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1. Batched Predecessor and Sorting with Size-Priced Information in External Memory

   Bender, Michael ; Goswami, Mayank ; Medjedovic, Dzejla ; Montes, Pablo ; Tsichlas, Kostas ( January 2021 , Latin American Theoretical Informatics Symposium)

   In the unit-cost comparison model, a black box takes an input two items and outputs the result of the comparison. Problems like sorting and searching have been studied in this model, and it has been general- ized to include the concept of priced information, where different pairs of items (say database records) have different comparison costs. These comparison costs can be arbitrary (in which case no algorithm can be close to optimal (Charikar et al. STOC 2000)), structured (for exam- ple, the comparison cost may depend on the length of the databases (Gupta et al. FOCS 2001)), or stochastic (Angelov et al. LATIN 2008). Motivated by the database setting where the cost depends on the sizes of the items, we consider the problems of sorting and batched predecessor where two non-uniform sets of items A and B are given as input. (1) In the RAM setting, we consider the scenario where both sets have n keys each. The cost to compare two items in A is a, to compare an item of A to an item of B is b, and to compare two items in B is c. We give upper and lower bounds for the case a ≤more »b ≤ c, the case that serves as a warmup for the generalization to the external-memory model. Notice that the case b = 1,a = c = ∞ is the famous “nuts and bolts” problem. ) In the Disk-Access Model (DAM), where transferring elements between disk and internal memory is the main bottleneck, we con- sider the scenario where elements in B are larger than elements in A. The larger items take more I/Os to be brought into memory, consume more space in internal memory, and are required in their entirety for comparisons. A key observation is that the complexity of sorting depends heavily on the interleaving of the small and large items in the final sorted order. If all large elements come after all small elements in the final sorted order, sorting each type separately and concatenating is optimal. However, if the set of predecessors of B in A has size k ≪ n, one must solve an associated batched predecessor problem in order to achieve optimality. We first give output-sensitive lower and upper bounds on the batched predecessor problem, and use these to derive bounds on the complexity of sorting in the two models. Our bounds are tight in most cases, and require novel generalizations of the classical lower bound techniques in external memory to accommodate the non-uniformity of keys.« less
2. Batched Predecessor and Sorting with Size-Priced Information in External Memory

   https://doi.org/10.1007/978-3-030-61792-9_13  

   Bender, Michael A ; Goswami, Mayank ; Medjedovic, Dzejla ; Montes, Pablo ; Tsichlas, Kostas ( January 2021 , Latin American Symposium on Theoretical Informatics)

   The fundamental problems of sorting and searching, traditionally studied in the unit-cost comparison model, have been generalized to include priced information, where different pairs of items have different comparison costs. These costs can be arbitrary (Charikar et al. STOC 2000), structured (Gupta et al. FOCS 2001), or stochastic (Angelov et al. LATIN 2008). Motivated by the database setting where the comparison cost depends on the sizes of the records, we consider the problems of sorting and batched predecessor where two non-uniform sets of items A and B are given as input. In the RAM model, pairwise comparisons (A-A, A-B and B-B) have respective comparison costs a, b and c. We give upper and lower bounds for the case a<= b <= c, which serves as a warmup for the generalization to the external-memory model. In the Disk-Access Model (DAM), where transferring elements between disk and RAM is the main bottleneck, we consider the scenario where elements in B are larger than elements in A. All items are required in their entirety for comparisons in RAM. A key observation is that the complexity of sorting depends on the interleaving of the small and large items in the final sorted order, andmore »with a high degree of interleaving, the lower bound is dominated by an associated batched predecessor problem. We give output-sensitive bounds on the batched predecessor and sorting; our bounds are tight in most cases. Our lower bounds require novel generalizations of lower bound techniques in external memory to accommodate non-uniform keys.« less
3. Streaming Verification for Graph Problems: Optimal Tradeoffs and Nonlinear Sketches

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.22  

   Chakrabarti, Amit ; Ghosh, Prantar ; Thaler, Justin ( August 2020 , Leibniz international proceedings in informatics)

   We study graph computations in an enhanced data streaming setting, where a space-bounded client reading the edge stream of a massive graph may delegate some of its work to a cloud service. We seek algorithms that allow the client to verify a purported proof sent by the cloud service that the work done in the cloud is correct. A line of work starting with Chakrabarti et al. (ICALP 2009) has provided such algorithms, which we call schemes, for several statistical and graph-theoretic problems, many of which exhibit a tradeoff between the length of the proof and the space used by the streaming verifier. This work designs new schemes for a number of basic graph problems - including triangle counting, maximum matching, topological sorting, and single-source shortest paths - where past work had either failed to obtain smooth tradeoffs between these two key complexity measures or only obtained suboptimal tradeoffs. Our key innovation is having the verifier compute certain nonlinear sketches of the input stream, leading to either new or improved tradeoffs. In many cases, our schemes, in fact, provide optimal tradeoffs up to logarithmic factors. Specifically, for most graph problems that we study, it is known that the product ofmore »the verifier’s space cost v and the proof length h must be at least Omega(n^2) for n-vertex graphs. However, matching upper bounds are only known for a handful of settings of h and v on the curve h*v = ~Theta(n^2). For example, for counting triangles and maximum matching, schemes with costs lying on this curve are only known for (h = ~O(n²), v = ~O(1)), (h = ~O(n), v = ~O(n)), and the trivial (h = ~O(1), v = ~O(n²)). A major message of this work is that by exploiting nonlinear sketches, a significant "portion" of costs on the tradeoff curve h*v=n^2 can be achieved.« less
4. Data-Independent Memory Hard Functions: New Attacks and Stronger Constructions

