arXiv subjects

Monika Henzinger

Publications and source records attributed to Monika Henzinger.

At least 73 records · Page 4Linked to original sources

arXiv2022

The Augmentation-Speed Tradeoff for Consistent Network Updates

Emerging software-defined networking technologies enable more adaptive communication infrastructures, allowing for quick reactions to changes in networking requirements by exploiting the workload's temporal structure. However, operating networks adaptively is algorithmically challenging, as meeting networks' stringent dependability requirements relies on maintaining basic consistency and performance properties, such as loop freedom and congestion minimization, even during the update process. This paper leverages an augmentation-speed tradeoff to significantly speed up consistent network updates. We show that allowing for a small and short (hence practically tolerable, e.g., using buffering) oversubscription of links allows us to solve many network update instances much faster, as well as to reduce computational complexities (i.e., the running times of the algorithms). We first explore this tradeoff formally, revealing the computational complexity of scheduling updates. We then present and analyze algorithms that maintain logical and performance properties during the update. Using an extensive simulation study, we find that the tradeoff is even more favorable in practice than our analytical bounds suggest. In particular, we find that by allowing just 10% augmentation, update times reduce by more than 32% on average, across a spectrum of real-world networks.

cs.NI↗

arXiv2022

Leximax Approximations and Representative Cohort Selection

Finding a representative cohort from a broad pool of candidates is a goal that arises in many contexts such as choosing governing committees and consumer panels. While there are many ways to define the degree to which a cohort represents a population, a very appealing solution concept is lexicographic maximality (leximax) which offers a natural (pareto-optimal like) interpretation that the utility of no population can be increased without decreasing the utility of a population that is already worse off. However, finding a leximax solution can be highly dependent on small variations in the utility of certain groups. In this work, we explore new notions of approximate leximax solutions with three distinct motivations: better algorithmic efficiency, exploiting significant utility improvements, and robustness to noise. Among other definitional contributions, we give a new notion of an approximate leximax that satisfies a similarly appealing semantic interpretation and relate it to algorithmically-feasible approximate leximax notions. When group utilities are linear over cohort candidates, we give an efficient polynomial-time algorithm for finding a leximax distribution over cohort candidates in the exact as well as in the approximate setting. Furthermore, we show that finding an integer solution to leximax cohort selection with linear utilities is NP-Hard.

cs.DS↗

arXiv2022

Fully Dynamic Four-Vertex Subgraph Counting

This paper presents a comprehensive study of algorithms for maintaining the number of all connected four-vertex subgraphs in a dynamic graph. Specifically, our algorithms maintain the number of paths of length three in deterministic amortized $\mathcal{O}(m^\frac{1}{2})$ update time, and any other connected four-vertex subgraph which is not a clique in deterministic amortized update time $\mathcal{O}(m^\frac{2}{3})$. Queries can be answered in constant time. We also study the query times for subgraphs containing an arbitrary edge that is supplied only with the query as well as the case where only subgraphs containing a vertex $s$ that is fixed beforehand are considered. For length-3 paths, paws, $4$-cycles, and diamonds our bounds match or are not far from (conditional) lower bounds: Based on the OMv conjecture we show that any dynamic algorithm that detects the existence of paws, diamonds, or $4$-cycles or that counts length-$3$ paths takes update time $Ω(m^{1/2-δ})$. Additionally, for $4$-cliques and all connected induced subgraphs, we show a lower bound of $Ω(m^{1-δ})$ for any small constant $δ> 0$ for the amortized update time, assuming the static combinatorial $4$-clique conjecture holds. This shows that the $\mathcal{O}(m)$ algorithm by Eppstein at al. for these subgraphs cannot be improved by a polynomial factor.

cs.DS↗

arXiv2022

Fast and Heavy Disjoint Weighted Matchings for Demand-Aware Datacenter Topologies

