Dynamic Matrix Inverse: Improved Algorithms and Matching Conditional Lower Bounds

Brand, Jan van den; Nanongkai, Danupon; Saranurak, Thatchaphol

doi:10.1109/focs.2019.00036

Cited by 53 publications

(51 citation statements)

References 36 publications

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“…There are two equivalent ways to prove Theorem 2.4: One could use the data-structures of [San04,vdBNS19] which maintain M −1 b for some non-singular matrix M and some vector b. Via a black-box reduction these data-structures would then be able to maintain P v and applying the tools of [CLS18] for optimizing the amortized complexity would then result in Theorem 2.4.…”

Section: Projection Maintenance (Details In Section 4)mentioning

confidence: 99%

“…Per iteration of the linear system solver, more than one entry of u and v may have to be changed. This can be interpreted as a so called batch-update, and the complexity for batch-updates was already analyzed in [vdBNS19], but again the focus was on worst-case complexity. Both datastructure from [San04] and [vdBNS19] had the property, that the data-structure would become slower the more updates they received.…”

Section: B Projection Maintenance Via Dynamic Linear System Solversmentioning

confidence: 99%

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A Deterministic Linear Program Solver in Current Matrix Multiplication Time

Brand¹

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Interior point algorithms for solving linear programs have been studied extensively for a long time [e.g. Karmarkar 1984; Lee, Sidford FOCS'14; Cohen, Lee, Song STOC'19]. For linear programs of the form min Ax=b,x≥0 c ⊤ x with n variables and d constraints, the generic case d = Ω(n) has recently been settled by Cohen, Lee and Song [STOC'19]. Their algorithm can solve linear programs in O(n ω log(n/δ)) expected time 1 , where δ is the relative accuracy. This is essentially optimal as all known linear system solvers require up to O(n ω ) time for solving Ax = b. However, for the case of deterministic solvers, the best upper bound is Vaidya's 30 years old O(n 2.5 log(n/δ)) bound [FOCS'89]. In this paper we show that one can also settle the deterministic setting by derandomizing Cohen et al.'s O(n ω log(n/δ)) time algorithm. This allows for a strict O(n ω log(n/δ)) time bound, instead of an expected one, and a simplified analysis, reducing the length of their proof of their central path method by roughly half. Derandomizing this algorithm was also an open question asked in Song's PhD Thesis.The main tool to achieve our result is a new data-structure that can maintain the solution to a linear system in subquadratic time. More accurately we are able to maintain √ U A ⊤ (AU A ⊤ ) −1 A √ U v in subquadratic time under ℓ 2 multiplicative changes to the diagonal matrix U and the vector v. This type of change is common for interior point algorithms. Previous algorithms [e.g. Vaidya STOC'89; Lee, Sidford FOCS'15; Cohen, Lee, Song STOC'19] required Ω(n 2 ) time for this task. In [Cohen, Lee, Song STOC'19] they managed to maintain the matrix √ U A ⊤ (AU A ⊤ ) −1 A √ U in subquadratic time, but multiplying it with a dense vector to solve the linear system still required Ω(n 2 ) time. To improve the complexity of their linear program solver, they restricted the solver to only multiply sparse vectors via a random sampling argument. In comparison, our data-structure maintains the entire productadditionally to just the matrix. Interestingly, this can be viewed as a simple modification of Cohen et al.'s data-structure, but it significantly simplifies their analysis of their central path method and makes their whole algorithm deterministic.1 Here O hides polylog(n) factors and O(n ω ) is the time required to multiply two n × n matrices. The stated O(n ω log(n/δ)) bound holds for the current bound on ω with ω ≈ 2.38 [V. Williams, STOC'12; Le Gall, ISSAC'14]. The upper bound for the solver will become larger than O(n ω log(n/δ)), if ω < 2 + 1/6.

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Section: Projection Maintenance (Details In Section 4)mentioning

confidence: 99%

Section: B Projection Maintenance Via Dynamic Linear System Solversmentioning

confidence: 99%

A Deterministic Linear Program Solver in Current Matrix Multiplication Time

Brand¹

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Finally, we show our algorithms for maintaining adjoint of polynomial matrices in Section 4.1, where we will also apply the reductions to get our distance and reachability oracles Theorems 1.1 and 1.2. 8 The current best algorithm [vdBNS19] takes O(n 1.407 ) operations to update one entry of a non-polynomial matrix. 9 The first limitation explains why there is only one application in [San05b], which is to maintain distances on unweighted graphs.…”

