CS代考程序代写 data structure algorithm Lecture 2:

Lecture 2:
Greedy algorithms
William Umboh
School of Computer Science
The University of Sydney
Page 1

General techniques in this course
– Greedy algorithms [today]
– Divide & Conquer algorithms [12 Mar]
– Dynamic programming algorithms [19 and 26 Mar] – Network flow algorithms [2 and 9 Apr]
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Page 2

Greedy algorithms
A greedy algorithm is an algorithm that follows the problem solving approach of making a locally optimal choice at each stage with the hope of finding a global optimum.
Greedy algorithms can be some of the simplest algorithms to implement, but they’re often among the hardest algorithms to design and analyse.
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Page 3

Greedy: Overview
Consider problems that can be solved using a greedy approach:
– Interval scheduling/partitioning – Scheduling to minimize lateness – Shortest path
– Minimum spanning trees
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Page 4

How to design algorithms
Step 1: Understand problem
Step 4: Better understanding of problem
Step 2: Start with simple alg. Step 5: Improved alg.
Step 3: Does it work? Is it fast?
No Yes
Problem
Algorithm
Analysis
Useful strategy:
– Try some simple examples to get
feel for algorithm.
– If none of them break algorithm,
see if there’s underlying structural property we can use to prove correctness.
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Page 5
DONE!

Interval Scheduling
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Interval Scheduling
– Interval scheduling.
– Input: Set of n jobs. Each job i starts at time sj and finishes at time fj.
– Two jobs are compatible if they don’t overlap in time.
– Goal: find maximum subset of mutually compatible jobs.
b
a
c
d
e
f
g
h
0 1 2 3 4 5 6 7 8 9 10 11 The University of Sydney
Time
Page 7

Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 10

Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 11

Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
A
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 12

Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
A
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 13

Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
A
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
A
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 15

Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
A
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
A
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
A
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
A
G
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Earliest start time]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
A
G
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling: Greedy Algorithms
Greedy template. Consider jobs in some order. Take each job provided it is compatible with the ones already taken.
breaks [Earliest start time]
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Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
C
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
C
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
C
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
C
H
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
C
H
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
C
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 29

Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
C
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 30

Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
C
H
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Shortest interval]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
C
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 32

Interval Scheduling: Greedy Algorithms
Greedy template. Consider jobs in some order. Take each job provided it is compatible with the ones already taken.
– [Earliest start time] Consider jobs in ascending order of start time sj.
– [Shortest interval] Consider jobs in ascending order of interval length fj – sj.
– [Fewest conflicts] For each job, count the number of conflicting jobs cj. Schedule in ascending order of conflicts cj.
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Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 36

Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 37

Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 38

Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 39

Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 40

Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 41

Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 42

Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 43

Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
E
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 44

Interval Scheduling – [Fewest Conflicts]
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
E
H
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
C
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
AB
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
E
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
DE
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 53

Interval Scheduling
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
E
F
0 1 2 3 4 5 6 7 8 9 10 11
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Interval Scheduling
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
E
G
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 55

Interval Scheduling
A
B
C
D
E
F
G
H
0 1 2 3 4 5 6 7 8 9 10 11
Time
B
E
H
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney Page 56

Interval Scheduling: Greedy Algorithm
Only [Earliest finish time] remains to be tested.
– Greedy algorithm. Consider jobs in increasing order of finish time.
Take each job provided it is compatible with the ones already taken.
Sort jobs by finish times so that f1 £ f2 £ … £ fn. jobs selected
A¬Æ
for j = 1 to n {
if (job j compatible with A) A ¬ A È {j}
}
return A
– Implementation. O(n log n).
– Rememberjobj*thatwasaddedlasttoA. – Job j is compatible with A if sj 3 fj*.
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Interval Scheduling: Analysis
High-Level General Idea:
At each step of Greedy, there is an optimal solution consistent with Greedy’s choices so far
One way to do this is by using an exchange argument.
1. Define your greedy solution.
2. Compare solutions. If Xgreedy 1 Xopt, then they must differ in some specific way.
3. Exchange Pieces. Transform Xopt to a solution that is “closer” to Xgreedy and prove cost doesn’t increase.
4. Iterate. By iteratively exchanging pieces one can turn Xopt into Xgreedy without impacting the quality of the solution.
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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 i1, i2, … ik denote the set of jobs selected by greedy.
– Let J1, J2, … Jm denote the set of jobs in an optimal solution with i1 = J1, i2 = J2, …, ir = Jr for the largest possible value of r.
Greedy: OPT:
job ir+1 finishes before Jr+1
i1
i1
ir
ir+1
J1
J2
Jr
Jr+1
…
…
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Page 59
Why not replace job Jr+1 with job ir+1?

