Data Structure And Algorithm 2

What is a graph?

A set of nodes connected in some way by a set of edges

What's a simple cycle?

All edges and vertices are distinct except start & finish

What is a directed graph?

All edges are directed

What is the ith level?

All nodes of depth i

What does BFS use to find the shortest path?

A Heap

Write the line of code you'd need at the top of your source code file to use STL template functions e.g., sort

#include

What's the equation of MAD?\nWhat are the components?

((ay + b) mod P)\n\n\nP = prime number\na and b = non-negative integers [0, P-1]

When ordering complete binary tree nodes, what number is the root?

-

What is Stack topIndex when the stack is empty?

-1

λ should definately be less than or equal to this with quadratic probing

.5

λ should be approximately equal to this for separate chaining

1

What are the 3 conditions that linear probe searching stops at?

1. An item is found with the key\n2. an empty cell is found\n3. N Cells have been unsuccessfully probed

What are the two parts of a hash function?

1. Generate hash code (keys -> integers)\n2. Apply a compression fuction (integers -> [0, N-1])

What's Dijkstra's algorithm

1. Start at a node and calculate the lowest distance to opposite nodes\n2. Choose Lowest node to visit\n3. Repeat using all nodes

What is the algorithm for rotating a binary tree root to the right

1. copy root.left --> temp\n2. set root.left --> temp.right\n3. set temp.right --> root\n4 set root --> temp

What's the maximum number of nodes at level i?

2^i

Length of path ABCD in an unweighted graph

3

QuickSort

3 parts: less than pivot, pivot, greater than pivot\n\nrecursively sort\n\nrandom pivot selection: expected runtime nlogn\nworst is still n^2

The minimum number of items a leaf node of a B-Tree of order 7 can store

4

What does a comparator object return?

>0 if more than\n<0 if less than\n0 if equal to

What is an AVL tree?

A Binary tree with a height balance property so that the height of children of any node differs at most by 1

What is a cycle?

A circular path beginning & ending on the same vertex

What is a binary heap?

A complete binary tree ordered from top to bottom so that the children keys are larger than their parents

What is the edge-list structure?

A list of nodes\nA list of edges\nEach edge has a pointer to two nodes

What is a connected component?

A maximum connected subgraph

What is a map?

A model of a searchable collection of key-value entries

What does "Connected" mean?

A path between every pair of vertices

What is a path?

A sequence of alternating vertices and edges

What's a spanning tree?

A spanning subgraph that is a tree

This graph representation is space-efficient for sparse graphs

Adjacency Lists

This graph representation allows the presence or absence of an edge to be checked in constant time

Adjacency Matrix

What does Method Overloading do?

Allows there to be two methods with the same name but different signatures

What is an Interface?

An interface contains only the signatures of methods properties

Define a directed edge

An ordered pair of vertices

Why are we taking this course?

Because Bill said so.

Why are hashtables better at implementing maps than arrays?

Better performance

Strings algs

Brute Force\nBM - Last occurence function\nKMP - Failure Function

DAG

directed, acyclic graph\n\ncan find topological ordering

What's a spanning subgraph?

Contains all vertices of a bigger graph

A directed graph with no directed edges

DAG

List

DLL or SLL\nawareness of "before" and "after"\nfast to manipulate, slow to access

A way of organizing information in memory

Data Structure

What is a Hash Table

Data structure that maps keys to values

What does DAG stand for?

Directed Acyclic Graph

How does the hashing of a Hash Table work?

Distributes key/value pairs across an array of buckets\n\n\nhash = hashfunc(key)\nindex = hash % array_size\n\n\n`hash` is independent of array size, and is reduced to an index (number between 0 and array_size - 1) using modulo operator

What is separate chaining?

Each bucket stores a list-based map

Structures

Edge List\nAdjacency List\nAdjacency Matrix

What is an incident edge?

Edges belonging to a vertex

What's another name for a key-value pair?

Entry

(T/F) For a B-Tree stored on a disk, the maximum number of items stored in a leaf node should be chosen to be as small as possible

FALSE

(T/F) The subtrees of a B-Tree can differ in height, but by no more than 1

FALSE

(T/F) With collision resolution strategy f(i) = i, insertions can fail if λ = 0.95

FALSE

How does a Queue work? What are its main methods?

FIFO: first in, first out\nOR\nLILO: last in, last out\n\n\nMethods:\nOffer : adds element to back of queue\nPeek: returns element at front of queue\nPoll: takes element at front of q

Queue

FIFO\nO(1)

Stack

FILO\nO(1)

KMP

Failure Function:\n- length of prefix that is suffix of current position\n\ni and j counter, j updated by F(j-1) if j > 0 otherwise j = j + 1\n\nO(m + n)

A set of vertices and a set of edges

Graph

If you wanted a non-recursive sort with\nO(n log n) worst-case running time, which one would you use?

