How to find all combinations of a multiset in a string in linear time?

I am given a bag B (multiset) of characters with the size m and a string text S of size n. Is it possible to find all substrings that can be created by B (4!=24 combinations) in S in linear time O(n)?

Example:

S = abdcdbcdadcdcbbcadc (n=19)
B = {b, c, c, d} (m=4)
Result: {cdbc (Position 3), cdcb (Position 10)}

The fastest solution I found is to keep a counter for each character and compare it with the Bag in each step, thus the runtime is O(n*m). Algorithm can be shown if needed.

Thanks for the answer. The add() and remove() methods have to be changed to make the algorithm work correctly.

add(c):
    if hist[c] > 0 and histrun[c] < hist[c] then
        histrunsum++
    else
        histrunsum--

    histrun[c] = histrun[c] + 1


remove(c):
    if histrun[c] > hist[c] then
        histrunsum++
    else
        histrunsum--

    histrun[c] = histrun[c] - 1

Explanation: histrunsum can be seen as a score of how identical both multisets are.

add(c): when there are less occurrences of a char in the histrun multiset than in the hist multiset, the additional occurrence of that char has to be "rewarded" since the histrun multiset is getting closer to the hist multiset. If there are at least equal or more chars in the histrun set already, and additional char is negative.

remove(c): like add(c), where a removal of a char is weighted positively when it's number in the histrun multiset > hist multiset.

Sample Code (PHP):

function multisetSubstrings($sequence, $mset)
{
    $multiSet = array();
    $substringLength = 0;
    foreach ($mset as $char)
    {
        $multiSet[$char]++;
        $substringLength++;
    }

    $sum = 0;
    $currentSet = array();
    $result = array();

    for ($i=0;$i<strlen($sequence);$i++)
    {

        if ($i>=$substringLength)
        {
            $c = $sequence[$i-$substringLength];

            if ($currentSet[$c] > $multiSet[$c])
                $sum++;
            else
                $sum--;

            $currentSet[$c]--;
        }


        $c = $sequence[$i];

        if ($currentSet[$c] < $multiSet[$c])
            $sum++;
        else
            $sum--;

        $currentSet[$c]++;

        echo $sum."<br>";


        if ($sum==$substringLength)
            $result[] = $i+1-$substringLength;
    }


    return $result;
}