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 i1, i2, … ik denote the set of jobs selected by greedy.
– Let J1, J2, … Jm denote the set of jobs in an optimal solution with i1 = J1, i2 = J2, …, ir = Jr for the largest possible value of r.
Greedy: OPT’:
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solution still feasible and optimal
and agrees with larger prefix of greedy’s solution, contradicting definition of r
Page 60
job ir+1 finishes before Jr+1
i1
i1
ir
ir+1
…
J1
J2
Jr
Jr+1
…
Exchange argument!

Interval Scheduling: Recap of Exchange Argument
Greedy: OPT:
– We have an optimal solution that is “closer” to the greedy solution.
– Start the argument over again, but now the first (r+1) elements of the greedy solution and the optimal solution are identical.
– Continue iteratively until the optimal solution is transformed into the greedy solution without increasing the cost.
i1
i1
ir
ir+1
…
J1
J2
Jr
Jr+1
…
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Interval Scheduling
There exists a greedy algorithm [Earliest finish time] that computes an optimal solution in O(n log n) time.
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Scheduling to Minimize Lateness
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Scheduling to Minimizing Lateness
– Minimizing lateness problem. [No fix start time]
– Single resource processes one job at a time.
– Job i requires ti units of processing time and is due at time di.
– Due times are unique
– Ifistartsattimesi,itfinishesattimefi =si +ti.
– Lateness: !i =max{0, fi -di }.
– Goal: schedule all jobs to minimize maximum lateness L = max !j.
1
2
3
4
5
6
ti
3
2
1
4
3
2
di
6
8
9
10
14
15
– Ex: lateness = 0
jobs
processing time due time
lateness = 0
lateness = 0
lateness = 2
lateness = 0
max lateness = 5
12 13 14 15
d3 = 9
d2 = 8
d6 = 15
d1 = 6
d5 = 14
d4 = 10
0 1 2 3 4 5 6 7 8 9 10 11
The University of Sydney
Page 71

Minimizing Lateness: Greedy Algorithms
Greedy template. Consider jobs in some order.
– [Shortest processing time first] Consider jobs in ascending order of processing time ti.
– [Smallest slack] Consider jobs in ascending order of slack di – ti.
– [Earliest deadline first] Consider jobs in ascending order of deadline di.
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Minimizing Lateness: Greedy Algorithms
– Greedy template. Consider jobs in some order.
– [Shortest processing time first] Consider jobs in ascending order of
processing time ti.
counterexample
– [Smallest slack] Consider jobs in ascending order of slack di – ti. counterexample
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Page 73
ti
1
1
2
10
di
100
10
1
2
ti
1
10
di
2
10

Minimizing Lateness: Greedy Algorithm
– Greedy algorithm. [Earliest deadline first]
Sortnjobsbydeadlinesothatd1 £ d2 £ … £ dn
t¬0
for j = 1 to n
Assign job j to interval [t, t + tj] sj ¬ t, fj ¬ t + tj
t ¬ t + tj
output intervals [sj, fj]
jobs
processing time due time
1
2
3
4
5
6
ti
3
2
1
4
3
2
di
6
8
9
10
14
15
The University of Sydney
Page 74

Minimizing Lateness: Greedy Algorithm
– Greedy algorithm. [Earliest deadline first]
Sortnjobsbydeadlinesothatd1 £ d2 £ … £ dn
t¬0
for j = 1 to n
Assign job j to interval [t, t + tj] sj ¬ t, fj ¬ t + tj
t ¬ t + tj
output intervals [sj, fj]
jobs
processing time due time
d1 = 6
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14 15
1
2
3
4
5
6
ti
3
2
1
4
3
2
di
6
8
9
10
14
15
The University of Sydney
Page 75

Minimizing Lateness: Greedy Algorithm
– Greedy algorithm. [Earliest deadline first]
Sortnjobsbydeadlinesothatd1 £ d2 £ … £ dn
t¬0
for j = 1 to n
Assign job j to interval [t, t + tj] sj ¬ t, fj ¬ t + tj
t ¬ t + tj
output intervals [sj, fj]
jobs
processing time due time
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14 15
1
2
3
4
5
6
ti
3
2
1
4
3
2
di
6
8
9
10
14
15
d1 = 6
d2 = 8
The University of Sydney
Page 76

Minimizing Lateness: Greedy Algorithm
– Greedy algorithm. [Earliest deadline first]
Sortnjobsbydeadlinesothatd1 £ d2 £ … £ dn
t¬0
for j = 1 to n
Assign job j to interval [t, t + tj] sj ¬ t, fj ¬ t + tj
t ¬ t + tj
output intervals [sj, fj]
jobs
processing time due time
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14 15
1
2
3
4
5
6
ti
3
2
1
4
3
2
di
6
8
9
10
14
15
d1 = 6
d2 = 8
d3 = 9
The University of Sydney
Page 77

