Data Structurez

Stack Application Examples

- Parsing for delimiters [-] Convert an expression to postfix or prefix - Evaluate a converted expression - Convert infix to postfix or prefix

Theta

- can be used when a function f can use g as it's upper and lower bounds (with different multiples)

General tree

A (general) tree T=(V,E') is defined on graph G=(V,E) Where E' is a subset of the original set of edges (E) If the size of V = n, then |E'| = n - 1. Note: T is connected and acyclic.

Doubly-linked list

A doubly-linked list is a data structure for implementing a list ADT, where each node has data, a pointer to the next node, and a pointer to the previous node. The list structure typically points to the first node and the last node. The doubly-linked list's first node is called the head, and the last node the tail.

Almost complete tree

Allow missing a few nodes from the right hand end on the bottom level.

Linked List

- list's head node representing the queue's front, and the list's tail node representing the queue's end. Pushing an item to the queue is performed by appending the item to the list

Worst-case algorithm analysis

- To analyze how runtime of an algorithm scales as the input size increases, we first determine how many operations the algorithm executes for a specific input size, N. e.g. - f(N) = 1+1+N(2+2) [for a for loop with an if condition] variable initialization, for loop (var init, bool, operation), if condition, action which would reduce to O(N)

Binary Search Trees

- "Search Tree" characteristic defines that everything in the LST is <= the root of the subtree. And everything in the RST is >= the root of the subtree. - recursive characteristic - an in-order traversal will yield the values in a sorted order

Queues: Linked implementation

- :) no size limits - :( forces sequential access

Lists: Linked implementation

- :) no size limits, dynamic allocation - :( sequential access. but not a huge problem because the list is sequential for insert, search, delete - special care needs to be taken around behavior at the front and end of the lists, as well as if there are duplicate elements (esp in the case of search)

Multi-linked list

useful to follow more than one kind of path through the data. Could track towards to pointers (maybe a horizontal and a vertical one)

Base case

- A case for which the answer is known (and can be expressed without recursion) - At some point, a recursive algorithm must describe how to actually do something - The if branch ends the recursion, known as the base case. The else branch has recursive calls. Such an if-else pattern is common in recursive methods.

Procedural languages

- C++/Java. iterative. full of for loops, assignment statements, etc... - conversely, some languages (like Prolog and SML) are written in a recursive fashion

Bound comparison notes

- making decisions based on g() values can and will lead to difference decisions than f() values (especially closer to the origin where the behavior is less predictable)

Stack overflow

- meaning a stack frame extends beyond the memory region allocated for stack, Stack overflow usually causes the program to crash and report an error like: stack overflow error or stack overflow exception.

Module 3 - Recursion

Goals: Define recursion informally and give simple examples. Define the three characteristics of recursion. Illustrate those characteristics using a specific example of recursion. Summarize how recursion works behind the scenes. Trace recursive execution of an example piece of code. Write recursive code to solve a simple, example problem. Analyze a complex piece problem and recommend the use of recursion or iterations.

Algorithm

- a sequence of steps for accomplishing a task. Linear search is a search algorithm that starts from the beginning of a list, and checks each element until the search key is found or the end of the list is reached.

Interpolation Search

- attempts to calculate where a search item might be based on previous data points. - at peak, can be O(log log n) if the keys are uniformly distributed, if not, it can deteriorate to a sequential search O(n)

Lower Bound

- big omega notation

Binary Search

- faster algorithm for searching a list if the list's elements are sorted and directly accessible (such as an array). Binary search first checks the middle element of the list. If the search key is found, the algorithm returns the matching location. If the search key is not found, the algorithm repeats the search on the remaining left sublist (if the search key was less than the middle element) or the remaining right sublist (if the search key was greater than the middle element). - suitable for small files (that can fit in memory) - O(log n) performance on average

Encapsulation

- information hiding and access control, - supported by data abstraction

Indexed Sequential Search

- jumps to specific indices (hopefully evenly split depending on the dataset) sequentially and then locates from there - optimizes sequential search

Linear/Sequential Search

- naive/brute force. O(n/2) on average

Counting sort

- not a comparison based sort, so can reach complexity of O(n), whereas floor for most comparison based sorts is O(nlogn)

