Problem reduction and classes of equivalent problems:
The scientific advances frequently follow from the successful attempt to establish similarities and relationships among seemingly various problems. In mathematics, these relationships are formalized in two manners:
a) Problem A can be reduced to the problem B, A ≤ B, implies that when we can solve B, we can as well solve A
b) Problems A and B are equal, A ≤ B and B ≤ A, implies that when we can solve either problem, we can as well solve the other, generally with an equivalent investment of resources. While here we use “≤” in an intuitive sense, definition of polynomial reducibility, represented by “≤p”.
This section is all about reducing certain problems to others, and to characterizing various prominent classes of equivalent problems. Consider some illustrations.
Sorting as the key data management operation. Answering nearly any query regarding an unstructured set D of data items needs, at least, looking at each data item. When the set D is organized according to certain helpful structure, on other hand, many significant operations can be done more rapidly than in linear time. Sorting D according to some helpful total order, for illustration, enables binary search to determine, in logarithmic time, any query item given by its key. Since a data management operation of main significance, sorting has been studied extensively. In RAM model of computation, the worst case complexity of sorting is Θ(n log n). This fact can be employed in two ways to bind the complexity of many other troubles:
A) To show that some other trouble P can be resolved in time O(n log n), as P reduces to sorting, and
B) To show that some other trouble Q needs time Ω(n log n), as sorting reduces to the Q. Two illustrations:
i) Determining the median of n numbers x1, .., xn can be completed in time O(n log n). After sorting x1, .., xn in an array, we determine the median in constant time as x [n div 2]. The more sophisticated analysis exhibits that the median can be found out in linear time, however this doesn’t detract from the fact that we can rapidly and simply establish an upper bound of O(n log n).
ii) Determining the convex hull of n points in plane needs time Ω(n log n). The algorithm for computing the convex hull of n points in plane takes as input a set {(x1, y1), .. , (xn, yn)} of coordinates and delivers as output a sequence [(xi1, yi1), .. , (xik, yik)] of extremal points, that is, those on convex hull, ordered clock-wise or counter-clockwise. The figure below exhibits 5 unordered input points and ordered convex hull.
We now decrease the sorting problem of known complexity Ω(n log n) to convex hull problem as shown in next figure. Given a set {x1, .. , xn} of reals, we produce the set {(xi, xi2), .. , (xn, yn2)} of point coordinates. Due to convexity of function x2, all points lie on the convex hull. Therefore a convex hull algorithm returns the sequence of points ordered by rising x-coordinates. After dropping irrelevant y-coordinates, we get the sorted sequence [x1, .. , xn]. Therefore, a convex hull algorithm can be employed to get a ridiculously complicated sorting algorithm, however that is not our goal. The main aim is to show that the convex hull problem is at least as complicated as sorting - if it was significantly simpler, then sorting would be simpler, too.
The given figure describes problem reduction in common. Given a problem U of unknown complexity and a problem and algorithm K of the known complexity. The coding function c converts a given input I for one trouble to input I* for the other, and the decoding function d converts output O* to output O. This kind of problem reduction is of interest only if the complexity of c and of d is no bigger than the complexity of K, and if the input transformation I* = c(I) leaves the size n = | I | of the input data asymptotically unmodified: | I*| ∈ Θ ( n).| This is what we suppose in the following.
Deriving the upper bound. At left we build a solution to the problem U of the unknown complexity by detour U(I) = d( K(c(I))). Letting |c|, |K| and |d| signifies for the time needed for c, K and d, correspondingly, this detour confirms the upper bound |U| ≤ |c| + |K| + |d| . Beneath the usual supposition that |c| ≤ |K| and |d| ≤ |K|, this bound asymptotically decreases to |U| ≤ |K|. More explicitly: When we know that K(n), c(n), d(n) are all in O( f(n)), then as well U(n) ∈ O( f(n)).
Deriving the lower bound. At right we argue that the problem U of unknown complexity |U| is asymptotically at least as complicated as a problem K for which we know the lower bound, K(n) ∈ Ω( f(n)). Counter-intuitively, we decrease the known problem K to unknown problem U, obtaining the inequality |K| ≤ |c| + |U| + |d| and |U| ≥ |K| - |c | - |d|. When K(n) ∈ Ω( f(n)) and c and d are strictly less complex than K, that is, c(n) ∈ o(f(n)), d(n) ∈ o( f(n)), then we get the lower bound U(n) ∈ Ω( f(n)).
Latest technology based Theory of Computation Online Tutoring Assistance
Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in Theory of Computation help via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Theory of Computation, project ideas and tutorials. We provide email based Theory of Computation help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Theory of Computation. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Theory of Computation Homework help and assignment help services. They use their experience, as they have solved thousands of the Theory of Computation assignments, which may help you to solve your complex issues of Theory of Computation. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay.
tutorsglobe.com pitcher assignment help-homework help by online leaf modification tutors
We offer top-class Corporate Finance Assignment Help from PhD tutors at affordable prices to fetch top grades at fair prices.
www.tutorsglobe.com offers Imperative Language homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
Gravimetric Analysis tutorial all along with the key concepts of Types of Gravimeter analysis, Precipitation Gravimetric Analysis, Volatilization gravimetric analysis, Application of Gravimetry
the meaning of television is “to see from a distance” .the 1st illustration of actual television was provided through j.l.baird in uk (united kingdom) and c.f.jenkins in us (united states) around 1927.
tutorsglobe.com isogamy and heterogamy assignment help-homework help by online algae tutors
Theory and lecture notes of the Multiplier all along with the key concepts of the multiplier, Determining the Size of the Multiplier, planned expenditure, autonomous spending. Tutorsglobe offers homework help, assignment help and tutor’s assistance on the multiplier.
tutorsglobe.com standard entropy assignment help-homework help by online entropy tutors
Measurement of Temperature tutorial all along with the key concepts of Heat and Temperature, Types of Temperature Scales, Conversion Formulas, Constant Volume Gas Thermometer Scale, Standard Thermometer Scale, Types of Thermometer and Optical Pyrometers
tutorsglobe.com management of cash assignment help-homework help by online working capital management tutors
Synthesis and Reactions of Benzofuran and Benzothiophene tutorial all along with the key concepts of Physical and Chemical Properties, Synthesis of Benzofuran and Benzothiophenes
tutorsglobe.com atp as high energy compound assignment help-homework help by online high energy compounds tutors
The process that generates recombinations of gene through interchanging of corresponding segments among the nonsister chromatids of homologous chromosomes, is known as recombination of chromosomes.
Metabolisms and Behavior in the Environment tutorial all along with the key concepts of Metabolism of Insecticides, Botanicals, Chlorinated Hydrocarbons, Organophosphorus Compounds, Methyl-Dimethyl Carbamates, Behavior-Properties of Pesticides, Fate of Pesticide in the Atmosphere
Endocrine Coordination In Animals tutorial all along with the key concepts of Nature and functions of hormones, Hormone Secretors - Endocrine Glands, Thyroid, Gonads, Hormones from stomach and intestine and Pheromones
1954565
Questions Asked
3689
Tutors
1457196
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!