Minimizing Lateness: Greedy Algorithm
– Greedy algorithm. [Earliest deadline first]
Sortnjobsbydeadlinesothatd1 £ d2 £ … £ dn
t¬0
for j = 1 to n
Assign job j to interval [t, t + tj] sj ¬ t, fj ¬ t + tj
t ¬ t + tj
output intervals [sj, fj]
jobs
processing time due time
max lateness = 1
1
2
3
4
5
6
ti
3
2
1
4
3
2
di
6
8
9
10
14
15
d1 = 6
d2 = 8
d3 = 9
d4 = 10
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14 15
The University of Sydney
Page 78

Minimizing Lateness: Greedy Algorithm
– Greedy algorithm. [Earliest deadline first]
Sortnjobsbydeadlinesothatd1 £ d2 £ … £ dn
t¬0
for j = 1 to n
Assign job j to interval [t, t + tj] sj ¬ t, fj ¬ t + tj
t ¬ t + tj
output intervals [sj, fj]
jobs
processing time due time
max lateness = 1
1
2
3
4
5
6
ti
3
2
1
4
3
2
di
6
8
9
10
14
15
d1 = 6
d2 = 8
d3 = 9
d4 = 10
d5 = 14
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14 15
The University of Sydney
Page 79

Minimizing Lateness: Greedy Algorithm
– Greedy algorithm. [Earliest deadline first]
Sortnjobsbydeadlinesothatd1 £ d2 £ … £ dn
t¬0
for j = 1 to n
Assign job j to interval [t, t + tj] sj ¬ t, fj ¬ t + tj
t ¬ t + tj
output intervals [sj, fj]
jobs
processing time due time
max lateness = 0
1
2
3
4
5
6
ti
3
2
1
4
3
2
di
6
8
9
10
14
15
d1 = 6
d2 = 8
d3 = 9
d4 = 10
d5 = 14
d6 = 15
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14 15
The University of Sydney
Page 80
Algorithm ignores processing time!

Minimizing Lateness: No Idle Time
– Observation: There exists an optimal schedule with no idle time.
d = 4 d = 6 d = 12
0 1 2 3 4 5 6 7 8 9 10 11
0 1 2 3 4 5 6 7 8 9 10 11
– Observation: The greedy schedule has no idle time. The University of Sydney
Page 81
d=4
d=6
d = 12

