Download Algorithms and Data Structures 1. Give an algorithm to find the

Algorithms and Data Structures
1. Give an algorithm to find the minimum and maximum of a sequence of n
numbers using at most d3n/2e − 2 comparisons.
2. Give an algorithm to find the second largest element in a sequence of n
numbers using n + o(n) comparisons in the worst case. (Hint: 1. Start
with finding the maximum element. 2. Part of the exercise is also test
your understanding of the notation o(n)).
3. How do you determine whether a given graph is connected? How much
time does your algorithm take if you use
• an adjacency matrix to represent the given graph,
• an adjacency list to represent the given graph?
4. Give an algorithm to determine whether a given graph has a cycle. Your
algorithm should take O(|V |) time (regardless of the number of edges in
the graph).
5. How would you find a maximum matching in a bipartite graph? What is
the complexity of the algorithm?
6. Recall a binary minheap, and the time (number of comparisons, number of moves) to perform insert, deletemin operations and an O(n) time
algorithm to produce a heap ordered array from a given arbitrary array.
Suppose, instead of a binary heap, we use a d-ary heap for some d ≥ 2.
Discuss what happens to the running times (number of comparisons and
number of moves) of all these operations, and argue whether there is a
better optimal value of d (than 2).
In particular, suppose I am interested only in minimizing the swaps in the
insert and deletemin operations, what choice of d is good?
7. Consider the insertion sort algorithm to sort an array A of n elements:
Start with the sorted list A[1] of size 1: for i = 2 to n insert A[i] into the
already sorted list A[1..i − 1].
• How much time (number of comparisons and moves) does your algorithm make if you use linear search to insert the element at each step
of the for loop?
• What if you use binary search?
• Can you think of a data structure to organize the sorted list A[1..i−1]
so that the insertion takes O(log n) time and you can still retrieve
the sorted list (at the end) in O(n) time?
8. We would like to maintain a set of elements (from an ordered set) to
support, along with the standard insert, search and delete operations,
return(k) that will return the k-th smallest element in the given list for
the query value of k (between 1 and n). Explain what augmentation you
would use to a standard data structure (which one?) to support this
operation (along with others) in O(log n) time?
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9. Can you show the following problem N P -complete? Given two graphs
G and H, determine whether there is a subgraph of G isomorphic to H?
Suppose G has n vertices and H has m ≤ n vertices, can you give a
brute-force algorithm to answer this question? How much time does your
algorithm take?
10. Let n, k ≥ 2 be integers. Give an O(nk log k) algorithm for the following
tasks:
• Let A1 , A2 , . . . Ak be k sorted lists of n elements each. The problem
is to sort the nk elements of A1 ∪ A2 ∪ . . . Ak .
• Given a list of nk elements, determine the elements of rank in for
i = 1 to k. Rank of an element is its position in the sorted list.
Can you also prove corresponding Ω(nk log k) lower bound on the number
of comparisons to solve these problems?
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