X + Y sorting

In computer science, X + Y sorting is the problem of sorting pairs of numbers by their sum. Given two finite sets $X$ and $Y$ , the problem is to order all pairs $(x, y)$ in the Cartesian product $X \times Y$ by the key $x + y$ . The problem is attributed to Elwyn Berlekamp.[1][2]

Time for comparison algorithms

Unsolved problem in computer science:

Is there an X + Y sorting algorithm faster than

O(n^{2}\log n)

?

(more unsolved problems in computer science)

This problem can be solved using a straightforward comparison sort on the Cartesian product, taking time $O (nm log(nm))$ for sets of sizes $n$ and $m$ . When it is assumed that $m = n$ , the complexity of this method is $O (n 2 log n 2) = O (n 2 log n)$ , which is also the best upper bound known on the time for this problem. Whether X + Y sorting can be done strictly faster than sorting $n \cdot m$ arbitrary numbers is an open problem.[3][2]

Number of comparisons

The number of required comparisons is certainly lower than for ordinary comparison sorting: Fredman showed, in 1976, that X + Y sorting can be done using only $O (n 2)$ comparisons, though he did not show an algorithm.[4] The first actual algorithm that achieves this number of comparisons and $O (n 2 log n)$ total complexity was published sixteen years later.[5]

Non-comparison-based algorithms

On a RAM machine with word size $w$ and integer inputs $0 \leq {x, y} < n = 2 w$ , the problem can be solved in $O (n log n)$ operations by means of the fast Fourier transform.[1]

Applications

Skiena recounts a practical application in transit fare minimisation, an instance of the shortest path problem: given fares $x$ and $y$ for trips from departure A to some intermediate destination B and from B to final destination C, determine the least expensive combined trip from A to C.[3]

References

Harper, L. H.; Payne, T.H.; Savage, J. E.; Straus, E. (1975). "Sorting X + Y". Communications of the ACM. 18 (6): 347–349. doi:10.1145/360825.360869.
Demaine, Erik; Erickson, Jeff; O'Rourke, Joseph (20 August 2006). "Problem 41: Sorting X + Y (Pairwise Sums)". The Open Problems Project. Retrieved 23 September 2014.
Skiena, Steven (2008). The Algorithm Design Manual. Springer. p. 119. doi:10.1007/978-1-84800-070-4_4.
Fredman, Michael L. (1976). "How good is the information theory bound in sorting?". Theoretical Computer Science. 1 (4): 355–361. doi:10.1016/0304-3975(76)90078-5. hdl:10338.dmlcz/103933.
Lambert, Jean-Luc (1992). "Sorting the sums ( $x i$ + $y j$ ) in $O (n 2)$ comparisons". Theoretical Computer Science. 103 (1): 137–141. doi:10.1016/0304-3975(92)90089-X.

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[harper-1] Harper, L. H.; Payne, T.H.; Savage, J. E.; Straus, E. (1975). "Sorting X + Y". Communications of the ACM. 18 (6): 347–349. doi:10.1145/360825.360869.

[topp-2] Demaine, Erik; Erickson, Jeff; O'Rourke, Joseph (20 August 2006). "Problem 41: Sorting X + Y (Pairwise Sums)". The Open Problems Project. Retrieved 23 September 2014.

[skiena-3] Skiena, Steven (2008). The Algorithm Design Manual. Springer. p. 119. doi:10.1007/978-1-84800-070-4_4.

[fredman-4] Fredman, Michael L. (1976). "How good is the information theory bound in sorting?". Theoretical Computer Science. 1 (4): 355–361. doi:10.1016/0304-3975(76)90078-5. hdl:10338.dmlcz/103933.

[5] Lambert, Jean-Luc (1992). "Sorting the sums ( $x i$ + $y j$ ) in $O (n 2)$ comparisons". Theoretical Computer Science. 103 (1): 137–141. doi:10.1016/0304-3975(92)90089-X.