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<blockquote data-quote="FormerlyHemlock" data-source="post: 6551811" data-attributes="member: 6787650"><p>A famous problem in computer science. The regular knapsack problem is this: imagine you have a knapsack and you're planning a bank heist. You'll be able to carry out a certain amount of stuff from the bank, whatever you can fit in your knapsack. Each thing in the bank has a certain value and also a cost in space. Which things do you pick? The solution is trivial: choose the item with the best value-per-cubic-inch and put it in your knapsack. Then choose the next-best and take that too. Continue until you run out of space in your knapsack, and then take the thing that didn't quite fit in your knapsack and cut off bits of it until you fit as much as you can in that knapsack. Voila! You have the optimal value in your knapsack.</p><p></p><p>In the 0/1 knapsack, you can't cut things in two, you have to either take it or leave it. (Hence, "0/1".) You could make the problem even more complicated than 0/1, you could make it multidimensional such that there's not just a linear amount of abstract "space" in your knapsack, there's different dimensions and shapes that you need to worry about, and now you need to worry about the possibility that the "best" item may take up enough space that you can't take the third- and fifth-best items, which together add up to more value than the best item. The more constraints you add (length/width/weight), the more difficult the problem becomes, and the more you have to backtrack and try different things on your way to a solution--you can't just zoom straight to the optimal solution.</p><p></p><p>I see class constraints in this same light. Every class in 5E has some things in it that aren't mechanically all that great, but are part of the package. For the sake of example, I'll pick on the Shadow Monk's "turn invisible in darkness" ability that he gets at 11th level. You may ask, "what good is it to be invisible if it's already dark?" (I know, I know. Darkvision, etc.) In a point-build system, if you had the chance to drop that ability in exchange for an extra feat, you would probably do that. The extra feat is clearly <em>better</em>. But if you're going to be a high-level Shadow Monk, you're going to have that ability anyway even though it's really only the 11th or 12th-best thing you'd like to see on your "abilities I gain this level" chart, so you make use of it, and maybe it becomes something cool. For example, maybe it occurs to you that Shadow Monk invisibility <em>isn't a spell</em> and so <em>doesn't take concentration</em>, which means that you can cast Pass Without Trace and then turn invisible, therefore becoming the stealthiest dungeon scout in the whole game except for <em>maybe</em> a high-level rogue/druid multiclass. It may not be better than a feat, but since the opportunity cost <em>isn't</em> a feat--since there is no opportunity cost at all--it's pretty nifty.</p><p></p><p>Hopefully you can see the analogy to 0/1 knapsack, where "X levels of classes" is your knapsack and you have to fill it with contiguous levels of various classes instead of freely selecting levels/powers. Just like 0/1 knapsack, the more constraints you add, the less easily you can zoom in on a trivially-optimized single solution.</p></blockquote><p></p>
[QUOTE="FormerlyHemlock, post: 6551811, member: 6787650"] A famous problem in computer science. The regular knapsack problem is this: imagine you have a knapsack and you're planning a bank heist. You'll be able to carry out a certain amount of stuff from the bank, whatever you can fit in your knapsack. Each thing in the bank has a certain value and also a cost in space. Which things do you pick? The solution is trivial: choose the item with the best value-per-cubic-inch and put it in your knapsack. Then choose the next-best and take that too. Continue until you run out of space in your knapsack, and then take the thing that didn't quite fit in your knapsack and cut off bits of it until you fit as much as you can in that knapsack. Voila! You have the optimal value in your knapsack. In the 0/1 knapsack, you can't cut things in two, you have to either take it or leave it. (Hence, "0/1".) You could make the problem even more complicated than 0/1, you could make it multidimensional such that there's not just a linear amount of abstract "space" in your knapsack, there's different dimensions and shapes that you need to worry about, and now you need to worry about the possibility that the "best" item may take up enough space that you can't take the third- and fifth-best items, which together add up to more value than the best item. The more constraints you add (length/width/weight), the more difficult the problem becomes, and the more you have to backtrack and try different things on your way to a solution--you can't just zoom straight to the optimal solution. I see class constraints in this same light. Every class in 5E has some things in it that aren't mechanically all that great, but are part of the package. For the sake of example, I'll pick on the Shadow Monk's "turn invisible in darkness" ability that he gets at 11th level. You may ask, "what good is it to be invisible if it's already dark?" (I know, I know. Darkvision, etc.) In a point-build system, if you had the chance to drop that ability in exchange for an extra feat, you would probably do that. The extra feat is clearly [I]better[/I]. But if you're going to be a high-level Shadow Monk, you're going to have that ability anyway even though it's really only the 11th or 12th-best thing you'd like to see on your "abilities I gain this level" chart, so you make use of it, and maybe it becomes something cool. For example, maybe it occurs to you that Shadow Monk invisibility [I]isn't a spell[/I] and so [I]doesn't take concentration[/I], which means that you can cast Pass Without Trace and then turn invisible, therefore becoming the stealthiest dungeon scout in the whole game except for [I]maybe[/I] a high-level rogue/druid multiclass. It may not be better than a feat, but since the opportunity cost [I]isn't[/I] a feat--since there is no opportunity cost at all--it's pretty nifty. Hopefully you can see the analogy to 0/1 knapsack, where "X levels of classes" is your knapsack and you have to fill it with contiguous levels of various classes instead of freely selecting levels/powers. Just like 0/1 knapsack, the more constraints you add, the less easily you can zoom in on a trivially-optimized single solution. [/QUOTE]
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