Algorithm Homework2
- 1. 1. Please hand in your homework in classroom. 2. Please hand in your homework within first 10 minutes of class. No late homework is allowed 3. Please hand in your hand-writing homework. No computer print-out is accepted. Homework #2 (Due on 03/24/2017) 1. Do Exercise 6.3-3 in CLRS second or third edition. 2. Do Exercise 6.4-1 in CLRS second or third edition. 3. Do Exercise 7.2-1 in CLRS second or third edition. 4. Do Exercise 7.2-2 in CLRS second or third edition. 5. Do Exercise 8.1-4 in CLRS second or third edition. 6. (Median-of-3 Partition) One way to improve the RANDOMIZED-QUICKSORT is to choose the pivot for partitioning more carefully than by picking a random element from the array. One common approach is to choose the pivot as the median of a set of 3 elements randomly selected from the array. Assume that all elements in the array are distinct. Please answer following questions. (a) What is the probability of getting an OK split if the pivot is chosen at random? Explain. (A split is “OK” if the smaller piece has at least n/4 elements.) (b) Roughly, what is the probability of getting an OK split with the new median-of-3 method? Explain. (c) Let I be the indicator random variable for getting an OK split using the median-of-3 partition: 1 if the split is OK I = 0 otherwise What is the expectation of I? 7. You are given an array of n integers ranging from 1 to 999. Design an O(n) algorithm to rearrange elements of the array in place so that all 1-digit numbers precede all 2-digit numbers, and all 2-digit numbers precede all 3-digit numbers. Analyze the worst-case running time of your algorithm.