Runtime

- the time the algorithm takes to execute. If each comparison takes 1 µs (1 microsecond), a linear search algorithm's runtime is up to 1 s to search a list with 1,000,000 elements, 10 s for 10,000,000 elements, and so on

Searching

- retrieving a record to be updated or to be used in a computation - range in complexity from O(1) to O(n)

Radix sort

- sorting algorithm designed specifically for integers. The algorithm makes use of a concept called buckets. - Any array of integer values can be subdivided into buckets by using the integer values' digits. A bucket is a collection of integer values that all share a particular digit value. Ex: Values 57, 97, 77, and 17 all have a 7 as the 1's digit, and would all be placed into bucket 7 when subdividing by the 1's digit. - The algorithm processes one digit at a time starting with the least significant digit and ending with the most significant. Two steps are needed for each digit. First, all array elements are placed into buckets based on the current digit's value. Then, the array is rebuilt by removing all elements from buckets, in order from lowest bucket to highest.

More Big O Notation Notes

- when comparing algorithms and functions, we usually define 'better' as either faster or taking up less space. Given the low cost of memory these days, speed is generally the characteristic focused on. - can break a piece of code into little components and derive the cost of each section (N for linear expressions, N^2 for quadratic, etc...). Then simplify according to the O Notation rules.

Basic Graph Theory

Context for follow-on material on binary trees. Pictoral way to represent information. Modeling tool.

Regular m-ary tree

Every node that has children has to have the same number of children (o or m).

Regular graph

If the degree of each vertex is the same

remove: max-heap

always a removal of the root, and is done by replacing the root with the last level's last node, and swapping that node with its greatest child until no max-heap property violation occurs. Because upon completion that node will occupy another node's location (which was swapped upwards), the tree height remains the minimum possible.

Max-heap

binary tree that maintains the simple property that a node's key is greater than or equal to the node's childrens' keys. (Actually, a max-heap may be any tree, but is commonly a binary tree). Because x ≥ y and y ≥ z implies x ≥ z, the property results in a node's key being greater than or equal to all the node's descendants' keys. Therefore, a max-heap's root always has the maximum key in the entire tree.

Parent pointers

can also keep track of an additional piece of data in each node (the parent), effectively making the structure a doubly linked list

Singly-linked list

data structure for implementing a list ADT, where each node has data and a pointer to the next node. The list structure typically has pointers to the list's first node and last node. A singly-linked list's first node is called the head, and the last node the tail.

Linked Binary Tree representation

dynamic allocation node specified as for doubly linked lists, except for child nodes. |left|data|right|

Rooted tree

in-degree will always be 1 in a rooted tree, so when talking about degree we are talking about out-degree (#of child vertices)

Issues with recursion

- can be inefficient, redundant, and resource hogging

Lists

- ordered collection of data items - all locations are available for insertion and deletion

Binary op-trees

- heterogeneous binary trees with operands as the leaves and operators as non-leaves. - operator with least precedence is the root of the tree

Types of binary trees

- A binary tree is full if every node contains 0 or 2 children. - A binary tree is complete if all levels, except possibly the last level, are completely full and all nodes in the last level are as far left as possible. - A binary tree is perfect, if all internal nodes have 2 children and all leaf nodes are at the same level. Minimum spanning tree, Subset of the graph's edges that connect all vertices in the graph together with the minimum sum of edge weights. The graph must be weighted and connected. A connected graph contains a path between every pair of vertices.

Threaded trees

- A binary tree is made threaded by making all right child pointers that would normally be NULL point to the in-order successor of the node (if it exists). - eliminates some of the redundancy of recursive traversals

Towers of Hanoi

- Only one disk can be moved at a time. - Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack. - No disk may be placed on top of a smaller disk. Goal: move the stack from one of the three posts to a separate post. The minimal number of moves required to solve a Tower of Hanoi puzzle is 2^n − 1, where n is the number of disks.

Sorting Terminology

- Record: unit or piece of data. (what gets sorted) - File: collection of records - Key: identifier, searchable. Primary (unique identifier, SSN). Secondary (not necessarily unique, address). - External Key: outside the record. - External vs. Internal Sorting - internal sorting can occur when the amount of data is small enough that the sort can be handled in main memory - external is when the files are too large. can chunk parts of the file and sort and merge in pieces.

