Algorithms and DS

Excercises for princeton algorithm course from coursera

Algorithm-I

WK-I: Dynamic connectivity problem

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Assignment: Finding percolation threshold: Refer to specs

Question 1 Social network connectivity.

Given a social network containing n members and a log file containing m timestamps at which times pairs of members formed friendships, design an algorithm to determine the earliest time at which all members are connected (i.e., every member is a friend of a friend of a friend ... of a friend). Assume that the log file is sorted by timestamp and that friendship is an equivalence relation. The running time of your algorithm should be m log n or better and use extra space proportional to n.

Question 2 Union-find with specific canonical element.

Add a method find() to the union-find data type so that find(i) returns the largest element in the connected component containing i. The operations, union(), connected(), and find() should all take logarithmic time or better.For example, if one of the connected components is {1,2,6,9}, then the find() method should return 9 for each of the four elements in the connected components.

Question 3 Successor with delete.

Given a set of n integers S={0,1,...,n−1} and a sequence of requests of the following form:

• Remove x from S
• Find the successor of x: the smallest y in S such that y≥x.

design a data type so that all operations (except construction) take logarithmic time or better in the worst case.

Question 4 3-SUM in quadratic time.

Design an algorithm for the 3-SUM problem that takes time proportional to n^2 in the worst case. You may assume that you can sort the nn integers in time proportional to n^2 or better.

Hint: given an integer x and a sorted array a[] of n distinct integers, design a linear-time algorithm to determine if there exists two distinct indices i and j such that a[i] + a[j] == x.

Question 5 Search in a bitonic array.

An array is bitonic if it is comprised of an increasing sequence of integers followed immediately by a decreasing sequence of integers. Write a program that, given a bitonic array of nn distinct integer values, determines whether a given integer is in the array.

• Standard version: Use ∼ 3 lg n compares in the worst case.
• Signing bonus: Use ∼ 2 lg n compares in the worst case (and prove that no algorithm can guarantee to perform fewer than ∼ 2 lg n compares in the worst case).

Hints: Standard version. First, find the maximum integer using ∼1 lg n compares—this divides the array into the increasing and decreasing pieces.

Signing bonus. Do it without finding the maximum integer.

Question 6 Egg drop.

Suppose that you have an nn-story building (with floors 1 through n) and plenty of eggs. An egg breaks if it is dropped from floor T or higher and does not break otherwise. Your goal is to devise a strategy to determine the value of T given the following limitations on the number of eggs and tosses:

• Version 0: 1 egg, ≤T tosses.
• Version 1: ∼ 1 lg n eggs and ∼ 1 lg n tosses.
• Version 2: ∼ lg T eggs and ∼ 2 lg T tosses.
• Version 3: 2 eggs and 2 sqrt(n) tosses.
• Version 4: 22 eggs and ≤c sqrt(T) tosses for some fixed constant c.

Hints:

• Version 0: sequential search.
• Version 1: binary search.
• Version 2: find an interval containing T of size ≤2T, then do binary search.
• Version 3: find an interval of size sqrt(n), then do sequential search. Note: can be improved to ∼ 2 sqrt(n) tosses.
• Version 4: 1 + 2 + 3 + ... + t = 1/2 t^2, Aim for c = 2 sqrt(2).

WK-II: Stacks-Queue and Elementry sorts

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Assignment: Deque and Randomized queue: Refer to specs

Question 1 Queue with two stacks.

Implement a queue with two stacks so that each queue operations takes a constant amortized number of stack operations.

Hint: If you push elements onto a stack and then pop them all, they appear in reverse order. If you repeat this process, they're now back in order.

Question 2 Stack with max.

Create a data structure that efficiently supports the stack operations (push and pop) and also a return-the-maximum operation. Assume the elements are real numbers so that you can compare them.

Hint: Use two stacks, one to store all of the items and a second stack to store the maximums.

Question 3 Java generics.

Explain why Java prohibits generic array creation.

Hint: to start, you need to understand that Java arrays are covariant but Java generics are not: that is, String[] is a subtype of Object[], but Stack<String> is not a subtype of Stack<Object>.

Question 4 Intersection of two sets.

Given two arrays a[] and b[], each containing nn distinct 2D points in the plane, design a subquadratic algorithm to count the number of points that are contained both in array a[] and array b[].

Hint: shellsort (or any other subquadratic sort).

Question 5 Permutation.

Given two integer arrays of size n, design a subquadratic algorithm to determine whether one is a permutation of the other. That is, do they contain exactly the same entries but, possibly, in a different order.

Hint: sort both arrays

Question 6 Dutch national flag.

Given an array of n buckets, each containing a red, white, or blue pebble, sort them by color. The allowed operations are:

• swap(i,j): swap the pebble in bucket i with the pebble in bucket j.
• color(i): determine the color of the pebble in bucket i. The performance requirements are as follows:
• At most n calls to color().
• At most n calls to swap().
• Constant extra space.

Hint: 3-way partitioning.

WK-III

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Assignment: Collinear points Refer to specs