0-1 Knapsack Problem
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- A group of m people is considering sharing the cost of buying some items for their summer camp. There are a total of n possible items to purchase. Each person writes a list of the items they are interested in purchasing on a paper. For example, person one may write “boat, trampoline, hot tub,” whereas person two may write “ hot tub, kayak, paddleboard”. There is no limit to how many items a person registers on their sheet. The goal is to determine if at least k items can be purchased. The following rules apply 1. The cost of any purchased item must be split equally among all those who listed that item on their sheet. No person can refuse to contribute to an item chosen for purchase, which is listed on their sheet. 2. Since each person has a limited budget, they may only contribute to at most one item on their list. The problem of determining if at least k items can be purchased is called SummerCamp. Show that this problem is NP-complete.Given two strings needle and haystack, return the index of the first occurrence of needle in haystack, or -1 if needle is not part of haystack. Example 1: Input: haystack="sadbutsad", needle = "sad" Output: 0 Explanation: "sad" occurs at index 0 and 6. The first occurrence is at index 0, so we return 0. Example 2: Input: haystack="leetcode", needle = "leeto" Output: -1 Explanation: "leeto" did not occur in "leetcode", so we return -1. Write C# codeThe following iterative sequence is defined for the set of positive integers: ifn is even 3n +1 ifnis odd Using the rule above and starting with 13, we generate the following sequence: 20 u20 = 10 → U10 = 5 U5 = 16 u16 =8us = 4- 13 u13 40 u40= u=2 u2 = 1. It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. The below function takes as input an integer n and returns the number of terms generated by the sequence starting at n. function i-Seq (n) u-n; 1=1%B while u -1 if statement 1 u-u/2%3; else statement 2 end 1=i+1%3B end statement 1 and statemet 2 should e replacerd bv: statement 1 is 'mod(n.2)=30 and statement 2 is 'u - 3*n+1;" O None of the choices KO statement 1 is u%2 and statement 2 is u= 3*u+1;" O statement 1 is mod(u.2)==0 and statement 2 is u= 3*u+1," op
- Dingyu is playing a game defined on an n X n board. Each cell (i, j) of the board (1 2, he may only go to (2, n).) The reward he earns for a move from cell C to cell D is |value of cell C – value of cell D|. The game ends when he reaches (n, n). The total reward - is the sum of the rewards for each move he makes. For example, if n = 1 2 and A = 3 the answer is 4 since he can visit (1, 1) → (1, 2) → (2, 2), and no other solution will get a higher reward. A. Write a recurrence relation to express the maximum possible reward Dingyu can achieve in traveling from cell (1, 1) to cell (n, n). Be sure to include any necessary base cases. B. State the asymptotic (big-O) running time, as a function of n, of a bottom-up dynamic programming algorithm based on your answer from the previous part. Briefly justify your answer. (You do not need to write down the algorithm itself.)G E N E T I C - A L G O R I T H M OBJECT REWARD WEIGHT A 20 1 B 5 2 C 10 3 D 40 8 E 15 7 F 25 4 G 4 5 H 7 2 Here, Maximum weight = 12 The above problem is a 0/1 Knapsack problem. You have to carry the different objects in your bag in a way such that the reward is maximized without exceeding the weight limit. You can carry an object exactly once but you always have to carry the object labeled “H”. Encode the problem and create an initial population of 4 different chromosomes. Explain what would be an appropriate fitness function for this problem. Use the fitness function and perform natural selection to choose the 2 fittest chromosomes. Using the selected chromosomes perform a single point crossover to get 2 offspring. Perform mutation and check the fitness of the final offspring. Explain your work.If the first number in a sequence is a positive integer, x Let ao= x, an is defined as follows if an is even, then an+1 = an/2 if an is odd, then an+1 =3 *an+ 1 Then there exists an integer k, such that ak =1 For example, if: 75, then k = 14 and the numbers in the sequence are: 75, 226, 113, 340, 170, 85, 256, 128, 64, 32, 16, 8, 4, 2, 1. The largest number in the sequence is 340 and it is a position 4 in the sequence (assuming 75 is at position 1) Design and implement a complete C++ program that will • read a series of integers (greater than 0) from a file and for each integer display (to the screen) − the integer − the number of steps it takes to reach 1 − the largest value in the sequence and its position
