# Efficiently finding $k$ smallest elements of Cartesian product

Given lists $A_1, A_2, \dots, A_n$ of non-negative numbers, I want to find the $k$ smallest elements of the Cartesian product $A_1 \times A_2 \times \dots \times A_n$ ordered by the value $x_1 + x_2 + \dots + x_n$, where $x_i\in A_i$.

I can't simply create all combinations in memory and sort them because of large amount of data.

For example, I have 1000 drinks, 1000 main courses and 1000 desserts. I want to find the 100 cheapest meals but I can't store all 1,000,000,000 elements in memory.

1. I have Drinks, Main Courses and Desserts already sorted.
2. I've found some solutions like Efficient sorted cartesian product of 2 sorted array of integers but only for 2 lists. For greater amount of sets it still memory-consuming.
3. Lazy generating next "sorted" value will be useful.
4. If no efficient exact algorithm, it could be heuristic or approximation algorithm.
5. It's similar to X + Y sorting problem , efficient faster than O(n^2 * log n) solution is unknown

Question:

Are there an efficient ways to get only $k$ cheapest(*) elements of Cartesian product?
(*) Price of the result element is sum of all numbers in that element.

If $k$ is known in advance, we can do the following. I will assume that each list $A_j$ is sorted according to cost. What we will do is a weighted form of breadth-first exploration of index tuples $(i_1,\dots,i_n)$, storing the current candidates in a (bounded length) priority queue $Q$ according to their cost $A_1[i_1]+\dots+A_n[i_n]$. Initially $Q$ contains just the initial tuple $(0,\dots,0)$.

In each iteration, pop the first tuple $t=(i_1,\dots,i_n)$ off $Q$; this is a cheapest previously unexplored solution. Then generate its successors of the form $t^{(j)}=(i_1,\dots,i_j+1,\dots,i_n)$, compute their cost, and insert them into $Q$ accordingly.

We can avoid creating duplicates by only creating the successors $t^{(j)}$ up to the first index $j$ for which $i_j>0$; e.g. for $t=(0,1,0,2)$ we would only create $(1,1,0,2)$ and $(0,2,0,2)$ (the other tuples $(0,1,1,2)$ and $(0,1,0,3)$ instead get created as successors of $(0,0,1,2)$ and $(0,0,0,3)$, respectively). This eliminates the need for a closed list of explored tuples. We still need to store the priority queue $Q$, but it can be bounded to length $k-j$, where $j$ is the current iteration, with any entries beyond that discarded.

For example, suppose we have $A_1=(1,5,7),A_2=(1,2,5),A_3=(1,3,4)$, and $k=4$.

• Initially, $Q=[(0,0,0)]$.
• Pop $(0,0,0)$, with cost $3$, and push $(1,0,0),(0,1,0),(0,0,1)$. Their costs are $7,4,5$, respectively, and $Q$ is now $[(0,1,0),(0,0,1),(1,0,0)]$.
• Pop $(0,1,0)$ (cost $4$), and push $(1,1,0)$ (cost $8$) and $(0,2,0)$ (cost $7$). $Q$ is now $[(0,0,1),(1,0,0),(0,2,0),(1,1,0)]$; since we only need two more solutions, we can drop the last two, getting $[(0,0,1),(1,0,0)]$.
• Pop $(0,0,1)$ (cost $5$), and push $(1,0,1)$ (cost $9$), $(0,1,1)$ (cost $6$), and $(0,0,2)$ (cost $6$). $Q$ is now $[(0,1,1),(0,0,2),(1,0,0),(1,0,1)]$; again, we can drop unneeded entries since we only want one more, getting $[(0,1,1)]$.
• Pop $(0,1,1)$; done.

To see that this method is correct, we just need to prove the following:

1. Given the above restrictions, any index tuple $i=(i_1,\dots,i_k)$ (with $0\le i_j\le size(A_j)$) is reachable along the unique path $(0,\dots,0)\to\dots\to(0,\dots,0,i_k)\to\dots\to(0,\dots,0,i_{k-1},i_k)\to\dots\to i$.
2. The cost of the tuples along that path is nondecreasing, since the weights are nonnegative.
3. If there is another tuple $i'$ with a higher cost, and $i'$ is popped off $Q$ during the algorithm, then so are all tuples along the path to $i$ (using induction).
4. If a tuple $i'$ is discarded due to the length restriction, then there are at least $k$ tuples whose cost is no greater than that of $i'$.
• Nice. It may be useful to provide the core reason (inductive step) for correctness, though. One might think that one had to go from $t$ to $(1,\dots, 1, i_j + 1, 1, \dots, 1)$ because that's clearly cheaper than $t^{(j)}$. – Raphael Sep 7 '15 at 22:54
• @Raphael What do you mean "from $t$ to $(1,…,1,i_j+1,1,…,1)$" ? – Dmitry Yaremenko Sep 8 '15 at 8:50
• @DmitryYaremenko That one would have to add those tuples to $Q$. – Raphael Sep 8 '15 at 12:15

This is more of a median problem, so compared to sorting you may be able to shave some work off for large k (for small k, expanding from the origin is probably best).

Assuming all sets are sorted, the function $s(x) = \sum{}{x_i}$ has nonnegative partial derivative in all dimensions, so if you pick a random pivot $p$ such that $0 < p <= \sum{max(A_i)}$ then the boundary of $P(p) = \{y: s(y) <= p \}$ has dimension $n-1$ and can be traced in that time (ie if $|A_i| = a$ and there are $a^n$ tuples, you only have to evaluate $o(a^{n-1})$ points to count the volume of $P(p)$). During tracing it's a simple matter to keep a running count of $|P|$.

If you binary search on $p$ then I believe you have $o(a^{n-1}\log{a})$.