Improved Testing Al gorithms For Monotonicity By Range Reduction

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