Universal Sorting: Finding a DAG using Priced Comparisons

We resolve two open problems in sorting with priced information, introduced by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]. In this setting, different comparisons have different (potentially infinite) costs. The goal is to sort with small competitive ratio (algorithmic cost divided by cheapest proof). 1) When all costs are in $\{0,1,n,\infty\}$, we give an algorithm that has $\widetilde{O}(n^{3/4})$ competitive ratio. Our algorithm generalizes the algorithms for generalized sorting (all costs are either $1$ or $\infty$), a version initiated by [Huang, Kannan, Khanna, FOCS 2011] and addressed recently by [Kuszmaul, Narayanan, FOCS 2021]. 2) We answer the problem of bichromatic sorting posed by [CFGKRS]: The input is split into $A$ and $B$, and $A-A$ and $B-B$ comparisons are more expensive than an $A-B$ comparisons. We give a randomized algorithm with a O(polylog n) competitive ratio. These results are obtained by introducing the universal sorting problem, which generalizes the existing framework in two important ways. We remove the promise of a directed Hamiltonian path in the DAG of all comparisons. Instead, we require that an algorithm outputs the transitive reduction of the DAG. For bichromatic sorting, when $A-A$ and $B-B$ comparisons cost $\infty$, this generalizes the well-known nuts and bolts problem. We initiate an instance-based study of the universal sorting problem. Our definition of instance-optimality is inherently more algorithmic than that of the competitive ratio in that we compare the cost of a candidate algorithm to the cost of the optimal instance-aware algorithm. This unifies existing lower bounds, and opens up the possibility of an $O(1)$-instance optimal algorithm for the bichromatic version.Comment: 40 pages, 5 figure

An Algorithm for Bichromatic Sorting with Polylog Competitive Ratio

The problem of sorting with priced information was introduced by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]. In this setting, different comparisons have different (potentially infinite) costs. The goal is to find a sorting algorithm with small competitive ratio, defined as the (worst-case) ratio of the algorithm's cost to the cost of the cheapest proof of the sorted order. The simple case of bichromatic sorting posed by [CFGKRS] remains open: We are given two sets $A$ and $B$ of total size $N$, and the cost of an $A-A$ comparison or a $B-B$ comparison is higher than an $A-B$ comparison. The goal is to sort $A \cup B$. An $\Omega(\log N)$ lower bound on competitive ratio follows from unit-cost sorting. Note that this is a generalization of the famous nuts and bolts problem, where $A-A$ and $B-B$ comparisons have infinite cost, and elements of $A$ and $B$ are guaranteed to alternate in the final sorted order. In this paper we give a randomized algorithm InversionSort with an almost-optimal w.h.p. competitive ratio of $O(\log^{3} N)$. This is the first algorithm for bichromatic sorting with a $o(N)$ competitive ratio.Comment: 18 pages, accepted to ITCS 2024. arXiv admin note: text overlap with arXiv:2211.0460

External memory priority queues with decrease-key and applications to graph algorithms

We present priority queues in the external memory model with block size B and main memory size M that support on N elements, operation Update (a combination of operations Insert and DecreaseKey) in O(1/Blog_{M/B} N/B) amortized I/Os and operations ExtractMin and Delete in O(ceil[(M^epsilon)/B log_{M/B} N/B] log_{M/B} N/B) amortized I/Os, for any real epsilon in (0,1), using O(N/Blog_{M/B} N/B) blocks. Previous I/O-efficient priority queues either support these operations in O(1/Blog_2 N/B) amortized I/Os [Kumar and Schwabe, SPDP \u2796] or support only operations Insert, Delete and ExtractMin in optimal O(1/Blog_{M/B} N/B) amortized I/Os, however without supporting DecreaseKey [Fadel et al., TCS \u2799]. We also present buffered repository trees that support on a multi-set of N elements, operation Insert in O(1/Blog_M/B N/B) I/Os and operation Extract on K extracted elements in O(M^{epsilon} log_M/B N/B + K/B) amortized I/Os, using O(N/B) blocks. Previous results achieve O(1/Blog_2 N/B) I/Os and O(log_2 N/B + K/B) I/Os, respectively [Buchsbaum et al., SODA \u2700]. Our results imply improved O(E/Blog_{M/B} E/B) I/Os for single-source shortest paths, depth-first search and breadth-first search algorithms on massive directed dense graphs (V,E) with E = Omega (V^(1+epsilon)), epsilon > 0 and V = Omega (M), which is equal to the I/O-optimal bound for sorting E values in external memory