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3.3: Problem Size

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    The last example we’ll consider is removeAll; here’s the implementation in MyArrayList:

    public boolean removeAll(Collection<?> collection) {
        boolean flag = true;
        for (Object obj: collection) {
            flag &= remove(obj);
        return flag;

    Each time through the loop, removeAll invokes remove, which is linear. So it is tempting to think that removeAll is quadratic. But that’s not necessarily the case.

    In this method, the loop runs once for each element in collection. If collection contains m elements and the list we are removing from contains n elements, this method is in \( O(nm) \). If the size of collection can be considered constant, removeAll is linear with respect to n. But if the size of the collection is proportional to n, removeAll is quadratic. For example, if collection always contains 100 or fewer elements, removeAll is linear. But if collection generally contains 1% of the elements in the list, removeAll is quadratic.

    When we talk about problem size, we have to be careful about which size, or sizes, we are talking about. This example demonstrates a pitfall of algorithm analysis: the tempting shortcut of counting loops. If there is one loop, the algorithm is often linear. If there are two loops (one nested inside the other), the algorithm is often quadratic. But be careful! You have to think about how many times each loop runs. If the number of iterations is proportional to n for all loops, you can get away with just counting the loops. But if, as in this example, the number of iterations is not always proportional to n, you have to give it more thought.

    This page titled 3.3: Problem Size is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Allen B. Downey (Green Tea Press) .

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