Heapsort

An algorithm for finding optimal prefix codes is named after him

Huffman

What extra steps need to be taken for an AVL tree during insertion?

If the unbalance is right-right/left-left then you rotate the first node that becomes unbalanced and it's unbalanced child and shift the child node to balance\nIf the unbalance is left-right/right-left then you rotate the first unbalanced node and it's unbalanced child and then use the left-left/right-right balance

How does up-heap-bubbling work?

Insert as last node\nSwap node and parent\nRepeat until order is restored

If an array is nearly sorted, the best sorting algorithm to use is _____________

Insertion Sort

An algorithm for finding minimu spanning trees is named after him

Kruskal

How does the Stack work? What are its main methods?

LIFO: last in, first out \nOR \nFILO: first in, last out \n\n\nMethods: \nPush(T item): adds item to top of stack \nPop(): grabs item from top of stack \nPeek: returns top item of stack

A node with no children in a tree

Leaf

When ordering complete binary tree nodes, what are the values of the left and right children?

Left: 2*parent+1\nRight: 2* parent+2

What does MAD stand for?

Multiply, Add, Divide

The number of edges in a tree with N nodes

N-1

What's the problem with edge list structure?

Nodes can't find what edges they're connected to without scanning all edges

What are endpoints?

Nodes of an edge

Average-case run time for insert in a hash table

O(1)

What's the order of growth for insertions, removals, and searches of hash tables?

O(1)

DAG sorting

O(V+E)

DFS / BFS running time

O(V+E)

k-D tree building time

O(knlogn)

Average-case run time for contains/find in a binary search tree

O(log n)

What is the order of growth for up-heap and down-heap bubbling?

O(log n)

Worst-case run time for find in an AVL tree

O(log n)

Worst-case run time for find in an STL set

O(log n)

Bucket sort

O(n + N)!!!\n\nplace all elements into buckets\nthen remove from buckets

What is the order of growth for heap building when implementing a priority queue using a heap?

O(n log n)

What is the order of growth for heap sorting when implementing a priority queue using a heap?

O(n log n)

Worst-case run time for clone or makeEmpty (both are traversals) in a binary search tree

O(n)

Worst-case run time for insert in a binary search tree

O(n)

Worst-case run time for insert in a hash table

O(n)

Worst-case run time for insert in a hash table using linear probing

O(n)

standard trie

O(n) space where n is the total length of all strings\n\nheight is longest string\n\nfinding string: O(dm) - note not dependent on n

What is the Order of Growth for PQ-Sort?

O(n^2)

If the running time of an algorithm increases by a factor of 8 whenever the size of the input increases by a factor of 2, the growth rate of the algorithm is ___________

O(n³)

What is the adjacency list structure?

On top of the edge-list structure, edges have 2 more pointers: to their references in the sequences for both of the nodes\nEach node has a sequence of edges (Incidence Sequence)\nNodes also have a pointer to their edge collections

What is Polymorphism?

Permits parent class to dynamically take behaviour of descendent class

What is linear probing?

Placing colliding items in the next available cell (circular array)

The fastest sort using C++ on a large, randomly-ordered array (taking into account both growth rate and constant factors) is ______________

Quicksort

Solves an instance of a problem by first solving one or more smaller instances

Recursive Algorithm

How does down-heap bubbling work?

Replace root with last node on heap\nSwap node with smallest key child until order is restored

What are the main operations for maps?

Searching for things\nInserting new things\nDeleting things

What are the two types of PQ-Sort?

Selection Sort\nInsertion Sort

What are the two ways of handling collisions?

Separate Chaining\nLinear Probing

The mathematical entity a queue is based on

Sequence

What is the Adjacency Matrix Structure?

Similar to the edge list structure\nThere's also another table labelled with nodes that points to edges\nEdges do not point to these references

What is graph traversal?

Systematic procedure visiting all nodes

Give the formal definition of Big-Oh, i.e., what does T(n) = O(f(n)) mean precisely?

T(n) = O(f(n)) if there exists a constant c & another n₀ that when used with T(n) allows f(n) to be an upper bound for all n ≥ n₀.\ni.e. T(n) ≤ c ∗ f(n)

If T(n) is Big-Oh of n² but not Big-Theta of n², what can we conclude?

