My cake shop is so popular, I'm adding some tables and hiring wait staff so folks can have a cute sit-down cake-eating experience.
I have two registers: one for take-out orders, and the other for the other folks eating inside the cafe. All the customer orders get combined into one list for the kitchen, where they should be handled first-come, first-served.
Recently, some customers have been complaining that people who placed orders after them are getting their food first. Yikes—that's not good for business!
To investigate their claims, one afternoon I sat behind the registers with my laptop and recorded:
Given all three lists, write a function to check that my service is first-come, first-served. All food should come out in the same order customers requested it.
We'll represent each customer order as a unique integer.
Watch out for index out of bounds errors! Will your function ever try to grab the 0th item from an empty list, or the n^{th} item from a list with n elements (where the last index would be n-1)?
We can do this in time and additional space.
Did you come up with a recursive solution? Keep in mind that you may be incurring a hidden space cost (probably ) in the call stack! You can avoid this using an iterative approach.
How can we re-phrase this problem in terms of smaller subproblems?
Breaking the problem into smaller subproblems will clearly involve reducing the size of at least one of our lists of customer order numbers. So to start, let's try taking the first customer order out of served_orders.
What should be true of this customer order number if my service is first-come, first-served?
If my cake cafe is first-come, first-served, then the first customer order finished has to either be the first take-out order entered into the system or the first dine-in order entered into the system.
In terms of the lists, the first item in served_orders should either be same as the first item in take_out or dine_in.
Once we can check the first customer order, how can we verify the remaining ones?
Let's "throw out" the first customer order from served_orders as well as the customer order it matched with from the beginning of take_out or dine_in. That customer order is now "accounted for."
Now we're left with a smaller version of the original problem, which we can solve using the same approach! So we keep doing this over and over until we exhaust served_orders. If we get to the end and every customer order "checks out," we return True.
How do we implement this in code?
Now that we have a problem that's the same as the original problem except smaller, our first thought might be to use recursion. All we need is a base case. What's our base case?
We stop when we run out of customer orders in our served_orders. So that's our base case: when we've checked all customer orders in served_orders, we return True because we know all of the customer orders have been "accounted for."
This solution will work. But we can do better.
Before we talk about optimization, note that our inputs are probably pretty small. This function will take hardly any time or space, even if it could be more efficient. In industry, especially at small startups that want to move quickly, optimizing this might be considered a "premature optimization." Great engineers have both the skill to see how to optimize their code and the wisdom to know when those optimizations aren't worth it. At this point in the interview I recommend saying, "I think we can optimize this a bit further, although given the nature of the input this probably won't be very resource-intensive anyway...should we talk about optimizations?"
Okay, back to our show. This function will take time and additional space.
Whaaaaat? Yeah. Take a look at this snippet:
In particular this expression:
That's a slice, and it costs time and space, where m is the size of the resulting list. That's going to determine our overall time and space cost here—the rest of the work we're doing is constant space and time.
In our recursing we'll build up n frames on the call stack. Each of those frames will hold a different slice of our original served_orders (and take_out and dine_in, though we only slice one of them in each recursive call).
So, what's the total time and space cost of all our slices?
If served_orders has n items to start, taking our first slice takes n-1 time and space (plus half that, since we're also slicing one of our halves—but that's just a constant multiplier so we can ignore it). In our second recursive call, slicing takes n-2 time and space. Etc.
So our total time and space cost for slicing comes to:
(n - 1) + (n - 2) + ... + 2 + 1This is a common series that turns out to be .
We can do better than this time and space cost. One way we could to that is to avoid slicing and instead keep track of indices in the list:
So now we're down to time, but we're still taking space in the call stack because of our recursion. We can rewrite this as an iterative function to get that memory cost down to .
What's happening in each iteration of our recursive function? Sometimes we're taking a customer order out of take_out and sometimes we're taking a customer order out of dine_in, but we're always taking a customer order out of served_orders.
So what if instead of taking customer orders out of served_orders 1-by-1, we iterated over them?
We walk through served_orders, seeing if each customer order so far matches a customer order from one of the two registers. To check this, we:
time and additional space.
If you read the whole breakdown section, you might have noticed that our recursive function cost us extra space. If you missed that part, go back and take a look.
Be careful of the hidden space costs from a recursive function's call stack! If you have a solution that's recursive, see if you can save space by using an iterative algorithm instead.