Linear-Time Selection PDF

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Summary

Lecture 11 - Linear-Time Selection...

Description

Linear-Time*Selection

Selection*Problem • Input:(an(array(with(any(permutation(of(the(numbers(1..n • Output:(the(ith order(statistic((assume(base(1(for(this(problem) • The(ith smallest(number • If(i =(n,(then(the(ith smallest(number(is(n • If(i =(1,(then(the(ith smallest(number(is(1 • What(is(“i”(for(the(median?((an(expression(base(on(n) • If(n(is(odd,(then(the(median(is(the((n+1)/2(order(statistic • If(n(is(even,(then(the(median(is(the(n/2(order(statistic

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Selection*Problem Find%the%ith smallest%number%in%an%array • Recall:(it(takes(linear(time(just(to(read(an(array • What(would(be(a(O(n(lg n)(algorithm(for(this(problem? • Sort • Return(the(ith element

Reduction

Selection*Problem • Can(we(do(better(than(O(n(lg n)? • To(beat(O(n(lg n)(we(cannot(sort((cannot(use(reduction) • Note:(comparison-based(sorting(cannot(be(done(faster(than(O(n(lg n)( • (we’ll(prove(this(later)

• First(question:(do(we(even(need(to(look(at(all(of(the(elements?

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Randomized*Selection*(Quickselect) • (Expected)(linear-time(complexity • Same(as(scanning(through(the(list(once

• We(are(going(to(start(by(modifying(the(only(randomized(algorithm( that(we(we’ve(seen(so(far:(Quicksort

Key*Component*of*Quicksort:*Partitioning 3

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*P

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Key*Component*of*Quicksort:*Partitioning 3

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What(if(we(are(looking(for(the(5th order(statistic? • Do(we(need(to(recurse on(both(sides(of(the(pivot?

Quicksort QUICKSORT (A, left_index, right_index) if left_index >= right_index return

PARTITION (A, left_index, right_index) pivot = A[left_index] i = left_index + 1

// Can be done in different ways

// Up to and including right_index

// Needs to be at most a O(n) procedure,

for j = (left_index + 1) .. right_index

// but O(1) works pretty well MOVE-PIVOT-TO-LEFT (A)

if A[j] < pivot swap A[j] and A[i] i += 1

pivot_index = PARTITION (A, left_index, right_index) swap A[left_index] and A[i - 1] QUICKSORT (A, left_index, pivot_index – 1)

return i - 1

QUICKSORT (A, pivot_index + 1, right_index)

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Randomized*Selection*Algorithm QUICKSELECT (A, l, r, i) if l == r return A[l]

QUICKSORT (A, left_index, right_index) if left_index >= right_index return

MOVE-PIVOT-TO-LEFT (A)

pivot_index = PARTITION (A, left_index, right_index)

// move pivot to l p = PARTITION (A, l, r)

QUICKSORT (A, left_index, pivot_index – 1) QUICKSORT (A, pivot_index + 1, right_index)

if i == p then return A[i] if i < p then return QUICKSELECT (A, l, p - 1, i) else return QUICKSELECT (A, p + 1, r, i-p)

Iterative*Version QUICKSELECT (A, l, r, i) loop if l == r return A[l] // move pivot to l p = PARTITION (A, l, r)

1. Asymptotically,(do(you(expect( any(differences? 2. Practically,(which(do(you(think( has(better(performance?

if i == p then return A[i] if i < p then r = p - 1 else l = p + 1

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Running*time*of*Randomized*Selection • What(is(the(worst(possible(running(time? • What(is(the(worst(pivot(choice?

• What(is(the(best(possible(running(time? • You’re(more(likely(to(finish(the(algorithm(in(constant time(than(in( quadratic (worst-case)(time. • Under(normal(operation,(what(is(the(best(choice(for(a(pivot?

Running*time*of*Randomized*Selection • Under(normal(operation,(what(is(the(best(choice(for(a(pivot? • What(is(the(running(time(if(we(always(pick(the(median? • How(would(you(calculate(the(running(time(if(I(told(you(that(the( algorithm(always(picked(the(median?

𝑛 + 𝑂(𝑛) 𝑇 𝑛 ≤𝑇 2

𝑂(𝑛)

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Randomized*Selection • Hope:(that(the(choice(of(pivot(is(“good(enough”(“often(enough” • Theorem:(for(every(input(array,(the(average(running(time(of( randomized(selection(is(O(n)

Randomized*Selection*Algorithm QUICKSELECT (A, l, r, i) if l == r return A[l]

Where(is(all(of(the(work(done?

