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CISC621 Algorithms, final exam review sheet, 2015 Fall

• Final exam question types much like midterm
• fewer multiple choice, more short answer such as
  • Give the definition of NP completeness.
  • What is the expected runtime of QuickSelect?
• Here is a question that will be on the test:
  Choose one of the following problems and show that it is NP complete. <list of problems>.

more or less complete topics list

• Not covered: Bipartite matching, section (26.3)
• Not covered: Halting problem
• Not covered: Linear algebra reductions
• Covered: Convex hull: Graham scan and (package wrapping), section (33.3)
• Covered: Network flow: Ford-Fulkerson algorithm, max-flow min-cut theorem (26.2)
• Covered: NP-Completeness proofs (CH 34). (→ denotes polytime reduction)
  • 3-SAT → CLIQUE
  • CLIQUE → VERTEX-COVER
  • VERTEX-COVER → HAM-CYCLE. Not Covered in detail but you should be able to state in very general terms how it works, how it uses widgets:
  • HAM-CYCLE → TSP
  • 3-SAT → SubsetSum
  • SubsetSum → Partition
• Covered: definitions of P, NP, NP completeness
• Covered: Cook's Theorem (34.7) general idea how it works, what it says.
• Covered Definitions of Decision problems, languages, and Poly time reductions
• Covered: All pairs shortest paths in graphs (Ch 25.1,2)
• Covered: FFT polynomial multiplication (Ch 30)
• Covered: Topological Sort, Strongly connected components (Ch 22)
• Dynamic programming, eg. longest increasing subsequence.

First half of course, covered less intensely

• Prim's MST algorithm.(Ch 23)
• Kruskal's MST algorithm (Ch 23)
• Breadth and Depth first search (Ch 22)
• Dynamic Sets, Union-Find (ch 21.4) with union by weight and with path compression
• Graph notation, representation, paths, spanning trees, digraphs, weighted graphs. (Ch 22, 23.1)
• Binomial Heaps, reference: CLRS Problem 19.2
• Binary Search trees (Ch 12)
• Red-Black trees, (Sedgewick notes and Ch 13)
• hash functions,Application to string matching, the Rabin-Karp algorithm (CLRS Section 32.2)
• Hash functions: quick, uniform, random
• Hash tables: Direct addressing: linear, quadratic and double hash probing for collision resolution.
• Dynamic arrays, Reading 17.4
• Binary heap data structure for Priority Queue.
• Sorting
  • Putting it all together: IntrospectiveSort
  • QuickSort
  • HeapSort
  • InsertionSort
  • MergeSort
• Lower bound for sorting by comparison. Reading: Section 8.1
• The recursion tree method for recursions. Reading: Section 4.4
• QuickSelect
• Deterministic Median (groups of 5), Section 9.3
• block decomposition of matrices
• Strassen's algorithm for matrix multiplication.
• Karatsuba's algorithm for polynomial and integer multiplication
• substitution method for solution of recurrences
• Θ, big O, Ω
• Master Theorem