show that is nondecreasing along each row and column use this to show that all entries in can be computed in ( time which of the dynamic programming algorithms can be solved in ( using these techniqueswrite routine to reconstruct the shortest paths from the algorithm in section the binomial coefficients (nkcan be defined recursively as followsc( (nn andfor nc(nkc( kc( write function and give an analysis of the running time to compute the binomial coefficients as followsa recursively using dynamic programming write the routines to perform insertiondeletionand searching in skip lists give formal proof that the expected time for the skip list operations is (log na examine the random-number generator on your system how random is itb figure shows routine to flip coinassuming that random returns an integer (which is prevalent in many systemswhat is the expected performance of the skip list algorithms if the random-number generator uses modulus of the form (which is unfortunately prevalent on many systems) use the exponentiation algorithm to prove that (mod show how the randomized primality test works for with several choices of implement the turnpike reconstruction algorithm two point sets are homometric if they yield the same distance set and are not rotations of each other the following distance set gives two distinct point sets find the two point sets extend the reconstruction algorithm to find all homometric point sets given distance set show the result of - pruning of the tree in figure does the code in figure implement pruning or pruningb implement the complementary routine coinside flipifrandom = return headselse return tailsfigure questionable coin flipper