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Learn array-based sequences through recursion, Python functions, and cryptography. Dive into the three laws of recursion, writing Python functions for string reversal, sequence rearrangement, and integer sum. Explore array types and strategies for dynamic lists.
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Array-Based Sequences Hongfei Yan April 1, 2015 Array-Based Sequences
Recap: The Three Laws of Recursion • A recursive algorithm must have a base case. • A recursive algorithm must change its state and move toward the base case. • A recursive algorithm must call itself, recursively. Array-Based Sequences
Q: How many recursive calls are made when computing the sum of the list [2,4,6,8,10]? a)6 b)5 c)4 d)3
Q: Suppose you are going to write a recursive function to calculate the factorial of a number. fact(n) returns n * n-1 * n-2 * ... Where the factorial of zero is definded to be 1. What would be the most appropriate base case? a)n==0 b)n==1 c)n>=0 d)n<=1
Assignment #3: Recursion • C-4.16 Write a short recursive Python function that takes a character string s and outputs its reverse. For example, the reverse of 'pots&pans' would be 'snap&stop' . • C-4.19 Write a short recursive Python function that rearranges a sequence of integer values so that all the even values appear before all the odd values. • C-4.21 Suppose you are given an n-element sequence, S, containing distinct integers that are listed in increasing order. Given a number k, describe a recursive algorithm to find two integers in S that sum to k, if such a pair exists. What is the running time of your algorithm? Array-Based Sequences
Caesar: An interesting application of strings and lists is cryptography • Dynamic arrays and amortization • Sorting high scores for a game • Insertion sort • Tic tac toe Array-Based Sequences
5.5 Using Array-Based Sequences • 5.5.1 Storing High Scores for a Game • The first application we study is storing a sequence of high score entries for a video game. Array-Based Sequences
Python has built-in types, list, tuple, and str. Each of these sequence types supports indexing to access an individual element of a sequence, using a syntax such as A[i] Each of these types uses an array to represent the sequence. An array is a set of memory locations that can be addressed using consecutive indices, which, in Python, start with index 0. Python Sequence Classes A 0 1 2 n i Array-Based Sequences
Arrays of Characters or Object References • An array can store primitive elements, such as characters, giving us a compact array. • An array can also store references to objects. Array-Based Sequences
Compact Arrays • Primary support for compact arrays is in a module named array. • That module defines a class, also named array, providing compact storage for arrays of primitive data types. • The constructor for the array class requires a type code as a first parameter, which is a character that designates the type of data that will be stored in the array. Array-Based Sequences
Type Codes in the array Class • Python’s array class has the following type codes: Array-Based Sequences
Insertion • In an operation add(i, o), we need to make room for the new element by shifting forward the n - i elements A[i], …, A[n -1] • In the worst case (i =0), this takes O(n) time A 0 1 2 n i A 0 1 2 n i A o 0 1 2 n i Array-Based Sequences
A 0 1 2 n i A 0 1 2 n i A o 0 1 2 n i Element Removal • In an operation remove(i), we need to fill the hole left by the removed element by shifting backward the n - i -1 elements A[i +1], …, A[n -1] • In the worst case (i =0), this takes O(n) time Array-Based Sequences
Performance • In an array based implementation of a dynamic list: • The space used by the data structure is O(n) • Indexing the element at I takes O(1) time • addand removerun in O(n) time in worst case • In an addoperation, when the array is full, instead of throwing an exception, we can replace the array with a larger one… Array-Based Sequences
Growable Array-based Array List • In an add(o) operation (without an index), we could always add at the end • When the array is full, we replace the array with a larger one • How large should the new array be? • Incremental strategy: increase the size by a constant c • Doubling strategy: double the size Algorithmadd(o) ift=S.length 1then A new array of size … fori0ton-1do A[i] S[i] S A nn +1 S[n-1] o Array-Based Sequences
Comparison of the Strategies • compare the incremental strategy and the doubling strategy by analyzing the total time T(n) needed to perform a series of n add(o) operations • assume that we start with an empty stack represented by an array of size 1 • call amortized time of an add operation the average time taken by an add over the series of operations, i.e., T(n)/n Array-Based Sequences
Incremental Strategy Analysis • We replace the array k = n/c times • The total time T(n) of a series of n add operations is proportional to n + c + 2c + 3c + 4c + … + kc = n + c(1 + 2 + 3 + … + k) = n + ck(k + 1)/2 • Since c is a constant, T(n) is O(n +k2),i.e., O(n2) • The amortized time of an add operation is O(n) Array-Based Sequences
geometric series 2 4 1 1 8 Doubling Strategy Analysis • We replace the array k = log2n times • The total time T(n) of a series of n add operations is proportional to n + 1 + 2 + 4 + 8 + …+ 2k=n+2k + 1- 1= 3n - 1 • T(n) is O(n) • The amortized time of an add operation is O(1) Array-Based Sequences
Python Implementation Array-Based Sequences
5.6 Multidimensional Data Sets • For example, the two-dimensional data • might be stored in Python as follows. data = [ [22, 18, 709, 5, 33], [45, 32, 830, 120, 750], [4, 880, 45, 66, 61] ] Array-Based Sequences
Constructing a Multidimensional List • initialize a one-dimensional list, rely on a syntax such as data = [0] * n to create a list of n zeros. • list comprehension syntax for a 2D list of integers data = [ [0] c for j in range(r) ] Array-Based Sequences
Summary Array-Based Sequences