Minimizing Lateness: Inversions
– Definition: An inversion in schedule S is a pair of jobs i and k such that i < k (by deadline) but k is scheduled before i. inversion – Observation: Greedy schedule has no inversions. Moreover, Greedy is only such schedule (by uniqueness of deadlines). – Observation: If a schedule (with no idle time) has an inversion, it has one with a pair of inverted jobs scheduled consecutively. k i The University of Sydney Page 82 Minimizing Lateness: Inversions – Definition: An inversion in schedule S is a pair of jobs i and k such that i < k (by deadline) but k is scheduled before i. inversion fi f'j k i before swap after swap i k – Claim: Swapping two adjacent, inverted jobs reduces the number of inversions by one and does not increase the max lateness. The University of Sydney Page 83 Minimizing Lateness: Inversions – Definition:AninversioninscheduleSisapairofjobsiandksuch that i < k (by deadline) but k is scheduled before i. before swap after swap inversion fi f‘k k i i k – Claim: Swapping two adjacent, inverted jobs reduces the number of inversions by one and does not increase the max lateness. – Proof: Let ! be the lateness before the swap, and let !' be the lateness after the swap. – !‘x =!x forallx1i,k – !'i £ !i – Ifjobkislate: `0k = fk0 dk = fi dk  fi di = `i definition (i finishes at time fi) (i < k) (definition) The University of Sydney Page 84 Minimizing Lateness: Analysis of Greedy Algorithm – Theorem: Greedy schedule S is optimal. – Proof: Define S* to be an optimal schedule that has the fewest number of inversions, and let's see what happens. – Can assume S* has no idle time. – If S* has no inversions, then S = S*. – If S* has an inversion, let i-k be an adjacent inversion. • swapping i and k does not increase the maximum lateness and strictly decreases the number of inversions • this contradicts definition of S* ■ The University of Sydney Page 85 Minimizing Lateness The University of Sydney Page 86 There exists a greedy algorithm [Earliest deadline first] that computes the optimal solution in O(n log n) time. What if deadlines are not unique? Can show that all schedules with no idle time and no inversions have the same lateness (p.g. 128 of textbook) Interval Partitioning The University of Sydney Page 87 Interval Partitioning – Interval partitioning. – Lecture i starts at si and finishes at fi. Assume integers. – 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. c e j d g b h a f i 9 9:30 10 10:30 11 11:30 12 12:30 1 1:30 2 2:30 3 3:30 4 4:30 Time The University of Sydney Page 88 Interval Partitioning – Interval partitioning. – Lecture i starts at si and finishes at fi. – 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 only 3. c d f j b g i a e h 9 9:30 10 10:30 11 11:30 12 12:30 1 1:30 2 2:30 3 3:30 4 4:30 Time The University of Sydney Page 89 Interval Partitioning: Lower bound – Definition: The depth of a set of open intervals is the maximum number that contain any given time. – Observation: Number of classrooms needed 3 depth. – Example: Depth of schedule below is 3 Þ schedule below is optimal. (a, b, c all contain 9:30) – Question: Does there always exist a schedule equal to depth of intervals? c d f j b g i a e h The University of Sydney Page 90 9 9:30 10 10:30 11 11:30 12 12:30 1 1:30 2 2:30 3 3:30 4 4:30 Time Interval Partitioning: Greedy Algorithm – Greedy algorithm. Consider lectures in increasing order of start time: assign lecture to any compatible classroom. Sort intervals by starting time so that s1 £ s2 £ ... £ sn. d ¬ 0 number of allocated classrooms for i = 1 to n { if (lecture i is compatible with some classroom k) schedule lecture i in classroom k else allocate a new classroom d + 1 schedule lecture i in classroom d + 1 d¬d+1 } The University of Sydney Page 91 Interval Partitioning: Greedy Algorithm – Greedy algorithm. Consider lectures in increasing order of start time: assign lecture to any compatible classroom. Sort intervals by starting time so that s1 £ s2 £ ... £ sn. d ¬ 0 number of allocated classrooms for i = 1 to n { if (lecture i is compatible with some classroom k) schedule lecture i in classroom k else allocate a new classroom d + 1 schedule lecture i in classroom d + 1 d¬d+1 } – Implementation. O(n log n). – Foreachclassroomk,maintainthefinishtimeofthelastjobadded. – Keeptheclassroomsinapriorityqueue. The University of Sydney Page 92 Interval Partitioning: Greedy Analysis – Observation: Greedy algorithm never schedules two incompatible lectures in the same classroom so it is feasible. – Theorem: Greedy algorithm is optimal. – Proof: – d = number of classrooms that the greedy algorithm allocates. – Classroom d is opened because we needed to schedule a job, say i, that is incompatible with all d-1 other classrooms. – Since we sorted by start time, all these incompatibilities are caused by lectures that start no later than sj. just after – Thus, we have d lectures overlapping at time [sj ,sj + 1]. time si – Key observation Þ all schedules use 3 d classrooms. ■ The University of Sydney Page 93 Interval Partitioning There exists a greedy algorithm [Earliest starting time] that computes the optimal solution in O(n log n) time. The University of Sydney Page 94 Greedy Analysis Strategies – Exchange argument. Gradually transform any solution to the one found by the greedy algorithm without hurting its quality. – Structural. Discover a simple "structural" bound asserting that every possible solution must have a certain value. Then show that your algorithm always achieves this bound. The University of Sydney Page 95 Minimum Spanning Tree The University of Sydney Page 96 Minimum Spanning Tree – Minimum spanning tree (MST). Given a connected graph G = (V, E) with real-valued edge weights ce, an MST is a subset of the edges T Í E such that T is a spanning tree whose sum of edge weights is minimized. 