});

//Alternative declaration method - 4-liner, foreach() loop:
/*$x=0;
foreach($values_and_weights as $val=>$wgt){
    $limits_and_values[$x+=$wgt]=$val;
}*/
var_export($limits_and_values);

$limits_and_values looks like this:

array (
  1 => 'one',
  9 => 'two',
  19 => 'three',
  23 => 'four',
  26 => 'five',
  36 => 'six',
)

Now to generate the random $pick and select the value:

// $x (from walk/loop) is the same as writing: end($limits_and_values); $x=key($limits_and_values);
$pick=mt_rand(1,$x);  // pull random integer between 1 and highest limit/key
while(!isset($limits_and_values[$pick])){++$pick;}  // smallest possible loop to find key
echo $limits_and_values[$pick];  // this is your random (weighted) value

This approach is brilliant because isset() is very fast and the maximum number of isset() calls in the while loop can only be as many as the largest weight (not to be confused with limit) in the array.

FOR YOUR CASE, THIS APPROACH WILL FIND THE VALUE IN 10 ITERATIONS OR LESS!

Here is my Demo that will accept a weighted array (like $values_and_weights), then in just four lines:

• Restructure the array,
• Generate a random number,
• Find the correct value, and
• Display it.

Answer 2

I greatly appreciate the answers above. Please consider this answer, which does not require checking every item in the original array.

// Given $a as an array of items
// where $a[0] is the item name and $a[1] is the item weight.
// It is known that weights are integers from 1 to 100.
for($i=0; $i<sizeof($a); $i++) // Safeguard described below
{
    $item = $a[array_rand($a)];
    if(rand(1,100)<=$item[1]) break;
}

This algorithm only requires storage for two variables ($i and $item) as $a was already created before the algorithm kicked in. It does not require a massive array of duplicate items or an array of intervals.

In a best-case scenario, this algorithm will touch one item in the original array and be done. In a worst-case scenario, it will touch n items in an array of n items (not necessarily every item in the array as some may be touched more than once).

If there was no safeguard, this could run forever. The safeguard is there to stop the algorithm if it simply never picks an item. When the safeguard is triggered, the last item touched is the one selected. However, in millions of tests using random data sets of 100,000 items with random weights of 1 to 10 (changing rand(1,100) to rand(1,10) in my code), the safeguard was never hit.

I made histograms comparing the frequency of items selected among my original algorithm, the ones from answers above, and the one in this answer. The differences in frequencies are trivial - easy to attribute to variances in the random numbers.

EDIT... It is apparent to me that my algorithm may be combined with the algorithm pala_ posted, removing the need for a safeguard.

In pala_'s algorithm, a list is required, which I call an interval list. To simplify, you begin with a random_weight that is rather high. You step down the list of items and subtract the weight of each one until your random_weight falls to zero (or less). Then, the item you ended on is your item to return. There are variations on this interval algorithm that I've tested and pala_'s is a very good one. But, I wanted to avoid making a list. I wanted to use only the given weighted list and never touch all the items. The following algorithm merges my use of random jumping with pala_'s interval list. Instead of a list, I randomly jump around the list. I am guaranteed to get to zero eventually, so no safeguard is needed.

// Given $a as the weighted array (described above)
$weight = rand(1,100); // The bigger this is, the slower the algorithm runs.
while($weight>0)
{
    $item = $a[array_rand($a)];
    $weight-= $item[1];
}
// $item is the random item you want.

I wish I could select both pala_ and this answer as the correct answers.

Answer 3

This ones your huckleberry.

  $arr = array(
    array("val" => "one", "weight" => 1),
    array("val" => "two", "weight" => 2),
    array("val" => "three", "weight" => 3),
    array("val" => "four", "weight" => 4)
  );

  $weight_sum = 0;
  foreach($arr as $val)
  {
    $weight_sum += $val['weight'];
  }

  $r = rand(1, $weight_sum);
  print "random value is $r\n";

  for($i = 0; $i < count($arr); $i++)
  {
    if($r <= $arr[$i]['weight'])
    {
      print "$r <= {$arr[$i]['weight']}, this is our match\n";
      print $arr[$i]['val'] . "\n";
      break;
    }
    else
    {
      print "$r > {$arr[$i]['weight']}, subtracting weight\n";
      $r -= $arr[$i]['weight'];
      print "new \$r is $r\n";
    }
  }

No need to generate arrays containing an item for every weight, no need to fill an array with n elements for a weight of n. Just generate a random number between 1 and total weight, then loop through the array until you find a weight less than your random number. If it isn't less than the number, subtract that weight from the random and continue.

Sample output:

# php wr.php
random value is 8
8 > 1, subtracting weight
new $r is 7
7 > 2, subtracting weight
new $r is 5
5 > 3, subtracting weight
new $r is 2
2 <= 4, this is our match
four

This should also support fractional weights.

modified version to use array keyed by weight, rather than by item

  $arr2 = array(
  );

  for($i = 0; $i <= 500000; $i++)
  {
    $weight = rand(1, 10);
    $num = rand(1, 1000);
    $arr2[$weight][] = $num;
  }

  $start = microtime(true);

  $weight_sum = 0;
  foreach($arr2 as $weight => $vals) {
    $weight_sum += $weight * count($vals);
  }

  print "weighted sum is $weight_sum\n";

  $r = rand(1, $weight_sum);
  print "random value is $r\n";
  $found = false;
  $elem = null;

  foreach($arr2 as $weight => $vals)
  {
    if($found) break;
    for($j = 0; $j < count($vals); $j ++)
    {
      if($r < $weight)
      {
        $elem = $vals[$j];
        $found = true;
        break;
      }
      else
      {
        $r -= $weight;
      }
    }
  }
  $end = microtime(true);

  print "random element is: $elem\n";
  print "total time is " . ($end - $start) . "\n";

With sample output:

# php wr2.php
weighted sum is 2751550
random value is 345713
random element is: 681
total time is 0.017189025878906

measurement is hardly scientific - and fluctuates depending on where in the array the element falls (obviously) but it seems fast enough for huge datasets.