T(n) = o(n²)

(T/F) The insert operation takes O(log n) worst-case but only constant time on average in a heap.

TRUE

(T/F) The subtrees of an AVL Tree can differ in height, but by no more than 1

TRUE

(T/F) With collision resolution strategy f(i) = i², insertions can fail if λ = 0.51

TRUE

What happens if you insert a key value pair and an entry is found that already has the key in a map?

That entry's data is replaced and the old data is returned

What is the load factor?

The expected number of entries in a bucket\nDescribed the probability of a collision

What is height?

The longest path from the root to a leaf

What is depth?

The number of ancestors

What does the override keyword mean?

The override modifier is required to extend or modify the abstract or virtual implementation of an inherited method property

What is Insertion Sort?

The variant of PQ-Sort which uses a sorted sequence

What is Selection Sort?

The variant of PQ-Sort which uses an unsorted sequence

How are arrays like maps?

They have an integer key for the data they hold. THey are special cases of maps

What's an important part of keys in maps?

They must be unique

What is the term for insertion in a binary heap?

Up-Heap Bubbling

What is PQ-Sort?

Uses priority queue to sort

What is the simple implementation of a map?

Using a doubly linked list

What is a subgraph?

Vertices and edges that are a subset of a bigger graph

When is a search table the best implementation for an ordered map?

When searches are the primary operation and the maps are small

When is the simple implementation of a map good?

When the map is small

suffix trie

all suffixes of a potential trie, compactable\n\npattern matching O(dm) with d being length of alphabet

What is a simple path?

all vertices and edges are distinct

What does the remove function require for linear probing hash tables?

an AVAILABLE item

What is a tree?

an undirected connected graph with no cycles

What is a forest?

an undirected graph that has no cycles

Vector

array\nhas rank\ncircular?\nslow to change, fast to access

What is a bucket array?

arrays assigned to each array index and hold key value pairs

2D range search

auxiliary structures are subtrees root v sorted with y\n\nRange search: O(log^2 n + s)\n\nSpace nlogn (getting worse)

AVL

balanced tree\n\nlogn access

DFS BFS

both take O(V + E) time

DP example change of coins

building network from the bottom up\n\nstoring solutions and using them later

Adjacency List

builds on Edge list, \n\nVertex now contains container of edges to link to, and edges link to that container\n\nruntime now dependent on deg(v) or minimum of deg(u),deg(v)

Adjacency matrix

builds on edge list, \n\nedges are stored in a vertex matrix, each vertex has an index\n\nnull = no link, object = link present\n\nareAdjacent no O(1), rest in n and space O(n^2)

AVL deletion

delete: basic if leaf node, otherwise left most node of right subtree becomes node \n\nz is first unbalanced, y and x are children of subtree with LARGER HEIGHT\n\nlogn steps worst case

This STL container class supports random access with an index, and can grow/shrink at either end

deque

What can BFS do?

determine if a graph is connected\nComputer connected components\nCompute a spanning forest

What can DFS do?

determine whether G is connected\nfind and report connected components\ncompute a spanning forest

What are examples of maps?

dictionaries\nassociative memory

This collision resolution strategy uses a second hash function to probe at intervals that depend on the value being inserted

double hashing

What is the term for removal in a binary hep?

down-heap bubbling

topological ordering of DAG

enqueue with in-degree 0\n\ndequeue with 0, reduce all linked vertices in-degree and store its rank. Repeat with new vertices that are no 0\n\nO(V+E)

What is the ceiling entry in an ordered map?

entry with the smallest key >= k

1D range tree

findall in O(logn + s)\n\nO(n) space

Euler tour

general tour for traversing a tree

A function that converts a key value into an array index

hash

What two components do hash tables have?

hash function\nbucket array

Heapsort

implementing the sequence as a heap.\n\neach insertion is logn, and retrieving is also logn. For n items, becomes nlogn\n\nin place in an array shifting the "barrier" between heap and unsorted sequence

AVL insertion

insert in correct location\nfind z if unbalanced, y and z are children\ndetermine abc, switch around\nconstant time

A class MUST implement all methods defined in the

interface

A class can implement multiple...

interfaces

Graph important methods

isAdjacent(v,w), adjacentVertices(v)\n\ndeleteVertex(v)\n\nthese will be compared frequently

Tree implementation (vector)

layers from top to bottom, children are 2xrank and + 1 for right child

Binary Search Trees

left < parent < right\nsearch is O(h)\n\ninsertion: trivial\nremoval: take left most node of right-subtree and replace (if has leaf node, just delete)

