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A Small Debate of Power of Randomness

Abner Huang

2009-10-16

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Power of Randomness

M. O. RabinThe citation for the Turing Award, awarded in 1976 jointly to Rabin and Dana Scott for a paper written in 1959,[1] states that the award was granted:

For their joint paper "Finite Automata and Their Decision Problem," which introduced the idea of nondeterministic machines, which has proved to be an enormously valuable concept. Their (Scott & Rabin) classic paper has been a continuous source of inspiration for subsequent work in this field

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Power of Randomness

• Quick Sort: O(n lg n)– Worst case : O(n^2)

• Randomized Quick Sort– Expected : O(n lg n)

• C. A. R. Hoare + M. Blum– Quick Sort + Median : Worst-case O(n lg n)

• Is randomness really helpful?

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P = BPP?

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The Story Was Originated at

……..

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Closest pair of points problem

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Closest pair of points problem

• Classical Result is O(n lg n) by divide-and-conquer.

• Rabin proposed an expected linear time Las Vegas randomized algorithm.– Rabin, M. O., Traub, J. .F. (ed.) Probabilistic algorithms In Algorithms

and Complexity, New Directions and Recent Trends, Academic Press, 1976, 21-39

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Nearest Neighbor Problem

hash

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An Example

Let’s sample s points from the data set.

And compute the minimum distance d among those s points.

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An Example (cont.)

Using the distance d, we will partition the plane into cells.

Thus each cell has few points

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An Example (cont.)

• If we are lucky, the adjacent cells of each cell will contain few points. Hence we can check it one by one.

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However, Rabin’s proof is

complicated!

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A Simpler Version

• Ref.: Khuller, S. & Matias, Y. A simple randomized sieve algorithm for the closest-pair problem, Information and Computation, 1995, 118, 34-37

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1. Pick a point uniformly at random.

2. Compute the min. distance d to all others.

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3. Partition the plane into cells by interval d/3.

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3. Find out the elements whose neighborhood contain more than itself.

4. If no such element, then we get the termination distance d. Otherwise we recursive call this procedure on these selected elements.

5. Use the termination distance d to build a mesh.

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Suitable interval d is critical.Key idea of Khuller’s algorithm: Approximate It.

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Four Facts

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Proof

Lemma

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Deterministic Algorithm

• Ref.: Fortune, S. & Hopcroft, J. E. A note on Rabin’s nearest-neighbor algorithm Information Processing Letters, 1979, 8, 20-23

• How powerful the randomness in Rabin’s algorithm is?– It breaks the lower

bound!!! John E. HopcroftTuring Award in 1986

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Key idea: re-hash S if there is one bucket which has more than sqrt(n) elements.

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An Example

Order of input sequence

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An element might appear in recursive calls at both two lines, e.g.,

Note that : the input sequence is not sorted

B1 B2 B3

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An Example of Worse Case

Order of input sequence

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• This case occurs when points fall into the same bucket but are processed by different levels of line 5-9, where a level is defined by one iteration of the loop (line 5-9).

• For a call, since there at most sqrt(n) levels, there are at most sqrt(n) points in a bucket at line 19.

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Its (ideal) recursive partition

Each partition have no more than sqrt(n) elements.

If all sub-problems are disjoint, then it is easy to show that this algorithm is in O(n lg (lg n))

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PROOF

• In a level, after calling FINDINT to process a non-empty bucket B with b elements, we will increase (b-1) non-empty sets after the call.

Let’s also define it a level.

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• If we apply this idea to recursive calls at a level and all levels, then we have

where k is the number of calls, and bi is the size of input elements of the i-th call.

• The total cost involved those recursive calls is bounded by

where d is a big constant.

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• Since the slop of x log(log x) is strictly increasing, we have the following property for r s>2.≧

• It means to get the upper bound, more large bi are better.

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Discuss

• Rabin’s assumption:– Random number is available.

– Square-Root is constant time operation.

