If T₁ (n) and T₂(n) are the time complexities of two program fragments På and P₂, where T₁(n)= 0(f(n)) and T₂(n) = 0(g(n)), find T₁(n) +T₂(n) and T₁(n).T₂(n).
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- The binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written recursively.a) Consider a recursive function to return the Number of Binary Digits in the Binary Representation of a Positive Decimal Integer (n) using a recursive algorithm. int Process (int n) { if (n == 1) return 1; else return (Extra() + Process (n/4) + Process (n/4)); } Given that Extra(n) is a function of O(n)1) Find T(n) = number of arithmetic operations. 2) Calculate the complexity of this algorithm using Back Substitution.Compute for the time complexity (in milliseconds) of each letter of the given program segment below, then get the total time complexity. int main( ) { int a, b, c, d; for (a=0;a<10;a++) for (b=0;b<10;b++) for (c=0;c<10;c++) for (d=0;d<10;d++) { If ((a==0)&&(b==0)&&(c==0)&&(d==0)) { a=1, b=2,c=3,d=4; } cout<<a<<b<<c<<d; } }
- Determine the correct running times for each recursive functions 1. If T(n) = T([n/2]) + 1, then T(n) belongs to a) O(log2n) b) O(n) c) Ω(log2n) d) Ω(n) e) Ω (√n) 2. If T(n) = T(n-1) + n, then T(n) belongs to a)O(log2n) b)O(n) c) Ω(log2n) d) Ω(n) e) Θ(n2) 3. If T(n) = 2T([n/2] + √n, then T(n) belongs to a) O(n log2n) b) O(n) c) Ω(n log2n) d) Ω(n) e) Θ(n2) 4. If T(n) = 16T([n/4]) + n2 , then T(n) belongs to a) O(n log2n) b) O(n) c) Ω(n log2n) d) Ω(n) e) Θ(n2)Develop two algorithms, one based on a loop structure and the other on a recursive structure, to print the daily salary of a worker who each day is paid 2.5 times the previous day’s salary (starting with one penny for the first day’s work) for a 30-day period. What problems relating to number storage are you likely to encounter if you implement your solutions on an actual machine?Perform an experimental analysis on the two algorithms & Visualize their running times as a function of the input size with a log-log chart. 1. prefixAverage12. prefixAverage2. RPgramming language: Java
- Create an Algorithm for Synchronised Simulation of a Number. in: a group of human participants H; a group of computer users C out: winner w constant: nmin = minimum number; nmax = maximum numberpredicted number g locallyFor each of the following three program fragments:a. Give an analysis of the running time (Big-Oh will do).b. Implement the code in the language of your choice, and give the running time for several values of N (100, 500, 1000, 5000).c. Compare your analysis with the actual running times. (1) sum = 0; for( i = 0; i < n; ++i )++sum; (2) sum = 0; for( i = 0; i < n; ++i ) for( j = 0; j < n * n; ++j ) ++sum;(3) sum = 0; for( i = 0; i < n; ++i ) for( j = 0; j < i * i; ++j ) for( k = 0; k < j; ++k ) ++sum;Describe an algorithm that given two regular expressions R1 and R2 outputs YES if andonly if every string generated by R1 is also generated by R2. (I.e., output YES exactlywhen the set of strings generated by R1 is a subset of the set of strings generated by R2.)
- The fibonacci sequence of numbers are as follows: 0,1,1,2,3,5,8,13,21,34,... where F0=0, F1=1, F2=1, F3=2 and so on. The recursive function definition is F0=0 F1=1 Fn=F(n-2)+F(n-1) (if n>1) Write a recursive function, fib, to implement this definition in MATLAB. The function will receive a single input (integer), n, and return one integer alue that is the nth Fibonacci number.for C++ write a progam for the greatest common divisor of integers x and y is the largest integer that evenly divides both x and y. Write a recursive function gcd that returns the greatest common divisor of x and y. The gcd of x and y is defined recursively as follows. If y is 0 the answer is x; otherwise gcd(x, y) is gcd(y, x%y).Computer Science Find the correct asymptotic complexity of an algorithm with runtime T(n) and step-by-step workings where T(x) = O(n) + T((2*x)/8) Assume x and n are the same. If it is different, how will the working be like?