4 24 4 62396 9 16 18 511 511 8 14 7 8 7 10 21 G = (V, E) T, SeÎT ce = 50 – Cayley's Theorem. There are nn-2 spanning trees of Kn. can't solve by brute force The University of Sydney Page 97 Applications MST is fundamental problem with diverse applications. – Network design. – telephone, electrical, hydraulic, TV cable, computer, road – Approximation algorithms for NP-hard problems. – traveling salesperson problem, Steiner tree – Indirect applications. – max bottleneck paths – LDPC codes for error correction – image registration with Renyi entropy – learning salient features for real-time face verification – reducing data storage in sequencing amino acids in a protein –... The University of Sydney Page 98 Cycles and Cuts – Cycle. Set of edges of the form a-b, b-c, c-d, ..., y-z, z-a. 1 23 64 5 Cycle C = 1-2, 2-3, 3-4, 4-5, 5-6, 6-1 7 8 – Cutset. A cut is a subset of nodes S. The corresponding cutset D is the subset of edges with exactly one endpoint in S. 23 64 5 1 7 8 The University of Sydney Page 99 Cycles and Cuts – Cycle. Set of edges of the form a-b, b-c, c-d, ..., y-z, z-a. 1 23 64 5 Cycle C = 1-2, 2-3, 3-4, 4-5, 5-6, 6-1 7 8 – Cutset. A cut is a subset of nodes S. The corresponding cutset D is the subset of edges with exactly one endpoint in S. 1 23 6 4 Cut S = { 4, 5, 8 } 5 7 8 The University of Sydney Page 100 Cycles and Cuts – Cycle. Set of edges of the form a-b, b-c, c-d, ..., y-z, z-a. 1 23 64 5 Cycle C = 1-2, 2-3, 3-4, 4-5, 5-6, 6-1 7 8 – Cutset. A cut is a subset of nodes S. The corresponding cutset D is the subset of edges with exactly one endpoint in S. 23 64 5 1 7 8 CutS = {4,5,8} Cutset D = 5-6, 5-7, 3-4, 3-5, 7-8 The University of Sydney Page 101 Greedy Algorithm Design Strategy – In previous problems, many obvious greedy algorithms but most of them fail – For minimum spanning tree, many “obvious” greedy algorithms work! E.g. Prim’s, Boruvka’s, Kruskal’s, Reverse-delete algorithm – Single framework from which we can derive these 4 24 4 62396 9 18 511 511 8 14 7 8 7 16 10 21 The University of Sydney Page 102 G = (V, E) T, SeÎT ce = 50 Framework for MST algorithms – Notion of progress – Greedy algorithm picks cheapest choice to make progress – Prim’s algorithm – Start with some vertex s – Maintain a connected subgraph H containing s (initially, H = ({s},∅)) – Progress = # of vertices connected to s – Greedy step: Pick cheapest edge e s.t. adding e to H increases number of vertices connected to s Sanity check: does greedy step ever add an edge introducing a cycle? The University of Sydney Page 103 Framework for MST algorithms – Notion of progress – Greedy algorithm picks cheapest choice to make progress – Prim’s algorithm – Start with some vertex s – Maintain a connected subgraph H containing s (initially, H = ({s},∅)) – Progress = # of vertices connected to s – Greedy step: Pick cheapest edge (u,v) such that u is in V(H) but not v, i.e. cheapest edge in cut set corresponding to V(H) Sanity check: does greedy step ever add an edge introducing a cycle? No! The University of Sydney Page 104 Walk-Through Legend: - Red vertices = current cut - Bold edges = current solution - Dashed edges = intra-cut edges 2 23 3 18 9 s 15 7 14 6 6 11 4 19 2 30 5 5 20 16 6 44 t The University of Sydney Page 106 Walk-Through Legend: - Red vertices = current cut - Bold edges = current solution - Dashed edges = intra-cut edges 2 23 3 18 9 s 15 7 14 6 6 11 4 19 2 30 5 5 20 16 6 44 t The University of Sydney Page 107 Walk-Through Legend: - Red vertices = current cut - Bold edges = current solution - Dashed edges = intra-cut edges 2 23 3 18 9 s 15 7 14 6 6 11 4 19 2 30 5 5 20 16 6 44 8 The University of Sydney Page 108 Walk-Through Legend: - Red vertices = current cut - Bold edges = current solution - Dashed edges = intra-cut edges 2 23 3 18 9 s 15 7 14 6 6 11 4 19 2 30 5 5 20 16 6 44 8 The University of Sydney Page 109 Walk-Through Legend: - Red vertices = current cut - Bold edges = current solution - Dashed edges = intra-cut edges 2 23 3 18 9 s 15 7 14 6 6 11 4 19 2 30 5 5 20 16 6 44 8 The University of Sydney Page 110 Walk-Through Legend: - Red vertices = current cut - Bold edges = current solution - Dashed edges = intra-cut edges 2 23 3 18 9 s 15 7 14 6 6 11 4 19 2 30 5 5 20 16 6 44 8 The University of Sydney Page 111 Walk-Through Legend: - Red vertices = current cut - Bold edges = current solution - Dashed edges = intra-cut edges 2 23 3 18 9 s 15 7 14 6 6 11 4 19 2 30 5 5 20 16 6 44 8 The University of Sydney Page 112 Walk-Through Legend: - Red vertices = current cut - Bold edges = current solution - Dashed edges = intra-cut edges 2 23 3 18 9 s 15 7 14 6 6 11 4 19 2 30 5 5 20 16 6 44 8 The University of Sydney Page 113 Walk-Through Legend: - Red vertices = current cut - Bold edges = current solution - Dashed edges = intra-cut edges 2 23 3 18 9 s 15 7 14 6 6 11 4 19 2 30 5 5 20 16 6 44 8 The University of Sydney Page 114 MST properties – Simplifying assumption. All edge costs ce are distinct. – Cut property. Let S be any subset of nodes, and let e be the min cost edge with exactly one endpoint in S. Then the MST contains e. – Cycle property. Let C be any cycle, and let f be the max cost edge belonging to C. Then the MST does not contain f. S e f C The University of Sydney Page 115 e is in the MST f is not in the MST Cycle-Cut Intersection – Claim. A cycle and a cutset intersect in an even number of edges. 