Upper Bounds on Number of Steals in Rooted Trees

Inspired by applications in parallel computing, we analyze the setting of work stealing in multithreaded computations. We obtain tight upper bounds on the number of steals when the computation can be modeled by rooted trees. In particular, we show that if the computation with n processors starts with one processor having a complete k-ary tree of height h (and the remaining n − 1 processors having nothing), the maximum possible number of steals is ∑i=1n(k−1)ihi${\sum }_{i=1}^{n}(k-1)^{i}\binom {h}{i}$.

[1]  Sylvain Bouveret,et al.  Characterizing conflicts in fair division of indivisible goods using a scale of criteria , 2016, Autonomous Agents and Multi-Agent Systems.

[2]  H. Varian.  Equity, Envy and Efficiency , 1974 .

[3]  Evangelos Markakis,et al.  Truthful Allocation Mechanisms Without Payments: Characterization and Implications on Fairness , 2017, EC.

[4]  Ariel D. Procaccia,et al.  Spliddit: unleashing fair division algorithms , 2015, SECO.

[5]  Robert D. Blumofe,et al.  Scheduling multithreaded computations by work stealing , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[6]  Ariel D. Procaccia,et al.  The Computational Rise and Fall of Fairness , 2014, AAAI.

[7]  Evangelos Markakis,et al.  On Truthful Mechanisms for Maximin Share Allocations , 2016, IJCAI.

[8]  Eric Budish,et al.  The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes , 2010, Journal of Political Economy.

[9]  A. C. Berry.  The accuracy of the Gaussian approximation to the sum of independent variates , 1941 .

[10]  Ronald L. Rivest,et al.  Introduction to Algorithms, third edition , 2009 .

[11]  D. Foley.  Resource allocation and the public sector , 1967 .

[12]  Warut Suksompong,et al.  Asymptotic existence of proportionally fair allocations , 2016, Math. Soc. Sci..

[13]  Ryan O'Donnell,et al.  Noise stability of functions with low influences: Invariance and optimality , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[14]  Evangelos Markakis,et al.  Approximation Algorithms for Computing Maximin Share Allocations , 2015, ICALP.

[15]  Warut Suksompong,et al.  Bounds on Multithreaded Computations by Work Stealing , 2014 .

[16]  Pasin Manurangsi,et al.  Computing an Approximately Optimal Agreeable Set of Items , 2017, IJCAI.

[17]  Robert H. Halstead,et al.  Implementation of multilisp: Lisp on a multiprocessor , 1984, LFP '84.

[18]  Sriram Krishnamoorthy,et al.  Scalable work stealing , 2009, Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis.

[19]  Ariel D. Procaccia,et al.  When Can the Maximin Share Guarantee Be Guaranteed? , 2016, AAAI.

[20]  Steven J. Brams,et al.  The undercut procedure: an algorithm for the envy-free division of indivisible items , 2009, Soc. Choice Welf..

[21]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[22]  Lenard Marlow,et al.  Is the Agreement Fair , 1990 .

[23]  W. Stromquist.  How to Cut a Cake Fairly , 1980 .

[24]  F. Warren Burton,et al.  Executing functional programs on a virtual tree of processors , 1981, FPCA '81.

[25]  Richard M. Karp,et al.  Randomized parallel algorithms for backtrack search and branch-and-bound computation , 1993, JACM.

[26]  Ronald L. Rivest,et al.  Introduction to Algorithms, 3rd Edition , 2009 .

[27]  N. Tomizawa,et al.  On some techniques useful for solution of transportation network problems , 1971, Networks.

[28]  Matthias G. Raith,et al.  Bidding for envy-freeness: A procedural approach to n-player fair-division problems , 2002, Soc. Choice Welf..

[29]  Yuxiong He,et al.  An empirical evaluation of work stealing with parallelism feedback , 2006, 26th IEEE International Conference on Distributed Computing Systems (ICDCS'06).

[30]  C. Greg Plaxton,et al.  Thread Scheduling for Multiprogrammed Multiprocessors , 1998, SPAA.

[31]  Siddharth Barman,et al.  Approximation Algorithms for Maximin Fair Division , 2017, EC.

[32]  Steven J. Brams,et al.  Two-Person Fair Division of Indivisible Items: An Efficient, Envy-Free Algorithm , 2013 .

[33]  M. Murdocca,et al.  $$L^p({\mathbb{R }}^n)$$Lp(Rn)-continuity of translation invariant anisotropic pseudodifferential operators: a necessary condition , 2012, 1210.0694.

[34]  Charles E. Leiserson,et al.  On the efficiency of localized work stealing , 2016, Inf. Process. Lett..

[35]  C. Greg Plaxton,et al.  Thread Scheduling for Multiprogrammed Multiprocessors , 1998, SPAA '98.

[36]  Elchanan Mossel,et al.  On approximately fair allocations of indivisible goods , 2004, EC '04.