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CS6212 |
May 25, 2011 |
Homework 2
Due Date: June 6, 2011
Problem 1: (20 points)
a) Show step by step how the quicksort
algorithm sorts the following array: 12, 7, 5, 9, 3, 18, 22, 13, 1, 7.
b) Show step by step how the mergesort
algorithm sorts the same arrary.
c) Show step by step how the insertion sort algorithm
sorts the same arrary.
d) Show step by step how the selection sort algorithm
sorts the same arrary.
Problem 2: (20 points)
a) Write a divide-and-conquer recursive algorithm that
takes as input a real array a[1..n], and gives as output an array A[1..n] where
A[i]=a[1]*a[2]*...*a[i].
b) Give a non-recursive algorithm that performs the
same task. Analyze the time complexities of both algorithms.
Problem 3: (25 points)
a) Give a divide-and-conquer
recursive algorithm to compute xn for any
real number x and any positive integer n. Analyze the time complexity of your
algorithm.
b) Let a[0..n-1] be a real
array where n is a power of 2. Give a divide-and-conquer recursive algorithm to
compute the polynomial P(x)=a[0]+a[1]x+a[2]x2+...+a[n-1]xn-1for
any x.
c) Give a non-recursive algorithm that performs the
same task as in (b). Analyze the time complexites of
both algorithms.
Problem 4: (15 points)
Consider this instance of the knapsack problem:
Weights: 15, 63, 27, 15, 9, 45, 30, 63
Prices: 24, 48, 18, 12, 42, 30, 18, 45
Capacity: M=56.
Apply the greedy method algorithm to find the solution (that is, how much of
every item we are going to get). Show the process of deriving the solution step
by step.
Problem 5: (20 points)
Let G=(V,E) be the following weighted graph: V={1,2,3,4,5,6,7} and E={[(1,2)
7],[(1,4) 4],[(2,4) 4],[(1,3) 22], [(3,5) 5],[(4,5) 10],[(2,6) 4],[(4,6) 5],[(5,7)
3],[(6,7) 5]}, where [(i,j) a] means that (i,j) is an edge of weight a.
a) Find a minimum spanning tree of G, showing which
edge is being considered each step.
b) Using the greedy single source shortest path
algorithm, find the distances between node 1 and all the other nodes. Show the
values of the DIST array at every step.