Patterns of polynomial labels

- Sorts tend to fall into the frame of linear up to quadratic time. - Search techniques tend to fall in the constant up to linear time frame.

Queue

- an ADT in which items are inserted at the end of the queue and removed from the front of the queue. The queue push operation inserts an item at the end of the queue. FIFO. - can be implemented using a linked list, an array, or a vector.

Huffman Encoding

- application of binary trees - useful for data compression. can require fewer bits to represent information. - ciphers - sometimes substitution codes can be decoded via frequency analysis. huffman encoding cannot be decoded in this fashion. - binary code - more frequently used letters get shorter codes - less frequently used letters get longer codes

Stack Implementations

- array implementation - linked implementation - hybrid implementation - overflow (trying to insert into a full structure) vs. underflow (attempting to delete from an empty structure)

Queues: Array implementation

- can provide random access :) - options 1) move the front pointer upwards through the array when deleting (from the front of the queue) - fast - takes up more space because leaving the deleted item in the array, and can also lead to a false overflow because the garbage is still there 2) shift the array down every time the front pointer bumps - fast add O(1) - slow delete O(n) because the whole queue has to shift down !!3) Standard array implementation of a queue. better solution - move the pointers as in 1), and wrap around to the beginning of the array when out of space - array fully utilized - insert and delete both O(1)

List ADT

- common ADT for holding ordered data, having operations like append a data item, remove a data item, search whether a data item exists, and print the list

Binary trees

- each node has up to two children, known as a left child and a right child. - Leaf: A tree node with no children. - Internal node: A node with at least one child. - Parent: A node with a child is said to be that child's parent. A node's ancestors include the node's parent, the parent's parent, etc., up to the tree's root. - Root: The one tree node with no parent (the "top" node). - edge: the link from a node to a child. - depth: the number of edges on the path from the root to the node. The root node thus has depth 0. - level: all nodes with the same depth form a tree level. - height: the largest depth of any node. A tree with just one node has height 0.

Parsing for delimiters

- ex. - [(A+B)-C*{(2-3)C}]+5 - count left delimiters, count right delimiters. if equal - ok - also need to validate that the nesting is logical Stacks are a great way to solve this problem. Matching types is the same when you simply compare to the top of the stack, same with ending counts. Put left delimiters on the stack, drop everything else that isn't a right delimiter, for right delimiters make sure the top of the stack matches.

Abstract Data Types (ADTs)

- functional approach to describing information storage and the way we access that information. - helps us divorce the way we use the data and the way we store the data. - black box of what's inside (implementation), user can only see what goes in and comes out (interface).

TODO - define the below with the book definition Upper Bound

- g only has to be greater than f beyond a certain point - f will do no worse than g (its upper bound) - behavior harder to predict closer to the origin - O(g(n)) is an estimate, a label. Not perfectly accurate.

Convert Expression

- infix. standard PEMDAS that we are used to. operator is in the middle. - prefix. operator is pre. - postfix. operator is post.

Big O Notation

- mathematical way of describing how a function (running time of an algorithm) generally behaves in relation to the input size - all functions that have the same growth rate (as determined by the highest order term of the function) are characterized using the same notation - used in Computer Science to describe the performance or complexity of an algorithm. Big O specifically describes the worst-case scenario, and can be used to describe the execution time required or the space used (e.g. in memory or on disk) by an algorithm - If f(x) is a sum of several terms, the highest order term (the one with the fastest growth rate) - called the dominant term - is kept and others are discarded. - If f(x) has a term that is a product of several factors, all constants (those that are not in terms of x) are omitted. - using the above two rules. O(5+13N+7N^2) --> O(7N^2) --> O(N^2) [pronounce Oh N Squared]

Array-based implementation

- negatives: limits the size of the stack (static). so must know in advance what the space requirement is going to be. - neutral: requires the stack to be homogenous - positive: random access data: need top pointer & array for data notes: - arrays and stacks do not act the same always. need to differentiate between a structure (stack) and something that's serving as a home for the structure (array).

