void removeAll( E item )        // removes all occurrences of the given item
{
    while ( remove(item) ) {    // remove a single occurrence, try again
        ;
    }
}

Our hunch was that the running time of removeAll is O(n^2):

• at worst may need to call method remove n times
• at worst method remove may need to go through the whole list, i.e. examine n boxes
• overall: n cycles x n steps

We used a couple of estimates (no need to be precise, just convincing):

• 3 units to inspect a box: is it valid; does it have what we need; move on
• 6 units to remove a box: is it valid; does it have what we need; adjust links, possibly the tail

Single 8 at the end

cycle	work details	work	list
			7→7→7→...7→8
1.	inspects n-1 boxes with 7s: 3(n-1) units work deletes the last 8: 6 units of work	3(n-1)+6	7→7→7→...7
2.	inspects n-1 boxes with 7s: 3(n-1) units work done, no 8 found	3(n-1)	7→7→7→...7
Total work: 3(n-1) + 6 + 3(n-1) = 3n, so O(n). Did not reach our goal.

Only 8s in the list

cycle	work details	work	list
			8→8→8→...8→8
1.	deletes first 8 right away: 6 units of work	6	8→8→...8→8
2.	deletes first 8 right away: 6 units of work	6	8→...8→8
...
n-1.	deletes first 8 right away: 6 units of work	6	8
n.	deletes first 8 right away: 6 units of work	6	empty
last.	done, no 8 found	2	empty
Total work: 6n + 2, so O(n). Did not reach our goal.

1st half only 7s, 2nd half only 8s

cycle	work details	work	list
			7→7→7→...→7→8→8→8...→8
1.	inspects n/2 boxes with 7s: 3(n/2) units work deletes first 8: 6 units of work	3(n/2)+6	7→7→7→...→7→8→8...→8
2.	inspects n/2 boxes with 7s: 3(n/2) units work deletes first 8: 6 units of work	3(n/2)+6	7→7→7→...→7→8...→8
...
n/2.	inspects n/2 boxes with 7s: 3(n/2) units work deletes first 8: 6 units of work	3(n/2)+6	7→7→7→...→7
last.	done, no 8 found	2	empty
Total work: n/2*(3(n/2) + 6) + 2 = 3/4n^2 + 3n + 2, so O(n^2). Reached the goal.

1st half only 8s, 2nd half only 7s

cycle	work details	work	list
			8→8→8→...→8→7→7→7...→7
1.	?	?	?
2.	?	?	?
...
?	?	?	?
last.	?	?	?
Total work: ?

Alternating 7s and 8s (half of each)

cycle	work details	work	list
			7→8→7→8→7→8...→7→8
1.	?	?	?
2.	?	?	?
...
?	?	?	?
last.	?	?	?
Total work: ?