Course Review

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Course Review

• 15-211 Fundamental Structures of Computer
  Science

Ananda Guna May 04, 2006
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This course was about

• How to solve computing problems using
• Problem analysis, to abstract away details and
  divide into smaller subproblems.
• Mathematical foundations for precise formulations
  of problems and solutions.
• Data structures and algorithms to solve problems
  correctly and efficiently.
• Java programming and modular software
  construction for good implementations.

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Data Structures/Algorithms

• -Complexity
• big-O, little-O, Omega etc..
• Lists
• linked, doubly, singly, multi-linked
• Trees
• general, B-trees, BST, splay, AVL, etc.
• Stacks and Queues
• operations, analysis
• Heaps
• binary, binomial
• Hash Tables
• collisions, implementation
• Graphs
• traversals, shortest paths
• FSM
• regular expressions, KMP
• String Matching
• KMP, Boyer-Moore, Rabin-Karp

.
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Data Structures/Algorithms

• Dynamic Programming
• table(bottom-up), recursion(top-down)
• memoization
• Game Trees
• mini-max, alpha-beta, iterative deepening
• - Sorting
• bubble, quick, insertion, selection
• Algorithms
• findMax, findMedian, reverse, MST, compression
• Traversals, shortest path

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Java Programming
Object oriented programming. Correctness (loop
invariants, induction). Time and space
complexity. Debugging (test harness).
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Studying for the Final Exam

• Review all material first
• Lectures, programs, quizzes
• Do sample finals under time restrictions.
• Old finals are in blackboard/course material
• Types and difficulty of problems may be similar
  (some topics not covered)

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Complexity
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Complexity
f(n)n2 g(n) n h(n) log n
f(n) is O(g(n)) iff f(n) is o(g(n)) iff
f(n) is ?(g(n)) iff f(n) is ?(g(n)) iff
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Complexity
True or False? If true prove it, else give
counter example
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Space Complexity

• How much memory is needed to run the program
• quicksort
• merge Sort
• insertion sort
• Prims
• Dijkstras

Key Point consider the sizes of the data
structures used
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Solving Recurrences, Asymptotics

• Solve T(n) 3 T(n/3) 1
• Prove the correctness
• By induction
• Invariants
• Hold initially
• Preserved after each iteration of loop or method
  call
• Or other methods

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Complexity
A is an array of integers
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Complexity of known algorithms
Algorithm Complexity(runtime)
Bubble sort O(n2) average, worst, O(n) best
Insertion sort O(n2) average, worst, O(n) best
Selection sort O(n2) average, worst, best
Find Max, Find min O(n), cannot be done in o(n)
Find Median With sorting O(nlogn) but O(n) using quick select
Quick sort O(nlogn) average best, O(n2) worst
Merge Sort O(nlogn) average, best, worst
Heap Sort O(nlogn) best, worst, average
Build heap O(n)
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Complexity of known algorithms
Algorithm Complexity
Heap find min O(1)
Heap delete Min O(logn)
Heap -insert O(logn)
Hash insert, find, delete O(1)
Graph - build O(V2) space (adjacency matrix) O(VE) space (adjacency list)
Graph - Traverse O(E)
Graph Topological Sort O(V E)
Graph Dijkstras O((EV) logE)
MST prims O(EV2)
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Complexity of known algorithms
Algorithm Complexity
MST Kruskals O(E logE) sort the edges O(V logV) find and union operations
Dynamic Programming O(Table size) Linear with memoization
Disjoint sets - union O(logn)
Disjoint sets - find O(logn)
KMP O(nm)




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Sorting
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Sorting and lower bounds
Insertion Sort Selection Sort Heapsort Mergesort Q
uicksort Radix Sort
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Sorting and lower bounds
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Questions
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Trees
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Balanced Binary Trees
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AVL Trees

1. Traverse the tree inorder
2. Do exactly two rotations so that the tree is
   balanced

• 3. Consider a binary tree of height h
• What are the max and min number of nodes?
• What are the max and min number of leaves?
• 4. Perfect, full, complete trees

