Algorithms and Data
Structures
Lecture XV
Simonas Šaltenis
Aalborg University
[email protected]
December 4, 2003
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Results of the questionnaire
Problematic lectures:
60
50
40
30
20
10
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This Lecture
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What have we learned?
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Summary-Pensum
Solving exercises
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Recurrences
Graph exercises
ADT exercises
Dvide and conquer algorithms
December 4, 2003
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The Course – Pensum
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Toolbox of algorithmic techniques
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Divide and Conquer
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Dynamic programming
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Matrix chain multiplication, Longest Common Subsequence –
not in exam pensum, except basic understanding of LCS
Greedy algorithms
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Merge sort, Quicksort, Binary search
Prim’s, Kruskal’s, Dijkstra’s
Analysis of algorithms:
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Correctness of algorithms
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Loop invariants – not in exam pensum
Asymptotic notations
Recurrences
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The Course – Pensum (2)
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Building algorithms - concept of ADT
Toolbox of data structures:
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Simple data structures and ADTs
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array, all sorts of linked lists, stacks, queues, trees,
heaps
Dictionaries
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hash tables
binary search trees (unbalanced)
red-black trees
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The Course – Pensum (3)
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Toolbox of algorithms:
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Sorting
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insertion sort, selection sort
merge sort
quick sort
heap sort (priority queues)
Graphs
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memory representation
graph traversal
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breadth-first search
depth-first search (topological sort)
minimum spanning trees (Prim, Kruskal)
shortest path (Dijkstra, Bellman-Ford)
December 4, 2003
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The Exam
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The exam will be written, you will have
three hours.
Two parts:
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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
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To prepare for the exam:
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Concentrate on solving the exercises, not studying the
textbook (although you have to know the concepts!)
Solve exercises from last year exams
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On Jan 16, 13:00 come to B2-104 (here) and have
your questions answered.
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When solving write your solutions down!
Or pass by my office E1-215b!
Good news: solutions to hand-in exercises will be
posted on the web before Christmas!
December 4, 2003
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Solving Recurrences
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Repeated substitution procedure:
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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
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Number of substitutions i :
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If T(n) = …T(n - 1)… and base case T(k) = …
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then i = n – k
If T(n) = …T(n/b)… and base case T(k) = …
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then i = logb(n/k)
December 4, 2003
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Types of recurrences II
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T(n) = aT(n – b) + c
T(n) = aT(n/b) + cn + d
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T(n)=T(n – b) + cn + d
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Geometric series
Arithmetic series
Prove it yourself!
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Graph Exercises
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Types of graph exercises:
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Decide what are the vertices and what are the
edges
Un-weighted graphs
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“Traverse and check(do) something” => use DFS or
BFS
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Find shortest paths => use BFS
Schedule or order dependent activities or processes
=> use Topological sorting
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Graph Exercises
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Types of graph exercises:
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Weighted graphs
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Decide what are the weights
Shortest paths, shortest times => use Dijkstra’s
Minimum (maximum) spanning tree => use Prim’s
or Kruskal’s
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Finding a critical path in a schedule (longest path in
a DAG) => use Topological sorting on a DAG of
activities
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Do not forget:
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Write preconditions (INPUT) and
postconditions (OUTPUT) of your
algorithms
You can only use the ADT operations
provided in the exercise !!!
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