   https://doi.org/10.1007/978-3-030-26951-7_20  

   Blocki, J ; Harsha, B ; Kang, S ; Lee, S ; Xing, L ; Zhou, S. ( August 2019 , Crypto)

   Memory-hard functions (MHFs) are a key cryptographic primitive underlying the design of moderately expensive password hashing algorithms and egalitarian proofs of work. Over the past few years several increasingly stringent goals for an MHF have been proposed including the requirement that the MHF have high sequential space-time (ST) complexity, parallel space-time complexity, amortized area-time (aAT) complexity and sustained space complexity. Data-Independent Memory Hard Functions (iMHFs) are of special interest in the context of password hashing as they naturally resist side-channel attacks. iMHFs can be specified using a directed acyclic graph (DAG) $G$ with $N=2^n$ nodes and low indegree and the complexity of the iMHF can be analyzed using a pebbling game. Recently, Alwen et al. [CCS'17] constructed an DAG called DRSample which has aAT complexity at least $\Omega\left( N^2/\log N\right)$. Asymptotically DRSample outperformed all prior iMHF constructions including Argon2i, winner of the password hashing competition (aAT cost $\mathcal{O}\left(N^{1.767}\right)$), though the constants in these bounds are poorly understood. We show that the the greedy pebbling strategy of Boneh et al. [ASIACRYPT'16] is particularly effective against DRSample e.g., the aAT cost is $\mathcal{O}\left( N^2/\log N\right)$. In fact, our empirical analysis {\em reverses} the prior conclusion of Alwen et al. that DRSample providesmore »stronger resistance to known pebbling attacks for practical values of $N \leq 2^{24}$. We construct a new iMHF candidate (DRSample+BRG) by using the bit-reversal graph to extend DRSample. We then prove that the construction is asymptotically optimal under every MHF criteria, and we empirically demonstrate that our iMHF provides the best resistance to {\em known} pebbling attacks. For example, we show that any parallel pebbling attack either has aAT cost $\omega(N^2)$ or requires at least $\Omega(N)$ steps with $\Omega(N/\log N)$ pebbles on the DAG. This makes our construction the first practical iMHF with a strong sustained space-complexity guarantee and immediately implies that any parallel pebbling has aAT complexity $\Omega(N^2/\log N)$. We also prove that any sequential pebbling (including the greedy pebbling attack) has aAT cost $\Omega\left( N^2\right)$ and, if a plausible conjecture holds, any parallel pebbling has aAT cost $\Omega(N^2 \log \log N/\log N)$ --- the best possible bound for an iMHF. We implement our new iMHF and demonstrate that it is just as fast as Argon2. Along the way we propose a simple modification to the Argon2 round function which increases an attacker's aAT cost by nearly an order of magnitude without increasing running time on a CPU. Finally, we give a pebbling reduction which proves that in the parallel random oracle model (PROM) the cost of evaluating an iMHF like Argon2i or DRSample+BRG is given by the pebbling cost of the underlying DAG. Prior pebbling reductions assumed that the iMHF round function concatenates input labels before hashing and did not apply to practical iMHFs such as Argon2i, DRSample or DRSample+BRG where input labels are instead XORed together.« less
5. Parallel Algorithms for Asymmetric Read-Write Costs

   https://doi.org/10.1145/2935764.2935767  

   Ben-David, Naama ; Blelloch, Guy E ; Fineman, Jeremy T ; Gibbons, Phillip B ; Gu, Yan ; McGuffey, Charles ; Shun, Julian ( July 2016 , 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA))

   Motivated by the significantly higher cost of writing than reading in emerging memory technologies, we consider parallel algorithm design under such asymmetric read-write costs, with the goal of reducing the number of writes while preserving work-efficiency and low span. We present a nested-parallel model of computation that combines (i) small per-task stack-allocated memories with symmetric read-write costs and (ii) an unbounded heap-allocated shared memory with asymmetric read-write costs, and show how the costs in the model map efficiently onto a more concrete machine model under a work-stealing scheduler. We use the new model to design reduced write, work-efficient, low span parallel algorithms for a number of fundamental problems such as reduce, list contraction, tree contraction, breadth-first search, ordered filter, and planar convex hull. For the latter two problems, our algorithms are output-sensitive in that the work and number of writes decrease with the output size. We also present a reduced write, low span minimum spanning tree algorithm that is nearly work-efficient (off by the inverse Ackermann function). Our algorithms reveal several interesting techniques for significantly reducing shared memory writes in parallel algorithms without asymptotically increasing the number of shared memory reads.