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Randomized Algorithms

Pasi Fränti. Randomized Algorithms. 1.10.2014. Treasure island. Treasure worth 20.000 awaits. ?. Map for sale: 3000. 5000. 5000. 5000. ?. DAA expedition. To buy or not to buy. Buy the map:. 20000 – 5000 – 3000 = 12.000. Take a change:. 20000 – 5000 = 15.000.

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Randomized Algorithms

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  1. Pasi Fränti Randomized Algorithms 1.10.2014

  2. Treasure island Treasure worth 20.000 awaits ? Map for sale: 3000 5000 5000 5000 ? DAAexpedition

  3. To buy or not to buy Buy the map: 20000 – 5000 – 3000 = 12.000 Take a change: 20000 – 5000 = 15.000 20000 – 5000 – 5000 = 10.000

  4. To buy or not to buy Buy the map: 20000 – 5000 – 3000 = 12.000 Take a change: 20000 – 5000 = 15.000 20000 – 5000 – 5000 = 10.000 Expected result: 0.5 ∙ 15000 + 0.5 ∙ 10000 = 12.500

  5. Three type of randomization • Las Vegas • Output is always correct result • Result is not always found • Probability of success p • Monte Carlo • Result is always found • Result can be inaccurate (or even false!) • Probability of success p • Sherwood • Balancing the worst case behavior

  6. Las Vegas

  7. Eating philosophizes Who eats?

  8. Las Vegas Input: Binary vector A[1, n] Output: Index of any 1-bit from A LV(A, n) REPEAT k ← RAND(1, n); UNTIL A[k]=1; RETURN k Revise

  9. 8-Queens puzzle INPUT: Eight chess queens and an 8×8 chessboard OUTPUT: Setup where no queens attack each other

  10. 8-Queens brute force • Brute force • Try all positions • Mark illegal squares • Backtrack if dead-end • 114 setups in total • Random • Select positions randomly • If dead-end, start over • Randomized • Select k rows randomly • Rest rows by Brute Force Where next…? 8 5 4 …

  11. Pseudo code 8-Queens(k) FOR i=1 TO k DO // k Queens randomly r  Random[1,8]; IF Board[i,r]=TAKEN THEN RETURN Fail; ELSE ConquerSquare(i,r); FOR i=k+1 TO 8 DO // Rest by Brute Force r1; foundNO; WHILE (r≤8) AND (NOT found) DO IF Board[i,r] NOT TAKEN THEN ConquerSquare(i,r); foundYES; IF NOT found THEN RETURN Fail; ConquerSquare(i,j) Board[i,j]  QUEEN; FOR z=i+1 TO 8 DO Board[z,j]  TAKEN; Board[z,j-(z-i)]  TAKEN; Board[z,j+(z-i)]  TAKEN;

  12. Probability of success s = processing time in case of success e = processing time in case of failure p = probability of success q = 1-p = probability of failure Example: s=e=1, p=1/6  t=1+5/1∙1=6

  13. Experiments with varying k Fastest expected time

  14. Swap-based clustering

  15. Clustering by Random Swap P. Fränti and J. Kivijärvi, "Randomised local search algorithm for the clustering problem", Pattern Analysis and Applications, 3 (4), 358-369, 2000. • RandomSwap(X) → C, P • C ← SelectRandomRepresentatives(X); • P ← OptimalPartition(X, C); • REPEAT T times • (Cnew, j) ← RandomSwap(X, C); • Pnew← LocalRepartition(X, Cnew,P, j); • Cnew, Pnew ← Kmeans(X, Cnew, Pnew); • IF f(Cnew, Pnew) < f(C,P) THEN • (C, P) ← Cnew, Pnew; • RETURN (C, P); Select random neighbor Accept only if it improves

  16. Clustering by Random Swap 1. Random swap: 2. Re-partition vectors from old cluster: 3. Create new cluster:

  17. Choices for swap O(M) clusters to be removed  O(M) clusters where to add = O(M2) different choices in total

  18. Probability for successful Swap Select a proper centroid for removal: • M clusters in total: premoval=1/M. Select a proper new location: • N choices: padd=1/N • M of them significantly different: padd=1/M In total: • M2significantly different swaps. • Probability of each is pswap=1/M2 • Open question: how many of these are good • Theorem: α are good for add and removal.

  19. Clustering by Random Swap Probability of not finding good swap: Iterated T times Estimated number of iterations:

  20. Bounds for the iterations Upper limit: Lower limit similarly; resulting in:

  21. Total time complexity Time complexity of single step (t): Number of iterations needed (T): t = O(αN) Total time:

  22. Monte Carlo

  23. Monte Carlo Input: A bit vector A[1, n], iterations I Output: An index of any 1 bit from A LV(A, n, I) i ← 0; DO k ← RAND(1, n); i ← i + 1; WHILE (A[k]≠1 AND i ≤ I) RETURN k Revise

  24. Monte Carlo • Potential problems to be considered: • Detecting prime numbers • Calculating integral of a function To appear in 2014… maybe…

  25. Sherwood

  26. 1 1 1 N-1 N-2 Selection of pivot element Something about Quicksort and Selection: • Practical example of re-sorting • Median selection Add material for 2014 N-3 … O(N2)

  27. Simulated dynamic linked list • Sorted array • Search efficient: O(logN) • Insert and Delete slow: O(N) • Dynamically linked list • Insert and Delete fast: O(1) • Search inefficient: O(N)

  28. Simulated dynamic linked listExample Linked list: Head 7 1 2 4 5 15 21 Simulated by array: Head=4

  29. Simulated dynamic linked listDivide-and-conquer with randomization Value searched SEARCH (A, x) i :=A.HEAD; max :=A[i].VALUE; FOR k:=1 TO N DO j:=RANDOM(1, N); y:=A[j].VALUE; IF (max<y) AND (y≤x) THEN i:=j; max:=y; RETURN LinearSearch(A, x, i); N random breakpoints Biggest breakpoint ≤ x Full search from breakpoint i

  30. Analysis of the search max search for (on average) • Divide into N segments • Each segment has N/N = N elements • Linear search within one segment. • Expected time complexity = N + N = O(N)

  31. Experiment with students 1 2 3 4 99 100 Data (N=100) consists of numbers from 1..100: Select N breaking points:

  32. Searching for… 42

  33. Empty space for notes

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