- TM M = (Q, E, I, 6, 90, 9a, qr), where Q = {90, 91, 92, 9a, 9r}, Σ = {0, 1}, r = {0, 1, L}, and 8 is: 8(qo, U) = (qr, U, R) 8(90, 0) = 8(go, 1) (91, 0, R) (qo, 1, R) = 8(g1, L) = (ga, U, R) 8(91,0) = (91, 0, R) 8(91, 1) = (92, 1, R) = (92, U, R) 8(92, U) 8(92, 0) = (90, 0, R) 8(92, 1) = (92, 1, R) i. Prove that M is NOT a decider. ii. Mathematically describe the language A that M recognises. Prove that A ≤ L(M). iii. Prove A = L(M). iv. Is A Turing-decidable? [Give clear reasons for your answer. No need for a formal proof.]The Knapsack Problem is a famous computer science problem that is defined as follows: imagine you are carrying a knapsack with capacity to hold a total of weight C. You are selecting among n items with values A={a_1, a_2, ... , a_n} and associated weights W={w_1, w_2, ... , w_n}. Here the weights and values are all positive (but not necessarily unique). You wish to maximize the total value of the items you select not exceeding the given weight capacity, i.e. maximize sum_{a in A} such that sum_{w in W} <= C. Please note that you can only select your items once. a) We can reformulate this as a 2D bottom-up dynamic programming problem as follows. Define T_{i,j} as the highest possible value sum considering items 1 through i and total weight capacity j (j <= C). What is the base case i.e. T_{0,j} for all j and T_{i,0} for all i?, and What is the loop statement?A uwuified sentence is sentence that has been transformed using a made-up Internet language in which some of the letters in the words are replaced by something else. The exact scheme is described below: Any uppercase/lowercase R or L is replaced by w/w, respectively. • If we encounter an o/o in a word, check if the previous letter (if it exists) is an M/m or N/n. If the previous letter is one of these, insert the lowercase letter y in between them, regardless of the capitalization of the other letters. • All other characters are left unchanged. Some examples: Professor will be converted to Pwofessow (There are two r's that are replaced by w's. Since the two o's aren't proceeded by an M/n or N/n, no y will be inserted.) LLunoacyo will be converted to wwunyoacyo (The two L's will be replaced with two ws according to the first rule. Then the first o will have a y inserted in front of it between then and the o according to the second rule. The last o won't have a y inserted in between…
- Let L be a line in the xy plane. If L is a vertical line, its equation is x=afor some real number a. Suppose L is not a vertical line and its slope is m. Then the equation of L is y=mx + b, where b is the y-intercept. If L passes through the point (x0,y0), the equation of L can be written as y –y0=m(x –x0). If (x1,y1)and (x2,y2)are two points in the xy plane and x1≠x2, the slope of the line passing through these points is m = (y2-y1)/ (x2-x1). Write a program that prompts the user to enter two points in the xy plane. The program outputs the equation of theline and uses ifstatements to determine and output whether the line is vertical, horizontal, increasing, or decreasing. If L is a nonvertical line, output its equation in the form y=mx + b.Suppose a person can buy a chocolate bar from the vending machine for $1 each. Inside every chocolate bar is a coupon. A person can redeem 3 coupons for one chocolate bar from the machine. This means that once a person has started buying chocolate bars from the machine, he/she always has some coupons. A person would like to know how many chocolate bars can be bought, if a person starts with N dollars and always redeem coupons, if he/she has enough for an additional chocolate bar. For example: With 3 dollars a person could buy 4 chocolate bars after purchasing 3 bars giving him/her 3 coupons and then redeeming the 3 coupons for one bar. This would leave him/her with one extra coupon. Thus, will have 4 chocolate bars and still have one coupon leftover. For 11 dollars, a person can have 16 chocolate bars and still have one coupon leftover. For 12 dollars, a person can have 17 chocolate bars and have two coupons leftover. Write a complete Python program that prompts a buyer to input…Python Programming You are given a list of M positive numbers, Ao, A1. ..., AM-1. You have to answer Q queries. Each query has four parameters, I, r, n, k. Given a query with parameters I, r, n, k here is how to compute the answer: • Let (B1, B2 .. Ba) = (Aj, A1+1, ... „A,), where d = r-l+1. • Let S = set of all points (x1, X2, S Bi. • For x, y eS, define dist(x, y) = min, sisal Xi - YıI • The answer is number of subsets of S of size exactly n such that distance Xa) in d-dimensional space such that 0 < xi (denoted by above defined dist function) between any two points is at least k. Input The first line of input contains two integers, M and Q. The second line contains M space-separated integers Ao, A1, ..., Am-1 denoting the array A. The next Q lines contain one query each. Each query consists of a single line having the four space separated integers I, r, n, k in this order. Output For each query, output a single line containing the answer modulo 1000000007 (109 + 7). Test Case: 5 1 4…