Reconfigurable optical topologies promise to improve the performance in datacenters by dynamically optimizing the physical network in a demand-aware manner. State-of-the-art optical technologies allow to establish and update direct connectivity (in the form of edge-disjoint matchings) between top-of-rack switches within microseconds or less. However, to fully exploit temporal structure in the demand, such fine-grained reconfigurations also require fast algorithms for optimizing the interconnecting matchings. Motivated by the desire to offload a maximum amount of demand to the reconfigurable network, this paper initiates the study of fast algorithms to find k disjoint heavy matchings in graphs. We present and analyze six algorithms, based on iterative matchings, b-matching, edge coloring, and node-rankings. We show that the problem is generally NP-hard and study the achievable approximation ratios. An extensive empirical evaluation of our algorithms on both real-world and synthetic traces (88 in total), including traces collected in Facebook datacenters and in HPC clusters reveals that all our algorithms provide high-quality matchings, and also very fast ones come within 95% or more of the best solution. However, the running times differ significantly and what is the best algorithm depends on k and the acceptable runtime-quality tradeoff.

cs.DS↗

arXiv2021

On fully dynamic constant-factor approximation algorithms for clustering problems

Clustering is an important task with applications in many fields of computer science. We study the fully dynamic setting in which we want to maintain good clusters efficiently when input points (from a metric space) can be inserted and deleted. Many clustering problems are $\mathsf{APX}$-hard but admit polynomial time $O(1)$-approximation algorithms. Thus, it is a natural question whether we can maintain $O(1)$-approximate solutions for them in subpolynomial update time, against adaptive and oblivious adversaries. Only a few results are known that give partial answers to this question. There are dynamic algorithms for $k$-center, $k$-means, and $k$-median that maintain constant factor approximations in expected $\tilde{O}(k^{2})$ update time against an oblivious adversary. However, for these problems there are no algorithms known with an update time that is subpolynomial in $k$, and against an adaptive adversary there are even no (non-trivial) dynamic algorithms known at all. In this paper, we complete the picture of the question above for all these clustering problems. 1. We show that there is no fully dynamic $O(1)$-approximation algorithm for any of the classic clustering problems above with an update time in $n^{o(1)}h(k)$ against an adaptive adversary, for an arbitrary function $h$. 2. We give a lower bound of $Ω(k)$ on the update time for each of the above problems, even against an oblivious adversary. 3. We give the first $O(1)$-approximate fully dynamic algorithms for $k$-sum-of-radii and for $k$-sum-of-diameters with expected update time of $\tilde{O}(k^{O(1)})$ against an oblivious adversary. 4. Finally, for $k$-center we present a fully dynamic $(6+ε)$-approximation algorithm with an expected update time of $\tilde{O}(k)$ against an oblivious adversary.

cs.DS↗

arXiv2021

Upper and Lower Bounds for Fully Retroactive Graph Problems

Classic dynamic data structure problems maintain a data structure subject to a sequence S of updates and they answer queries using the latest version of the data structure, i.e., the data structure after processing the whole sequence. To handle operations that change the sequence S of updates, Demaine et al. (TALG 2007) introduced retroactive data structures. A retroactive operation modifies the update sequence S in a given position t, called time, and either creates or cancels an update in S at time t. A partially retroactive data structure restricts queries to be executed exclusively in the latest version of the data structure. A fully retroactive data structure supports queries at any time t: a query at time t is answered using only the updates of S up to time t. If the sequence S only consists of insertions, the resulting data structure is an incremental retroactive data structure. While efficient retroactive data structures have been proposed for classic data structures, e.g., stack, priority queue and binary search tree, the retroactive version of graph problems are rarely studied. In this paper we study retroactive graph problems including connectivity, minimum spanning forest (MSF), maximum degree, etc. We provide fully retroactive data structures for maintaining the maximum degree, connectivity and MSF in $\tilde{O}(n)$ time per operation. We also give an algorithm for the incremental fully retroactive connectivity with $\tilde{O}(1)$ time per operation. We compliment our algorithms with almost tight hardness results. We show that under the OMv conjecture (proposed by Henzinger et al. (STOC 2015)), there does not exist fully retroactive data structures maintaining connectivity or MSF, or incremental fully retroactive data structure maintaining the maximum degree with $O(n^{1-ε})$ time per operation, for any constant $ε> 0$.