23 64 5 Cycle C = 1-2, 2-3, 3-4, 4-5, 5-6, 6-1 Cutset D = 3-4, 3-5, 5-6, 5-7, 7-8 Intersection = 3-4, 5-6 1 7 – Proof: (by picture) S 8 C V-S The University of Sydney Page 116 Structural Property of MSTs – Simplifying assumption. All edge costs ce are distinct. – Cut property. Let S be any subset of nodes, and let e be the min cost edge with exactly one endpoint in S. Then every MST T* contains e. – Proof: (exchange argument) – Suppose e does not belong to T*, and let's see what happens. – Adding e to T* creates a cycle C in T*. – Edge e is both in the cycle C and in the cutset D corresponding to S Þ there exists another edge, say f, that is in both C and D. – T'=T*È{e}-{f}isalsoaspanningtree. – Since ce < cf, cost(T') < cost(T*). S – This is a contradiction. ■ f e The University of Sydney T* Page 117 Proof of Correctness of Prim’s – Cut property. Let S be any subset of nodes, and let e be the min cost edge with exactly one endpoint in S. Then every MST T* contains e. – Theorem: Prim’s algorithm finds an MST. – Proof: – Let T be Prim’s solution. – Every edge of Prim’s is the cheapest edge of some cutset. – By cut property, every MST T* contains T. – T and T* have the same number of edges so T = T*.■ – Corollary: If edge costs are distinct, then MST is unique. – Alternative viewpoint: – At every step of Prim’s, current tree T is contained in every MST T*. – T is always consistent with every optimal solution. The University of Sydney Page 118 Implementation: Prim's Algorithm – Implementation. Use a priority queue. – Maintain set of explored nodes S. – For each unexplored node v, maintain attachment cost a[v] = cost of cheapest edge v to a node in S. – O(m log n) with a priority queue. [O(n2) with an array] Prim(G, c) { foreach (v Î V) d[v] ¬ ¥ Initialize an empty priority queue Q foreach (v Î V) insert v onto Q Initialize set of explored nodes S ¬ Æ while (Q is not empty) { u ¬ delete min element from Q S¬SÈ{u} foreach (edge e = (u, v) incident to u) } if ((v Ï S) and (ce < d[v])) decrease priority d[v] to ce The University of Sydney Page 135 Framework for MST algorithms – Notion of progress – Greedy algorithm picks cheapest choice to make progress – Prim’s algorithm – Start with some vertex s – Maintain a connected subgraph H containing s (initially, H = ({s},∅)) – Progress = # of vertices connected to s – Greedy step: Pick cheapest edge e s.t. adding e to H increases number of vertices connected to s – Kruskal’s algorithm – Start with empty graph – Maintain a subgraph H (possibly disconnected) – Progress = # of connected components in current subgraph – Greedy step: Pick cheapest edge that reduces # of connected components The University of Sydney Page 136 Sanity check: does Kruskal’s ever add an edge introducing a cycle? Naïve implementation: O(nm) time Need a Union-Find data structure to efficiently keep track of connected components Kruskal's Algorithm Kruskal's algorithm. [Kruskal, 1956] Consider edges in ascending order of weight. Case 1: If adding e to T creates a cycle, discard e according to cycle property. Case 2: Otherwise, insert e = (u, v) into T according to cut property where S = set of nodes in u's connected component. v u S e e The University of Sydney Case 1 Case 2 Page 137 Lexicographic Tiebreaking To remove the assumption that all edge costs are distinct: perturb all edge costs by tiny amounts to break any ties. Impact. Kruskal and Prim only interact with costs via pairwise comparisons. If perturbations are sufficiently small, MST with perturbed costs is MST with original costs. e.g., if all edge costs are integers, perturbing cost of edge ei by i / n2 boolean less(i, j) { if (cost(ei) < cost(ej)) return true else if (cost(ei) > cost(ej)) return false else if (i < j) return true else return false } Implementation. Can handle arbitrarily small perturbations implicitly by breaking ties lexicographically, according to index. The University of Sydney Page 138 Shortest Paths in a Graph Shortest path from SIT to the Sydney Opera house The University of Sydney Page 139 Shortest Path Problem – Shortest path network. – Directed graph G = (V, E). – Source s, destination t. – Length !e = length of edge e. – Shortest path problem: find shortest directed path from s to t. cost of path = sum of edge costs in path 9 s 15 2 23 3 18 The University of Sydney 5 7 44 t Page 140 14 6 6 11 4 19 5 20 16 6 2 Cost of path s-2-3-5-t = 9+23+2+16 = 50 30 Recall: How to design algorithms Step 1: Understand problem Step 4: Better understanding of problem Step 2: Start with simple alg. Step 5: Improved alg. Step 3: Does it work? Is it fast? No Yes Problem Algorithm Analysis The University of Sydney Page 141 DONE! What is simple alg. for Shortest Paths Problem? – Previous problems have easy-to-guess simple algorithms – A good algorithms design strategy is to first consider special cases The University of Sydney Page 142 Shortest Path Problem (Unit Length Edges, Undirected) – Shortest path network. – Undirected graph G = (V, E). – Source s, destination t. – Length!e =lengthofedgee=1. Can solve this with BFS! – Shortest path problem: find shortest path from s to t. cost of path = number of edges in path 23 s Cost of path s-7-t = 1+1 =2 6 5 4 The University of Sydney 7t Page 143 Shortest Path Problem (Unit-Length Edges) – Shortest path network. – Directed graph G = (V, E). – Source s, destination t. – Length!e =lengthofedgee=1. Can solve this with directed BFS! – Shortest path problem: find shortest directed path from s to t. cost of path = number of edges in path 23 s Cost of path s-7-t = 1+1 =2 6 5 4 The University of Sydney 7t Page 144 Shortest Path Problem (First try) – Shortest path network. – Directed graph G = (V, E). – Source s, destination t. – Length !e = length of edge e. – Algo: – Subdivide each edge e into !e edges of unit length – Run directed BFS 9 s 15 2 23 3 18 14 6 6 11 4 19 2 30 The University of Sydney 5 7 44 t Page 145 5 20 16 6 Shortest Path Problem (First try) – Shortest path network. – Directed graph G = (V, E). – Source s, destination t. – Length !e = length of edge e. Input size = O(n + m log