Linked implementation

- negatives: sequential access. (potentially not that negative, since stacks always are accessed from the top) - neutral: homogenous stack - positive: no size limits (dynamic allocation) data: need top pointer

Degree of a vertex

- number of incident (describes the relationship between an edge and a vertex) edges - if a directed graph, can do (in, out) ordered pair of edges wrt a specific node

Memoization

- reduces redundancy in recursive methods - creates a table and stores recursively calculated values in the table (which can save time/resources in future calculations of a recursive call - e.g. the fibonacci calculations) - check the table before calculating a new value

Data Abstraction

- requires splitting the data itself and how it is stored (implementation) from the way in which it is used (functionality). - the goal is to separate the two and have the functionality unaffected by the implementation. - user of the data doesn't need to know how the data is stored or organized, simply how information can be used and accessed and what manipulations can be performed. - programmer should be able to wear both hats. understand what the user may want/need, and also the specifics of how to achieve the best implementation. - Trade offs: typically the best solution is situationally dependent, and different situations will call for different implementations. Matching an implementation to a problem to maximize utility.

Quicksort

- sorting algorithm that repeatedly partitions the input into low and high parts (each part unsorted), and then recursively sorts each of those parts. To partition the input, quicksort chooses a pivot to divide the data into low and high parts. The _pivot_ can be any value within the array being sorted, commonly the value of the middle array element. - quicksort algorithm's runtime is typically O(N log N). Quicksort has several partitioning levels , the first level dividing the input into 2 parts, the second into 4 parts, the third into 8 parts, etc. At each level, the algorithm does at most N comparisons moving the l and h indices. If the pivot yields two equal-sized parts, then there will be log N levels, requiring the N * log N comparisons.

$

- using this symbol for exponentiation

Stack Uses

- when processing order is not important - when application reflects LIFO order - Operating Systems use: - to support recursion - to facilitate program execution

Converting expression to postfix

- while there is an input, read the expression from the left - if it's an operand, send to the output - else //operation - while stacktop has precedence, pop - push operator // no more input pop stack until empty

Recursive Binary Trees

A tree is a collection of three things - root - left subtree (LST) - right subtree (RST) Traversal - Pre-order traversal (ROOT LST RST), visit root, traverse left, traverse right - In-order traversal (LST ROOT RST) - Post-order traversal (LST RST ROOT)

ADT Format

ADT Name is Data Describe the nature of the data and any initialization Methods Method1 Input: Data from the client Precondition: necessary state of the system (what needs to be true) before execution of the method Process: actions performed with the data Postcondition: state of the system after execution Output: data values returned to the client Method2 etc.. end ADT Name

Stack

ADT in which items are only inserted on or removed from the top of a stack. a LIFO ADT - linked list, array, or a vector. stack of plates analogy. - push() operation inserts an item on top of the stack - pop() operation removes and returns the item at the top of the stack other common operations: - peek(stack) - returns but does not remove item - isEmpty(stack) - getLength(stack)

Recursive Algorithm

An algorithm that is defined by repeated applications of the same algorithm on smaller problems

Module 4: Queues and Lists

Goals: Define a queue and the standard methods. Differentiate between the rules of the queue definition and implementation imposed restrictions. Characterize problems that have a FIFO characteristic, and give examples. Assess problems for appropriate use of queues, and make implementation recommendations. Discuss the pros and cons, including costs, of the three array based approaches to implementing a queue. Write a Queue ADT and Implement queues using arrays. Define a list and the standard methods. Differentiate between the rules of the list definition and implementation imposed restrictions. Characterize problems are appropriate to using a general list and a sorted list, and give examples. Assess problems for appropriate use of lists, and make implementation recommendations. Write a List ADT and implement lists using plain arrays, arrays using a "marked delete" strategy, and dynamic references (links). Rewrite list methods by imposing access restrictions, to obtain the standard stack and queue methods. Implement stacks and queues with dynamic references. Discuss the pros and cons, including costs, of each implementation type and each access restriction type and match to applications.

Structures Summary

List Type, - General List - sorted - unsorted - Stack - Queue - Priority Queue - Deque - Other Access Restricted structures - Tree Implementations, - Array based - traditional - using Marked deletes - Linked (references) - Header - Circular - Doubly-linked - Multiply-linked - Hybrid Methods - insert - delete - search - copy - print - peek - etc...