cs.DS↗

arXiv2021

Constant-Time Dynamic Weight Approximation for Minimum Spanning Forest

We give two fully dynamic algorithms that maintain a $(1+\varepsilon)$-approximation of the weight $M$ of a minimum spanning forest (MSF) of an $n$-node graph $G$ with edges weights in $[1,W]$, for any $\varepsilon>0$. (1) Our deterministic algorithm takes $O({W^2 \log W}/{\varepsilon^3})$ worst-case update time, which is $O(1)$ if both $W$ and $\varepsilon$ are constants. Note that there is a lower bound by Patrascu and Demaine (SIAM J. Comput. 2006) which shows that it takes $Ω(\log n)$ time per operation to maintain the exact weight of an MSF that holds even in the unweighted case, i.e. for $W=1$. We further show that any deterministic data structure that dynamically maintains the $(1+\varepsilon)$-approximate weight of an MSF requires super constant time per operation, if $W\geq (\log n)^{ω_n(1)}$. (2) Our randomized (Monte-Carlo style) algorithm works with high probability and runs in worst-case $O(\log W/ \varepsilon^{4})$ update time if $W= O({(m^*)^{1/6}}/{\log^{2/3} n})$, where $m^*$ is the minimum number of edges in the graph throughout all the updates. It works even against an adaptive adversary. This implies a randomized algorithm with worst-case $o(\log n)$ update time, whenever $W=\min\{O((m^*)^{1/6}/\log^{2/3} n), 2^{o({\log n})}\}$ and $\varepsilon$ is constant. We complement this result by showing that for any constant $\varepsilon,α>0$ and $W=n^α$, any (randomized) data structure that dynamically maintains the weight of an MSF of a graph $G$ with edge weights in $[1,W]$ and $W = Ω(\varepsilon m^*)$ within a multiplicative factor of $(1+\varepsilon)$ takes $Ω(\log n)$ time per operation.

cs.DS↗

arXiv2021

New Techniques and Fine-Grained Hardness for Dynamic Near-Additive Spanners

Maintaining and updating shortest paths information in a graph is a fundamental problem with many applications. As computations on dense graphs can be prohibitively expensive, and it is preferable to perform the computations on a sparse skeleton of the given graph that roughly preserves the shortest paths information. Spanners and emulators serve this purpose. This paper develops fast dynamic algorithms for sparse spanner and emulator maintenance and provides evidence from fine-grained complexity that these algorithms are tight. Under the popular OMv conjecture, we show that there can be no decremental or incremental algorithm that maintains an $n^{1+o(1)}$ edge (purely additive) $+n^δ$-emulator for any $δ<1/2$ with arbitrary polynomial preprocessing time and total update time $m^{1+o(1)}$. Also, under the Combinatorial $k$-Clique hypothesis, any fully dynamic combinatorial algorithm that maintains an $n^{1+o(1)}$ edge $(1+ε,n^{o(1)})$-spanner or emulator must either have preprocessing time $mn^{1-o(1)}$ or amortized update time $m^{1-o(1)}$. Both of our conditional lower bounds are tight. As the above fully dynamic lower bound only applies to combinatorial algorithms, we also develop an algebraic spanner algorithm that improves over the $m^{1-o(1)}$ update time for dense graphs. For any constant $ε\in (0,1]$, there is a fully dynamic algorithm with worst-case update time $O(n^{1.529})$ that whp maintains an $n^{1+o(1)}$ edge $(1+ε,n^{o(1)})$-spanner. Our new algebraic techniques and spanner algorithms allow us to also obtain (1) a new fully dynamic algorithm for All-Pairs Shortest Paths (APSP) with update and path query time $O(n^{1.9})$; (2) a fully dynamic $(1+ε)$-approximate APSP algorithm with update time $O(n^{1.529})$; (3) a fully dynamic algorithm for near-$2$-approximate Steiner tree maintenance.

