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5.4 randomized datastructures

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Data Structure-randomized datastructures

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5.4 randomized datastructures

  1. 1. CSE 326 Randomized Data Structures David Kaplan Dept of Computer Science & Engineering Autumn 2001
  2. 2. Randomized Data Structures CSE 326 Autumn 2001 2 Motivation for Randomized Data Structures We’ve seen many data structures with good average case performance on random inputs, but bad behavior on particular inputs  e.g. Binary Search Trees  Instead of randomizing the input (since we cannot!), consider randomizing the data structure  No bad inputs, just unlucky random numbers  Expected case good behavior on any input
  3. 3. Randomized Data Structures CSE 326 Autumn 2001 3 Average vs. Expected Time Average (1/N) • ∑ xi Expectation ∑ Pr(xi) • xi Deterministic with good average time  If your application happens to always (or often) use the “bad” case, you are in big trouble! Randomized with good expected time  Once in a while you will have an expensive operation, but no inputs can make this happen all the time Like an insurance policy for your algorithm!
  4. 4. Randomized Data Structures CSE 326 Autumn 2001 4 Randomized Data Structures  Define a property (or subroutine) in an algorithm  Sample or randomly modify the property  Use altered property as if it were the true property Can transform average case runtimes into expected runtimes (remove input dependency). Sometimes allows substantial speedup in exchange for probabilistic unsoundness.
  5. 5. Randomized Data Structures CSE 326 Autumn 2001 5 Randomization in Action  Quicksort  Randomized data structures  Treaps  Randomized skip lists  Random number generation  Primality testing
  6. 6. Randomized Data Structures CSE 326 Autumn 2001 6 Treap Dictionary Data Structure Treap is a BST  binary tree property  search tree property Treap is also a heap  heap-order property  random priorities 15 12 10 30 9 15 7 8 4 18 6 7 2 9 priority key
  7. 7. Randomized Data Structures CSE 326 Autumn 2001 7 Treap Insert  Choose a random priority  Insert as in normal BST  Rotate up until heap order is restored 6 7 insert(15) 7 8 2 9 14 12 6 7 7 8 2 9 14 12 9 15 6 7 7 8 2 9 9 15 14 12
  8. 8. Randomized Data Structures CSE 326 Autumn 2001 8 Tree + Heap… Why Bother?  Insert data in sorted order into a treap … What shape tree comes out? 6 7 insert(7) 6 7 insert(8) 7 8 6 7 insert(9) 7 8 2 9 6 7 insert(12) 7 8 2 9 15 12
  9. 9. Treap Delete  Find the key  Increase its value to ∞  Rotate it to the fringe  Snip it off delete(9) 6 7 7 8 2⇒∞ 9 9 15 15 12 7 8 6 7 ∞ 9 9 15 15 12 7 8 6 7 9 15 ∞ 9 15 12 7 8 6 7 9 15 15 12 ∞ 9 7 8 6 7 9 15 15 12
  10. 10. Randomized Data Structures CSE 326 Autumn 2001 10 Treap Summary  Implements Dictionary ADT  insert in expected O(log n) time  delete in expected O(log n) time  find in expected O(log n) time  but worst case O(n)  Memory use  O(1) per node  about the cost of AVL trees  Very simple to implement  little overhead – less than AVL trees
  11. 11. Randomized Data Structures CSE 326 Autumn 2001 11 Perfect Binary Skip List  Sorted linked list  # of links of a node is its height  The height i link of each node (that has one) links to the next node of height i or greater 8 2 11 10 19 13 20 22 29 23
  12. 12. Randomized Data Structures CSE 326 Autumn 2001 12 Find in a Perfect Binary Skip List  Start i at the maximum height  Until the node is found or i is one and the next node is too large:  If the next node along the i link is less than the target, traverse to the next node  Otherwise, decrease i by one Runtime?
  13. 13. Randomized Data Structures CSE 326 Autumn 2001 13 Randomized Skip List Intuition It’s far too hard to insert into a perfect skip list But is perfection necessary? What matters in a skip list?  We want way fewer tall nodes than short ones  Make good progress through the list with each high traverse
