Course Review

1
Course Review
COMP171 Spring 2009
2
Elementary Data Structures

• Linked lists
• Types singular, doubly, circular
• Operations insert, find, delete O(N)
• Stack ADT (array or linked-list)
• First-in-Last-out
• Operations push, pop
• Queue ADT (array or linked-list)
• First-in-First-out
• Operations enqueue, dequeue
• Priority queue (heap)
• Heap-order property
• Operations insert and deleteMin (both O(log N)
  using array)

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Analysis of Algorithms

• Worst case analysis
• time and memory space
• depends on the input (size, special features) and
  algorithms
• growth rate of the functions
• Asymptotic notations
• O(.) upper bound, at most, worst case,
• O(.) lower bound, at least, best case,
• T(.) tight bound, O(.) O(.), asymptotically
  growth rate

4
Comparison-based Sorting Algorithms
Algorithms Method Worst time Average time Memory
Insertion sort A1p-1 are sorted, then insert Ap O(N²) O(N Inversions) O(1)
Mergesort Divide, conquer and merge O(N logN) O(N logN) O(N)
Quicksort Pick pivot, partition, recursion O(N²) O(N logN) O(logN)
Heapsort Build heap, deleteMin O(N logN) O(N logN) O(1)
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Other Sorting Algorithms

• Theorem Any comparison based sorting algorithm
  takes O(n logn) comparisons to sort a list of n
  distinct elements.
• Non-comparison sorting
• Rely on some special features of the input
• Counting sort O(N Range)
• Radix sort O(N digits)

6
Overview of the Forrest

• Binary Trees
• Binary Search Trees
• AVL trees
• M-ary Trees
• B-Trees
• Terms
• root, leaves, child parent, siblings, internal
  nodes,
• height, paths length, depth,
• subtrees
• Main theme search, insertion, deletion.

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Binary Search Trees

• Traversal
• Pre-order, in-order, post-order
• Average depth O(log N), max depth O(N),
• Operations
• Search, findMin, findMax
• Insertion
• Deletion 3 cases, recursive
• All takes O(height of the tree).

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AVL Trees

• Balanced BST
• Height of an AVL tree is O(log N).
• Search O(log N)
• Insertion 2 types of rebalancing O(log N)
• Trace from the new node to the root, locate the
  unbalanced node, and at most one rotation.
• Single rotation for outside insertion
• Double rotation for inside insertion
• Deletion O(log N)
• Trace the deletion path, and more than one node
  may need rotation.

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B-Trees

• M-ary trees
• The root is either a leaf or has 2 to M children
  (1 to M-1 keys).
• Each internal node, except the root, has between
  ?M/2? and M children ( ?M/2? - 1 and M - 1
  keys).
• Each leaf has between ?L/2? and L keys and
  corresponding data items.
• The data items are stored at leaves
• Search
• Insertion always insert to a leaf
• Possibly splitting leaf (and internal nodes)
• Update the key in a internal node if necessary
• Deletion
• Borrow a key from your siblings
• Merge two leaves (or internal nodes) opposite
  to splitting

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Graph Essentials

• Graph Vertices Edges
• Representation
• Adjacency matrix O(N2) space, O(1) access time
• Adjacency list O(MN) space, O(N) access time
• Keywords
• subgraph, tree, acyclic graph
• path, length, cycle, simple
• directed, undirected
• incident, degree, indegree / outdegree
• connected components

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Graph Traversal BFS

• Connectivity and shortest path (from source)
• Time O(MN)
• Use visited table,
• predecessor list,
• and distance table
• BFS tree

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Graph Traversal DFS

• Connectivity and cycle detection
• Time O(MN)
• Recursive function
• DFS tree

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Topological Sort

• Topological ordering on DAG and connectivity
• Time O(MN)
• Repeatedly remove
• zero-degreed vertices
• and the outgoing arcs
• Linear ordering

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Hashing

• Hashing is a function that maps a key (K) into an
  entry in a table (hash table), Th(K). It only
  supports operations Find, insertion, deletion
• The hashing function should be
• Computationally fast, and efficient with O(1)
  complexity
• Minimize the collisions, when Th(K1) Th)K2)
  and K1 and K2 are two distinctive keys
• Collision Resolution 1 Separate chaining
• Collision Resolution 2 Open Addressing
• Linear Probing
• Quadratic Probing
• Double hashing