Algorithms and Data Structures Lecture XV

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Algorithms and Data StructuresLecture XV

• Simonas Šaltenis
• Aalborg University
• simas_at_cs.auc.dk

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Results of the questionnaire

• Problematic lectures

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This Lecture

• What have we learned?
• Summary-Pensum
• Solving exercises
• Recurrences
• Graph exercises
• ADT exercises
• Dvide and conquer algorithms

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The Course Pensum

• Toolbox of algorithmic techniques
• Divide and Conquer
• Merge sort, Quicksort, Binary search
• Dynamic programming
• Matrix chain multiplication, Longest Common
  Subsequence not in exam pensum, except basic
  understanding of LCS
• Greedy algorithms
• Prims, Kruskals, Dijkstras
• Analysis of algorithms
• Correctness of algorithms
• Loop invariants not in exam pensum
• Asymptotic notations
• Recurrences

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The Course Pensum (2)

• Building algorithms - concept of ADT
• Toolbox of data structures
• Simple data structures and ADTs
• array, all sorts of linked lists, stacks, queues,
  trees, heaps
• Dictionaries
• hash tables
• binary search trees (unbalanced)
• red-black trees

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The Course Pensum (3)

• Toolbox of algorithms
• Sorting
• insertion sort, selection sort
• merge sort
• quick sort
• heap sort (priority queues)
• Graphs
• memory representation
• graph traversal
• breadth-first search
• depth-first search (topological sort)
• minimum spanning trees (Prim, Kruskal)
• shortest path (Dijkstra, Bellman-Ford)

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The Exam

• The exam will be written, you will have three
  hours.
• Two parts
• Multiple-choice, quiz-style part tests basic
  knowledge, understanding of the core concepts
• Creative exercises come up with algorithms,
  analyze them, argue for your solutions.

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Preparation for the Exam

• To prepare for the exam
• Concentrate on solving the exercises, not
  studying the textbook (although you have to know
  the concepts!)
• Solve exercises from last year exams
• When solving write your solutions down!
• On Jan 16, 1300 come to B2-104 (here) and have
  your questions answered.
• Or pass by my office E1-215b!
• Good news solutions to hand-in exercises will be
  posted on the web before Christmas!

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Solving Recurrences

• Repeated substitution procedure
• Substitute, expand, substitute, expand
• Observe a pattern and write how your expression
  looks after the i-th substitution
• Find out what the value of i (e.g., lgn or n-1)
  should be to get the base case of the recurrence
  (e.g., T(1))
• Insert the value of T(1) and the expression of i
  into your expression
• Compute your recurrence for small values and
  check if your general solution gives the same
  results!

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Types of Recurrences

• Number of substitutions i
• If T(n) T(n - 1) and base case T(k)
• then i n k
• If T(n) T(n/b) and base case T(k)
• then i logb(n/k)

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Types of recurrences II

• T(n) aT(n b) c
• T(n) aT(n/b) cn d
• Geometric series
• T(n)T(n b) cn d
• Arithmetic series
• Prove it yourself!

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Graph Exercises

• Types of graph exercises
• Decide what are the vertices and what are the
  edges
• Un-weighted graphs
• Traverse and check(do) something gt use DFS or
  BFS
• Find shortest paths gt use BFS
• Schedule or order dependent activities or
  processes gt use Topological sorting

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Graph Exercises

• Types of graph exercises
• Weighted graphs
• Decide what are the weights
• Shortest paths, shortest times gt use Dijkstras
• Minimum (maximum) spanning tree gt use Prims or
  Kruskals
• Finding a critical path in a schedule (longest
  path in a DAG) gt use Topological sorting on a
  DAG of activities

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Do not forget

• Write preconditions (INPUT) and postconditions
  (OUTPUT) of your algorithms
• You can only use the ADT operations provided in
  the exercise !!!