Randomized Algorithm
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Introduction to Randomized algorithms

Introduction to Randomized algorithms

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Randomized Algorithm Randomized Algorithm Presentation Transcript

  • An IntroductionKanishka Khandelwal-BCSE IV , JU 3/20/2012
  • Flip a coinAn algorithm which flip coins is called a randomized algorithm. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  • Making decisions could be complicated. A randomized algorithm is simpler.Consider the minimum cut problemRandomized algorithm? Pick a random edge and contract. And Continue until two vertices are left Kanishka Khandelwal-BCSE IV , JU 3/20/2012 View slide
  • Making good decisions could be expensive. A randomized algorithm is faster.Consider a sorting procedure.Picking an element in the middle makes the procedure very efficient,but it is expensive (i.e. linear time) to find such an element. 5 9 13 11 8 6 7 10 5 6 7 8 9 13 11 10 Picking a random element will do. Kanishka Khandelwal-BCSE IV , JU 3/20/2012 View slide
  •  Avoid worst-case behavior: randomness can (probabilistically) guarantee average case behavior Efficient approximate solutions to intractable problems In many practical problems,we need to deal with HUGE input,and don’t even have time to read it once.But can we still do something useful? Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  • DeterministicInput Output Computer Random Bits www.lavarnd.org (doesn’t use lava lamps anymore) Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  Randomized algorithms make random rather than deterministic decisions. The main advantage is that no input can reliably produce worst-case results because the algorithm runs differently each time. These algorithms are commonly used in situations where no exact and fast algorithm is known. Behavior can vary even on a fixed input. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  Minimum spanning trees A linear time randomized algorithm, but no known linear time deterministic algorithm. Primality testing A randomized polynomial time algorithm, but it takes thirty years to find a deterministic one. Volume estimation of a convex body A randomized polynomial time approximation algorithm, but no known deterministic polynomial time approximation algorithm. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  • Monte Carlo Las Vegas Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  Always gives the true answer. Running time is random. Running time is variable whose expectation is bounded(say by a polynomail). E.g. Randomized QuickSort Algorithm Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  It may produce incorrect answer! We are able to bound its probability. By running it many times on independent random variables, we can make the failure probability arbitrarily small at the expense of running time. E.g. Randomized Mincut Algorithm Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  Suppose we want to find a number among n given numbers which is larger than or equal to the median. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  • Suppose A1 < … < An.We want Ai, such that i ≥ n/2.It’s obvious that the best deterministic algorithm needs O(n) time to produce the answer.n may be very large!Suppose n is 100,000,000,000 ! Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  Choose 100 of the numbers with equal probability. find the maximum among these numbers. Return the maximum. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  The running time of the given algorithm is O(1). The probability of Failure is 1/(2100). Consider that the algorithm may return a wrong answer but the probability is very smaller than the hardware failure or even an earthquake! Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  QUICKSORT Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  QuickSort is a simple and efficient approach to sorting: Select an element m from unsorted array c and divide the array into two subarrays: csmall - elements smaller than m and clarge - elements larger than m. Recursively sort the subarrays and combine them together in sorted array csorted Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  • 1. QuickSort(c)2. if c consists of a single element3. return c4. m  c15. Determine the set of elements csmall smaller than m6. Determine the set of elements clarge larger than m7. QuickSort(csmall)8. QuickSort(clarge)9. Combine csmall, m, and clarge into a single array, csorted10. return csorted Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  Runtime is based on our selection of m: -A good selection will split c evenly such that |csmall | = |clarge |, then the runtime is O(n log n). -For a good selection, the recurrence relation is: T(n) = 2T(n/2) + const ·n The time it takes to Time it takes to split the sort two smaller array into 2 parts where arrays of size n/2 const is a positive constant Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  • However, a poor selection will split c unevenly and in the worst case, all elements will be greater or less than m so that one subarray is full and the other is empty. In this case, the runtime is O(n2).For a poor selection, the recurrence relation is: T(n) = T(n-1) + const · nThe time it takes to sort Time it takes to split the arrayone array containing n-1 into 2 parts where const is aelements positive constant Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  QuickSort seems like an ineffecient MergeSort To improve QuickSort, we need to choose m to be a good ‘splitter.’ It can be proven that to achieve O(nlogn) running time, we don’t need a perfect split, just reasonably good one. In fact, if both subarrays are at least of size n/4, then running time will be O(n log n). This implies that half of the choices of m make good splitters. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  To improve QuickSort, randomly select m. Since half of the elements will be good splitters, if we choose m at random we will get a 50% chance that m will be a good choice. This approach will make sure that no matter what input is received, the expected running time is small. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  • 1. RandomizedQuickSort(c)2. if c consists of a single element3. return c4. Choose element m uniformly at random from c5. Determine the set of elements csmall smaller than m6. Determine the set of elements clarge larger than m7. RandomizedQuickSort(csmall)8. RandomizedQuickSort(clarge)9. Combine csmall, m, and clarge into a single array, csorted10. return csorted Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  Worst case runtime: O(m2) Expected runtime: O(m log m). Expected runtime is a good measure of the performance of randomized algorithms, often more informative than worst case runtimes. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  Making a random choice is fast. An adversary is powerless; randomized algorithms have no worst case inputs. Randomized algorithms are often simpler and faster than their deterministic counterparts. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  In the worst case, a randomized algorithm may be very slow. There is a finite probability of getting incorrect answer. However, the probability of getting a wrong answer can be made arbitrarily small by the repeated employment of randomness. Getting true random numbers is almost impossible. Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  • Assignments Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  Input: a set of 2D points Determine the closest pair (and its dist) Input points are stored in an array Suppose we have a strange storage data structure D : When we give a point to D, it stores the point and outputs the closest pair of points stored in D Our knowledge: Insertion time depends on whether the closest pair is changed or not. If output is the same: 1 clock tick If output is not the same: |D| clock ticks Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  •  With random insertion order, show that the expected total number of clock ticks used by D is O(n) Kanishka Khandelwal-BCSE IV , JU 3/20/2012
  • Suppose you are given a directed graph with n verticesand m unit-length edges. Consider the problem ofestimating the number of vertices within distance d ofeach vertex. Give a fully polynomial approximationscheme that solves this problem simultaneously for allvertices for any fixed d. Kanishka Khandelwal-BCSE IV , JU 3/20/2012