Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Suppose you are given a set L of n line segments in the plane, where each segment has one endpoint on the vertical line x = 0 and one endpoint on the vertical line x = 1 , and all 2n endpoints are distinct. Describe and analyze an algorithm to compute the largest subset of L in which no pair of segments intersects.
If five integers are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}, must there be at least two integers with the property that the larger minus the smaller is 2? Write an answer that would convince a good but skeptical fellow student who has learned the statement of the pigeonhole principle but not seen an application like this one. Describe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes.
Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies.
Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…
Chapter 16 Solutions
Introduction to Algorithms
Ch. 16.1 - Prob. 1ECh. 16.1 - Prob. 2ECh. 16.1 - Prob. 3ECh. 16.1 - Prob. 4ECh. 16.1 - Prob. 5ECh. 16.2 - Prob. 1ECh. 16.2 - Prob. 2ECh. 16.2 - Prob. 3ECh. 16.2 - Prob. 4ECh. 16.2 - Prob. 5E
Ch. 16.2 - Prob. 6ECh. 16.2 - Prob. 7ECh. 16.3 - Prob. 1ECh. 16.3 - Prob. 2ECh. 16.3 - Prob. 3ECh. 16.3 - Prob. 4ECh. 16.3 - Prob. 5ECh. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.4 - Prob. 1ECh. 16.4 - Prob. 2ECh. 16.4 - Prob. 3ECh. 16.4 - Prob. 4ECh. 16.4 - Prob. 5ECh. 16.5 - Prob. 1ECh. 16.5 - Prob. 2ECh. 16 - Prob. 1PCh. 16 - Prob. 2PCh. 16 - Prob. 3PCh. 16 - Prob. 4PCh. 16 - Prob. 5P
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- Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you ll in the values is the correct one.arrow_forwardConsider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Prove that the coin changing problem exhibits optimal substructure. Design a recursive backtracking (brute-force) algorithm that returns the minimum number of coins needed to make change for n cents for any set of k different coin denominations. Write down the pseudocode and prove that your algorithm is correct.arrow_forwardImplement an algorithm for a competition on hills. in: n unranked opponents in set P (1 n)set R of ranked individuals (R, 0) with a champion ranking.arrow_forward
- Java Programming Consider a set of n intervals I1,...,In, each given as an integer tuple (si,fi) with si<fi, such that the ith interval starts at si and ends at fi. We want to determine the maximum number of nonoverlapping intervals. More formally, we want to determine the size of a largest subset S⊆{1,...,n} such that for i, j∈S, with i≠j we have fj≤si or fi≤sj. Note that the intervals [1,2] and [2,3] are not considered to be overlapping. Input The first line of input consists of an integer n∈{1,...,105}, the number of intervals. Each of the following n lines consists of two space-separated integers ss and ff, indicating an interval starting at s and ending at f, with 0≤s<f≤109. requirements: CPU Time limit: 4 seconds Memory limit: 1024 MB Output Output a single integer, the largest number of nonoverlapping intervals.arrow_forwardThe algorithm of Euclid computes the greatest common divisor (GCD) of two integer numbers a and b. The following pseudo-code is the original version of this algorithm. Algorithm Euclid(a,b)Require: a, b ≥ 0Ensure: a = GCD(a, b) while b ̸= 0 do t ← b b ← a mod b a ← tend whilereturn a We want to estimate its worst case running time using the big-Oh notation. Answer the following questions: a. Let x be a integer stored on n bits. How many bits will you need to store x/2? b. We note that if a ≥ b, then a mod b < a/2. Assume the values of the input integers a and b are encoded on n bits. How many bits will be used to store the values of a and b at the next iteration of the While loop? c. Deduce from this observation, the maximal number iterations of the While loop the algorithm will do.arrow_forwardLet f and g be functions from the set of integers or the set of real numbers to the set of real numbers. We say that f ( x ) is O ( g ( x ) ), read as "f ( x ) is big-oh of g ( x )", if there are constants C and k such that | f ( x ) | ≤ C | g ( x ) | whenever x > k. KINDLY SHOW YOUR SOLUTION. 7. Generating sequences of random-like numbers in a specific range. Xi+1 = aXi + c Mod m where, X, is the sequence of pseudo-random numbers m, ( > 0) the modulus a, (0, m) the multiplier c, (0, m) the increment X0, [0, m) – Initial value of sequence known as seed m, a, c, and X0 should be chosen appropriately to get a period almost equal to m For a = 1, it will be the additive congruence method. For c = 0, it will be the multiplicative congruence methodarrow_forward
- Imagine that you have a problem P that you know is N P-complete. For this problem you have two algorithms to solve it. For each algorithm, some problem instances of P run in polynomial time and others run in exponential time (there are lots of heuristic-based algorithms for real N P-complete problems with this behavior). You can’t tell beforehand for any given problem instance whether it will run in polynomial or exponential time on either algorithm. However, you do know that for every problem instance, at least one of the two algorithms will solve it in polynomial time. (a) What should you do? (b) What is the running time of your solution? 564 Chap. 17 Limits to Computation (c) What does it say about the question of P = N P if the conditions described in this problem existed?arrow_forwardGiven a non-empty list items of positive integers in strictly ascending order, find and return thelongest arithmetic progression whose all values exist somewhere in that sequence. Return the answer as a tuple (start, stride, n) of the values that define the progression. To ensure unique results to facilitate automated testing, if there exist several progressions of the same length, this function should return the one with the lowest start. If several progressions of equal length emanate from the lowest start, return the progression with the smallest stride. INTRUCTIONS: Input a comment or a solution on how you solve the problem, write a psuodocode and flow chart about th problemarrow_forwardImplement Algorithm for Hill-climbing tournament. in: set P of n unranked players (1 ≤ n)out: set R of ranked players which has a champion ranked(R, 0arrow_forward
- Problem Alice was given a number line containing every positive integer, x, where 1<=x<=n. She was also given m line segments, each having a left endpoint, l (1<=l<=n), a right endpoint, r (1<=r<=n), and an integer weight, w. Her task is to cover all numbers on the given number line using a subset of the line segments, such that the maximum weight among all the line segments in the chosen subset is minimal. In other words, let S=(l1,r1,w1),(l2,r2,w2),....,(lk,rk,wk), represent a set of k line segments chosen by Alice, max(w1,w2,...,wk) should be minimized. All numbers 1,2,....n should be covered by at least one of k the chosen line segments. It is okay for the chosen line segments to overlap. You program should output the minimized maximum weight in the chosen subset that covers all the numbers on the number line, or -1 if it is not possible to cover the number line. Input format The first line of the input contains an integer, n - denoting the range of numbers…arrow_forwardIf 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 positionarrow_forwardPLEASE HELP ME. kindly show all your work 1. Prove that∀k ∈ N, 1k + 2k + · · · + nk ∈ Θ(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 ∈ Θ(g1) and f2 ∈ Θ(g2).Prove that (f1 + f2) ∈ Θ(max{g1, g2}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}. 3. Let n ∈ N \ {0}. Describe the largest set of values n for which you think 2n < n!. Use induction toprove that your description is correct.Here m! stands for m factorial, the product of first m positive integers. 4. Prove that log2 n! ∈ O(n log2 n). Thank you. But please show all work and all stepsarrow_forward
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