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Introduction to Robotics

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The Fourier Transform and its Applications

Introduction to Linear Dynamical Systems

Convex Optimization I

Convex Optimization II

<?
## 0-1 Knapsack Problem Solve (with memoization optimization)
# $w = weight of item
# $v = value of item
# $i = index
# $aW = Available Weight
# $m = memo array

function knapSolveFast($w,$v,$i,$aW,$m) {

    global $numcalls, $m;
    $numcalls ++;
    // echo "Called with i=$i, aW=$aW";

    // Return memo if we have one
    if (isset($m[$i][$aW])) {
        return $m[$i][$aW];
    } else {

        if ($i == 0) {
            if ($w[$i]  $aW) {
            $m[$i][$aW] = $without_i;
            return $without_i;
        } else {
            $with_i = $v[$i] + knapSolveFast($w, $v, ($i-1), ($aW - $w[$i]),$m);
            $res = max($with_i, $without_i); 
            $m[$i][$aW] = $res;
            return $res;
        }   
    }
}


$w3 = array(1, 1, 1, 2, 2, 2, 4, 4, 4, 44, 96, 96, 96);
$v3 = array(1, 1, 1, 2, 2, 2, 4, 4, 4, 44, 96, 96, 96);

$numcalls = 0;
$m = array();
$m3 = knapSolveFast($w3, $v3, sizeof($v3) -1, 54,$m);
print_r($w3); echo "";
echo "Max: $m3 ($numcalls calls)";
?>

For those who are interested, here is a direct translation of the first (Python) version above in PHP:

0-1 Knapsack Problem Solve

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$w = weight of item
$v = value of item
$i = index
$aW = Available Weight
$numcalls = 0;
function maxVal($w,$v,$i,$aW) {
    global $numcalls;
    $numcalls ++;
    echo "Called with i=$i, aW=$aW";
    if ($i == 0) {
        if ($w[$i]  $aW) {
            return $without_i;
        } else {
            $with_i = $v[$i] + maxVal($w, $v, ($i-1), ($aW - $w[$i]));
            return max($with_i, $without_i);
        }
}
$w1 = array(5,3,2);
$v1 = array(9,7,8);
$w2 = array( 1,1,5,5,3,3,4,4);
$v2 = array(15,15,10,10,4,4,5,5);
$m1 = maxVal($w1, $v1, sizeof($v1)-1, 5 );
echo "<b>Max: $m1</b> ($numcalls calls)";
$numcalls = 0;
$m2 = maxVal($w2, $v2, sizeof($v2) -1, 8 );
echo "<b>Max: $m2</b> ($numcalls calls)";

This may be an unusual request, but I’m looking for a “dumbed down” explanation of this algorithm so that I can make a PHP implementation of the 0-1 Knapsack solve.

I’ve found a number of solutions in various other languages, but none that I am familiar with being primarily a web programmer. One in particular that was ideal for my application was written in Java, and although I recognize some of the control structures, the language primitives are still too foreign for me to make sense of.

http://www.alesdar.org/oldSite.....pplet.html

I need to be able to take a list of items with a weight and value (both positive integers) and return the best optimal combination for a specified maximum weight. The above linked applet works perfectly for my needs (though I don’t need the additional GA algorithm that it’s comparing against).

Any help understanding the code or a simplified explanation would be of tremendous assistance. I will happy post the final PHP code for others encountering the same issue…

Brian