Having dealt with the lanternfish we now set to work to escape a potentially grisly encounter with a whale. For various plot reasons we have to align a collection of crab submarines – which can obviously only move horizontally – onto a single point. The submarines incur a cost for moving from its starting position to the alignment point, so we want to find such a point that minimises the total cost incurred by the crabs.
I found the two problems easy to solve, but after some consideration I noticed that while I got the correct answers quickly they were actually very inefficient. It makes for a nice discussion and learning point, so I will first introduce some notation then proceed by discussing my solutions to the puzzles and how they changed.
Our puzzle’s input is a list \(x_1,\ldots,x_n\) of the crab submarines’ starting positions. We want to find a point \(y\) on which to align the crabs, and on placing the alignment point at \(y\) then crab \(i\) will incur a cost \(c_i(x_i,y)\) to move from his starting position to that point. For the first part we are told that each crab incurs the cost \(c_i(x_i,y) = |x_i-y|\) for placing the point at \(y\). In other words, a crab’s cost is simply the distance he must move from his starting position to the alignment point. The second part slightly altered the crabs’ cost functions, but we will talk about that in due course.
We will denote the cost for choosing alignment point \(y\) as \(c(y)\), and for both parts we are looking to choose a point that minimises this value. We will denote the optimal alignment point as \(y^* = \arg \min_y c(y)\). In our case the value \(c(y)\) is simply the sum of costs for each individual crab, so \(c(y) = \sum_i c_i(x_i,y)\), but note that this need not necessarily have been the case: we might have wanted to minimise the maximum cost, minimise the average cost, etc.
Now let’s discuss the solutions.
One way to minimise our objective function \(c(y)\), whatever it is, is to iterate over all possible solutions and simply keep the point \(y\) that minimises cost, and this is the way I solved both parts initially. That is, for each point \(y\) between the minimum and maximum crab starting positions given in the input, calculate its cost \(c(y)\), and keep it if it is smaller than the smallest value we currently have stored. The problem with this is that it is inefficient, and in fact takes exponential time in the size of the input, since if numbers are represented by \(b\) bits then we have to check \(2^b\) numbers in the worst case. I will now argue for a better method of computing the optimal alignment point \(y^*\) with the following claim.^{1}
Claim: Any point \(y\) in the interval \([m_1,m_2]\), where \(m_1,m_2\) are the median values in the input, is an optimal alignment point.
Proof: First observe that placing the point at \(m_1\) or \(m_2\) results in the same cost. This is trivial when \(n\) is odd since there is only one median point, so suppose \(n\) is even. Throughout this short proof we will use the phrases “to the left/right of” a point \(p\), which we will take to mean the points to the left/right of \(p\) when the input is written as a non-decreasing list. Let \(c_L(y)\) denote the sum of costs of all points in the input to the left of \(y\), and \(c_R(y)\) denote the analogous value for all points to the right of \(y\). Thus the cost of aligning at \(y\) is \(c(y)=c_L(y)+c_R(y)\), and we want to show \(c(m_1)=c(m_2)\). Since \(m_1\) and \(m_2\) are both median points and \(n\) is even then the number of points to the left of \(m_1\) is equal to the number of points to the right of \(m_2\), and this is \(\frac{n}{2}\). Therefore, the cost for all points left of \(m_2\) of placing the alignment point at \(m_2\) is equal to the cost of placing it at \(m_1\) plus \(\frac{n}{2}\) times the distance between \(m_1\) and \(m_2\), or:
\[c_L(m_2) = c_L(m_1) + \frac{n}{2}(m_2-m_1)\]Similarly we may rewrite \(c_R(m_2)\) as \(c_R(m_1) - \frac{n}{2}(m_2-m_1)\) and thus we get the cost of placing the alignment point at \(m_2\) as:
\[\begin{align} c(m_2) & = c_L(m_2) + c_R(m_2) \\ & = c_L(m_1) + \frac{n}{2}(m_2-m_1) + c_R(m_1) - \frac{n}{2}(m_2-m_1) \\ & = c_L(m_1) + c_R(m_1) \\ & = c(m_1) \end{align}\]It turns out that placing the alignment point \(y\) anywhere in the interval \([m_1,m_2]\) has the same cost as placing it at either median point. Performing the same trick as before, for any \(y \in [m_1,m_2]\):
\[\begin{align} c(y) & = c_L(y) + c_R(y) \\ & = c_L(m_1) + \frac{n}{2}(y-m_1) + c_R(m_1) - \frac{n}{2}(m_2-y) \\ & = c(m_1) \end{align}\]It remains to show that \(c(y)\) is optimal for any \(y \in [m_1,m_2]\). Suppose for the sake of contradiction that \(y\) is an optimal alignment point outside of the interval \([m_1,m_2]\) and without loss of generality^{2} assume \(y > m_2\). The cost of aligning at \(y\) is:
\[\begin{align} c(y) & = c(m_2) + (1 + \frac{n}{2})(y-m_2) - \frac{n}{2}(y-m_2) \\ & = c(m_2)+(y-m_2) \\ & > c(m_2) \end{align}\]So placing the alignment point outside the interval \([m_1,m_2]\) results in a strictly larger total cost than placing it at \(m_2\), contradicting our assumption. Thus any point \(y \in [m_1,m_2]\) is optimal. QED
(flet ((total-cost (position numbers)
"Sum of absolute differences between POSITION and NUMBERS"
(loop for n in numbers
sum (abs (- position n)))))
(let* ((input (get-numbers filename))
(y (median input)))
(total-cost y input)))
The entire code for the first part given above. Although the input is small enough to not make a practical difference, this solution is much more efficient. In the brute force solution we take time \(O(2^b)\) to calculate the optimal alignment point \(y^*\), since we iterate over all \(2^b\) numbers representable by \(b\) bits in the worst case^{3} and compute its cost. When using the median we instead take time \(O(n \log n)\), since this is the time required to order the list and find the median, followed by a single calculation of the total resulting cost.
The second part changed the crabs’ cost functions, so instead of being the absolute difference between \(y\) and \(x_i\) it was now 1 if the crab was 1 away from \(y\), 1+2 if the crab was 2 away from \(y\), 1+2+3 if it was 3 away from \(y\), and so on. In other words, with \(d=|x_i-y|\), when placing the alignment point at \(y\) each crab \(i\) now incurs the cost:
\[c_i(x_i,y) = \sum_{k=1}^d k = \frac{d(d+1)}{2}\]I currently don’t have a better solution than the brute force one – that is, try all possible positions between the minimum and maximum starting position and return the one that results in the smallest total cost. The code implementing this is given below:
(flet ((total-cost (position numbers)
(loop for n in numbers
for difference = (abs (- position n))
sum (/ (* difference (1+ difference)) 2))))
(loop with input = (get-numbers filename)
for i from (reduce #'min input) upto (reduce #'max input)
minimize (total-cost i input)))
As I’ve mentioned neither part of this puzzle was difficult to get right, since the brute force solution is simple to understand and implement. The low difficulty was not lost on the community as despite this being the quickest I had solved both parts of a problem it has not led to my highest ranking in the leaderboard. It was an interesting exercise to prove the median points are optimal in the first part; now it remains to improve the efficiency of my solution for the second part.
A brief note: normally when the input list has an even number of elements then the (single) median value is the arithmetic mean between elements \(x_\frac{n}{2}\) and \(x_{1+\frac{n}{2}}\). In our discussion we simply use the term “median points” to refer to these elements. ↩
We can follow symmetric reasoning to prove the claim when \(y < m_1\). ↩
i.e., the starting positions 0 and \(2^b-1\) appear in the input. ↩