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Improved Monotonicity Testing Algorithms via Range Reduction

This article presents a novel approach to monotonicity testing for functions defined on partially ordered sets. The main theorem focuses on an efficient tester with specific query and time complexity, utilizing the concept of range reduction. It explores the process of transforming a function into a monotone function while ensuring minimal changes. The work discusses key definitions, lemmas, and the proof of the theorem regarding range reduction, establishing a robust framework for validating the monotonicity of functions based on probabilistic methods.

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Improved Monotonicity Testing Algorithms via Range Reduction

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  1. Improved Testing Algorithms For Monotonicity By Range Reduction Presented By Daniel Sigalov

  2. Introduction • The main idea of the article is to prove that there exist a tester of monotonicity with query and time complexity

  3. The theorem of range reduction • Consider the task of checking monotonicity of functions defined over partially ordered set S. • Suppose that for some distribution on pairs • with and for every function • where C defends on S only. Then for every and every function for pairs selected according to the same distribution

  4. Basic definitions • For each 2 functions • - the fraction of instances • On which • - the minimum distance between function and any other monotone function • - the probability that a pair selected according to witnesses that is not monotone.

  5. Monotonicity How we do it? • Incrementally transform into a monotone function, while insuring that for each repaired violated edge, the value of the function changed only in a few points.

  6. Operators (1) MON(f) - arbitrary monotone function at distance from

  7. Operators(2) SQUASH

  8. Operators (3) CLEAR Claim: Proof: by the definition of CLEAR by the definition of MON

  9. More definitions.. • Interval of a violated edge with • respect to function - • two intervals cross if they intersect in • more than one point. [1,6] example: [2,3], [4,6] 0 1 2 3 4 0 1 2 3 4 5 6

  10. Lemma 1 - Clear • Lemma: The function has the following properties: • 1. • 2. has no violated edges whose intervals cross . • 3. The interval of a violated edge with respect to is contained in the interval of this edge with respect to .

  11. Proof of the Lemma • Define • Note: 1. is monotone and takes values from • 2. • 3. • We will check the 4 possibilities for : • - not possible. Why? • - agree on is violated by and . Proves (1) & (3). • If cross • Contradiction to the monotonicity of CLEAR definition

  12. Proof of the Lemma (cont.) 3. - is violated Therefore intersects in one point only - . This proves (2) In case (1) & (3) follows. If not then (1) & (3) follows. 4. - symmetric to case 3.

  13. Lemma 2 - Range reduction Defining the functions • Lemma: given define: • Those functions have the following properties: • 1. • 2. • 3.

  14. Proof of the Range reduction lemma (1) • The SQUASH operator never adds new violated edges

  15. Proof of the Range reduction lemma (2) Note:

  16. Proof of the Range reduction lemma (3) • 3. Note: Why? • the distance from to the set of monotone functions is at most the distance to a particular monotone function :

  17. Proof of The theorem of range reduction • We will prove by induction on • that for every function • the following hypothesis: • Base case : • In the theorem we assumed - • By the definition of detect we get the hypothesis.

  18. Proof of The theorem of range reduction (cont.) • Lets assume the hypothesis holds for and prove it for :

  19. Questions? Testing monotonicity

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