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Splay Trees
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Full, Complete and Perfect Trees

• Definitions
• Full, complete and perfect
• Ex Show that a full binary tree with n leaves
  has (n-1) internal nodes. (use induction on n)
  Therefore a full binary tree with n leaves has
  (2n-1) nodes
• Ex Start from the root and do a preorder
  traversal as follows. Write 1 for a node with 2
  children. Visit left child, then right child.
  Write 0 if the node is a leaf. Therefore a full
  binary tree with 3 nodes can be represented by
  bit sequence 100.
• What is the bit sequence representing a
  full binary tree with 8 leaves?

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Code Examples
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Dictionaries
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Dictionaries
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Hashing

• Describe why hashing a string to sum of its
  characters is not a good idea.
• Assume S a1a2.an is a string
• Define H(S) ? ai2i-1 (i1..n)
• Describe a way to efficiently calculate H(S)

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Heaps
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Priority Queues and Heaps

• Suppose you implement a priority queue using
  following
• Unsorted array
• Linked list (smallest at front)
• Heap
• What is the complexity in each case for
• deleteMin
• insert

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Binary heaps
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Compression
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Compression
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LZW compression
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Data Compression

• Encode I AM SAM SAM I AM SAM SAM using
• Huffman compression
• LZW
• In each case calculate the compression ratio
• Is it possible to apply Huffman after applying
  LZW?
• If so apply Huffman to output generated by LZW
  above

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Graphs
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Graphs
Adjacency lists and matrices Traversal (DFS,
BFS) Reachability (DFS, BFS, Warshall) Shortest
Paths (Dijkstra, Bellman-Ford) MST (Prim,
Kruskal)
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Graphs

• What is breadth first traversal(BFS)?
• What is depth first traversal(DFS)?
• What data structures can be used to implement
  each one?
• Reachability
• Connected components
• Topological sort

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Graphs
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Greedy Algorithms
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60
60
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20
50
50
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60
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40
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100
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30
20
20
50
40
150
Find the Shortest Path from Node 1 to every other
node
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Dijkstras Algorithm
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MST

• Find the MST using
• a. Prims Algorithm
• b. Kruskals algorithm

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Homework

• A graph G is bipartite if the vertex set V can be
  partitioned into two subsets L and R, such that
  every edge has one vertex in L the other in R.
  Give an example of a bipartite graph.
• G is a tree if it is connected and acyclic. What
  is the number is edges in a tree with n nodes?
• 3. Draw all possible connected graphs G on 4
  vertices and 3 edges.
• 4. Draw all possible connected graphs G on 4
  vertices and 4 edges.
• 5. Suppose G is not connected. What the maximum
  number of edges in G?
• 5. Prove or give a counterexample to the
  following claim Any tree is bipartite.

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Dynamic Programming
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Dynamic Programming

• Dependent subproblems, recursive solutions,
  memoizing, explicit tables.
• Knapsack
• Longest Common Subsequence
• Matrix Multiplication

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Game Trees
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Game trees
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Games
2-person, deterministic, complete information,
... Backtracking MiniMax Alpha-beta
pruning Heuristics, iterative deepening, move
order, tricks, ...
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Disjoint Sets
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Union-find
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Regular Expressions
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Exercises
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Homework

• Write a regular expression of the alphabet a,
  b, c, d that describes all strings consisting of
  either an odd number of a's or an even number of
  b's, followed by any number (and ordering) of c's
  and d's. Zero is an even number and any number of
  x's can be zero x's. Draw a FSM to recognize any
  such string

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String matching
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Final Exam

• Monday, May 8, 800 1100 am
• Make sure not to be late.

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Final Exam

• Closed book, no calculators, cell phones,
  pagers, IM,
• You can bring 1 (one) page of notes.
• Bring fluids

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Conclusion

• Review all material
• Do all sample quizzes and finals
• Good luck!!