Ancestor/Descendant

Nodes in the same lineage but separated by more than one level (as you should call that parent/child).

weighted graph

associates a weight with each edge. A graph edge's weight, or cost, represents some numerical value between vertex items, such as flight cost between airports, connection speed between computers, or travel time between cities. A weighted graph may be directed or undirected.

Linked code

class SNode { Datatype Data; // any appropriate type SNode Next; } public class LStack { private SNode Top; private int size; // optional size param } public boolean Empty () { return (Top==null); } public void push (Datatype Item) { SNode Temp = new SNode; // allocate space Temp.Data = Item; // stuff in info Temp.Next = Top; // attach to top of stack Top = Temp; // reset Top pointer } public Datatype pop () { Datatype Item; SNode Temp = Top; if Empty() "error" // do appropriate error handling else { Item = Top.Data; // Extract info Top = Top.Next; // Reset Stack top Temp.Next = null; // disconnect node Temp.Data = "a string of blanks"; // optional return Item; } }

Hybrid-implementation

ex - two stacks coming from either end of an array towards the middle. Idea is to generalize the solution, support the concept of multiple data structures sharing the same space. another ex - could have a series of linked lists stored in an array. first piece is value, second piece is pointer to index of the next value.

Source

in-degree zero and positive out-degree

(graph) cycle

path that starts and ends at the same vertex. A directed graph is cyclic if the graph contains a cycle, and acyclic if the graph does not contain a cycle.

Evaluate a converted expression

postfix/postfix (will just have a different reading / execution order): - if it's an operand, push it onto the stack - if it's an operator, pop the stack twice and perform an operation with those values and push the result back onto the stack

Complete m-ary tree

regular and has all the leaves at same level k (which is the height)

M-ary tree

rooted tree, with m or fewer children per vertex. m describes the maximum (out) degree of the tree

Sequential Array Binary Tree representation

works well if the tree is complete or almost complete (so that a lot of space is not wasted in allocating space for the complete version of the array)

Stack ADT

Stack is Data A list of items with a reference for the top of the stack, initialized to empty Methods Push Pop Is_Empty Constructor -- Copy Display Peek etc.. end Stack

Graph

data structure for representing connections among items, and consists of vertices connected by edges. - A vertex (or node) represents an item in a graph. - An edge represents a connection between two vertices in a graph. - Two vertices are adjacent if connected by an edge. - A path is a sequence of edges leading from a source (starting) vertex to a destination (ending) vertex. The path length is the number of edges in the path. - The distance between two vertices is the number of edges on the shortest path between those vertices.

Kruskal's minimum spanning tree algorithm

determines subset of the graph's edges that connect all vertices in an undirected graph with the minimum sum of edge weights. Kruskal's minimum spanning tree algorithm uses 3 collections: - An edge list initialized with all edges in the graph. - A collection of vertex sets that represent the subsets of vertices connected by current set of edges in the minimum spanning tree. Initially, the vertex sets consists of one set for each vertex. - A set of edges forming the resulting minimum spanning tree.

Sparse graph

graph with far fewer edges than the maximum possible. Many graphs are sparse, like those representing a computer network, flights between cities, or friendships among people (every person isn't friends with every other person). Thus, the adjacency list graph representation is very common. directed graph - or digraph, consists of vertices connected by directed edges. A directed edge is a connection between a starting vertex and a terminating vertex. In a directed graph, a vertex Y is adjacent to a vertex X, if there is an edge from X to Y.

Sink

when a vertex has an out-degree of 0

insert: max-heap

starts by inserting the node in the tree's last level, and then swapping the node with its parent until no max-heap property violation occurs. Inserts fill a level (left-to-right) before adding another level, so the tree's height is always the minimum possible. The upward movement of a node in a max-heap is sometime called percolating.

Lists: Array Implementation

- :( limits size, static - :) random access standard implementation: shifting. - i.e. - front of list at 0, end of list at k-1 where k is length (even if k, k+1, etc... in the array have value because they were previously deleted members of the list which have yet to be overwritten) - shifting up of contents always occurs after deletion, unless it was the last item in the list that was deleted alternatively: - can use flags or bits to mark an index in the array as 0/1 not/part of the array, which makes it faster to insert/delete, but harder to get the benefits of random access. So will need to garbage collect the list (and remove all the garbage and consolidate the useful values) which is basically what the standard implementation is doing, but here we would be putting off the pain for a while and doing it in a large batch.

