Algorithms Q&A - Recent questions in Dynamic Programming

Selecting lowest cost hotel rooms, with no adjacent rooms selected

Sat, 10 Dec 2022 20:43:44 +0000

You and your friends (k friends in total, including yourself) are going to Hawaii for a vacation. In Hawaii, you are given a choice of n hotel rooms, in a straight line, numbered from 1 to n. The cost of the i-th room is c[i]. You can select any k hotel rooms, but you want to ensure that no two selected hotel rooms selected are adjacent to each other. Your objective is to minimize the cost. Describe an efficient algorithm for this problem.

"SpicyBitter" Sequence of Sauces

Tue, 16 Jun 2020 22:25:31 +0000

“SpicyBitter” Some people can handle spicy. Some people can tolerate bitter foods. You have access to n different kinds of sauces, with different levels of “spicyness” and bitterness. You are organizing a food festival and want to create a sequence of sauces such that each sauce is both SPICIER AND MORE BITTER than the sauce prior to that. You want to create as long a sequence as possible using the given list of sauces. Consider this kind of structure: “Sauce [spice: double, bitterness: double]”. You are given an array a[1..] of Sauce objects. Give an algorithm to create the longest possible sequence, and analyze the time complexity of your algorithm in terms of n.

Maximize value of coins from k stacks, of n coins each

Wed, 10 Apr 2019 05:05:07 +0000

You are given k stacks, of n coins each. The value of i-th coin in j-th stack is given by v[i][j].

You play a game against an opponent by alternating turns. In each turn, a player selects one of the top coins (the coin at the highest location of the stack), removes it permanently, and receives the value of the coin. In each move, each person removes one coin only, and can not remove a coin that is not at the top of its stack.

Design an algorithm to maximize the money value you can win if you move first.

Gold is a precious metal

Fri, 29 Mar 2019 19:07:09 +0000

You are given a gold wire that is n feet long.

You can cut the wire and sell the smaller pieces. The value of smaller pieces of wire is not always proportional to the length of the wire. You are given a mapping, p[1..m] of smaller lengths and the sell price of each, for example:

p[1].length = 4, p[1].value = 300, // A piece of wire 4 feet long sells for 300$

p[2].length = 6, p[2].value = 400, // A piece of wire 6 feet long sells for 400$

....

p[m].length = 11, p[m].value = 800 // A piece of wire 11 feet long sells for 800 $

Give an algorithm to maximize the value that you can extract from the gold wire. Keep in mind that gold is very expensive, so you want to ensure the absolute highest value that you can extract.

1-D k-means clustering

Thu, 08 Mar 2018 18:35:27 +0000

Let x1, . . . , xn be an input array of n numbers sorted in non-descending order. The problem of 1-D k-means clustering is defined as assigning elements of the input 1-D array into k clusters so that the sum of squares of within-cluster distances from each element to its corresponding cluster mean is minimized.

DNA sequences

Sun, 04 Mar 2018 19:45:12 +0000

Given two DNA sequences (sequences of A, C, T, G characters), the longest common subsequence (LCS) problem is to find the longest subsequence (not nescessarily contiguous) that exists in both of the input strings. For example, given strings "ACTGACT" and "CAAGCATA", the subsequence "CGCT" is common in both. Given two DNA sequences of sizes n1 and n2 respectively, find a dynamic programming algorithm to find the longest common subsequence in O(n1n2) time.

From the midterm practice

If you are happy and you know it, jump up high!

Sun, 04 Mar 2018 03:25:54 +0000

You are standing on step 0 of a staircase. Your goal is to reach the step n. At each step i, you have three choices: hop to next step i+1, i+2 or i+3 Give an algorithm to count the number of possible paths to reach n.

Please provide the Notation, optimality, recurrence, and algorithm.

Canoeing on the cheap

Sun, 04 Mar 2018 03:18:45 +0000

You are canoeing down a river and there are n renting posts along the way. Before starting your journey, you are given, for each 1 <= i <= j <= n, the fee f(i,j) for renting a canoe from post i to post j. These fees are arbitrary. For example it is possible that f(1,3) = 10 and f(1,4) = 5. You begin at trading post 1 and must end at trading post n (using rented canoes). Your goal is to minimize the rental cost. Give the most efficient algorithm you can for this problem. Prove that your algorithm yields an optimal solution and analyze the time complexity.

Please provide the Notation, optimality, recurrence, and algorithm.

Around the block party planning

Sun, 04 Mar 2018 03:14:53 +0000

Consider a row of n houses represented as an array: A[1..n], where the phrase "next door neighbor" having its natural meaning. Each resident is assigned a "fun factor" F[1..n], which represents how much fun they bring to a party. Your goal is to maximize the fun of a party that you are arranging, but with the constraint that you cannot select three consecutive neighbors. (So for example, if you invite the A[5] and A[6] family, you cannot invite the A[4] or A[7] families.) Give an efficient algorithm to select the guest list.

Please provide the Notation, optimality, recurrence, and algorithm.

Love (Skip) thy neighbor

Sun, 04 Mar 2018 03:10:07 +0000

Given a list of n positive numbers, your objective is to select the set of numbers that maximized the sum of selected numbers, given the constraint that we cannot select two numbers that are located adjacent to each other. Describe a linear time algorithm for this problem.

Please provide the Notation, optimality, recurrence, and algorithm.

