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Randomized Algorithms for Selection and Sorting Prepared by John Reif, Ph.D. Analysis of Algorithms

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Randomized Algorithms for Selection and Sorting a)Randomized Sampling b)Selection by Randomized Sampling c)Sorting by Randomized Splitting: Quicksort and Multisample Sorts

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Readings Main Reading Selections: CLR, Chapters 9

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Comparison Problems Input set X of N distinct keys total ordering < over X Problems 1)For each key x X rank(x,X) = |{x’ X| x’ < x}| + 1 2)For each index i {1, …, N) select (i,X) = the key x X where i = rank(x,X) 3)Sort (X) = (x 1, x 2, …, x n ) where x i = select (i,X)

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Randomized Comparison Tree Model

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Algorithm Samplerank s (x,X) begin Let S be a random sample of X-{x} of size s output 1+ N/s [rank(x,S)-1] end

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Algorithm Samplerank s (x,X) (cont’d) Lemma 1 –The expected value of samplerank s (x,X) is rank (x,X)

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Algorithm Samplerank s (x,X) (cont’d)

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S is Random Sample of X of Size s

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More Precise Bounds on Randomized Sampling (cont’d)

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More Precise Bounds on Randomized Sampling Let S be a random sampling of X Let r i = rank(select(i,S),X)

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More Precise Bounds on Randomized Sampling (cont’d) Proof We can bound r i by a Beta distribution, implying Weak bounds follow from Chebychev inequality The Tighter bounds follow from Chernoff Bounds

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Subdivision by Random Sampling Let S be a random sample of X of size s Let k 1, k 2, …, k s be the elements of S in sorted order These elements subdivide X into

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Subdivision by Random Sampling (cont’d) How even are these subdivisions?

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Subdivision by Random Sampling (cont’d) Lemma 3 If random sample S in X is of size s and X is of size N, then S divides X into subsets each of

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Subdivision by Random Sampling (cont’d) Proof The number of (s+1) partitions of X is The number of partitions of X with one block of size v is

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Subdivision by Random Sampling (cont’d) So the probability of a random (s+1) partition having a block size v is

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Randomized Algorithms for Selection “canonical selection algorithm” Algorithm can select (i,X) input set X of N keys index i {1, …, N} [0] if N=1 then output X [1] select a bracket B of X, so that select (i,X) B with high prob. [2] Let i 1 be the number of keys less than any element of B [3] output can select (i-i 1, B) B found by random sampling

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Hoar’s Selection Algorithm Algorithm Hselect (i,X) where 1 i N begin if X = {x} then output x else choose a random splitter k X let B = {x X|x < k} if |B| i then output Hselect(i,B) else output Hselect(i-|B|, X-B) end

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Hoar’s Selection Algorithm (cont’d) Sequential time bound T(i,N) has mean

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Hoar’s Selection Algorithm (cont’d) Random splitter k X Hselect (i,X) has two cases

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Hoar’s Selection Algorithm (cont’d) Inefficient: each recursive call requires N comparisons, but only reduces average problem size to ½ N

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Improved Randomized Selection By Floyd and Rivest Algorithm FRselect(i,X) begin if X = {x} then output x else choose k 1, k 2 X such that k 1 < k 2 let r 1 = rank(k 1, X), r 2 = rank(k 2, X) if r 1 > i then FRselect(i, {x X|x < k 1 }) else if r 2 > i then FRselect(i-r 1, {x X|k 1 x k 2 }) else FRselect(i-r 2, {x X|x > k 2 }) end

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Choosing k 1, k 2 We must choose k 1, k 2 so that with high likelihood, k 1 select(i,X) k 2

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Choosing k 1, k 2 (cont’d) Choose random sample S X size s Define

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Choosing k 1, k 2 (cont’d) Lemma 2 implies

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Expected Time Bound

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Small Cost of Recursions Note With prob 1 - 2N - each recursive call costs only O(s) = o(N) rather than N in previous algorithm

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Randomized Sorting Algorithms “canonical sorting algorithm” Algorithm cansort(X) begin if x={x} then output X else choose a random sample S of X of size s Sort S S subdivides X into s+1 subsets X 1, X 2, …, X s+1 output cansort (X 1 ) cansort (X 2 )… cansort(X s+1 ) end

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Using Random Sampling to Aid Sorting Problem: –Must subdivide X into subsets of nearly equal size to minimize number of comparisons –Solution: random sampling!

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Hoar’s Randomized Sorting Algorithm Uses sample size s=1 Algorithm quicksort(X) begin if |X|=1 then output X else choose a random splitter k X output quicksort ({x X|x k}) end

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Expected Time Cost of Hoar’s Sort Inefficient: Better to divide problem size by ½ with high likelihood!

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Improved Sort using Better choice of splitter is k = sample select s (N/2, N) Algorithm samplesort s (X) begin if |X|=1 then output X choose a random subset S of X size s = N/log N k select (S/2,S) cost time o(N) output samplesort s ({x X|x k}) end

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Randomized Approximant of Mean By Lemma 2, rank(k,X) is very nearly the mean:

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Improved Expected Time Bounds Is optimal for comparison trees!

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Open Problems in Selection and Sorting 1)Improve Randomized Algorithms to exactly match lower bounds on number of comparisons 2)Can we de randomize these algorithms – i.e., give deterministic algorithms with the same bounds?

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Randomized Algorithms for Selection and Sorting Prepared by John Reif, Ph.D. Analysis of Algorithms

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