LOWER BOUNDS FOR EXTERNAL MEMORY INTEGER SORTING VIA NETWORK CODING

Sorting extremely large datasets is a frequently occurring task in practice. These datasets are usually much larger than the computer's main memory; thus external memory sorting algorithms, first introduced by Aggarwal and Vitter [Commun. ACM, 31 (1988), pp. 1116-1127], are often used. The complexity of comparison based external memory sorting has been understood for decades by now, but the situation remains elusive if we assume the keys to be sorted are integers. In internal memory, one can sort a set of n integer keys of Theta(lgn) bits each in O(n) time using the classic radix sort algorithm, but in external memory, there are no faster integer sorting algorithms known than the simple comparison based ones. Whether such algorithms exist has remained a central open problem in external memory algorithms for more than three decades. In this paper, we present a tight conditional lower bound on the complexity of external memory sorting of integers. Our lower bound is based on a famous conjecture in network coding by Li and Li [Proceedings of the 42nd Allerton Annual Conference on Communication, Control and Computing, 2004], who conjectured that network coding cannot help anything beyond the standard multicommodity flow rate in undirected graphs. The only previous work connecting the Li and Li conjecture to lower bounds for algorithms is due to Adler et al. [Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm, 2006, pp. 241-250]. Adler et al. indeed obtain relatively simple lower bounds for oblivious algorithms (the memory access pattern is fixed and independent of the input data). Unfortunately obliviousness is a strong limitation, especially for integer sorting: we show that the Li and Li conjecture implies an Omega(n lg n) lower bound for internal memory oblivious sorting when the keys are Theta(lgn) bits. This is in sharp contrast to the classic (nonoblivious) radix sort algorithm. Indeed going beyond obliviousness is highly nontrivial; we need to introduce several new methods and involved techniques, which are of their own interest, to obtain our tight lower bound for external memory integer sorting.