Barbie's Array of Diamonds

Sun, 04 Mar 2018 03:07:27 +0000

Barbie has n diamonds. Each diamond has two attributes: shiny value and weight value. Barbie wants to create a "diamond line" in which each diamond is both shinier and heavier than the previous one. She may not be able to use all her diamonds, but wants to maximize the number of diamonds in this diamond line. Give a polynomial time algorithm for creating a diamond line with the maximum number of diamonds. Assume that her initial list of diamonds is not in any specific order.

Please provide the Notation, optimality, recurrence, and algorithm.

Cost Minimizing Ice Cream Shop

Thu, 08 Feb 2018 23:37:17 +0000

An ice cream shop is looking to minimize their operation costs, under the given constraints:

They are faced with costs of purchasing ice cream from a local distributor, where each order has a (potentially large) fixed delivery cost shipping _ cost as well as the price price for each pint

to minimize delivery costs, they can buy more ice cream than needed for a given day, and keep it stored it in their freezer overnight

their freezer can store max capacity pints of ice cream at any time (includes both the shipment as well as any pints stored overnight)

However, storing extra ice cream overnight is not free, and it costs them overnight cost dollars per pint (per night) to keep any additional ice cream frozen if they buy in advance

Given a list of daily demand (in pints) of ice cream for the next num_days days, what is the minimum cost for the shop to be stocked up ready to meet their demands each day?

Inputs

An integer num _ days , for the number of days

A list of n integers demands, separated by a space, for the pints of ice cream demanded each day

An integer max _ capacity , for the max capacity of their freezer

An integer shipping _ cost , for the fixed cost of each shipment taken

An integer price, for the price they pay for each pint of ice cream

An integer overnight_cost, the cost of storing a pint of ice cream for a day

Maximum Value Contiguous Subregion

Mon, 24 Jul 2017 05:25:12 +0000

You are given a two dimentional array A[1:n, 1..n] of real numbers (possibly containing both positive and negative real numbers). You want to find a rectangular sub-region of the array that maximizes the sum of that region. For example, the subarray S(1,2,5,8) consists of contiguous rectangle between A[1,2] to A[5,8], both corners inclusive, therefore, it has 35 total numbers.

Give the most efficient algorithm for this problem that you can.

Maximum Value But Limited Neighbors

Tue, 20 Jun 2017 17:39:04 +0000

You are given an array a[1..n] of positive numbers and an integer k. You have to produce an array b[1..n], such that: (i) For each j, b[j] is 0 or 1, (ii) Array b has adjacent 1s at most k times, and (iii) sum_{j=1 to n} a[j]*b[j] is maximized. For example, given an array [100, 300, 400, 50] and integer k = 1, the array b can be: [0 1 1 0], which maximizes the sum to be 700. Or, given an array [10, 100, 300, 400, 50, 4500, 200, 30, 90] and k = 2, the array b can be [1, 0, 1, 1, 0, 1, 1, 0, 1] which maximizes the sum to 5500.

Calculate g(1000) given the following definition of g function..

Thu, 08 Jun 2017 23:22:25 +0000

g(1) = g(2) = .. g(10) = 1

g(n) = g(n-1) + 3 *g(n-3) + 7*g(n-10)

Make change using smallest number of coins of given denominations

Sat, 03 Dec 2016 13:25:56 +0000

Given a set of k coins with values v[1], v[2], … v[k] such that v[1] = 1, v[1] < v[2] < v[3] < v[4] < … < v[k]. We want to make change for a given value n using as few coins as possible. Present an algorithm to do so. The algorithm must be optimal in terms of the number of coins.

Minimizing Weight of a Linear Partition

Mon, 21 Nov 2016 01:27:02 +0000

Let A[1:n] be a real array, and let k be an integer, 1 ≤ k ≤ n. A linear k-partition of A is a sequence of subarrays of the form [1:x1],[x1+1:x2],[x2+1:x3],…,[x(k-1)+1:n], for some x1< x2 <… < x(k−1). That is, the array A got divided into k subarrays. From this, we calculate the sums of each of k subarrays. Finally, we take the maximum of those k sums, and that is defined as the weight of this linear k-partition.

Give an O(nk) dynamic programming algorithm to minimize the weight of the linear k-partition of the given array.

Magical eggs and tiny floors

Thu, 30 Jun 2016 01:56:13 +0000

You are given 4 eggs and a 30 floor building. You need to figure out the highest floor an egg can be dropped without breaking, assuming that (i) all eggs are identical, (ii) if an egg breaks after being dropped from one floor, then the egg will also break if dropped from all higher floors, and (iii) if an egg does not break after being thrown from a certain floor, it retains all of its strength and you can continue to use that egg. Your goal is to minimize the number of throws. From which floor do you drop the first egg? How do you handle this problem given generally m eggs and an n-floor building?

“Teleportation” across astro-haunted galaxies

Sat, 05 Mar 2016 03:29:22 +0000

“Teleportation”: You have a teleporter that can take you from galaxy i to galaxy j. Cost to teleport is given by c(i,j) > 0, which can be arbitrary. Some galaxies are “astro-haunted” - this is specified by a matrix A, where A[i] can be 0 or 1 (1 means that that galaxy is “astro-haunted”). Give a polynomial time algorithm that minimizes the cost of going from galaxy 1 to galaxy n, such that you pass through a maximum of m astro-haunted galaxies. (You can assume that galaxies 1 and n are not astro-haunted.)