cs.DS↗

arXiv2021

Cut-Toggling and Cycle-Toggling for Electrical Flow and Other p-Norm Flows

We study the problem of finding flows in undirected graphs so as to minimize the weighted $p$-norm of the flow for any $p > 1$. When $p=2$, the problem is that of finding an electrical flow, and its dual is equivalent to solving a Laplacian linear system. The case $p = \infty$ corresponds to finding a min-congestion flow, which is equivalent to max-flows. A typical algorithmic construction for such problems considers vertex potentials corresponding to the flow conservation constraints, and has two simple types of update steps: cycle toggling, which modifies the flow along a cycle, and cut toggling, which modifies all potentials on one side of a cut. Both types of steps are typically performed relative to a spanning tree $T$; then the cycle is a fundamental cycle of $T$, and the cut is a fundamental cut of $T$. In this paper, we show that these simple steps can be used to give a novel efficient implementation for the $p = 2$ case and to find near-optimal $p$-norm flows in a low number of iterations for all values of $p > 1$. Compared to known faster algorithms for these problems, our algorithms are simpler, more combinatorial, and also expose several underlying connections between these algorithms and dynamic graph data structures that have not been formalized previously.

cs.DS↗

arXiv2021

Random Rank-Based, Hierarchical or Trivial: Which Dynamic Graph Algorithm Performs Best in Practice?

Fully dynamic graph algorithms that achieve polylogarithmic or better time per operation use either a hierarchical graph decomposition or random-rank based approach. There are so far two graph properties for which efficient algorithms for both types of data structures exist, namely fully dynamic (Delta + 1) coloring and fully dynamic maximal matching. In this paper we present an extensive experimental study of these two types of algorithms for these two problems together with very simple baseline algorithms to determine which of these algorithms are the fastest. Our results indicate that the data structures used by the different algorithms dominate their performance.

cs.DS↗

arXiv2021

Symbolic Time and Space Tradeoffs for Probabilistic Verification

We present a faster symbolic algorithm for the following central problem in probabilistic verification: Compute the maximal end-component (MEC) decomposition of Markov decision processes (MDPs). This problem generalizes the SCC decomposition problem of graphs and closed recurrent sets of Markov chains. The model of symbolic algorithms is widely used in formal verification and model-checking, where access to the input model is restricted to only symbolic operations (e.g., basic set operations and computation of one-step neighborhood). For an input MDP with $n$ vertices and $m$ edges, the classical symbolic algorithm from the 1990s for the MEC decomposition requires $O(n^2)$ symbolic operations and $O(1)$ symbolic space. The only other symbolic algorithm for the MEC decomposition requires $O(n \sqrt{m})$ symbolic operations and $O(\sqrt{m})$ symbolic space. A main open question is whether the worst-case $O(n^2)$ bound for symbolic operations can be beaten. We present a symbolic algorithm that requires $\widetilde{O}(n^{1.5})$ symbolic operations and $\widetilde{O}(\sqrt{n})$ symbolic space. Moreover, the parametrization of our algorithm provides a trade-off between symbolic operations and symbolic space: for all $0<ε\leq 1/2$ the symbolic algorithm requires $\widetilde{O}(n^{2-ε})$ symbolic operations and $\widetilde{O}(n^ε)$ symbolic space ($\widetilde{O}$ hides poly-logarithmic factors). Using our techniques we present faster algorithms for computing the almost-sure winning regions of $ω$-regular objectives for MDPs. We consider the canonical parity objectives for $ω$-regular objectives, and for parity objectives with $d$-priorities we present an algorithm that computes the almost-sure winning region with $\widetilde{O}(n^{2-ε})$ symbolic operations and $\widetilde{O}(n^ε)$ symbolic space, for all $0 < ε\leq 1/2$.