Section: Organizationmentioning

confidence: 99%

Sensitive Distance and Reachability Oracles for Large Batch Updates

Brand

Saranurak

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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In the sensitive distance oracle problem, there are three phases. We first preprocess a given directed graph G with n nodes and integer weights from [−W, W ]. Second, given a single batch of f edge insertions and deletions, we update the data structure. Third, given a query pair of nodes (u, v), return the distance from u to v. In the easier problem called sensitive reachability oracle problem, we only ask if there exists a directed path from u to v.Our first result is a sensitive distance oracle withÕ(W n ω+(3−ω)µ ) preprocessing time, O(W n 2−µ f 2 + W nf ω ) update time, andÕ(W n 2−µ f + W nf 2 ) query time where the parameter µ ∈ [0, 1] can be chosen. The data-structure requires O(W n 2+µ log n) bits of memory. This is the first algorithm that can handle f ≥ log n updates. Previous results (e.g. [Demetrescu et al. SICOMP'08; Bernstein and Karger SODA'08 and FOCS'09; Duan and Pettie SODA'09; Grandoni and Williams FOCS'12]) can handle at most 2 updates. When 3 ≤ f ≤ log n, the only non-trivial algorithm was by [Weimann and Yuster FOCS'10]. When W =Õ(1), our algorithm simultaneously improves their preprocessing time, update time, and query time.In particular, when f = ω(1), their update and query time is Ω(n 2−o(1) ), while our update and query time are truly subquadratic in n, i.e., ours is faster by a polynomial factor of n. To highlight the technique, ours is the first graph algorithm that exploits the kernel basis decomposition of polynomial matrices by [Jeannerod and Villard J.Comp'05; Zhou, Labahn and Storjohann J.Comp'15] developed in the symbolic computation community.As an easy observation from our technique, we obtain the first sensitive reachability oracle can handle f ≥ log n updates. Our algorithm has O(n ω ) preprocessing time, O(f ω ) update time, and O(f 2 ) query time. This data-structure requires O(n 2 log n) bits of memory. Efficient sensitive reachability oracles were asked in [Chechik, Cohen, Fiat, and Kaplan SODA'17]. Our algorithm can handle any constant number of updates in constant time. Previous algorithms with constant update and query time can handle only at most f ≤ 2 updates. Otherwise, there are non-trivial results for f ≤ log n, though, with query time Ω(n) by adapting [Baswana, Choudhary and Roditty STOC'16].

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“…The main differences of Lemma 2.4 compared to previous dynamic algebraic algorithms for distances (e.g. [San05b,vdBNS19]) is that our algorithm maintains (M −1 ) [k] for k ∈ S for any set S ⊂ [n s ], whereas previous algebraic algorithms for distances maintain (M −1 ) [k] for all k = 1, 2, ..., n s , i.e. they were restricted to the special case S = [n s ].…”

Section: Proof Sketch Of Lemma 24mentioning

confidence: 99%

Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

Brand

Nanongkai

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Consider the following distance query for an n-node graph G undergoing edge insertions and deletions: given two sets of nodes I and J, return the distances between every pair of nodes in I × J. This query is rather general and captures several versions of the dynamic shortest paths problem. In this paper, we develop an efficient (1 + ǫ)-approximation algorithm for this query using fast matrix multiplication. Our algorithm leads to answers for some open problems for Single-Source and All-Pairs Shortest Paths (SSSP and APSP), as well as for Diameter, Radius, and Eccentricities. Below are some highlights. Note that all our algorithms guarantee worstcase update time and are randomized (Monte Carlo), but do not need the oblivious adversary assumption.Subquadratic update time for SSSP, Diameter, Centralities, ect.: When we want to maintain distances from a single node explicitly (without queries), a fundamental question is to beat trivially calling Dijkstra's static algorithm after each update, taking Θ(n 2 ) update time on dense graphs. A better time complexity was not known even with amortization. It was known to be improbable for exact algorithms and for combinatorial any-approximation algorithms to polynomially beat the Ω(n 2 ) bound (under some conjectures) [Roditty, Zwick, ESA'04; Abboud, V. Williams, FOCS'14]. 1 Our algorithm with I = {s} and J = V (G) implies a (1+ǫ)-approximation algorithm for this, guaranteeingÕ(n 1.823 /ǫ 2 ) worst-case update time for directed graphs with positive real weights in [1, W ]. 2 With ideas from [Roditty, V. Williams, STOC'13], we also obtain the first subquadratic worst-case update time for (5/3 + ǫ)-approximating the eccentricities and (1.5 + ǫ)-approximating the diameter and radius for unweighted graphs (with small additive errors). We also obtain the first subquadratic worst-case update time for (1 + ǫ)-approximating the closeness centralities for undirected unweighted graphs.Worst-case update time for APSP: When we want to maintain distances between all-pairs of nodes explicitly, theÕ(n 2 ) amortized update time by Demetrescu and Italiano [STOC'03] already matches the trivial Ω(n 2 ) lower bound. A fundamental question is whether it can be made worst-case. The state-of-the-art algorithm takesÕ(n 2+2/3 ) worst-case update time to maintain the distances exactly [Abraham, Chechik, Krinninger, SODA'17; Thorup STOC'05]. When it comes to (1 + ǫ) approximation, this bound is still higher than calling theÕ(n ω /ǫ)-time static algorithm of Zwick [FOCS'98], where ω ≈ 2.373. Our algorithm with I = J = V (G) implies nearly tight bounds for this, namelyÕ(n 2 /ǫ 1+ω ) for undirected unweighted graphs and O(n 2.045 /ǫ 2 ) for directed graphs with positive real weights. Besides this, we also obtain the first dynamic APSP algorithm with subquadratic update time and sublinear query time.

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Dynamic Matrix Inverse: Improved Algorithms and Matching Conditional Lower Bounds

Cited by 53 publications

References 36 publications

A Deterministic Linear Program Solver in Current Matrix Multiplication Time

A Deterministic Linear Program Solver in Current Matrix Multiplication Time

Sensitive Distance and Reachability Oracles for Large Batch Updates

Dynamic Approximate Shortest Paths and Beyond: Subquadratic and Worst-Case Update Time

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