Dynamic Programming

Let $A$ be an array containing integer values. The distance of $A$ is defined as the minimum number of elements in $A$ that must be replaced with another integer so that the resulting array is sorted in non-decreasing or...

Define R n to be the maximum amount earned by cutting a rod of length n meters into one or more pieces of integer length and selling them. For i > 0, let p[i] denotes the selling price of a rod whose length is i meters....

Assume that multiplying a matrix $${G_1}$$ of dimension $$p \times q$$ with another matrix $${G_2}$$ of dimension $$q \times r$$ requires $$pqr$$ scalar multiplications. Computing the product of $$n$$ matrices $${G_1}{G_...

Consider the weights and values of items listed below. Note that there is only one unit of each item. Item number Weight (in Kgs) Value (in Rupees) 1 10 60 2 7 28 3 4 20 4 2 24 The task is to pick a subset of these items...

Let $${A_1},\,{A_2},\,{A_3}$$ and $${A_4}$$ be four matrices of dimensions $$10 \times 5,\,5 \times 20,\,20 \times 10,$$ and $$10 \times 5,$$ respectively. The minimum number of scalar multiplications required to find th...

Let $${A_1},{A_2},{A_3},$$ and $${A_4}$$ be four matrices of dimensions $$10 \times 5,\,\,5 \times 20,\,\,20 \times 10,$$ and $$10 \times 5,\,$$ respectively. The minimum number of scalar multiplications required to find...

The Floyd-Warshall algorithm for all-pair shortest paths computation is based on

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre specified pair of integers i, j with i < j. Given a sho...

Consider two strings A = “qpqrr” and B = “pqprqrp”. Let x be the length of the longest common subsequence (not necessarily contiguous) between A and B and let y be the number of such longest common subsequences between A...

Four matrices $${M_1},\,\,\,{M_2},\,\,\,{M_3}$$ and $${M_4}$$ of dimensions $$p\,\,x\,\,q,\,\,\,\,\,q\,\,x\,\,e,\,\,\,\,\,r\,\,x\,\,s$$ and $$\,\,\,\,s\,\,x\,\,t$$ respectively can be multiplied in sevaral ways with diff...

Four matrices M 1 , M 2 , M 3 and M 4 of dimensions p $$\times$$ q, q $$\times$$ r, r $$\times$$ s and s $$\times$$ t respectively can be multiplied is several ways with different number of total scalar multiplications....

An algorithm to find the length of the longest monotonically increasing sequence of numbers in an array A[0:n−1] is given below. Let L i , denote the length of the longest monotonically increasing sequence starting at in...

A sub-sequence of a given sequence is just the given sequence with some elements (possibly none or all) left out. We are given two sequences X[m] and Y[n] of lengths m and n, respectively with indexes of X and Y starting...

A sub-sequence of a given sequence is just the given sequence with some elements (possibly none or all) left out. We are given two sequences X[m] and Y[n] of lengths m and n, respectively with indexes of X and Y starting...

The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a 1 ,a 2 ,a 3 ,…,a n } and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this pro...

The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a 1 ,a 2 ,a 3 ,…,a n } and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this pro...

Consider the following C functions: int f1(int n){ if(n == 0 || n == 1){ return n; } return (2 * f1(n - 1) + 3 * f1(n - 2)); } int f2(int n){ int i; int X[N], Y[N], Z[N]; X[0] = Y[0] = Z[0] = 0; X[1] = 1; Y[1] = 2; Z[1]...

Consider the following C - function: double foo(int n){ int i; double sum; if(n == 0) return 1.0; sum = 0.0; for (i = 0; i < n; i++){ sum += foo(i); } return sum; } Suppose we modify the above function foo() and store th...

Obtain the optimal binary search tree with equal probabilities for the identifier set (a 1 , a 2 , a 3 ) = ( if, stop, while)