Pop Quiz 2: Answers. Data structures Recursion Big-O

Transcription

1 Pop Quiz 2: Answers Data structures Recursion Big-O

2 Data Structures What are the 4 main data structures? Set, sequence, tree, graph Which data structures are ordered? sequence, tree Which are unordered? set, tree, graph Which may be both? tree What is a bag? a set with duplicate values What is a multiset? a set with duplicate values What is a deque? a double ended queue 19/06/2015 DFR / Pop Quiz 2 2

3 Data Structures Define a set an unordered collection of unique entities Define a sequence. an ordered collection of entities Can a tree with maximum out degree 2 be unordered? Yes Define a BST. an ordered binary tree (i.e. an ordered tree with maximum 2 children) where all the values to the left of a node are less than the value at the node and all the values to the right of the node are greater than the value at the node Define an AVL tree. a BST tree where the balance factor is defined to be height(lc) height(rc) < 2 for any node Define a Queue. a sequence where values are added at the end of the sequence and removed from the beginning of the sequence (FIFO. First in first out) 19/06/2015 DFR / Pop Quiz 2 3

4 Recursion Give a recursive definition of a sequence. sequence head tail ::= head tail empty ::= element (entity) ::= sequence Give a recursive definition of a Binary Tree. BT Node LeftChild RightChild ::= LeftChild Node RightChild empty ::= element (entity) ::= BT ::= BT What is the point of recursion? since some entities (sequence, BT above) may be view as inherently recursive, recursive definitions may be stated which in turn define how operations on these structures may be encoded, leading to more compact code. In the case of the BT it is more difficult to give non-recursive definitions and code implementations of operations. Recursion is a tool used in certain areas oc computer science. What is one of the main operations used in recursive functions? cons Define recursion! A recursive definition is a definition PARTIALLY defined in terms of itself e.g. sequence and BT above A recursive function is a function which CONDITIONALLY calls itself 19/06/2015 DFR / Pop Quiz 2 4

5 Big-O What is Big-O? An INDICATOR of the performance of given algorithms, usually expressed as a function of the number of entities in the collection with respect to the time required for an operation. Usually expressed as a worstcase upper bound. What is the big-o for the following? for (i = 1 step 1 to n) g(i); n times the big-o for the finction g(i) What is the best case? Example? the LEAST amount of time required for an operation. E.g. add at the beginning of a sequence O(1) - constant What is the average case? Example? the AVERAGE amount of time required for an operation. E.g. add at the middle of a sequence O(n) What is the worst case? Example? the MOST amount of time required for an operation. E.g. add at the end of a sequence O(n) if there is an end reference, this may take O(1) constant time What factors affect big-o? Whether specific examples deal with best, average, worst case scenarios 19/06/2015 DFR / Pop Quiz 2 5

6 Big-O What is the average case big-o for the cardinality function on a sequence? O(n) How would you change this to O(1)? Add a counter to the system O(1) What is the advantage of lastel? makes adding at the end of a sequence O(1) Is for/for always O(n 2 )? NO What is the big-o for add using hashing? O(1) possibly slightly higher depending on the collision resolution used 19/06/2015 DFR / Pop Quiz 2 6

7 Big-O What are the 5 useful big-o s? O(log n), O(n), O(n log n), O(n 2 ), O(n 3 ) What are typical big-o s for search functions? O(n log n), O(n2) Which algorithms are O(n 3 )? Floyd, Warshall Which sort functions are O(n 2 )? Sorts using swaps; worst case quicksort How would you calculate big-o just by looking at the code? Examine each line of code and assign values to each operation: basic operations, assignment & comparisons O(1); loops O(n); if-then-else max (O(then part), O(else part). Functions as above. 19/06/2015 DFR / Pop Quiz 2 7

8 Abstraction Name the 3 forms of abstraction defined in this course. modelling; implementation; collection What is the point of implementation abstraction? to separate the implementation details (arrays/indexes or structures/pointers from the rest of the code when then becomes abstract operations. Usually all the get/set and create_element are the implementation dependent parts. An ADT is a mathematical model what is the point of this? This allows code implementation to be discussed in the abstract, independently of the programming language (and hence implementation structures). This also provides a set of conceptual tools for programming. What are the most common operations on a collection? add, remove, find element; cardinality (size), is_empty (predicate) What are the most common operations on an entity? create the entity; read and write attribute values for each attribute 19/06/2015 DFR / Pop Quiz 2 8