Sets, Maps, implementing them (lists, trees, hashing, etc.)



Balanced BST
Tree balancing:
  AVL: imbalance condition, rotations to re-balance
       rotations alter tree structure but maintain the order property
       worst case time complexity for insert, remove, contains
       show how AVL tree is altered by insert, remove

  Splay: another approach to balancing
         splaying done any time a node is accessed
         splaying is applying preserving rotations until accesses node
           is moved to the root
         amortized worst case time complexity
         show how splaying works in a tree for insert, remove, find, etc.


Hashing, Hash Tables, Hash Maps
  Pigeon hole principle
  Hash Table, hash function
    how to pick table size
    how to make an effective hash function
    what to do about collisions
      hash into linked lists (or other collection data structures)
        table size should be prime, and same the size as # elements expected
      probing to find an open cell in the table
        linear probing function
        quadratic probing function
        "expo" probing function
      table size should be prime, and twice the # elements expected
    Rehasing to extend the table size

  Hash Maps... storing associated information/object along with a hashed key

Blockchain
  SHA-256 Cryptographic hash function
  basic block chain construction and operation
  nonce
  "mining" what it is, how it's done
  complexity of the hashing used in mining blocks (for Bitcoin application)


Basic graph theory
  vertex (node), edge (arc), edge weight
  undirected graph
  directed graph
  path in a graph
  cycle in a graph
  bi-partite graph
    a tree is a bi-partite graph
  complete graph
  planar graph
  
  depth-first search (traversal) in a graph (use a stack)
  breadth-first search (traversal) in a graph (use a queue)
  
  representation of a graph
    adjacency matrix
      useful when graph is dense ( |E| approx.= |V|^2 )
    adjacency lists
      useful when graph is sparse ( |E| << |V|^2 )
    although an undirected edge between nodes A and B is not the 
      same theoretically as two directed edges A --> B and B --> A, 
      we often represent them that way in adjacency lists for 
      an undirected graph

  Topological sort

  Single Source Shortest Path algorithm for shortest paths in unweighted digraphs
  Dijkstra's algorithm for shortest paths in weighted digraphs

  Spanning tree
  Minimum spanning tree
    Kruskal's algorithm for MST
    Prim's algorithm for MST

  Uses for graphs: course prerequisite structure, maps,
    modeling digital networks, airline and auto routes, 
    flow of traffic, work flow in an organization, 
    finite state machines, document structure, hypermedia,
    linguistic representations, all tree uses (trees are graphs)
    tons and tons of stuff 

  Euler path 
    path through a graph that crosses each edge exactly once
  Euler circuit
    EP that begins and ends on the same node
  We have an efficient algorithm for finding Euler path/circuit

  Hamiltonian path
    path through a graph that visits each node exactly once
  Hamiltonian circuit
    HP that begins and ends on the same node
  Traveling salesman is example of Hamiltonian path/circuit
  No efficient (polynomial time, like O(N^2) ) algorithm is known for solving it


Sorting
  we can prove that no sort that uses only pairwise comparisons
    can do better than N log N best case.  This means that all
    comparison sorts are Big-Omega(N log N), bounded below.
    They might be worse, however, and some are.
  bubble sort
    O(N^2) worst case
  insertion sort
    O(N^2) worst case
    doesnt swap, it inserts
    sometimes used as the "finisher" in quicksort
  selection sort
    O(N^2) worst case
    similar in concept to insertion sort but operationally it 
      manipulates the array by swaps
  heap sort
    O(N log N) best case
    O(N log N) worst case
    O(N log N) average case
  merge sort
    O(N log N) best case
    O(N log N) wosrt case
    O(N log N) average case
    cost: often uses lots of extra memory, needs a second array as large
          as the one you are sorting
  quicksort
    O(N log N) best case
    O(N^2) worst case
    O(N log N) average case
    does not need much extra space
    get extra efficiency by cutting off recursion when the lists
      become smaller than, say, length 15 and then doing some other
      sort like insertion sort on the small lists to finish up
  bogosort
    random shuffling until you get the one that is in order
    O(infinity) worst case
    an extreme, just an illustration, not a serious shuffle

  sorting into buckets
    a way to sort in O(N) time
    can be used when the full range of possible input values is known
    allocate an array of int A and initialize all slots to 0
      then when you see an input number n, do A[n]++
    this seems to violate the Big-Omega(N log N) proof, but this
      sort does not do only pairwise comparisons.  It effectively
      does lots of comparisons at once by putting a number right
      into an array slot.  Since we believe TANSTAAFL, the tradeoff
      is that is uses lots of memory (not all array slots are filled)
      and you have to know in advance the range of inputs.
 
  Other O(N) sorts
    we can also sort N items in N time if we use parallel hardware
      the Big-Oh complexity results for our sort algorithms have
      all assumed we were running them on a single processor.

  sorts are either stable or unstable
    stable means it preserves the original order of items with 
      the same value when there are duplicate values in the input.
    often a sort can be implemented to be stable or unstable, and 
      in many cases making it stable sacrifices some efficiency
    bubble sort is stable
    insertion sort is stable
    selection sort as we did it in class is not stable
      it can be made stable with no change in Big-Oh complexity
      but at a cost in practical efficiency
    merge sort is not necessarily stable but is easy to make so
      (just make sure the merge takes from the left-most array if
       it can for duplicates)
    quick sort is not stable
    heap sort is not stable (heap is the issue)


Skip Lists
  like a linked list except items are kept in order
  alternative to balanced binary search trees
  multi-level linked cells
    each higher level skips over more of the items at the lower levels
  probabilistic data structure... 
    meaning we roll dice to decide aspects of the structure as we build
    also means if we build a skip list twice with the same input data sequence
      we will almost certainly get two different linked structures
  average time complexity: find O(log N), insert O(log N)
  worst case time complexity: find O(N), insert O(N)
    however the worst case is probabilistically very very unlikely
    average case is probabilistically very likely



Trie 
  different form of tree used to do prefix-based storage and searching for text strings
  used in doing "look ahead" for typing in apps like google searched, phone lists
  what is longest path in trie?
  what is worst case time complexity of finding a word in a trie?
  emphasizes that the time complexity of an algorithm will depend on
    -- cost of building the data structure
    -- plus cost of using the data structure