Best-order streaming model

We study a new model of computation, called best-order stream, for graph problems. Roughly, it is a proof system where a space-limited verifier has to verify a proof sequentially (i.e., it reads the proof as a stream). Moreover, the proof itself is just a specific ordering of the input data. This model is closely related to many models of computation in other areas such as data streams, communication complexity, and proof checking, and could be used in applications such as cloud computing. In this paper we focus on graph problems where the input is a sequence of edges. We show that even under this model, checking some basic graph properties deterministically requires linear space in the number of nodes. We also show that, in contrast with this, randomized verifiers are powerful enough to check many graph properties in polylogarithmic space.

[1]  Oded Goldreich Property testing in massive graphs , 2002 .

[2]  Sudipto Guha,et al.  Lower Bounds for Quantile Estimation in Random-Order and Multi-pass Streaming , 2007, ICALP.

[3]  Feifei Li,et al.  Proof-Infused Streams: Enabling Authentication of Sliding Window Queries On Streams , 2007, VLDB.

[4]  Joan Feigenbaum,et al.  On graph problems in a semi-streaming model , 2005, Theor. Comput. Sci..

[5]  Jaikumar Radhakrishnan,et al.  Finding duplicates in a data stream , 2009, SODA.

[6]  Prabhakar Raghavan,et al.  Computing on data streams , 1999, External Memory Algorithms.

[7]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[8]  T. S. Jayram,et al.  Tight lower bounds for selection in randomly ordered streams , 2008, SODA '08.

[9]  Tak Wah Lam,et al.  Results on Communication Complexity Classes , 1992, J. Comput. Syst. Sci..

[10]  Feifei Li,et al.  Randomized Synopses for Query Assurance on Data Streams , 2008, 2008 IEEE 24th International Conference on Data Engineering.

[11]  Yin Yang,et al.  CADS: Continuous Authentication on Data Streams , 2007, VLDB.

[12]  Richard J. Lipton,et al.  Efficient Checking of Computations , 1990, STACS.

[13]  Matthias Ruhl,et al.  Efficient algorithms for new computational models , 2003 .

[14]  Thomas Lengauer VLSI Theory , 1990, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[15]  Graham Cormode,et al.  Robust lower bounds for communication and stream computation , 2008, Theory Comput..

[16]  Michael Sipser,et al.  Introduction to the Theory of Computation , 1996, SIGA.

[17]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[18]  J. Ian Munro,et al.  Selection and sorting with limited storage , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[19]  Noga Alon,et al.  The Space Complexity of Approximating the Frequency Moments , 1999 .

[20]  Jan van Leeuwen,et al.  Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity , 1994 .

[21]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[22]  Wolfgang Maass,et al.  On the communication complexity of graph properties , 1988, STOC '88.

[23]  Avi Wigderson,et al.  The Randomized Communication Complexity of Set Disjointness , 2007, Theory Comput..

[24]  Yael Tauman Kalai,et al.  Delegating computation: interactive proofs for muggles , 2008, STOC.

[25]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[26]  Graham Cormode,et al.  Streaming Graph Computations with a Helpful Advisor , 2010, Algorithmica.

[27]  Mahesh Viswanathan,et al.  Testing and spot-checking of data streams (extended abstract) , 2000, ACM-SIAM Symposium on Discrete Algorithms.

[28]  Sudipto Guha,et al.  Approximate quantiles and the order of the stream , 2006, PODS '06.

[29]  Georg Schnitger,et al.  The communication complexity of several problems in matrix computation , 1991, J. Complex..

[30]  Mayur Datar,et al.  On the streaming model augmented with a sorting primitive , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[31]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume 4, Fascicle 2: Generating All Tuples and Permutations (Art of Computer Programming) , 2005 .

[32]  M. Aigner,et al.  Proofs from "The Book" , 2001 .

[33]  Camil Demetrescu,et al.  Trading off space for passes in graph streaming problems , 2006, SODA 2006.

[34]  Joan Feigenbaum,et al.  Graph distances in the streaming model: the value of space , 2005, SODA '05.

[35]  Graham Cormode,et al.  Annotations in Data Streams , 2009, ICALP.

[36]  Mahesh Viswanathan,et al.  Testing and Spot-Checking of Data Streams , 2000, SODA '00.