# randomized algorithms

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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. - PowerPoint PPT Presentation

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• Randomized Algorithms9.10.2015

• Treasure islandTreasure worth 20.000 awaits5000DAA expedition50005000??Map for sale: 3000

• To buy or not to buyBuy the map:Take a change:20000 5000 3000 = 12.00020000 5000 = 15.00020000 5000 5000 = 10.000

• To buy or not to buyBuy the map:Take a change:20000 5000 3000 = 12.00020000 5000 = 15.00020000 5000 5000 = 10.000Expected result:0.5 15000 + 0.5 10000 = 12.500

• Three type of randomizationLas VegasOutput is always correct resultResult is not always foundProbability of success pMonte CarloResult is always foundResult can be inaccurate (or even false!)Probability of success pSherwoodBalancing the worst case behavior

• Las Vegas

• Dining philosophersWho eats?

• Las Vegas Input:Bit-vector A[1,n]Output:Index of any 1-bit from A

LasVegas(A, n) indexREPEATk Random(1, n);UNTIL A[k]=1;RETURN k

6

• 8-Queens puzzleINPUT: Eight chess queens and an 88 chessboardOUTPUT: Setup where no queens attack each other

• 8-Queens brute forceBrute force Try all positions Mark illegal squares Backtrack if dead-end 114 setups in totalRandom Select positions randomly If dead-end, start overRandomized Select k rows randomly Rest rows by Brute Force864Where next?

• Pseudo code8-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 Forcer1; foundNO;WHILE (r8) AND (NOT found) DOIF Board[i,r] NOT TAKEN THEN ConquerSquare(i,r); foundYES;IF NOT found THEN RETURN Fail;

ConquerSquare(i,j)Board[i,j] QUEEN;FOR z=i+1 TO 8 DOBoard[z,j] TAKEN;Board[z,j-(z-i)] TAKEN;Board[z,j+(z-i)] TAKEN;

• Probability of successs = processing time in case of successe = processing time in case of failurep = probability of successq = 1-p = probability of failureExample:s=e=1, p=1/6t=1+5/11=6

• Experiments with varying kFastestexpectedtime

• Random SwapClustering

• Swap-based clustering

• Clustering by Random SwapRandomSwap(X) C, PC 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);P.Frnti and J.Kivijrvi, "Randomised local search algorithm for the clustering problem", Pattern Analysis and Applications, 3 (4), 358-369, 2000.Select random neighborAccept only if it improves

• 1. Random swap:

2. Re-partition vectors from old cluster:

3. Create new cluster:

Clustering by Random Swap

• Choices for swapO(M) clusters to be removedO(M) clusters where to addO(M2) different choices in total=

• Select a proper centroid for removal:M clusters in total: premoval=1/M.Select a proper new location:N choices: padd=1/NM of them significantly different: padd=1/MIn total:M2 significantly different swaps.Probability of each is pswap=1/M2Open question: how many of these are goodTheorem: are good for add and removal.

Probability for successful Swap

• Probability of not finding good swap:Estimated number of iterations:Clustering by Random SwapIterated T times

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

• Number of iterations needed (T):t = O(N)Total time:Time complexity of single step (t):Total time complexity

• Probability of success (p)depending on T

• Time-distortion performance

• Monte Carlo

• Input:Bit-vector A[1,n], max iterations zOutput:An index of any 1-bit in A.MonteCarlo(A, n, z) True/Falsei 0;REPEATk Random(1, n);i i+1;UNTIL (A[k]=1) OR (i>z);RETURN kMonte Carlo

• Monte CarloPotential problems to be considered: Detecting prime numbers Calculating integral of a functionTo appear in future

• Sherwood

• Used in QuicksortData is already sortedWorst can happens always if sorted dataN-3O(N2)Selection of pivot elementNave: first or last item

• Selection of pivot elementRandom item75%O(NlogN)Worst can still happensBut with probability (1/n)n1125%75%25%25%

• Simulated dynamic linked listSorted arraySearch efficient: O(logN)Insert and Delete slow: O(N)Dynamically linked listInsert and Delete fast: O(1)Search inefficient: O(N)