In this paper we study the space requirement of algorithms that make only one (or a small number of) pass(es) over the input data. We study such algorithms under a model of data streams that we introduce here. We give a number of upper and lower bounds for problems stemming from queryprocessing, invoking in the process tools from the area of communication complexity.

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 single-level memories, are increasing in importance. In this paper we introduce the Memory Hierarchy Game, a multi-level pebble game simulating data movement in memory hierarchies for straight-line 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.

This paper introduces communicating branching programs, and develops a general technique for demonstrating communication-space tradeoffs for pairs of communicating branching programs. This technique is then used to prove communication-space tradeoffs for any pair of communicating branching programs that hashes according to a universal family of hash functions. Other tradeoffs follow from this result. As an example, any pair of communicating Boolean branching programs that computes matrix-vector products over GF(2) requires communication-space product Ω(n 2), provided the space used is o(n / log n). These are the first examples of communication-space tradeoffs on a completely general model of communicating processes.

In this paper, we examine the problem of incrementally evaluating algebraic functions. In particular, if f(x 1 ; x 2 ; \Delta \Delta \Delta ; xn ) = (y 1 ; y 2 ; \Delta \Delta \Delta ; ym ) is an algebraic problem, we consider answering on-line requests of the form "change input x i to value v" or "what is the value of output y j ?" We first present lower bounds for some simply stated algebraic problems: multipoint polynomial evaluation, polynomial reciprocal, and extended polynomial GCD, proving an \Omega\Gamma n) lower bound for the incremental evaluation of these functions. In addition, we prove two time-space trade-off theorems that apply to incremental algorithms for almost all algebraic functions. We then derive several general-purpose algorithm design techniques and apply them to several fundamental algebraic problems. For example, we give an O( p n) time per request algorithm for incremental DFT. We also present a design technique for serving incremental requests using a para...

A time-space tradeoff is established in the branching program model for the problem of computing the product of two n x n matrices over the semiring ((0, l}, V, A). It is a.ssumed that ea.ch element of each nxn input matrix is chosen independently to be 1 with probability n-ll2 and to be 0 with probability 1- n-1/2. Letting S and T denote expected space and time of a deterministic algorithm, the tradeoff is ST = R(n3.5) for T < cln2.5 and ST = R(n3) for T> where c1, c2> 0. The lower bounds are matched to within a logarithmic factor by upper bounds in the branching program model. Thus, the tradeoff possesses a sharp break a.t T = O(n2.5). These expected case lower bounds are also the best known lower bounds for the worst case.

Directed and undirected st-connectivity are important problems in computing. There are algorithms for the undirected case that use O (n) time and algorithms that use O (log n) space. The first result of this thesis proves that, in a very natural structured model, the JAG (Jumping Automata for Graphs), these upper bounds are not simultaneously achievable. This uses new entropy techniques to prove tight bounds on a game involving a helper and a player that models a computation having precomputed information about the input stored in its bounded space. The second result proves that a JAG requires a time-space tradeoff of T \Theta S 1 2 2\Omega i mn 1 2 j to compute directed st-connectivity. The third result proves a time-space tradeoff of T \Theta S 1 3 2\Omega i m 2 3 n 2 3 j on a version of the...

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 resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs. 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 trade-offs between proof size and proof space.

Abstract. We investigate methods for providing easy-to-check proofs of computational effort. Originally intended for discouraging spam, the concept has wide applicability as a method for controlling denial of service attacks. Dwork, Goldberg, and Naor proposed a specific memory-bound function for this purpose and proved an asymptotically tight amortized lower bound on the number of memory accesses any polynomial time bounded adversary must make. Their function requires a large random table which, crucially, cannot be compressed. We answer an open question of Dwork et al. by designing a compact representation for the table. The paradox, compressing an incompressible table, is resolved by embedding a time/space tradeoff into the process for constructing the table from its representation. 1

The following is a second proof of (basically) the same undirected st-connectivity result using recursive flyswatters as given in my thesis and in STOC-93 [Ed93a, Ed-PHD]. The input graph and the reduction techniques in the two proofs are similar. The main difference is that JAG result is reduced to a different game. In this paper, the game consists of a pebble walking on a line. The movements of the pebble are directed by a player and a random input. The conjecture is that the player cannot get the pebble across the line much faster than that done by a random walk. Likely, however, this is hard to prove. What can be proven is that this game becomes equivalent to the game in the original paper, if the player who is directing the pebble always knows where in the line pebble is. Therefore, the lower bound for the original game applies to this new game. Hence, the JAG lower bound proved in this paper is the same as that proven before. Two advantages of this new proof are that it is a litt...

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 resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs. 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 black-white 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 black-white pebbling (not at all true in general) or which admit simulations of black-white pebblings in resolution. This paper contributes to both these approaches. First, we design a restricted form of black-white 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 black-only pebbling, and in particular that the space lower bounds on pebbling formulas in [Ben-Sasson and Nordström 2008] are tight. Second, we present a versatile parametrized graph family with essentially the same properties for black and black-white pebbling, which gives sharp simultaneous trade-offs for black and black-white pebbling for various parameter settings. Both of our contributions have been instrumental in obtaining the time-space trade-off results for resolution-based proof systems in [Ben-Sasson and Nordström 2009]. 1