Stacks using linked lists

- A stack is often implemented using a linked list, with the list's head node being the stack's top. A push is performed by prepending the item to the list. A pop is performed by pointing a local variable to the list's head node, removing the head node from the list, and then returning the local variable.

Polynomial time and Nondeterministically Polynomial Time

- Is P a subset of NP? Most theoretical scientists believe so. Helps answer the question - are there faster algorithms that exist?

Other recursive examples

- POW (x, N) - Binary Search - Max array problem - Kth Smallest Value problem

Specifying an ADT

- commonly represented by a class - use preconditions, postconditions, and invariants - precondition: something that must be true before the event is to take place - postcondition: something that must be true after the event takes place - invariance: something that is true both before and after - no function bodies are needed. only need the interfaces.

Merge sort

- sorting algorithm that divides a list into two halves, recursively sorts each half, and then merges the sorted halves to produce a sorted list. The recursive partitioning continues until a list of 1 element is reached, as list of 1 element is already sorted. - partitioning - merge sort algorithm uses three index variables to keep track of the elements to sort for each recursive function call. The index variable i is the index of first element in the list, and the index variable k is the index of the last element. The index variable j is used to divide the list into two halves. Elements from i to j are in the left half, and elements from j + 1 to k are in the right half. - Merge sort merges the two sorted partitions into a single list by repeatedly selecting the smallest element from either the left or right partition and adding that element to a temporary merged list. Once fully merged, the elements in the temporary merged list are copied back to the original list. - merge sort algorithm's runtime is O(N log N). Merge sort divides the input in half until a list of 1 element is reached, which requires log N partitioning levels. At each level, the algorithm does about N comparisons selecting and copying elements from the left and right partitions, yielding N * log N comparisons. - Merge sort requires O(N) additional memory elements for the temporary array of merged elements. For the final merge operation, the temporary list has the same number of elements as the input. Some sorting algorithms sort the list elements in place and require no additional memory, but are more complex to write and understand.

Heap sort

- sorting algorithm that takes advantage of a max-heap's properties by repeatedly removing the max and building a sorted array in reverse order. An array of unsorted values must first be converted into a heap. The heapify operation is used to turn an array into a heap. Since leaf nodes already satisfy the max heap property, heapifying to build a max-heap is achieved by percolating down on every non-leaf node in reverse order. - Heapsort uses 2 loops to sort an array. The first loop heapifies the array using MaxHeapPercolateDown. The second loop removes the maximum value, stores that value at the end index, and decrements the end index, until the end index is 0.

Shell sort

- sorting algorithm that treats the input as a collection of interleaved lists, and sorts each list individually with a variant of the insertion sort algorithm. - uses gap values to determine the number of interleaved lists - gap value positive integer representing the distance between elements in an interleaved list. For each interleaved list, if an element is at index i, the next element is at index i + gap value. - Shell sort begins by choosing a gap value K and sorting K interleaved lists in place. Shell sort finishes by performing a standard insertion sort on the entire array. Because the interleaved parts have already been sorted, smaller elements will be close to the array's beginning and larger elements towards the end. Insertion sort can then quickly sort the nearly-sorted array. - Shell sort begins by picking an arbitrary collection of gap values. For each gap value K, K calls are made to the insertion sort variant function to sort K interleaved lists. Shell sort ends with a final gap value of 1, to finish with the regular insertion sort. - Shell sort tends to perform well when choosing gap values in descending order. A common option is to choose powers of 2 minus 1, in descending order. Ex: For an array of size 100, gap values would be 63, 31, 15, 7, 3, and 1. This gap selection technique results in shell sort's time complexity being no worse than O(N3/2).

Insertion sort

- sorting algorithm that treats the input as two parts, a sorted part and an unsorted part, and repeatedly inserts the next value from the unsorted part into the correct location in the sorted part. - runtime: Insertion sort's typical runtime is O(N^2). If a list has N elements, the outer loop executes N - 1 times. For each outer loop execution, the inner loop may need to examine all elements in the sorted part. Thus, the inner loop executes on average N2 times. So the total number of comparisons is proportional to (N−1)⋅(N2), or O(N^2).