cs.LO↗

arXiv2021

A Deamortization Approach for Dynamic Spanner and Dynamic Maximal Matching

Many dynamic graph algorithms have an amortized update time, rather than a stronger worst-case guarantee. But amortized data structures are not suitable for real-time systems, where each individual operation has to be executed quickly. For this reason, there exist many recent randomized results that aim to provide a guarantee stronger than amortized expected. The strongest possible guarantee for a randomized algorithm is that it is always correct (Las Vegas), and has high-probability worst-case update time, which gives a bound on the time for each individual operation that holds with high probability. In this paper we present the first polylogarithmic high-probability worst-case time bounds for the dynamic spanner and the dynamic maximal matching problem. 1. For dynamic spanner, the only known $o(n)$ worst-case bounds were $O(n^{3/4})$ high-probability worst-case update time for maintaining a 3-spanner and $O(n^{5/9})$ for maintaining a 5-spanner. We give a $O(1)^k \log^3(n)$ high-probability worst-case time bound for maintaining a $(2k-1)$-spanner, which yields the first worst-case polylog update time for all constant $k$. (All the results above maintain the optimal tradeoff of stretch $2k-1$ and $\tilde{O}(n^{1+1/k})$ edges.) 2. For dynamic maximal matching, or dynamic $2$-approximate maximum matching, no algorithm with $o(n)$ worst-case time bound was known and we present an algorithm with $O(\log^5(n))$ high-probability worst-case time; similar worst-case bounds existed only for maintaining a matching that was $(2+ε)$-approximate, and hence not maximal. Our results are achieved using a new black-box reduction that converts any data structure with worst-case expected update time into one with a high-probability worst-case update time: the query time remains the same, while the update time increases by a factor of $O(\log^2(n))$.

cs.DS↗

arXiv2021

Practical Fully Dynamic Minimum Cut Algorithms

We present a practically efficient algorithm for maintaining a global minimum cut in large dynamic graphs under both edge insertions and deletions. While there has been theoretical work on this problem, our algorithm is the first implementation of a fully-dynamic algorithm. The algorithm uses the theoretical foundation and combines it with efficient and finely-tuned implementations to give an algorithm that can maintain the global minimum cut of a graph with rapid update times. We show that our algorithm gives up to multiple orders of magnitude speedup compared to static approaches both on edge insertions and deletions.

cs.DS↗

arXiv2020

Tight Bounds for Online Graph Partitioning

We consider the following online optimization problem. We are given a graph $G$ and each vertex of the graph is assigned to one of $\ell$ servers, where servers have capacity $k$ and we assume that the graph has $\ell \cdot k$ vertices. Initially, $G$ does not contain any edges and then the edges of $G$ are revealed one-by-one. The goal is to design an online algorithm $\operatorname{ONL}$, which always places the connected components induced by the revealed edges on the same server and never exceeds the server capacities by more than $\varepsilon k$ for constant $\varepsilon>0$. Whenever $\operatorname{ONL}$ learns about a new edge, the algorithm is allowed to move vertices from one server to another. Its objective is to minimize the number of vertex moves. More specifically, $\operatorname{ONL}$ should minimize the competitive ratio: the total cost $\operatorname{ONL}$ incurs compared to an optimal offline algorithm $\operatorname{OPT}$. Our main contribution is a polynomial-time randomized algorithm, that is asymptotically optimal: we derive an upper bound of $O(\log \ell + \log k)$ on its competitive ratio and show that no randomized online algorithm can achieve a competitive ratio of less than $Ω(\log \ell + \log k)$. We also settle the open problem of the achievable competitive ratio by deterministic online algorithms, by deriving a competitive ratio of $Θ(\ell \lg k)$; to this end, we present an improved lower bound as well as a deterministic polynomial-time online algorithm. Our algorithms rely on a novel technique which combines efficient integer programming with a combinatorial approach for maintaining ILP solutions. We believe this technique is of independent interest and will find further applications in the future.