  14. 14. Randomized Data Structures CSE 326 Autumn 2001 14 Randomized Skip List  Sorted linked list  # of links of a node is its height  The height i link of each node (that has one) links to the next node of height i or greater  There should be about 1/2 as many height i+1 nodes as height i nodes 2 19 23 8 13 29 20 10 22 11
  15. 15. Randomized Data Structures CSE 326 Autumn 2001 15 Find in a RSL  Start i at the maximum height  Until the node is found or i is one and the next node is too large:  If the next node along the i link is less than the target, traverse to the next node  Otherwise, decrease i by one Same as for a perfect skip list! Runtime?
  16. 16. Randomized Data Structures CSE 326 Autumn 2001 16 Insertion in a RSL  Flip a coin until it comes up heads; that takes i flips. Make the new node’s height i.  Do a find, remembering nodes where we go down  Put the node at the spot where the find ends  Point all the nodes where we went down (up to the new node’s height) at the new node  Point the new node’s links where those redirected pointers were pointing
  17. 17. RSL Insert Example 2 19 23 8 13 29 20 10 11 insert(22) with 3 flips 2 19 23 8 13 29 20 10 22 11 Runtime?
  18. 18. Randomized Data Structures CSE 326 Autumn 2001 18 Range Queries and Iteration Range query: search for everything that falls between two values  find start point  walk list at the bottom  output each node until the end point Iteration: successively return (in order) each element in the structure  start at beginning, walk list to end  Just like a linked list! Runtimes?
  19. 19. Randomized Data Structures CSE 326 Autumn 2001 19 Randomized Skip List Summary  Implements Dictionary ADT  insert in expected O(log n)  find in expected O(log n)  Memory use  expected constant memory per node  about double a linked list  Less overhead than search trees for iteration over range queries
  20. 20. Randomized Data Structures CSE 326 Autumn 2001 20 Random Number Generation Linear congruential generator  xi+1 = Axi mod M Choose A, M to minimize repetitions  M is prime  (Axi mod M) cycles through {1, … , M-1} before repeating  Good choices: M = 231 – 1 = 2,147,483,647 A = 48,271 Must handle overflow!
  21. 21. Randomized Data Structures CSE 326 Autumn 2001 21 Some Properties of Primes If:  P is prime  0 < X < P Then:  XP-1 = 1 (mod P)  the only solutions to X2 = 1 (mod P) are: X = 1 and X = P - 1
  22. 22. Randomized Data Structures CSE 326 Autumn 2001 22 Checking Non-Primality Exponentiation (pow function, Weiss 2.4.4) can be modified to detect non-primality (composite-ness) // Computes x^n mod(M) HugeInt pow(HugeInt x, HugeInt n, HugeInt M) { if (n == 0) return 1; if (n == 1) return x; HugeInt squared = x * x % M; // 1 < x < M - 1 && squared == 1 --> M isn’t prime! if (isEven(n)) return pow(squared, n/2, M); else return (pow(squared, n/2, M) * x); }
  23. 23. Randomized Data Structures CSE 326 Autumn 2001 23 Checking Primality Systematic algorithm For prime P, for all A such that 0 < A < P Calculate AP-1 mod P using pow Check at each step of pow and at end for primality conditions Randomized algorithm Use one random A  If the randomized algorithm reports failure, then P isn’t prime.  If the randomized algorithm reports success, then P might be prime.  At most (P-9)/4 values of A fool this prime check  if algorithm reports success, then P is prime with probability > ¾ Each new A has independent probability of false positive  We can make probability of false positive arbitrarily small
  24. 24. Randomized Data Structures CSE 326 Autumn 2001 24 Evaluating Randomized Primality Testing Your probability of being struck by lightning this year: 0.00004% Your probability that a number that tests prime 11 times in a row is actually not prime: 0.00003% Your probability of winning a lottery of 1 million people five times in a row: 1 in 2100 Your probability that a number that tests prime 50 times in a row is actually not prime: 1 in 2100
  25. 25. Randomized Data Structures CSE 326 Autumn 2001 25 Cycles Prime numbers Random numbers Randomized algorithms

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