Download Randomized algorithms

Randomized algorithms
Hiring problem
We always want the best hire for a job!
• Using employment agency to send one candidate at a time
• Each day, we interview one candidate
• We must decide immediately if to hire candidate and if so, fire previous
In other words …
• Always want the best hire…
• Cost to interview (low)
• Cost to fire/hire… (expensive)
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Hire-Assistant(n)
1. best = 0 //least qualified candidate
2. for i = 1 to n
3. interview candidate i
4. if candidate i better than best
5.
best = i
6.
hire candidate i
• Always want the best hire… fire if better candidate comes along
• Cost to interview (low) Ci
• Cost to fire/hire… (expensive) Ch
n total number of candidates
m total number of hires
Ch > Ci
O (?)
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Hire-Assistant(n)
1. best = 0 //least qualified candidate
2. for i = 1 to n
3. interview candidate i
4. if candidate i better than best
5.
best = i
6.
hire candidate i
• Always want the best hire… fire if better candidate comes along
• Cost to interview (low) Ci
• Cost to fire/hire… (expensive) Ch
n total number of candidates
m total number of hires
Ch > Ci
O (?) different cost of hiring
Not run time but cost of hiring, but flavor
of max problem
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• Always want the best hire… fire if better candidate comes along
• Cost to interview (low) Ci
• Cost to fire/hire… (expensive) Ch
Ch > Ci
n total number of candidates
m total number of hires
O (cin + chm)
Different type of cost – not run time, but cost of hiring.
But when is it most expensive?
When candidates interview in reverse order, worst first…
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Hiring problem
1. best = 0 //least qualified candidate
2. for i = worst_candidate to best_candidate…
O (cin + chn) = O(chn)
When is this most expensive?
When candidates interview in reverse order, worst first…
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• Always want the best hire… fire if better candidate comes along
• Cost to interview (low) Ci
• Cost to fire/hire… (expensive) Ch
Ch > Ci
n total number of candidates
m total number of hires
When is it least expensive?
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Hiring problem
1. best = 0 //least qualified candidate
2. for i = best_candidate to worst_candidate…
O (cin + ch)
When is this leat expensive?
When candidates interview in order, best first…
But we don’t know who is best a priori
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Hiring problem
•
•
•
•
Cost to interview (low Ci)
Cost to fire/hire … (expensive Ch)
n number of candidates
m hired
O (cin + chm)
depends on order of candidates
Independent of order of candidates
What about if we randomly invite candidates?
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Hiring problem and randomized algorithms
O (cin + chm)
depends on order of candidates
• Can’t assume in advance that candidates are in random order – bias might
exist.
• We can add randomization to our algorithm – by asking the agency to send
names in advance and randomize
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Hiring problem - randomized
We always want the best hire for a job!
• Employment agency sends list of n candidates
in advance
• Each day, we choose randomly a candidate
from the list to interview
• We do not rely on the agency to send us
randomly, but rather take control in algorithm
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Randomized hiring problem(n)
1. randomly permute the list of candidates
2. Hire-Assistant(n)
What do we mean by randomly permute
(we need a random number generator)?
Random(a,b)
Function that returns an integer between a and b,
inclusive, where each integer has equal probability
Most programs: pseudorandom-number generator
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Including random number generator
Random(a,b)
Function that returns an integer between a and b,
inclusive, where each integer has equal probability
So, Random(0,1)?
0 with P = 0.5
1 with P = 0.5
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Including random number generator
Random(a,b)
Function that returns an integer between a and b,
inclusive, where each integer has equal probability
So, Random(3,7)?
3 with P = 0.2
4 with P = 0.2
5 with P = 0.2
6 with P = 0.2
7 with P = 0.2
Like rolling a (7-3+1) dice
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How do we random permutations?
Randomize-in-place(A,n)
1. For i = 1 to n
2. swap A[i] with A[Random(i,n)]
i
Random choice from A[i,…,n]
A=123456
A=321456
A=361452
A=361452
…
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Probability review
• Sample space S: space of all possible outcomes (e.g. that can happen in a
coin toss)
• Probability of each outcome: p(i) ≥ 0
å p(i) =1
iÎS
• Example: coin toss S = {H, T}
1
p(H ) = p(T ) =
2
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Probability review
• Event: a subset of all possible outcomes; subset of sample space S
EÍS
• Probability of event:
å p(i)
iÎE
• Example: coin toss that give heads E = {H}
• Example: rolling two dice
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Probability review
Example: rolling two dice
• List all possible outcomes
S={(1,1},(1,2),(1,3), …(2,1),(2,2),(2,3), …(6,6)}
• List the event or subset of outcomes for which the sum of the dice is 7
E = {(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)}
• Probability of event that sum of dice 7: 6/36 = 1/6
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Probability review
• Random variable: attaches value to an outcome/s
Example: value of one dice
Example: sum of two dice
P(X = x): Probability that random variable X takes on value x
random variable
value of random variable
P(X = x): Probability that sum of two dice has value x, e.g. P(X = 7) = 1/6
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Probability review
• Expectation or Expected Value of Random Variables: Average
E[X] = å p(X = x)x
x
sum over each possible values, weighted by probability of that value
5
3
1
2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
5
6
5
E[X] = 2 + 3 + 4 + 5 + 6 + 7 + 8 +
36 36
36
36
36
36
36
4
3
2
1 252
+9 +10 +11 +12 =
=7
36
36
36
36 36
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Easier way?
• Linearity of Expectation
X1,… , Xn
random variables, then:
én ù n
E êå Xi ú = å E [ Xi ]
ë i=1 û i=1
Does not require independence of random variables.
Example: X1 and X2 represent the values of a dice
1
1
1
1
1
1 21
E[X1 ] = E[X2 ] = 1 + 2 + 3 + 4 + 5 + 6 =
6
6
6
6
6
6 6
42
E[X1 + X2 ] = E[X1 ]+ E[X2 ] =
=7
6
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Probability review
• Indicator Random Variable (indicating if occurs…)
ì 1
X A = I { A} = í
î 0
if A occurs
vice versa
Lemma: For event A
E[X A ] = Pr { A}
E[X A ] =1Pr(A) + 0(1- Pr(A)) = Pr(A)
Example: Expected number Heads when flipping fair coin
XH = I {H }
1 1 1
E [ X H ] = 0 +1 = = Pr(H )
2 2 2
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Probability review
• Indicator Random Variable & Linearity of Expectation
ì 1
X A = I { A} = í
î 0
if A occurs
vice versa
én ù n
X A = å Xi , E êå Xi ú = å E[Xi ]
ë i=1 û i=1
i=1
n
Example: Expected number Heads when flipping 3 fair
coins
1
E[3X H ] = 3E[X H ] = 3 =1.5
2
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