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

Charting the Algorithmic Complexity of Waypoint Routing

Modern computer networks support interesting new routing models in which traffic flows from a source sto a destination t can be flexibly steered through a sequence of waypoints, such as (hardware) middleboxes or (virtualized) network functions (VNFs), to create innovative network services like service chains or segment routing. While the benefits and technological challenges of providing such routing models have been articulated and studied intensively over the last years, less is known about the underlying algorithmic traffic routing problems. The goal of this paper is to provide the network community with an overview of algorithmic techniques for waypoint routing and also inform about limitations due to computational hardness. In particular, we put the waypoint routing problem into perspective with respect to classic graph theoretical problems. For example, we find that while computing a shortest path from a source s to a destination t is simple (e.g., using Dijkstra's algorithm), the problem of finding a shortest route from s to t via a single waypoint already features a deep combinatorial structure.</jats:p

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