Selection sort

- sorting algorithm that treats the input as two parts, a sorted part and an unsorted part, and repeatedly selects the proper next value to move from the unsorted part to the end of the sorted part.

Encapsulation Inheritance and Polymorphism

- the three pillars of object oriented Programming

Converting to prefix

same as before except: - read from right - older items in the stack are further to the right

(graph) path

sequence of directed edges leading from a source (starting) vertex to a destination (ending) vertex.

min-heap

similar to a max-heap, but a node's key is less than or equal to its children's keys.

Binary Tree ADT

Constructor - initialize empty tree MakeTree - create tree containing 1 value as root with no children SetLeft - attach value as left child SetRight - attach value as right child Display Tree - optional TraverseTree - move through tree in orderly fashion SearchTree - look for a specified value DeleteNode - deletes specified node Data section: - reference to tree (root) - NumNodes // optional - Height // optional - Actual data

Sort Types

Exchange Sorts - Bubble Sorts - simple, inefficient - good for small data files because it needs to be done in an array - simple double nested for-loop - pattern: compare across 2 indeces at a time (increasing as you go 1 v 2, 2 v 3 etc..) with swaps in between if needed to have the higher value be the higher index. _bubbling_ up the higher values. repeat the process until sorted. - # of comparisons stays constant, # of exchanges goes down with each sort. - BUT we can start to ignore the last index of each progressive comparison (since the top value will bubble up), which will reduce the # of comparisons by one at each level - can also improve by having a test flag to check the # of exchanges, which if it ever reaches 0 can exit the sort (say the list happens to be sorted with 4 or 5 cycles left) - cost: - If just counting the comparisons, we are quadratic. (n(n+1))/2. - If also counting the exchanges, we are still quadratic but with different modifiers. n(n+1). - Not well suited for reverse order files, as every exchange / comparison will occur. In order files are great, would give linear performance. - Quick Sort - quite efficient. comes close to the 'all occasion' sort - primarily recursive - select a pivot, partition, repeat on partitions until sorted - many ways to partition / select pivot - sometimes pivots are selected arbitrarily (beginning, middle, etc...), and sometimes picking the middle makes the most sense if the file happens to be sorted. - could also do something like check beginning/middle/end and pick the median. leads to O(nlogn) performance for all data orders. - could also select the mean of all values in the file. MeanSort. O(n) to get the average. then O(nlogn) - may also not be a value in the file. - cost: - n(k) or n log n. simplified way to look at cost all the way down if assuming fairly even splits. (1x2^k)+(2x2^[k-1]+...+(Nx2^0)). Selection Sorts + select something and put it in it's final position. - Simple Selection Sort - Binary Tree Sort - Simple BT Sort - HeapSort - nlogn regardless or order of data - Quadratic Selection Sort Insertion Sorts + performs well on small data files that tend to be ordered - Simple Insertion Sort - Shell Sort - overall cost: O(n(logn)^2) Merge Sorts + basis for external sorting - Straight Merge - cost: O(nlogn) regardless or order. logn number of levels split, n sorted on each level. - requires O(2n) space - Natural Merge - base the first pass not on splitting every single node into its own file, but rather group an array until the next element is smaller than the last, then start a new array there. (can work better with arrays) - doesn't really help in fairly random files, but hugely helpful in ordered files. would be O(n). reverse order would be O(n^2) Radix Sorts - based on columns

Module 1 - Complexity and ADTs

For Complexity, we will introduce the notation we will use to talk about the cost of algorithms or code, for the rest of the term. It's just notation, a convenient tool that facilitates comparison. It is also something that most of you will see again when you take Foundations of Algorithms. In ADTs, we grapple with the idea of looking at structures in which we store information, from a functional standpoint, without worrying about the implementation. This perspective helps us understand the important aspects of the structure Goals: - Define Big-Oh notation and use to label functions. - Calculate and label the cost of simple algorithms. - Explain why empirical results may not match the theoretical predictions. - Use Big O Notation to choose one algorithm over another for a specific application. - Explain the conditions under which a more costly algorithm might be preferable. - Give examples that differentiate between the implementation and the functionality of a module of code. Explain how the implementation can affect the functionality. - Practice modularity in code design by consciously considering software from a user's perspective as well as from a programmer's perspective. - Write ADTs for specified applications using at least one of the styles discussed in class.