L) Issue: Running time = O(L(m+ n)) where L is max edge length – Algo: – Subdivide each edge e into !e edges of unit length – Run directed BFS – Exercise: Shortest path in subdivided graph ↔ Shortest path in original graph 9 s 15 2 23 3 18 1 14 6 86 30 1 11 4 19 5 5 7 44 t 20 16 6 The University of Sydney Page 146 Dijkstra's Algorithm – Compute shortest path distances to every vertex. – Dijkstra's algorithm. – Maintain a set of explored nodes S for which we have determined the shortest path distance d(u) from s to u and the shortest path tree T rooted at s spanning S – InitializeS={s},d(s)=0,T={} – Repeatedly choose unexplored node v and edge e which minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, e to T, and set d(v) = p(v). – Observe: s-v path in T has length d(v) !e u shortest path to some u in explored part, followed by a single edge (u, v) v d(u) S s The University of Sydney Page 147 Dijkstra's Algorithm – Compute shortest path distances to every vertex. – Dijkstra's algorithm. – Maintain a set of explored nodes S for which we have determined the shortest path distance d(u) from s to u and the shortest path tree T rooted at s spanning S – InitializeS={s},d(s)=0,T={} – Repeatedly choose unexplored node v and edge e which minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, e to T, and set d(v) = p(v). – Observe: s-v path in T has length d(v) !e u shortest path to some u in explored part, followed by a single edge (u, v) v d(u) S s The University of Sydney Page 148 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). ¥ shortest path to some u in explored part, followed by a single edge (u, v) ¥ 09 s 2 24 3 14 6 6 ¥ 18 2 ¥ 30 ¥ 11 4 19 15 5 5 20 6 16 7 44 t The University of Sydney ¥¥ Page 149 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 shortest path to some u in explored part, followed by a single edge (u, v) ¥ 09 s 2 24 3 14 14 6 18 2 6 ¥ 30 ¥ 11 4 19 15 5 5 20 6 16 7 44 t The University of Sydney 15 ¥ Page 150 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 shortest path to some u in explored part, followed by a single edge (u, v) ¥ 09 s 2 24 3 14 14 6 18 2 6 ¥ 30 ¥ 11 4 19 15 5 5 20 7 44 16 6 t ¥ The University of Sydney 15 Page 151 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 shortest path to some u in explored part, followed by a single edge (u, v) 33 09 s 2 24 3 14 14 6 18 2 6 ¥ 30 ¥ 11 4 19 15 5 5 20 7 44 16 6 t ¥ The University of Sydney 15 Page 152 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 shortest path to some u in explored part, followed by a single edge (u, v) 33 09 s 2 24 3 14 14 6 18 2 6 ¥ 30 ¥ 11 4 19 15 5 5 20 7 44 16 6 t ¥ The University of Sydney 15 Page 153 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 shortest path to some u in explored part, followed by a single edge (u, v) X 32 33 09 s 2 24 3 14 14 6 18 2 ¥ 6 3044 114 15 5 5 19 6 t ¥ 20 7 44 16 The University of Sydney 15 Page 154 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 shortest path to some u in explored part, followed by a single edge (u, v) X 32 33 09 s 2 24 3 14 14 6 18 2 ¥ 6 3044 114 15 5 5 19 6 t ¥ 20 7 44 16 The University of Sydney 15 Page 155 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 shortest path to some u in explored part, followed by a single edge (u, v) X 32 33 09 s 2 24 3 14 14 6 18 2 ¥ 6 30 4X435 11 4 15 5 5 19 6 t 59 20 7 44 16 The University of Sydney 15 Page 156 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 2 24 3 shortest path to some u in explored part, followed by a single edge (u, v) X 32 33 09 s 18 2 ¥ 30 44 35 4 X 11 19 14 14 6 6 15 5 5 20 7 44 16 6 t 59 Page 157 The University of Sydney 15 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 2 24 3 14 14 6 shortest path to some u in explored part, followed by a single edge (u, v) X 32 33 09 s 18 2 6 XX11 19 34 ¥ 30 44 35 4 15 5 5 20 7 44 16 6 t The University of Sydney 15 59 51 X Page 158 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 2 24 3 14 14 6 shortest path to some u in explored part, followed by a single edge (u, v) X 32 33 09 s 18 2 6 XX11 19 34 ¥ 30 44 35 4 15 5 5 20 7 44 16 6 t The University of Sydney 15 59 51 X Page 159 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 2 24 3 14 14 6 shortest path to some u in explored part, followed by a single edge (u, v) X 32 33 09 s 18 2 6 XX11 19 45 30 44 35 4 34 15 5 5 20 7 44 The University of Sydney 15 16 6 t 5 9 X5 1 5 0 X Page 160 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 2 24 3 shortest path to some u in explored part, followed by a single edge (u, v) X 32 33 09 s 18 2 14 14 6 6 X11 19 34 20 7 44 The University of Sydney 15 45 4 30 4X435 15 5 5 6 t 59 5X1 50 X 16 Page 161 Dijkstra's Shortest Path Algorithm In each iteration, choose v not in S that minimizes p(v) = min d(u) + !e , e = (u,v) : uÎS add v to S, and set d(v) = p(v). 9 2 24 3 shortest path to some u in explored part, followed by a single edge (u, v) X 32 33 09 s 18 2 14 14 6 6 X11 19 34 20 7 44 The University of Sydney 15 45 4 30 4X435 15 5 5 6 t 16 5X9 51 50 X Page 162 Dijkstra's Algorithm: Proof of Correctness – Invariant: For each node u Î S, d(u) is the length of the shortest s-u path. – Proof: (by induction on |S|) Base case: |S| = 1 is trivial. Inductive hypothesis: Assume true for |S| = k > 1.
– Let v be next node added to S, and let (u,v) be the chosen edge.
– Claim The shortest s-u path plus (u, v) is an s-v path of length p(v).
– Pf Consider any s-v path P. We’ll show that it’s no shorter than p(v).
– Let x-y be the first edge in P that leaves S, and let P’ be the subpath to x.
– PisalreadytoolongassoonasitleavesS. s !(P) 3 !(P’) + !(x,y) 3 d(x) + !(x,y) 3 p(y) 3 p(v)
y
P
v
P’
x u
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inductive hypothesis
S instead of y
defn of p(y)
Dijkstra chose v

