Problem Set 2 Solutions - Massachusetts Institute of Technology
... order. Therefore, remove all the bad elements from the array. We now argue that the remaining elements are in sorted order. Consider any two of the remaining good elements, key1 and key2 , where key1 is at index i and key2 is at index j. If i < j, then Part(c) shows that key1 � key2 . Similarly, if ...
... order. Therefore, remove all the bad elements from the array. We now argue that the remaining elements are in sorted order. Consider any two of the remaining good elements, key1 and key2 , where key1 is at index i and key2 is at index j. If i < j, then Part(c) shows that key1 � key2 . Similarly, if ...
MATHEMATICS WITHOUT BORDERS 2015
... many ways can we do this, if we use all three bottle sizes? ...
... many ways can we do this, if we use all three bottle sizes? ...
decision analysis - CIS @ Temple University
... Linear Programming (LP) problems can be solved on the computer using Matlab, and many others. There are special classes of LP problems such as the assignment problem (AP). Efficient solutions methods exist to solve AP. AP can be formulated as an LP and solved by general purpose LP codes. However, th ...
... Linear Programming (LP) problems can be solved on the computer using Matlab, and many others. There are special classes of LP problems such as the assignment problem (AP). Efficient solutions methods exist to solve AP. AP can be formulated as an LP and solved by general purpose LP codes. However, th ...
decision analysis - Temple University
... Linear Programming (LP) problems can be solved on the computer using Matlab, and many others. There are special classes of LP problems such as the assignment problem (AP). Efficient solutions methods exist to solve AP. AP can be formulated as an LP and solved by general purpose LP codes. However, th ...
... Linear Programming (LP) problems can be solved on the computer using Matlab, and many others. There are special classes of LP problems such as the assignment problem (AP). Efficient solutions methods exist to solve AP. AP can be formulated as an LP and solved by general purpose LP codes. However, th ...
Here
... Now notice that the left side is a multiple of 3, so the right side must be also. Thus the least positive amount that can be measured is a multiple of 3. One can also reason this as follows: each fill adds a multiple of 3 units of water to the total amount on hand, each pour leaves the total number ...
... Now notice that the left side is a multiple of 3, so the right side must be also. Thus the least positive amount that can be measured is a multiple of 3. One can also reason this as follows: each fill adds a multiple of 3 units of water to the total amount on hand, each pour leaves the total number ...
Problem 1. If we increase the length of each edge of a cube by 100
... then there exists a step that decreases it. In the beginning, H is at most 12 · 41 · 42 (if every pair is switched, i.e. if the people are in the opposite order). Thus, the “worst” ordering requires 861 steps. Problem 27. Parsley lives in Vegetable State where one can pay only by coins with values 7 ...
... then there exists a step that decreases it. In the beginning, H is at most 12 · 41 · 42 (if every pair is switched, i.e. if the people are in the opposite order). Thus, the “worst” ordering requires 861 steps. Problem 27. Parsley lives in Vegetable State where one can pay only by coins with values 7 ...
Document
... Jug’s problem: Given a 5-liter jug filled with wine and empty 3liter and 2-liter jugs. Let’s obtain precisely 1 liter wine in the 2liter jug. Missioner - cannibal problem: n missionaries and n cannibals want to cross a river in a boat that can hold h people in such a way that cannibals never outnumb ...
... Jug’s problem: Given a 5-liter jug filled with wine and empty 3liter and 2-liter jugs. Let’s obtain precisely 1 liter wine in the 2liter jug. Missioner - cannibal problem: n missionaries and n cannibals want to cross a river in a boat that can hold h people in such a way that cannibals never outnumb ...
The Pigeonhole Principle
... Problem 3 (Putnam) Here, it is more a matter of counting the elements of a set in two different ways, one of them using the drawer principle. Consider the set V of all the vertices of a hypercube in n space, V = {(ε1, ε2, …, εn ) : εi = ±1}. Prove that, for any subset S of V such that |S| > (2 n+1)/ ...
... Problem 3 (Putnam) Here, it is more a matter of counting the elements of a set in two different ways, one of them using the drawer principle. Consider the set V of all the vertices of a hypercube in n space, V = {(ε1, ε2, …, εn ) : εi = ±1}. Prove that, for any subset S of V such that |S| > (2 n+1)/ ...
Exact discovery of length-range motifs
... The algorithm starts by selecting a random set of Nr reference points. The algorithm works in two phases: The first phase (called hereafter referencing phase) is used to calculate both the upper limit on best motif distance and a lower limit on distances of all possible pairs. During this phase, dis ...
... The algorithm starts by selecting a random set of Nr reference points. The algorithm works in two phases: The first phase (called hereafter referencing phase) is used to calculate both the upper limit on best motif distance and a lower limit on distances of all possible pairs. During this phase, dis ...
Lecture Notes (pptx)
... Use the size of the input rather than the input itself – n Count the number of “basic steps” rather than computing exact time Ignore multiplicative constants and small inputs (order-of, big-O) Determine number of steps for either worst-case expected-case These assumptions allow us to analyze algor ...
... Use the size of the input rather than the input itself – n Count the number of “basic steps” rather than computing exact time Ignore multiplicative constants and small inputs (order-of, big-O) Determine number of steps for either worst-case expected-case These assumptions allow us to analyze algor ...
Document
... Siepel A. Exact algorithms for the reversal median problem. Master Thesis, University of New Mexico, 2001. Yancopoulos S., Attie O., Friedberg R. Efficient sorting of genomic permutations by translocation, inversion and block exchange. Bioinformatics ...
... Siepel A. Exact algorithms for the reversal median problem. Master Thesis, University of New Mexico, 2001. Yancopoulos S., Attie O., Friedberg R. Efficient sorting of genomic permutations by translocation, inversion and block exchange. Bioinformatics ...
Travelling salesman problem
The travelling salesman problem (TSP) asks the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science.TSP is a special case of the travelling purchaser problem and the Vehicle routing problem.In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether the graph has any tour shorter than L) belongs to the class of NP-complete problems. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially (perhaps, specifically, exponentially) with the number of cities.The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved completely and even problems with millions of cities can be approximated within a small fraction of 1%.The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments. The TSP also appears in astronomy, as astronomers observing many sources will want to minimise the time spent slewing the telescope between the sources. In many applications, additional constraints such as limited resources or time windows may be imposed.