Graph definition

G=(V,E), an ordered pair, where V is a set of vertices (nodes), & E is a set of edges on the vertices. V is not empty. E may be empty.

Module 9 - Huffman Encoding and Sorting

Goals: - Given an alphabet, a set of protocols, and a frequency table, construct a Huffman Encoded Tree. Code and decode example messages. - Discuss the viability of using Huffman Encoding as a compression technique. - Discuss the costs of Huffman Encoding, as a function of various possible implementations. - Enumerate at least 10 factors that need to be considered in selecting a sort. - Discuss the pros and cons, including costs, of each of the standard sorting strategies. Consider best, average, and worst cases, and implementation and application needs, where appropriate. - Apply material on List Structures (Modules 2,4, and 6) to the discussion.

Module 2 - Stacks

Goals: - Define a stack and the standard methods. - Differentiate between the rules of the stack definition and various implementation imposed restrictions. - Characterize problems that have a LIFO characteristic, and give examples. - Assess problems for appropriate use of stacks, and make implementation recommendations. - Write a Stack ADT and implement stacks using arrays.

Module 6 - More on lists (then Graphs)

Goals: - Discuss the pros and cons of each list variation, including costs, and match to applications, separately and in combination. - Implement each access restriction type and list variation using the hybrid implementation. Discuss the impact on costs. - Categorize and label graphs accurately using basic graph terminology.

Module 10: Sorting and Searching Sorted Data

Goals: - Enumerate at least 10 factors that need to be considered in selecting a sort. - Discuss the pros and costs, including costs, of each of the standard sorting strategies. Consider best, average, and worst cases, and implementation and application needs, where appropriate. - Identify sorts that optimize a simpler strategy. Name the underlying, simpler sorting strategy and explain what aspects of the sort are leveraged. - Evaluate multiple sort strategies and recommend an appropriate sort for a specified example application. Justify your recommendation based on the strengths and weaknesses of the selected sort and the correspondence to the strengths and weaknesses of the application. - Enumerate at least 10 factors that need to be considered in selecting a search strategy. - Distinguish between searches based on sorted data and those based on other ordering schemes. Give examples of applications appropriate for each type. - Distinguish search trees from ordinary m-ary trees.

Module 6 - Trees

Goals: Define an m-ary tree in terms of an underlying graph. Restrict an m-ary tree into a rooted tree. Build a general tree using a random list of data. Define and build a heap using an example data set. Define and implement priority queues Implement a tree using both array based and linked implementations Define a Binary Tree recursively. Specify the three traversal orders and generate the traversal orders on example trees. Define and implement threaded trees. Appreciate Binary-op trees as an application of Binary trees and tying in earlier material from Stacks. Build a balanced tree using a sorted list of data, then compare and contrast with the balanced trees built earlier. Define the search tree characteristic Perform basic insertions and deletions on search trees, explaining why the rational behind the traditional strategy.

Heaps

Heaps are typically stored using arrays. Given a tree representation of a heap, the heap's array form is produced by traversing the tree's levels from left to right and top to bottom. The root node is always the entry at index 0 in the array, the root's left child is the entry at index 1, the root's right child is the entry at index 2, and so on.

Data Structures Course Overview

This course investigates abstract data types (ADTs), recursion, algorithms for searching and sorting, and basic algorithm analysis. ADTs to be covered include lists, stacks, queues, priority queues, trees, sets, and dictionaries. The emphasis is on the trade-offs associated with implementing alternative data structures for these ADTs. Course Objectives: - Accurately and appropriately use Big "O" notation to describe and compare algorithms, while differentiating algorithm functionality from implementation. - Accurately detail at least two implementations for each standard data structure: stacks, queues, lists, and binary trees; and analyze the pros and cons of each implementation to match applications in programming assignments. - Explain the recursive paradigm through practice with non-trivial programming assignments and assess practical applications of recursion. - Analyze the pros and cons of the standard sorting and searching strategies to select an optimal strategy for a specific programming application.