Converting high probability into nearly-constant time—with applications to parallel hashing

We present a new paradigm for efficient randomized parallel algorithms that needs ~(log’ n) time, where ~(x) means ‘O(Z) expected’. It leads to: (1) constructing a perfect hash function for n elements in ~(log” n log(log” n)) time and ~(n) operations; (2) an algorithm for generating a random per— mutation in O(log* n) time, using n processors or in ~(log” n Iog(log” n)) time and ~(n) operations; and (3) an efficient optimize~ consider a parallel algorithm that runs in t time using p processors; since at each time unit some of the processors may be idle, we let z, the total number of actual operations, be the sum over all non-idle processors at every time unit; assuming the algorithm belongs to a certain kind, it can be adapted to run in ~(t+log” n log(log” n)) time (additive cwerhead!) using z/(t + log* n log(log* n)) processors. We ak get an optimal integer sorting adgorithm. Givel~ ~ntegers from the domain [1, .n], it runs in O( log log n K ) time.

[1]  Yijie Han,et al.  Matching partition a linked list and its optimization , 1989, SPAA '89.

[2]  Gregory E. Shannon.  Optimal on-line load balancing , 1989, SPAA '89.

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  Sanguthevar Rajasekaran,et al.  Optimal and Sublogarithmic Time Randomized Parallel Sorting Algorithms , 1989, SIAM J. Comput..

[5]  Prabhakar Ragde,et al.  The Parallel Simplicity of Compaction and Chaining , 1990, J. Algorithms.

[6]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[7]  Friedhelm Meyer auf der Heide,et al.  A New Universal Class of Hash Functions and Dynamic Hashing in Real Time , 1990, ICALP.

[8]  Torben Hagerup.  Constant-time parallel integer sorting , 1991, STOC '91.

[9]  Joseph JáJá,et al.  Some triply-logarithmic parallel algorithms , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[10]  Richard J. Anderson.  Parallel algorithms for generating random permutations on a shared memory machine , 1990, SPAA '90.

[11]  Uzi Vishkin,et al.  On Parallel Hashing and Integer Sorting , 1991, J. Algorithms.

[12]  Torben Hagerup.  Fast Parallel Generation of Random Permutations , 1991, ICALP.

[13]  Friedhelm Meyer auf der Heide,et al.  An optimal parallel dictionary , 1989, SPAA '89.

[14]  János Komlós,et al.  Storing a sparse table with O(1) worst case access time , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[15]  Uzi Vishkin,et al.  Randomized speed-ups in parallel computation , 2015, STOC '84.

[16]  Clyde P. Kruskal,et al.  Searching, Merging, and Sorting in Parallel Computation , 1983, IEEE Transactions on Computers.

[17]  M. V. Wilkes,et al.  The Art of Computer Programming, Volume 3, Sorting and Searching , 1974 .

[18]  Friedhelm Meyer auf der Heide,et al.  Not all keys can be hashed in constant time , 1990, STOC '90.

[19]  P. Ragde.  The parallel simplicity of compaction and chaining , 1990 .

[20]  Larry Rudolph,et al.  Counting and Packing in Parallel , 1986, ICPP.

[21]  Joseph JáJá,et al.  An Introduction to Parallel Algorithms , 1992 .

[22]  John H. Reif,et al.  An optimal parallel algorithm for integer sorting , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[23]  Gary L. Miller,et al.  Parallel tree contraction and its application , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[24]  Uzi Vishkin,et al.  Recursive *-tree parallel data-structure , 1989, 30th Annual Symposium on Foundations of Computer Science.

[25]  Johan Håstad,et al.  Optimal bounds for decision problems on the CRCW PRAM , 1987, STOC.

[26]  Rajeev Raman,et al.  The Power of Collision: Randomized Parallel Algorithms for Chaining and Integer Sorting , 1990, FSTTCS.

[27]  Friedhelm Meyer auf der Heide,et al.  Dynamic perfect hashing: upper and lower bounds , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[28]  Torben Hagerup,et al.  Parallel Retrieval of Scattered Information , 1989, ICALP.

[29]  Richard Cole,et al.  Deterministic Coin Tossing with Applications to Optimal Parallel List Ranking , 2018, Inf. Control..

[30]  Richard Cole,et al.  Faster Optimal Parallel Prefix Sums and List Ranking , 1989, Inf. Comput..