The deterministic space complexity of approximating the length of the longest increasing subsequence of a stream of N integers is known to be Theta~(sqrt N). However, the randomized complexity is wide open. We show that the technique used in earlier work to establish the Omega(sqrt N) deterministic lower bound fails strongly under randomization: specifically, we show that the communication problems on which the lower bound is based have very efficient randomized protocols. The purpose of this note is to guide and alert future researchers working on this very interesting problem

Chakrabarti, Amit

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Dartmouth College Dartmouth Digital Commons Computer Science Technical Reports Computer Science 5-1-2010 A Note on Randomized Streaming Space Bounds for the Longest Increasing Subsequence Problem Amit Chakrabarti Dartmouth College Follow this and additional works at: https://digitalcommons.dartmouth.edu/cs_tr  Part of the Computer Sciences Commons Dartmouth Digital Commons Citation Chakrabarti, Amit, "A Note on Randomized Streaming Space Bounds for the Longest Increasing Subsequence Problem" (2010). Computer Science Technical Report TR2010-667. https://digitalcommons.dartmouth.edu/cs_tr/337 This Technical Report is brought to you for free and open access by the Computer Science at Dartmouth Digital Commons. It has been accepted for inclusion in Computer Science Technical Reports by an authorized administrator of Dartmouth Digital Commons. For more information, please contact dartmouthdigitalcommons@groups.dartmouth.edu. A Note on Randomized Streaming Space Bounds for theLongest Increasing Subsequence ProblemAmit Chakrabarti∗AbstractThe deterministic space complexity of approximating the length of the longest increasing subse-quence of a stream of N integers is known to be Θ˜(√N). However, the randomized complexity is wideopen. We show that the technique used in earlier work to establish theΩ(√N) deterministic lower boundfails strongly under randomization: specifically, we show that the communication problems on which thelower bound is based have very efficient randomized protocols. The purpose of this note is to guide andalert future researchers working on this very interesting problem.1 IntroductionFor a sequence σ of integers, let lis(σ) denote the length of the longest (strictly) increasing subsequence ofσ . In the APPROXIMATE-LIS problem (abbreviated ALISMN,ε ), we are given streaming access to a sequenceσ of length N, with entries in [M] := {1,2, . . . ,M}, and must report a (1± ε)-approximation to lis(σ).The goal, as is usual in data stream algorithms [Mut03], is to minimize the space (i.e., amount of workingmemory) and the per-item processing time used to do so. We are concerned here primarily with the spacecomplexity of this problem; both deterministic and randomized algorithms are of interest.The vast majority of sublinear space streaming algorithms use randomization as a key technique and, infact, provably need to do so [AMS99]. The ALIS problem presents one of the very few instances where (1) anatural problem has a sublinear space deterministic streaming algorithm, (2) randomization is not known toprovide any extra space savings, and (3) randomization could provide an exponential improvement, basedon current knowledge. We believe that this makes the randomized space complexity of ALIS an extremelyinteresting theoretical question about data stream algorithms.A sublinear upper bound for ALIS was given by Gopalan et al. [GJKK07], who showed the following.Theorem 1.1. There is a deterministic O(√N/ε · logM)-space streaming algorithm for ALISMN,ε .An essentially matching lower bound was then given by Ga´l and Gopalan [GG07] and also — inde-pendently and via a different argument — by Ergu¨n and Jowhari [EJ08]. These lower bounds applied onlyto deterministic algorithms and used reductions from certain communication problems on which suitable“direct sum” theorems could be proven. In this note, we show that these techniques do not generalize to giverandomized streaming lower bounds, because the underlying communication problems do have randomizedprotocols with cost exponentially lower than the best deterministic protocol.2 PreliminariesFor a communication problem f , let Dmax( f ) denote the maximum number of bits sent by any single playerin a deterministic protocol that computes f , minimized over all such protocols. Let Rmax( f ) denote theanalogous quantity for constant-error randomized protocols.∗Dartmouth College. ac@cs.dartmouth.edu. Supported in part by NSF Grants CCF-0448277 and IIS-0916565.1In the HIDDEN-INCREASING-SEQUENCE problem (abbreviated HISmt,n,k), the input is a t×n matrix X ={xi j}i∈[t], j∈[n] with entries in [m], with the promise that each column X j of X satisfies one of the following:1. The column is non-increasing, i.e., lis(X j) = 1.2. The column has a “long” increasing subsequence, i.e., lis(X j)≥ k.This input is divided amongst t players, named PLR1, . . . , PLRt , with player i receiving the ith row of X . Thegoal is to compute the predicate HISmt,n,k(X) :=∨nj=1(lis(X j)≥ k), in the following model of communication:PLR1 sends a (private) message to PLR2, who then sends a message to PLR3 and so on, until we reach PLRt ,who then announces the output.Let σX denote the length-(tn) sequence, with elements in [tm], obtained by applying the mapping xi j 7→( j−1)m+ xi j to the entries of X and reading off the result in row-major order. It is easy to see thatHISmt,n,k(X) = 0 =⇒ lis(σX) = n , and HISmt,n,k(X) = 1 =⇒ lis(σX)≥ n+ k−1 . (1)This was first formally observed by Gopalan