Answer 4

This way requires two random calculations but they should be faster and require about 1/4 of the memory but with some reduced accuracy if weights have disproportionate counts. (See Update for increased accuracy at the cost of some memory and processing)

Store a multidimensional array where each item is stored in the an array based on its weight:

$array[$weight][] = $item;
// example: Item with a weight of 5 would be $array[5][] = 'Item'

Generate a new array with the weights (1-10) appearing n times for n weight:

foreach($array as $n=>$null) {
  for ($i=1;$i<=$n;$i++) {
    $weights[] = $n;
  }
}

The above array would be something like: [ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4 ... ]

First calculation: Get a random weight from the weighted array we just created

$weight = $weights[mt_rand(0, count($weights)-1)];

Second calculation: Get a random key from that weight array

$value = $array[$weight][mt_rand(0, count($array[$weight])-1)];

Why this works: You solve the weighted issue by using the weighted array of integers we created. Then you select randomly from that weighted group.

---

Update: Because of the possibility of disproportionate counts of items per weight, you could add another loop and array for the counts to increase accuracy.

foreach($array as $n=>$null) {
  $counts[$n] = count($array[$n]);
}

foreach($array as $n=>$null) {
  // Calculate proportionate weight (number of items in this weight opposed to minimum counted weight)
  $proportion = $n * ($counts[$n] / min($counts));
  for ($i=1; $i<=$proportion; $i++) {
    $weights[] = $n;
  }
}

What this does is if you have 2000 10's and 100 1's, it'll add 200 10's (20 * 10, 20 because it has 20x the count, and 10 because it is weighted 10) instead of 10 10's to make it proportionate to how many are in there opposed the minimum weight count. So to be accurate, instead of adding one for EVERY possible key, you are just being proportionate based on the MINIMUM count of weights.

Answer 5

ere is my offer in case I've understand you right. I offer you take a look and if there are some question I'll explain. Some words in advance:

My sample is with only 3 stages of weight - to be clear - With outer while I'm simulating your main loop - I count only to 100. - The array must to be init with one set of initial numbers as shown in my sample. - In every pass of main loop I get only one random value and I'm keeping the weight at all.

<?php
$array=array(
    0=>array('item' => 'A', 'weight' => 1),
    1=>array('item' => 'B', 'weight' => 2),
    2=>array('item' => 'C', 'weight' => 3),
);
$etalon_weights=array(1,2,3);
$current_weights=array(0,0,0);
$ii=0;
while($ii<100){ // Simulates your main loop
    // Randomisation cycle
    if($current_weights==$etalon_weights){
        $current_weights=array(0,0,0);
    }
    $ft=true;
    while($ft){
        $curindex=rand(0,(count($array)-1));
        $cur=$array[$curindex];
        if($current_weights[$cur['weight']-1]<$etalon_weights[$cur['weight']-1]){
            echo $cur['item'];
            $array[]=$cur;
            $current_weights[$cur['weight']-1]++;
            $ft=false;
        }
    }
    $ii++;
}
?>

Answer 6

I'm not sure if this is "faster", but I think it may be more "balance"d between memory usage and speed.

The thought is to transform your current implementation (500000 items array) into an equal-length array (100000 items), with the lowest "origin" position as key, and origin index as value:

<?php
$set=[["a",3],["b",5]];
$current_implementation=["a","a","a","b","b","b","b","b"];
// 0=>0 means the lowest "position" 0
// points to 0 in the set;
// 3=>1 means the lowest "position" 3
// points to 1 in the set;
$my_implementation=[0=>0,3=>1];

And then randomly picks a number between 0 and highest "origin" position:

// 3 is the lowest position of the last element ("b")
// and 5 the weight of that last element
$my_implemention_pick=mt_rand(0,3+5-1);

Full code:

<?php
function randomPickByWeight(array $set)
{
    $low=0;
    $high=0;
    $candidates=[];
    foreach($set as $key=>$item)
    {
        $candidates[$high]=$key;
        $high+=$item["weight"];
    }
    $pick=mt_rand($low,$high-1);
    while(!array_key_exists($pick,$candidates))
    {
        $pick--;
    }
    return $set[$candidates[$pick]];
}
$cache=[];
for($i=0;$i<100000;$i++)
{
    $cache[]=["item"=>"item {$i}","weight"=>mt_rand(1,10)];
}
$time=time();
for($i=0;$i<100;$i++)
{
    print_r(randomPickByWeight($cache));
}
$time=time()-$time;
var_dump($time);

3v4l.org demo
3v4l.org have some time limitation on codes, so the demo didn't finished. On my laptop the above demo finished in 10 seconds (i7-4700 HQ)