Chapter 2 Getting Started

Chapter 2 Getting Started

2.1 Insertion sort

    Input: A sequence of n numbers <a1, a2, ... , an>.

    Output: A permutation (reordering) <a1', a2', ... , an'> of the input sequence such that a1' <= a2' <= ... <= an'.

        Insertion-Sort(A) for j <- 2 to length(A) do key = A[j] //Insert A[j] into the sorted sequence A[1...j-1] i <- j - 1 while i > 0 and A[i] > key do A[i+1] = A[i] i <- i - 1 A[i + 1] <- key

    Loop invariants and the correctness of insertion sort.(Initialization, Maintennance, Termination)

    Pseudocode conventions.

2.2 Analyzing algorithms

        We shall assume a generic one-processor, random-access machine (RAM) model of computation as our implementation technology.

    Analysis of insertion sort

         The best case: (the array is already sorted) O(n)

         The worst case: (the array is in reverse sorted) O(n²)

    Order of growth

         We usually consider one algorithm to be more efficient than another if its worst-case running time has a  lower order of growth.

2.3 Designing algorithms

    2.3.1 The divide-and-conquer approach

        Divide the problem into a number of subproblems.

        Conquer the subproblems by solving them recursively.

        Combine the solutions to the subproblems into the solution for the original problem.