et al. [GJKK07], and it immediately implies the following.Lemma 2.1 (Lemma 4.4 of [GJKK07]). The deterministic and randomized streaming space complexities ofALISMN,ε , with M = tm, N = tn and ε = (k−1)/n, are at least Dmax(HISmt,n,k) and Rmax(HISmt,n,k), respectively.Thus, a natural approach to establishing space lower bounds on ALIS is to prove communication complexitylower bounds for HIS. Using this approach, Ga´l and Gopalan proved a tight determinstic lower bound:Theorem 2.2 (Theorems 1.1 and 4.1 of [GG07]). Let Hb denote the binary entropy function. Then, we haveDmax(HISmt,n,k) ≥ n((1− kt)log(mk−1)−Hb(kt))− log t .In particular, setting k− 1 = t/2 = εn, we have Dmax(HISmt,n,k) = Ω(n log(m/εn)). Combining this boundwith Lemma 2.1 shows that the deterministic space complexity of ALISMN,ε is Ω(√N/ε · log(M/εN)).In fact, they also generalized this theorem to apply to multi-pass, but still deterministic, streaming al-gorithms by extending the communication lower bound to multi-round protocols. Our concern here is witha different potential generalization: can one generalize Theorem 2.2 to randomized protocols, and thus,randomized streaming algorithms? Our main result is a negative one, showing that this is not possible.3 A Randomized Communication Upper BoundTheorem 3.1 (Main Theorem). We haveRmax(HISmt,n,k) = O(nt logmk2).In particular, for the setting k = Θ(t) = Θ(εn), which was used for the lower bound in Theorem 2.2, wehave Rmax(HISmt,n,k) = O(ε−1 logm).Proof. Let r = 2(t−1)/(k−1). Consider the following protocol. Each player goes through a receive-and-compute phase (skipped by PLR1) followed by a transmit phase (skipped by PLRt). In the transmit phase,PLRi chooses a subset Ji⊆ [n] of size |Ji|= d2n/(k−1)e uniformly at random from amongst all such subsets.He then sends to PLRi+1 the following data.1. The set Si = {xi j : j ∈ Ji}; for each j ∈ Ji, we say that PLRi samples column j.2. The sets Sh for max{1, i− r+1} ≤ h< i, which he obtains from PLRi−1.2In the receive-and-compute phase (which precedes the transmit phase), PLRi receives from PLRi−1 thesets Sh, for all h ∈ Hi := {h : max{1, i− r} ≤ h< i}. He checks whether the following condition holds:∃h ∈ Hi ∃ j ∈ Jh : xh j < xi j . (2)He can do so because xh j is available to him from the message he receives and xi j is within the part ofthe input he knows to begin with. If (2) holds, he terminates the protocol and outputs 1. Notice that he iscorrect to do so, because he has discovered that lis(X j)> 1, which, by the promise, means that lis(X j)≥ k.Otherwise, if (2) does not hold, he moves on to the transmit phase as described above.If the protocol reaches PLRt , and no player (including himself) has output 1 in their receive-and-computephase, then he outputs 0. This completes the description of the protocol. Clearly, one can tweak it so thatthe output is always announced by PLRt , as in our definition.We now argue that this protocol is correct. Suppose that HISmt,n,k(X) = 0. Then (2) never holds, and theprotocol always reaches PLRt , who correctly outputs 0.Suppose that HISmt,n,k(X) = 1, and suppose that the jth column of X contains a “hidden increasing sub-sequence” 〈xi j〉i∈I , where I ⊆ [t] and |I|= k. Call the player PLRi critical if the following condition holds:i ∈ I ∧ (∃ i′ ∈ I : 0 < i′− i≤ r) . (3)Also, in this case, call PLRi′ the follower of PLRi, where i′ is the minimum integer such that 0 < i′− i≤ r.A straightforward estimation argument shows that there exist at least (k−1)/2 critical players. Noticethat if a critical player samples column j, then his follower correctly announces the output to be 1. Thus,the probability that the protocol fails to output 1 is at mostPr[no critical player samples column j] ≤(1− 2k−1)(k−1)/2≤ e−1 .Finally, note that in order to achieve this constant error probability, each player had to send at mostr · d2n/(k−1)e entries of X , which required O((nr/k) · logm) = O((nt/k2) · logm) bits.4 Concluding RemarksWe remark that a similar randomized communication upper bound can be shown to hold for the slightly dif-ferent communication problem used by Ergu¨n and Jowhari [EJ08] in their alternate proof of the deterministiclower bound for ALIS.These protocols raise an obvious question: does the idea extend to give a polylogarithmic-space stream-ing upper bound for ALIS? We leave this question open.References[AMS99] Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequencymoments. J. Comput. Syst. Sci., 58(1):137–147, 1999. Preliminary version in Proc. 28th Annu. ACMSymp. Theory Comput., pages 20–29, 1996.[EJ08] Funda Ergu¨n and Hossein Jowhari. On distance to monotonicity and longest increasing subsequence of adata stream. In Proc. 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 730–736, 2008.[GG07] Anna Ga´l and Parikshit Gopalan. Lower bounds on streaming algorithms for approximating the length ofthe longest increasing subsequence. In Proc. 48th Annual IEEE Symposium on Foundations of ComputerScience, pages 294–304, 2007.[GJKK07] Parikshit Gopalan, T. S. Jayram, Robert Krauthgamer, and Ravi Kumar. Estimating the sortedness of adata stream. In Proc. 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 318–327, 2007.[Mut03] S. Muthukrishnan. Data streams: Algorithms and applications. In Proc. 14th Annual ACM-SIAM Sympo-sium on Discrete Algorithms, page 413, 2003.3

A Note on Randomized Streaming Space Bounds for the Longest Increasing Subsequence Problem

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A Note on Randomized Streaming Space Bounds for the Longest Increasing Subsequence Problem

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