Applied Algorithms

1
Applied Algorithms

• Lecture 3
• Data Structures

2
Homework Grading Notes

• You must turn in your homework in class each
  Friday.
• Give me a paper copy in class. Email is not
  convenient for me.
• You must turn in the report from the judge.
• Homework without a Judge report can get a maximum
  of 85.
• Always turn in a page or two of output. Try and
  make the input for the output you turn in be
  non-standard. Test the gotchas!
• Include some sort of high level description of
  the strategy your program uses to solve the
  problem.

3
Quiz

• We have a 10 minute quiz today on the reading
  Chapter 2.

4
Representing Graphs

• Graphs are ubiquitous, they appear in many
  problems.
• Best to have some canned strategies for thinking
  about graphs.
• Separate in your mind the logical descriptions of
  graphs, form the physical representations
• Functions
• Arrays
• Matrices
• Pointers

5
Graphs

• A graph is a pair (V,E) where
• V is a set of vertices
• E is a set of edges u,v, where u,v are distinct
  vertices from V.
• For example
• G (a,b,c,d, a,b, a,c, a,d, b,d)
• Examples computer networks, street layout, etc

6
Variations

• There are many variations on this theme. For
  example, in some cases we may want to allow
• Self loops v,v
• Multiple edges between two vertices
  (multigraphs)
• Labels attached to vertices by a function V ? A
• Labels attached to edges by a function E ? B
• Hyperedges that connect multiple vertices
  (hypergraphs)
• etc

7
Representing graphs

• Function Define a function that when applied to
  vertex v, returns a set of children (or a set of
  parents). Each child is a node where (v,c) is in
  the set of edges.
• Adjacency list for each vertex v, we store a
  linked list of the vertices u that it connects to
  by a single edge.
• Adjacency matrix a two dimensional array
  gij. An entry of 1 means that there is an
  edge between vertices i and j.
• Pointers actually construct a heap object where
  edges are implemented by pointers

8
Adjacency matrix representation

• A simple example
• Uses O(V2) space, much of which will be wasted
  if the graph is sparse (i.e., relatively few
  edges).
• Easily adapted to store information about each
  edge in the entries of the matrix.
• Alternatively, if all we need is 0/1, then a
  single bit will do!

9
Adjacency list representation

• A simple example
• Uses O(VE) space, good for sparse graphs,
  more expensive for dense case (i.e., many edges).
• Easily adapted to store information about each
  edge in each part of the linked lists.
• Testing to see if there is an edge (u,v) is not
  O(1) we must search the adjacency list of u for
  v.

10
Function representation

• Best when we have directed graphs. I.e. edges
  have an orientation.
• A simple example
• list graph(node x)
• if (nodea) return b,c,d
• else if (nodeb) return d
• else if (nodec) return
• else if (noded) return
• else return

11
Arrays

• When a graph has fixed in-degree (or out degree)
  the function representation has an especially
  nice implementation as a set of parallel arrays.
• list graph(node x)
• if (nodea) return b,c,d
• else if (nodeb) return d
• else if (nodec) return
• else if (noded) return
• else return

int count 5
node child1 5
node child2 5
node child3 5
12
In Class Problems

• Jolly Jumper 2.8.1
• Page 42 of the text
• We will write this together as a class
• Hartals 2.8.3
• Page 45 of the text
• Keeping in mind the rules we discussed in class,
  break into teams of two, and solve the problem.
• Pick a different partner than you used last time.
• Use pair programming. One person drives the
  other observes.

13
Todays Assignments

• Read for next time
• Chapter 3 of the text. pp 56-77
• Be prepared to answer questions in class next
  Friday from the reading.
• Programming assignment
• 2.8.2 Poker Hands
• Page 43-44
• Write a solution
• Submit your solution (until you get it right)
• Hand in both your program, and the judge output.
• Those who volunteer to discuss their program get
  class participation points. Email me solutions
  before noon on Friday, April 22.