Algorithm Accuracy and Execution Time, CNF and DNF, Derivatives and Exponential Time Hypothesis

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Algorithm Accuracy and Execution Time, CNF and DNF, Derivatives and Exponential Time Hypothesis


Algorithm accuracy and execution time

The correctness of the algorithm depends on the input. Therefore, to get the desired result, you need to provide the input that the algorithm expects. Otherwise, it will take a long time to run. Even though the execution time of the algorithm is correct, it usually increases or increases with the input size, but it can also change. Examples of execution times include constant, linear, exponential, logarithmic execution time, and quadratic execution time.

(I). Constant execution time – This means that regardless of the input size provided to the function, its time complexity remains the same.

(ii) Linear execution time increases linearly with input size.

(iii) Its operation is expressed as 2^n, where n is the input size, so the exponential execution time increases exponentially.


n≧3…. (2^3)((3^o(1))

Since the distance traveled is the input and the time taken is the output, it is possible to draw a distance versus time curve. In the above seller scenario, you should know that this algorithm implements both exponential and constant execution time. Therefore, time is not constant for the traveling salesman, and exponential travel time prevails. Therefore, runtime as output is not effectively false or trusted input. The number of cities he visits does not adjust the travel time, resulting in an exponential curve. Still, the exponential plot suggests that the execution time is the worst case, since the time complexity is not easily compromised regardless of the number of cities visited. a) Create a CNF that computes d(x,y,z).

In computer science, most problems are solved using Boolean formulas in associative and disjunctive normal form. The normal form of the subjunctive expresses expressions as a combination of clauses using AND or OR. When the negation of a false truth assignment occurs, we assume a conjunction (^). This is a negation that represents the usual form of disjunction, representing the disjunction (v) of all satisfied truth assignments.

Let’s express it as d(x,y,z)=xy+yz

You’ll need to find DNF here and disable it later…

-d=(x^-y^z)v(x^-y^-z)v(-x^-y^z) DNF

∆CNF of d= -(DNF)…

Deny DNF and find CNF

CNF=(-x+y-z)^(-x+y+z)^(x+y-z)…then solve to find the subjunctive literal.


B). The first step is to derive a function of ‘f’ whose variables are (y,z,x).

Next, let “x” be a constant.



/ ̄d(xy)=/ ̄ydx

Since x.y=y _/ ̄dx, x.y is the final output of d(y,x).

Δy_/ ̄dx=-(y_/ ̄dx)

Derivation: y_/ ̄dx=yx^x


Here we need to find the satisfiability element and equate it to that variable.


CNF=x^y….Vi=_/ ̄dxy

_/ ̄{d,0,3}



Sun 3-Sat… Possibility: Var(d3)=(x1+x3+y1)^n/3. 3. Note that the size of a CNF is the number of occurrences of its literal. The size of the smallest CNF that computes the Boolean function f is the smallest

a).For fixed Boolean inputs, show that f(x,y,z)=0 if VLj=0 for some j=1…m.

Find the derivative of f(y,z,x).


d(z,y,z))….This happens to be the first equation (i).

We need to decompose and solve its satisfiable values ​​using f(jklm).

Let f(jklm)=jk+lm…expand.




B). (x.-(y+z)+u=f(x,y,z).

However, f(x,y,z)

DNF= -1(



The logical product of (x,y,z) is (zx+yz)-x(y+z).

Solution = (z.x+y.z)-1(x.y+x.z)

In your own words, what does the exponential time hypothesis mean?

Time complexity is an algorithm used to represent the time it takes for an algorithm to complete its execution. Exponential time is a computational assumption that applies when finding literal Boolean expressions in associative and disjunctive normal form when polynomial-time problems are not equivalent to nondeterministic problems.

The exponential time hypothesis assumes that, in the worst case, nondeterministic polynomial completeness problems cannot be solved in quasi-exponential time. Therefore, if the exponential time hypothesis is true, it means that the polynomial is not equal to non-deterministic, but it is a strong enough statement to solve and prove the “three satisfiability” problem. will be called. This can be used to show that many computational problems are of comparable complexity, that is, they all work as long as one of them has a time less than or equal to an exponential function.

Similarly, to account for the exponential time hypothesis, we apply the satisfiability problem by bringing the Boolean expression ø into associative normal form when the clause is a conjunction of clauses that are disjuncts of literals . Each literal is either a boolean variable (x) or negation (-x). This algorithm consists of a “K-satisfiability problem”. K-satisfiability tests whether a Boolean expression in associative normal form with at most k variables per clause can be made true by assigning a Boolean value to that variable.

Similarly, K-satisfiability is a nondeterministic polynomial that is hard when k is greater than or equal to 3, so P≠PN indicates that there is no polynomial-time algorithm for k-satisfiability and k≥3. A satisfactory assignment to ø is one with a value of true or false.

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