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For all divide-and-conquer algorithms follow these steps:
1. Describe the steps of your algorithm in plain English.
2. Write a recurrence equation for the runtime complexity.
3. Solve the equation by the master theorem.
For all dynamic programming algorithms follow these steps:
1. Define (in plain English) subproblems to be solved.
2. Write the recurrence relation for subproblems.
3. Write pseudo-code to compute the optimal value
4. Compute its runtimecomplexity in terms of the input size.
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1. Suppose we define a new kind of directed graph in which positive weights are
assigned to the vertices but not to the edges. If the length of a path is defined
by the total weight of all nodes on the path, describe an algorithm that finds
the shortest path between two given points A and B within this graph.
2. For each of the following recurrences, give an expression for the runtime T(n)
if the recurrence can be solved with the Master Theorem. Otherwise, indicate
that the Master Theorem does not apply.
• T(n) = 3T(n/2) + n
2
• T(n) = 4T(n/2) + n
2
• T(n) = T(n/2) + 2n − n
• T(n) = 2nT(n/2) + n
n
• T(n) = 16T(n/4) + n + 10
• T(n) = 2T(n/2) + nlogn
• T(n) = 2T(n/4) + n
0.51
• T(n) = 0.5T(n/2) + 1/n
• T(n) = 16T(n/4) + n!
• T(n) = 10T(n/3) + n
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3. Suppose that we are given a sorted array of distinct integers A[1, …, n] and we
want to decide whether there is an index i for which A[i] = i. Describe an
efficient divide-and-conquer algorithm that solves this problem and explain the
time complexity.
4. We know that binary search on a sorted array of size n takes O(log n) time.
Design a similar divide-and-conquer algorithm for searching in a sorted singly
linked list of size n. Describe the steps of your algorithm in plain English.
Write a recurrence equation for the runtime complexity. Solve the equation by
the master theorem
5. Given n balloons, indexed from 0 to n−1. Each balloon is painted with a number on it represented by array nums. You are asked to burst all the balloons. If
the you burst balloon i you will get nums[i − 1] × nums[i] × nums[i + 1] coins.
Here left and right are adjacent indices of i. After the burst, the left and right
then becomes adjacent. You may assume nums[−1] = nums[n] = 1 and they
are not real therefore you cannot burst them. Design a dynamic programming
algorithm to find the maximum coins you can collect by bursting the balloons
wisely. Analyze the running time of your algorithm.
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Example. If you have the nums = [3, 1, 5, 8]. The optimal solution would
be 167, where you burst balloons in the order of 1, 5, 3 and 8. The array nums
after each step is:
[3, 1, 5, 8] → [3, 5, 8] → [3, 8] → [8] → []
And the number of coins you get is 3×1×5+3×5×8+1×3×8+1×8×1 = 167.
6. Given a non-empty string str and a dictionary containing a list of unique words,
design a dynamic programming algorithm to determine if str can be segmented
into a sequence of dictionary words. If str =“algorithmdesign” and your dictionary contains “algorithm” and “design”. Your algorithm should answer Yes as
str can be segmented as “algorithmdesign”. You may assume that a dictionary
lookup can be dome in O(1) time.
7. A tourism company is providing boat tours on a river with n consecutive segments. According to previous experience, the profit they can make by providing
boat tours on segment i is known as ai. Here, ai could be positive (they earn
money), negative (they lose money), or zero. Because of the administration
convenience, the local community requires that the tourism company do their
boat tour business on a contiguous sequence of the river segments (i.e., if the
company chooses segment i as the starting segment and segment j as the ending
segment, all the segments in between should also be covered by the tour service,
no matter whether the company will earn or lose money). The company’s goal
is to determine the starting segment and ending segment of boat tours along
the river, such that their total profit can be maximized. Design a dynamic
programming algorithm to achieve this goal and analyze its runtime.
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