Radix Sort

lexicographically sorting pairs\n\nfrom back to front\n\nruns O(d(n+N)) with d being the length or depth of sorting

This ADT manages data by position

list

The number of items in a hash table divided by the table size

load factor

AVL proof of height

looking at number of nodes stored when balanced with respect to height.\n\nn(h) > 2 * n(h-2) (always more nodes than this)\n\nso logn

This STL container class always stores key-value pairs ordered by key

map

This STL container class can use a string like an array index (with [ ]) for retrieving a value

map

The main advantage of more sophisticated (than a heap) priority queue implementations like binomial queues and fibonnacci heaps is that they support the ____________ operation efficiently (less than linear time).

meld

Sequences

methods of List and vector, implementation can vary

Spanning tree

of a connected, undirected graph is a subgraph with all vertecies but no cycles and is fully connected

a class can only have...

one parent class

A function object (a.k.a functor) is an instance of a struct or class that must define which member function?

operator()

Heap

parent < children\nalways balanced -> O(logn)\nbinary tree implementation or array\nbottom up construction takes O(n) because of maximum path

in place quicksort

pivot and flipping elements over the pivot if smaller

Insertion Sort (for PQ)

places each item sortedly in the sequence\n\nruns in O(n^2) but not always

k-D tree

point or region based. point is balanced, height logn\n\nnearest neighbour is O(n) worst but O(logn) expected

This phenomenon makes linear probing take longer than random probing as λ increases

primary clustering

QuadTree

region in quads, can be greatly impalanced\n\nO(sqrt(n) + s) for 2D

Selection Sort (for PQ)

retrieves the smallest item from a sequence. \n\nRuns in O(n^2)

What does remove return in a map?

returns the value if it's found\nreturns null if not

Boyer Moore

scanning from back of pattern\n\nlast Occurence function for each letter in alphabet \n\njumpy by according m to align or entire length if not in pattern or 1 if neither\n\nstill O(mn)

The mathematical entity the dictionary ADT is based on

set

This STL container class can store any type for which an ordering (e.g., operator<) is defined and traverse the container in sorted order using an iterator

set

This STL container class lets you efficiently insert, find, and remove items based on their value, not position, and requires them to be comparable using operator

set

Dynamic programming

similar to DC bt more powerful\n\nsubproblems can overlap - store solutions\n\n3 properties:\n- must be able to break problem down\n- optimal subproblems must give optimal solution\n- overlapping is GOOD

Sets (merging)

smallest element from the front of both sorted parts, when one runs out, just let the other one finish

What is the higher entry?

smallest key > k

divide and conquer

solving distinct sub-problems and then combining

Lookup Table

sorted array\nwith binary search: logn access

priority queue

sorted queue with rank or weight\ntotal order relation\ncan return smallest key

compact trie

space is O(s) where s is the number of strings

Merge Sort

splitting into two parts, recursively sorting, then merging\n\nMerging: smallest object of each, proceed, merge takes O(n1+n2)\n\ntotal nlogn

integer multiplication (D&C)

splitting large and small orders\n\nexpanding the multiplication\n\nusing a substitution for the mixed part, then storing answers and O(n^1.585)

Trie

standard -> compressed -> compact\n\nuse special character to deal with prefix

What does the static keyword mean?

static means that the method or class belongs to the TYPE and not the INSTANCE of a class. A static member cannot be referenced through an instance. Instead it is referenced through the type name

a child class exhibits AT LEAST...

the same behaviour as the parent class

Dictionary

umbrella for sorted search methods\nKEYS\nhas methods for key before and key after

Which STL class lets you store key-value pairs using primitive types or strings as keys in a hash table without needing to define a structured type or your own hash function.

unordered_map

Graph

vertices and edges\n\ndirected\n\ncycles

Edge List

vertices are objects in list\n\nedges are objects in container, link to vertices\n\nruntime mostly O(E)

What are 3 BFS properties?

visits all vertices and edges\ntraverse edges frmo a spanning tree\nthe shortest path from a node to another node

Why is does the hash table use modulo with prime numbers?

when you're repeating over a set space, you're going to provide an even distribution across your hash space.

If an algorithm can reduce the size of a problem by a constant factor (e.g. divide it by 2) in constant time, the growth rate of the algorithm is __________

θ(log n)

A tight, worst-case Big-Oh time bound for the buildHeap operation is ____________

θ(n)

What is the worst-case running time of Quicksort? Is there a simple ordering of the data that could cause this worst case, using the book's implementation of Quicksort?

θ(n²), No

What is the average-case growth rate for Shellsort using Hibbard's increments?

θ(n⁵/⁴)