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Discuss• Rabin’s assumption:

– Random number is available.

– Square-Root is constant time operation.

– Hashing function, i.e., floor is a constant time operation.• It is important! In Rabin’s original paper, it

only achieves O(n lg n) without hashing method.

• In Hopcroft’s paper, it costs O(n lglg n) deterministically.

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Q: Is Unit-Cost Floor Function Reasonable?

• Recall The Sorting Algorithm.– Well known lower bound: O(n lg n)

– What does it mean?

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Q: Is Unit-Cost Floor Function Reasonable?

• Recall The Sorting Algorithm.– Well known lower bound: O(n lg n)

– What does it mean?

– We can sort n numbers by c*n lg n computation steps?

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Q: Is Unit-Cost Floor Function Reasonable?

• Recall The Sorting Algorithm.– Well known lower bound: O(n lg n)– What does it mean?

– We can sort n numbers by c*n lg n computation steps?

– Is it reasonable, if the numbers are in the interval from 2^{n} to 2^{2^{2^{n}}}?

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Q: Is Unit-Cost Floor Function Reasonable?

• Recall The Sorting Algorithm.– Well known lower bound: O(n lg n)– What does it mean?

– We can sort n numbers by c*n lg n comparisons.

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Even x+y, x-y, xy, x/y are not in constant time.

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Unit-Cost Assumptions

Moore's Law

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Q: Is Unit-Cost Floor Function Reasonable in Theory?

• Rabin's randomized algorithm for closest pairs [1976].

• Schönhage 1979: If we could compute

x+y, x-y, xy, x/y, xfor any real x,y in a single step, then we could solve NP- and even PSPACE-complete problems in polynomial time

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Furthermore

• A few years later, Bertoni et al. [bms-scram-85] generalized the same approach to the #P-complete problem #SAT: Howmany satisfying assignments does this Boolean formula have?

• Peter van Emde Boas has a great discussion of "the unreasonable power of integermultiplication" in his survey of models of computation [e-mms-90].

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We now have a deterministic O (n lg lg n)

algorithm and anexpected O(n) algorithm.

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We now have a deterministic O (n lg n) algorithm

and anexpected O(n) algorithm.

Happy Ending?

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Curse of Dimensionality!!!

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A Simple Calculus

• Given: 1 million points in 100-dimension space.

– Clever algorithm– 2100 ~1030

– 1030 x ( 106 lg 106)

– Brute-force method– 106 x 106

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High Dimensional Geometry

How to beat the brute-force method?

Indyk, Piotr.; Motwani, Rajeev. (1998). ", Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality.". Proceedings of 30th Symposium on Theory of Computing.

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It is another story.

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THANKS FOR YOUR ATTENTION

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Ref.

• Rabin, M. O., Traub, J. .F. (ed.) Probabilistic algorithms In Algorithms and Complexity, New Directions and Recent Trends, Academic Press, 1976, 21-39

• Khuller, S. & Matias, Y. A simple randomized sieve algorithm for the closest-pair problem, Information and Computation, 1995, 118, 34-37

• Fortune, S. & Hopcroft, J. E. A note on Rabin’s nearest-neighbor algorithm Information Processing Letters, 1979, 8, 20-23

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Ref.

• Schönhage, A. On the power of random access machines Automata, Languages and Programming: Sixth Colloquium, Graz, Austria, July 16–20, 1979, 1979, 71, 520-529

• Bertoni, A.; Mauri, G. & Sabadini, N. Ausiello, G. & Lucertini, M. (ed.) Simulations Among Classes of Random Access Machines and Equivalence Among Numbers Succinctly Represented Analysis and Design of Algorithms for Combinatorial Problems, North-Holland, 1985, 109, 65 - 89

• van Emde Boas, P. van Leeuwen, J. (ed.) Machine models and simulation Handbook of Theoretical Computer Science, Elsevier, 1990, A, 1-66