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A TimeSpace Tradeoff for Sorting on NonOblivious Machines
, 1981
"... This paper adopts the latter strategy in order to pursue the complexity of sorting ..."
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This paper adopts the latter strategy in order to pursue the complexity of sorting
PEBBLE GAMES, PROOF COMPLEXITY AND TIMESPACE TRADEOFFS
, 2010
"... Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when compari ..."
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Cited by 10 (5 self)
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Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing sizespace tradeoffs. This is a survey of research in proof complexity drawing on results and tools from pebbling, with a focus on proof space lower bounds and tradeoffs between proof size and proof space.
SpaceTime Tradeoffs in Memory Hierarchies
, 1993
"... The speed of CPUs is accelerating rapidly, outstripping that of peripheral storage devices and making it increasingly difficult to keep CPUs busy. Multilevel memory hierarchies, scaled to simulate singlelevel memories, are increasing in importance. In this paper we introduce the Memory Hierarchy ..."
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The speed of CPUs is accelerating rapidly, outstripping that of peripheral storage devices and making it increasingly difficult to keep CPUs busy. Multilevel memory hierarchies, scaled to simulate singlelevel memories, are increasing in importance. In this paper we introduce the Memory Hierarchy Game, a multilevel pebble game simulating data movement in memory hierarchies for straightline computations. This game provides a framework for deriving upper and lower bounds on computation time and the I/O time at each level in a memory hierarchy. We apply this framework to a representative set of problems including matrix multiplication and the Fourier transform. We also discuss conditions on hierarchies under which they act as fast flat memories.
TimeSpace Tradeoffs for BacktoBack FFT Algorithms
 IEEE Trans. Computing C32
, 1983
"... [15] N. Wirth, "Modula: A language for modular multiprogramming," ..."
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[15] N. Wirth, &quot;Modula: A language for modular multiprogramming,&quot;
Catena: A MemoryConsuming Password Scrambler
"... Abstract. It is a common wisdom that servers should better store the oneway hash of their clients ’ passwords, rather than storing the password in the clear. This paper introduces Catena, a new oneway function for that purpose. Catena is memoryhard, which can hinder massively parallel attacks on ..."
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Abstract. It is a common wisdom that servers should better store the oneway hash of their clients ’ passwords, rather than storing the password in the clear. This paper introduces Catena, a new oneway function for that purpose. Catena is memoryhard, which can hinder massively parallel attacks on cheap memoryconstrained hardware, such as recent “graphical processing units”, GPUs. Furthermore, Catena has been designed to resist cachetiming attacks. This distinguishes Catena from scrypt, which may be sequentially memoryhard, but which we show to be vulnerable to cachetiming attacks. Additionally, Catena supports (1) clientindependent updates (the server can increase the security parameters and update the password hash without user interaction or knowing the password), (2) a server relief protocol (saving the server’s resources at the cost of the client), and (3) a variant CatenaKG for secure key derivation (to securely generate many cryptographic keys of arbitrary lengths such that compromising some keys does not help to break others).
On the Relative Strength of Pebbling and Resolution
, 2010
"... The last decade has seen a revival of interest in pebble games in the context of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing sizespace trade ..."
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The last decade has seen a revival of interest in pebble games in the context of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing sizespace tradeoffs. The typical approach has been to encode the pebble game played on a graph as a CNF formula and then argue that proofs of this formula must inherit (various aspects of) the pebbling properties of the underlying graph. Unfortunately, the reductions used here are not tight. To simulate resolution proofs by pebblings, the full strength of nondeterministic blackwhite pebbling is needed, whereas resolution is only known to be able to simulate deterministic black pebbling. To obtain strong results, one therefore needs to find specific graph families which either have essentially the same properties for black and blackwhite pebbling (not at all true in general) or which admit simulations of blackwhite pebblings in resolution. This paper contributes to both these approaches. First, we design a restricted form of blackwhite pebbling that can be simulated in resolution and show that there are graph families for which such restricted pebblings can be asymptotically better than black pebblings. This proves that, perhaps somewhat unexpectedly, resolution can strictly beat blackonly pebbling, and in particular that the space lower bounds on pebbling formulas in [BenSasson and Nordström 2008] are tight. Second, we present a versatile parametrized graph family with essentially the same properties for black and blackwhite pebbling, which gives sharp simultaneous tradeoffs for black and blackwhite pebbling for various parameter settings. Both of our contributions have been instrumental in obtaining the timespace tradeoff results for resolutionbased proof systems in [BenSasson and Nordström 2009].
High Parallel Complexity Graphs and MemoryHard Functions
"... Abstract. Motivated by growing importance of parallelism in modern computational systems, we introduce a very natural generalization to a parallel setting of the powerful (sequential) black pebbling game over DAGs. For this new variant, when considering pebbling graphs with with multiple disconnecte ..."
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Abstract. Motivated by growing importance of parallelism in modern computational systems, we introduce a very natural generalization to a parallel setting of the powerful (sequential) black pebbling game over DAGs. For this new variant, when considering pebbling graphs with with multiple disconnected components (say when modelling the computation of multiple functions in parallel), we demonstrate a significant shortcoming of the two most common types of complexity measures for DAGs inherited from the sequential setting (namely Scomplexity and STcomplexity). Thus, to ensure the applicability of the new pebbling game as a tool for proving results about say the amortized hardness of functions being repeatedly evaluated, we introduce a new complexity measure for DAGs called cumulative complexity (CC) and show how it overcomes this problem. With the aim of facilitating the new complexity lowerbounds in parallel settings we turn to the task of finding high CC graphs for the parallel pebbling game. First we look at several types of graphs such as certain stacks of superconcentrators, permutation graphs, bitreversal graphs and pyramid graphs, which are known to have high (even optimally so) complexity in the sequential setting. We show that all of them have much lower parallel CC then one could hope for from a graph of equal size. This motivates our first main technical result, namely the construction of a new family of constant indegree graphs whose parallel CC approaches maximality to within a polylogarithmic factor.
Hardness of Approximation in PSPACE and Separation Results for Pebble Games
"... We consider the pebble game on DAGs with bounded fanin introduced in [Paterson and Hewitt ’70] and the reversible version of this game in [Bennett ’89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove ..."
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We consider the pebble game on DAGs with bounded fanin introduced in [Paterson and Hewitt ’70] and the reversible version of this game in [Bennett ’89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove that the problem of deciding whether s pebbles suffice to reversibly pebble a DAG G is PSPACEcomplete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and Tarjan ’80]. Via two different graph product constructions we then strengthen these results to establish that both standard and reversible pebbling space are PSPACEhard to approximate to within any additive constant. To the best of our knowledge, these are the first hardness of approximation results for pebble games in an unrestricted setting (even for polynomial time). Also, since [Chan ’13] proved that reversible pebbling is equivalent to the games in [Dymond and Tompa ’85] and [Raz and McKenzie ’99], our results apply to the Dymond–Tompa and Raz–McKenzie games as well, and from the same paper it follows that resolution depth is PSPACEhard to determine up to any additive constant. We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space. This improves on the additive logarithmic separation previously known and could plausibly be tight, although we are not able to prove this. We leave as an interesting open problem whether our additive hardness of approximation result could be strengthened to a multiplicative bound if the computational resources are decreased from polynomial space to the more common setting of polynomial time. Keywords pebbling; reversible pebbling; DymondTompa game; RazMcKenzie game; PSPACEcomplete; separation; PSPACEhardness of approximation; resolution depth I.