cs.DS↗

arXiv2020

Fully-Dynamic Coresets

With input sizes becoming massive, coresets -- small yet representative summary of the input -- are relevant more than ever. A weighted set $C_w$ that is a subset of the input is an $\varepsilon$-coreset if the cost of any feasible solution $S$ with respect to $C_w$ is within $[1 {\pm} \varepsilon]$ of the cost of $S$ with respect to the original input. We give a very general technique to compute coresets in the fully-dynamic setting where input points can be added or deleted. Given a static $\varepsilon$-coreset algorithm that runs in time $t(n, \varepsilon, λ)$ and computes a coreset of size $s(n, \varepsilon, λ)$, where $n$ is the number of input points and $1 {-}λ$ is the success probability, we give a fully-dynamic algorithm that computes an $\varepsilon$-coreset with worst-case update time $O((\log n) \cdot t(s(n, \varepsilon/\log n, λ/n), \varepsilon/\log n, λ/n) )$ (this bound is stated informally), where the success probability is $1{-}λ$. Our technique is a fully-dynamic analog of the merge-and-reduce technique that applies to insertion-only setting. Although our space usage is $O(n)$, we work in the presence of an adaptive adversary, and we show that $Ω(n)$ space is required when adversary is adaptive. As a consequence, we get fully-dynamic $\varepsilon$-coreset algorithms for $k$-median and $k$-means with worst-case update time $O(\varepsilon^{-2}k^2\log^5 n \log^3 k)$ and coreset size $O(\varepsilon^{-2}k\log n \log^2 k)$ ignoring $\log \log n$ and $\log(1/\varepsilon)$ factors and assuming that $\varepsilon, λ= Ω(1/$poly$(n))$. These are the first fully-dynamic algorithms for $k$-median and $k$-means with worst-case update times $O($poly$(k, \log n, \varepsilon^{-1}))$. We also give conditional lower bound on update/query time for any fully-dynamic $(4 - δ)$-approximation algorithm for $k$-means.

cs.DS↗

arXiv2020

Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers

We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1) A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in $\tilde{O}(n^{2/3})$ amortized time against an oblivious adversary, and $\tilde{O}(m^{3/4})$ time against an adaptive adversary. (2) An incremental data structure that maintains $O(1)$-approximate shortest path in $n^{o(1)}$ time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in $\tilde{O}(n^{2/3+o(1)})$ amortized time per operation. (3) A fully-dynamic algorithm that approximates all-pair effective resistance up to an $(1+ε)$ factor in $\tilde{O}(n^{2/3+o(1)} ε^{-O(1)})$ amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as j-trees) and approximately captures the cut-flow and metric structure of the graph. The $O(1)$-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM `05]. Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC `19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy.

cs.DS↗

arXiv2020

Faster Parallel Multiterminal Cuts

We give an improved branch-and-bound solver for the multiterminal cut problem, based on the recent work of Henzinger et al.. We contribute new, highly effective data reduction rules to transform the graph into a smaller equivalent instance. In addition, we present a local search algorithm that can significantly improve a given solution to the multiterminal cut problem. Our exact algorithm is able to give exact solutions to more and harder problems compared to the state-of-the-art algorithm by Henzinger et al.; and give better solutions for more than two third of the problems that are too large to be solved to optimality. Additionally, we give an inexact heuristic algorithm that computes high-quality solutions for very hard instances in reasonable time.

cs.DS↗

arXiv2020

Dynamic Maintenance of Low-Stretch Probabilistic Tree Embeddings with Applications

We give the first non-trivial fully dynamic probabilistic tree embedding algorithm for weighted graphs undergoing edge insertions and deletions. We obtain a trade-off between amortized update time and expected stretch against an oblivious adversary. At the two extremes of this trade-off, we can maintain a tree of expected stretch $ O (\log^4 n) $ with update time $ m^{1/2 + o(1)} $ or a tree of expected stretch $ n^{o(1)} $ with update time $ n^{o(1)} $ (for edge weights polynomial in $ n $). A guarantee of the latter type has so far only been known for maintaining tree embeddings with average (instead of expected) stretch [Chechik/Zhang, SODA '20]. Our main result has direct implications to fully dynamic approximate distance oracles and fully dynamic buy-at-bulk network design. For dynamic distance oracles, our result is the first to break the $ O (\sqrt{m}) $ update-time barrier. For buy-at-bulk network design, a problem which also in the static setting heavily relies on probabilistic tree embeddings, we give the first non-trivial dynamic algorithm. As probabilistic tree embeddings are an important tool in static approximation algorithms, further applications of our result in dynamic approximation algorithms are conceivable. From a technical perspective, we obtain our main result by first designing a decremental algorithm for probabilistic low-diameter decompositions via a careful combination of Bartal's ball-growing approach [FOCS '96] with the pruning framework of Chechik and Zhang [SODA '20]. We then extend this to a fully dynamic algorithm by enriching a well-known 'decremental to fully dynamic' reduction with a new bootstrapping idea to recursively employ a fully dynamic algorithm instead of a static one in this reduction.

cs.DS↗