Algorithms: Depth First Search, Greedy, Dynamic Programming, Network Flows, Summaries of Algorithms and Programming

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COMP9007 - Algorithms

Summary of course contents

The University of Sydney Page 2 2

Progressive Marks - 40% Quizzes 20% Assignments 20%

Final Exam - 60% (minimum 40% required to pass)

One-2-One Helpdesk Next Friday 5-7pm ..? (SCS Building J12 level 2)

Assessment Summary

The University of Sydney Page 4 4

• 6 questions of 10 marks each
• Q1 will have 4 sub-questions comprising topics on stable matching and

running time

• Pay attention to all sub questions asked like describe/explain, write

pseudocode, running time, justifying running time, etc.

• Advice for lower and upper bound students

Exam Questions Overview

Content overview

1. Basic knowledge, concepts, notations,…

a) Asymptotic notations

b) Data structures

c) Graphs

2. Five techniques

a) Stable Matching

b) Greedy

c) Divide & Conquer

d) Dynamic Programming

e) Flow Networks

3. Advance Topics: P & NP and Coping with Hardness

Given functions f ( n ) and g ( n ), we say that f ( n ) is O ( g ( n )) if there are positive constants c and n 0 such that

f ( n ) ≤ cg ( n ) for n ≥ n 0

Example: 2 n + 10 is O ( n )

2 n + 10 ≤ cn ( c − 2) n ≥ 10 n ≥ 10/( c − 2) Pick c = 3 and n 0 = 10

Asymptotic Order of Growth

Data Structures

› Queues

• Enqueue, dequeue, first and size operations in O(1) time.

› Stacks

• Push, pop, top and size operations in O(1) time

› Balanced binary trees (e.g. AVL trees)

• Insert, delete and find operations in O(log n) time
• Recursion
• The Master Theorem
• Mergesort

Graph Representation: Adjacency Matrix

Adjacency matrix. n-by-n matrix with Auv = 1 if (u, v) is an edge.

• Two representations of each edge.
• Space proportional to n 2.
• Checking if (u, v) is an edge takes Θ(1) time.
• Identifying all edges takes Θ(n 2 ) time.

1 2 3 4 5 6 7 8 1 0 1 1 0 0 0 0 0 2 1 0 1 1 1 0 0 0 3 1 1 0 0 1 0 1 1 4 0 1 0 0 1 0 0 0 5 0 1 1 1 0 1 0 0 6 0 0 0 0 1 0 0 0 7 0 0 1 0 0 0 0 1 8 0 0 1 0 0 0 1 0

Graph Representation: Adjacency List

Adjacency list. Node indexed array of lists.

• Two representations of each edge.
• Space proportional to m + n.
• Checking if (u, v) is an edge takes O(deg(u)) time.
• Identifying all edges takes Θ(m + n) time.

1 2 3 2 3 4 2 5 5 6 7 3 8 8

1 3 4 5 1 2 5 7 8

2 3 4 6 5

3 7

DFS – Depth first search

• Pick a starting vertex, follow outgoing edges that lead to new

vertices, and backtrack whenever “stuck”.

Technique 1: Greedy algorithms

A greedy algorithm is an algorithm that follows the

problem solving heuristic of making the locally optimal

choice at each stage with the hope of finding a global

optimum.

Interval Scheduling

• Interval scheduling.
  • Input: Set of n jobs. Each job i starts at time s (^) j and finishes at time f (^) j.
  • Two jobs are compatible if they don't overlap in time.
  • Goal: find maximum subset of mutually compatible jobs.

0 1 2 3 4 5 6 7 8 9 10 11 Time

f g h

e

a b c d

Interval Scheduling

• Interval scheduling.
  • Input: Set of n jobs. Each job i starts at time s (^) j and finishes at time f (^) j.
  • Two jobs are compatible if they don't overlap in time.
  • Goal: find maximum subset of mutually compatible jobs.

0 1 2 3 4 5 6 7 8 9 10 11 Time

f g h

e

a b c d

...

Interval Scheduling: Analysis

• Theorem: Greedy algorithm [Earliest finish time] is optimal.
• Proof: (by contradiction)
  • Assume greedy is not optimal, and let's see what happens.
  • Let i 1 , i 2 , ... i (^) k denote the set of jobs selected by greedy.
  • Let J 1 , J 2 , ... J (^) m denote the set of jobs in an optimal solution with i 1 = J 1 , i 2 = J 2 , ..., i (^) r = Jr for the largest possible value of r.

J 1 J 2 J (^) r

i 1 i 1 i (^) r i (^) r+

...

Greedy:

OPT: Jr+

Why not replace job J (^) r+ with job i (^) r+1?

job i (^) r+1 finishes before J (^) r+

Interval Partitioning

• Interval partitioning.
  • Lecture i starts at s (^) i and finishes at f (^) i.
  • Goal: find minimum number of classrooms to schedule all lectures so that no two occur at the same time in the same room.
• Ex: This schedule uses 4 classrooms to schedule 10 lectures.

9 9:30 10 10:30 11 11:30 12 12:30 1 1:30 2 2:30 Time

h

c b a

e

d g

f i

j

3 3:30 4 4: