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Sorting and Selection with Imprecise Comparisons

Sorting and Selection with Imprecise Comparisons. Miklós Ajtai , Vitaly Feldman, Avinatan Hassidim, Jelani Nelson Presented by Dan Garber. Introduction. Sorting max-finding algorithms are based on performing comparisons between pairs of elements.

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Sorting and Selection with Imprecise Comparisons

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  1. Sorting and Selection with Imprecise Comparisons MiklósAjtai, Vitaly Feldman, Avinatan Hassidim, Jelani Nelson Presented by Dan Garber

  2. Introduction • Sorting \ max-finding algorithms are based on performing comparisons between pairs of elements. • Given two elements to compare, we assume that we can tell which is of greater value. • In some scenarios it is not always possible to assume the above because we don’t always know the “real” values of the elements.

  3. Introduction • Which car do we prefer to buy?

  4. Introduction • How to decide which sports team is champion? • Is team A better than team B? • Is team D better than team B? D A D A B C D

  5. Model • We are given a group of n elements; each of them is associated with an unknown “true” numerical value. • Given two elements: • Corollary: its not possible to find the exact maximum or the correct permutation.

  6. Model • The error of a max-finding algorithm A which outputs xis k if: • The error of a sorting algorithm A which outputs a permutation π is k if:

  7. Goal • Explore the tradeoff between the error of an algorithm and the number of comparisons. • An algorithm that performs all possible comparisons “knows everything” and can minimize the error. • Can we find algorithms that can achieve the same error bound with less comparisons?

  8. Motivation • Experimental Psychology & Sociology • Ranking of elements by human subjects. • Marketing Research • Information Retrieval • Training algorithms using human evaluators. • Designing Sports tournaments • Minimize error while reducing the number of games required.

  9. Examples • Max finding: • Sorting (bubble sort): max = array[0]; for (i=1; i < 10; i++) if (array[i] > max) max = array[i]; do swapped = false; for (i=0; i < 9; i++) if (array[i] > array[i+1]) { swap(array[i], array[i+1]); swapped = true; } while swapped

  10. Agenda • Lower error bounds • Maximum finding • Error 2 algorithm • Error k algorithm • Sorting • Sorting with error 2 • Selection with error k • Sorting with error k • Lower bounds

  11. Lower error bounds • Theorem 1. sorting according to the number of wins in a round-robin tournament yields error 2. • Proof. • Let x,y such that: val(y)+2 < val(x). • For any z: y defeats z  x defeats z. • x defeats y. • x has strictly more wins than y.

  12. Lower error bounds • Theorem 2. no deterministic max-finding algorithm has error less than 2. • Proof. • Assume three elements: a,b,c. • The comparator can claim: a>b>c>a. • w.l.o.g assume the algorithm outputs a as max. • The values of a,b,c could be 0,1,2.

  13. Max finding with error 2 • Algorithm A₂(s): • Label all elements as candidates. • while there are more than s candidate elements: • Pick an arbitrary set of s candidate elements and play them in a RR tournament. Let x have the most number of wins. • Compare x against all candidate elements and eliminate all elements that lose to x. • Play the final (at most s) candidate elements in a RR tournament and return the element with the most wins.

  14. Max finding with error 2 • Lemma 1.A₂ has error 2 and makes at most ns+(n^2)/s comparisons. With s=sqrt(n) we get at most 2n^(3/2) comparisons. • Proof. • If x* is never eliminated  x* participates in Step 3. Theorem 1 ensures the error. • If x* was eliminated, it was by an element x s.t. x*-x<=1 any element with value less then X*-2 was also eliminated in this iteration. • Comparisons bound: In each iteration at least (s-1)/2 elements are eliminated.

  15. Max finding with error k • k-max-set is a set of elements that contains an element x such that x*-x≤k. • Lemma 2. the following algorithm performs a RR tournament and outputs a 1-max-set of size at most log(n). • After performing RR, the algorithm greedily picks an element which defeats as many thus-far undefeated elements as possible.

  16. Max finding with error k • Algorithm1-Cover: • RunA₂(s) with s=sqrt(n)/8. • Return the union of the x that were chosen in any iteration of Step 2(a), in addition to the output of Lemma 2 on the elements in the final tournament in Step 3. • Lemma 3.1-Cover finds a 1-max-set of size at most sqrt(n)/4 using O(n^(3/2)) comparisons.

  17. Max finding with error k • Algorithm - Returns a k-max for k≥3 • return • Algorithm -Returns a (k-1)-max set of size for k ≥ 2 • if k=2 return • else • Equipartition the n elements into t(n,k) sets. • Recursively call on each set to recover (k-2)-max–set . • Return the output of 1-COVER with as input.

  18. Max finding with error k • k=5 • k=4 • k=3 • k=2

  19. Max finding with error k • Theorem 3.For every 3≤k ≤ loglogn , the algorithm finds a k-max element using comparisons. • Corollary. There exists a max-finding algorithm using O(n) comparisons with error loglogn.

  20. Sorting with error 2 • Lemma 5.In a RR tournament on n elements, the element with the median number of wins has at least (n-2)/4 wins and at least (n-2)/4 losses.

  21. Sorting with error 2 • Algorithm B₂: • Modify A₂ so that the x found in Step 2(a) is a pivot in the sense of Lemma 5. • Compare x against all elements and pivot into two sets. • Recursively sort each of the two sets. • Lemma 6.Algorithm B₂ sorts with error 2 and requires at most O(n^(3/2)) comparisons. • Error bound - trivial

  22. Sorting with error 2 • Analysis: • Every recursive call contains at least elements  at most iterations. • In each iteration at most comparisons to find a median. • Pivoting in each iteration: at most n comparisons. • Sorting the base step: at most comparisons.

  23. Selection with error k • Defenition 1.Element in a set of n elements is of k-ordreri if there exists a partition S₁, S₂ of [n] such that: • A k-median is an element of k-order floor(n/2).

  24. Selection with error k • Lemma 7. There exists a deterministic algorithm such that for any i in [n] and 2 ≤ k ≤ loglogn, the algorithm finds an element of k-order i in comparisons.

  25. Selection with error k • Algorithm - Returns an element of k-order i • If k≤3, sort using B₂ and return the element with index i. • Equipartition the elements into sets • Recursively call on each set to get a (k-2)-median • Play the in a RR tournament and let y be the element with the median number of wins. • Partition the elements according to y into X₁,X₂ • If | X₁| = i-1 return i. • Else if i≤| X₁| recursively find a k-order i in X₁. • Else recursively find a k-order (i- | X₁| -1) in X₂.

  26. Sorting with error k • Theorem 5. For any 2 ≤ k ≤ 2loglogn, there exists a deterministic sorting algorithm with error k using comparisons. • Algorithm: • Find an element x that is a k-median. • Equipartition the elements into sets S₁,S₂ such that every element in S₂ is k-greater than every element in S₁∪{x}. • Recursively sort each partition.

  27. Lower Bounds • Theorem 6.Every deterministic max-finding algorithm with error k requires comparisons. • We saw an algorithm with • Theorem 7.Every deterministic algorithm which k-sorts n elements requires comparisons. • We saw an algorithm with

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