Dijkstra’s Algorithm: Implementation
– For each unexplored node, explicitly maintain p(v)= min d(u)+”e .
!! e = (u,v) : uÎS
– Next node to explore = node with minimum p(v).
– When exploring v, for each incident edge e = (v, w), update p(w) = min { p(w), p(v)+”e }.
– Efficient implementation. Maintain a priority queue of unexplored nodes, prioritized by p(v).
!!
Priority Queue
PQ Operation
Dijkstra
Array
Binary heap
d-way Heap
Fib heap †
Insert
n
n
log n
d log d n
1
ExtractMin
n
n
log n
d log d n
log n
ChangeKey
m
1
log n
logd n
1
IsEmpty
n
1
1
1
1
Total
n2
m log n
m log m/n n
m + n log n
The University of Sydney † Individual ops are amortized bounds Page 165

Shortest Path
The shortest path between two vertices in a graph G with n vertices and m nodes can be computed in O(m+n log n)
time.
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Summary: 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.
Problems
– Interval scheduling/partitioning
– Scheduling: minimize lateness
– Shortest path in graphs (Dijkstra’s algorithm) – Minimum spanning tree (Prim’s algorithm) –…
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This Week
–Quiz 1:
– Posted tonight
– Due Sunday 8 March 23:59:00 • One try
• 20 minutes from the time quiz is opened
• No late submissions accepted – Assignment 1:
– Released today 12:00
– Due Wednesday 11 March 23:59:00
– No submissions later than Friday 13 March 23:59:00 accepted
– Assignment 2:
– Released Thursday 12 March
– Due Wednesday 25 March 23:59:00
– No submissions later than Friday 27 March 23:59:00 accepted
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