// move pivot to l p = PARTITION (A, l, r) if i == p then return A[i] if i < p then return QUICKSELECT (A, l, p - 1, i) else return QUICKSELECT (A, p + 1, r, i-p)

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Randomized*Selection • Hope:(that(the(choice(of(pivot(is(“good(enough”(“often(enough” • Theorem:(for(every(input(array,(the(average(running(time(of( randomized(selection(is(O(n) • All(of(the(work(is(done(in(partition(and(the(total(amount(of(work(done( in(a(single(call(to(partition(is(≤ 𝑐𝑛

Notation*for*our*proof • Quickselect is(said(to(be(in(phasej if(the(current(array(size(is(between 3 3 ( ),-.𝑛/𝑎𝑛𝑑 /( ), 𝑛 4 4 • j(=(0(:(0.75n(to(1.00n • j(=(1(:(0.56n(to(0.75n • j(=(2(:(0.42n(to(0.56n • j(=(3(:(0.32n(to(0.42n

Multiple(recursive(calls(can( be(in(the(same(phasej

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Proof • Multiple(recursive(calls(can(be(in(the(same(phasej • Let(Xj denote(the(number(of(recursive(calls(made(during(phasej • What(is(the(total(running(time(of(Quickselect? 

3 𝑇 𝑛 ≤ 2 𝑋, 𝑐/( ), 𝑛 4 56789:



3 𝑇 𝑛 ≤ 2 𝑋, 𝑐/( ), 𝑛 4 56789:

• Xj •c • (¾)jn • c(¾)jn

:(number(of(calls(in(phasej :(constant(for(amount(of(work(done(by(partition :(upper(bound on(array(size(during(phasej :(total(amount(of(work(during(a(single(call(in(phasej

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Probability*of*leaving*a*phase? • We(leave(phasej when(our(partition(element(is(within(the(25-75( middle(part(of(the(array.

*P

• So,(if(either(the(left(or(the(right(side(contains(at(most(75%(of(the( elements. • What(is(the(probability(that(we(choose(a(partition(in((25(..(75]?

Probability*of*leaving*a*phase? • We(have((at(worst)(a(50%(chance(to(pick(a(“good”(partition(element. • Which(means(that(we(leave(the(current(phasej . • So,(we(have(reduced(our(problem(to(the(coin(flip(problem: 𝐸 𝑋, ≤ 𝐸 #/𝑜𝑓/𝑐𝑜𝑖𝑛/𝑓𝑙𝑖𝑝𝑠/𝑡𝑜/𝑔𝑒𝑡/ℎ𝑒𝑎𝑑𝑠 Heads:(good(pivot Tails(:(bad pivot

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Coin*Flips • Let(N(=(the(number(of(flips(until(you(get(a(heads( (a(geometric(random(variable) E[N](=(1(+(½(E[N] E[N](=(1(+(½(+(¼(E[N] E[N](=(1(+(½(+(¼(+(1/8(E[N] E[N](=(1(+(½(+(¼(+(1/8(+(1/16(E[N] …

Need(at(least(one.(Then(we( have(a(50%(chance(that(it(was( tails(and(we(need(to(flip(again.

Coin*Flips • Let(N(=(the(number(of(flips(until(you(get(a(heads( (a(geometric(random(variable) E[N](=(1(+(½(E[N] E[N](=(1(+(½(+(¼(E[N] E[N](=(1(+(½(+(¼(+(1/8(E[N] E[N](=(1(+(½(+(¼(+(1/8(+(1/16(E[N] E[N]*=*2

Need(at(least(one.(Then(we( have(a(50%(chance(that(it(was( tails(and(we(need(to(flip(again.

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Coin*Flips • Let(N(=(the(number(of(flips(until(you(get(a(heads( (a(geometric(random(variable) • Alternatively,(what(is(the(expected(number(of(heads(per(coin(flip? 1 1 1 𝐸 #𝑜𝑓/ℎ𝑒𝑎𝑑𝑠/𝑝𝑒𝑟/𝑓𝑙𝑖𝑝 = / J 1 + J 0 = 2 2 2 𝑇𝑜𝑡𝑎𝑙ℎ𝑒𝑎𝑑𝑠 = 𝐸 #𝑜𝑓/ℎ𝑒𝑎𝑑𝑠/𝑝𝑒𝑟/𝑓𝑙𝑖𝑝 J 𝑁 M 1 = à N=2 N

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Back*to*the*proof 

3 𝑇 𝑛 ≤ 2 𝑋, 𝑐/( ), 𝑛 4 56789:

𝐸 𝑇(𝑛) ≤ 𝐸[∑ 56789, 𝑋, /𝑐/

Q , R

𝑛]

Expected*Value*of*T 𝐸 𝑇(𝑛) ≤ 𝐸



2 𝑋, /𝑐/

3

56789, 

𝐸 𝑇(𝑛) ≤ 𝑐𝑛 2 𝐸[𝑋, ] 56789,

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4 3

𝑛 ,

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Expected*Value*of*T 𝐸 𝑇(𝑛) ≤ 𝐸



3

2 𝑋, /𝑐/

56789,

𝐸 𝑇(𝑛) ≤ 2𝑐𝑛 2



3

56789,

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4 ,

𝑛 Converges(to(4

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𝐸 𝑇(𝑛) ≤ 2𝑐𝑛𝑐8TU = 𝑂(𝑛)

Running*time*of*Quickselect 𝐸 𝑇(𝑛) ≤ 2𝑐𝑛𝑐8TU = 𝑂 𝑛 Thus,(the(running(time(of(Quickselect T(n)(...