New Online Algorithms for Story Scheduling in Web Advertising - - PowerPoint PPT Presentation

New Online Algorithms for Story Scheduling in Web Advertising

Susanne Albers Achim Paßen TU Munich HU Berlin

Online advertising

Worldwide online ad spending 2012/13: $ 100 billion Expected to surpass print ad spending soon Display advertising: images, videos, animations Content shown depending on browsing history of user

Story boarding

Story boarding

• Maintain ad position of a web site during browsing session of a user.
• Position depicts image sequences of advertisers.

Advertiser pay unit shown.

• Depending on history/state user becomes interesting for advertisers.
• Maximize revenue of session.

Model

• Session time is slotted

Time t: user continues surfing with probability β 0 < β < 1

• I = J1, . . . , JN

Ji = (ai, li, vi) ai = arrival time li = length vi = per-unit value

• Preemption: Job Ji may be scheduled for < li time units
• max
• t≥0

βtv(t) v(t) = per-unit value of job scheduled at time t Dasgupta, Ghosh, Nazerzadeh, Raghavan SODA’09

Competitive analysis

A(I) A: Online algorithm OPT(I) OPT: Offline algorithm A is c-competitive if for all I A(I) ≥ 1 c · OPT(I).

Previous results

• Lower bound: c ≥ 2 for general β

c ≥ β + β2

• Upper bound: c = 7

ALG: ∀t vi of current Ji ⇐ ⇒ loss in delaying Jk with vk > vi for 1 time unit Dasgupta, Ghosh, Nazerzadeh, Raghavan SODA’09

Previous results

• Lower bound: c ≥ 2 for general β

c ≥ β + β2

• Upper bound: c = 7
• Jobs to be scheduled immediately upon arrival

c = Ω(

• log µ/ log log µ)

µ = max{lmax/lmin, vmax/vmin}

• Total value of vi gained only if entire Ji is shown

c = O(log(vmax/vmin)) Dasgupta, Ghosh, Nazerzadeh, Raghavan SODA’09

Our contribution

• Upper bound: c = 4/(2 − β)
• Upper bound: c = 1 + Φ ≈ 2.62

Φ = 1+

√ 5 2

Golden Ratio

• Problem extension: Web page with m ad positions, where stories can be

shown simultaneously; job migration not allowed max

• t≥0

m

• j=1

βtv(t, j) v(t, j) = per-unit value of job scheduled on ad position j at time t Upper bound: 3.414 ≈ 2/(2 − √ 2) ≤ c ≤ 1/(3 − 2 √ 2) ≈ 5.828

Algorithmic approach

k 2k

Time: P1 P2 P3

2k − 1 k − 1 3k − 1

• Phases P1, P2, P3, . . . of k consecutive time steps.
• Scheduling decisions are made at the beginning of the phase.
• Jobs arriving during the phase are ignored.

Simple algorithm

Time:

(n − 1)k nk − 1

Qn : Pn

ALG1: Phase Pn: Qn = {unscheduled jobs Ji with ai ≤ (n − 1)k} Schedule jobs of Qn in order of non-increasing per-unit value Preempt job at the end of Pn Thm: c = 1/(βk−1(1 − βk)) c = 4/(2 − β) for k = ⌈− logβ 2⌉ c = 1/(1 − β) for k = 1

Refined algorithm

Time:

(n − 1)k nk − 1

Qn : Pn

ALG2: Phase Pn: Jn = remainder of last job in Pn−1 Qn = {Jn + unscheduled jobs Ji with ai ≤ (n − 1)k} Schedule jobs of Qn in order of non-increasing per-unit value If Jn contained in schedule, move it to the front Thm: c = 1/βk−1 · max{1/βk−1, 1/(1 − β2k), β3k/(1 − βk)} c = 1 + Φ ≈ 2.618, where Φ = (1 + √ 5)/2 for k = ⌊− 1

2 logβ(1 + Φ)⌋ + 1

Refined algorithm

Time:

(n − 1)k nk − 1

Qn : Pn

ALG2: Phase Pn: Jn = remainder of last job in Pn−1 Qn = {Jn + unscheduled jobs Ji with ai ≤ (n − 1)k} Schedule jobs of Qn in order of non-increasing per-unit value If Jn contained in schedule, move it to the front Thm: c = 1/βk−1 · max{1/βk−1, 1/(1 − β2k), β3k/(1 − βk)} c = 1 + Φ ≈ 2.618, where Φ = (1 + √ 5)/2 for k = ⌊− 1

2 logβ(1 + Φ)⌋ + 1

Algorithm for m ad positions

(n − 1)k nk − 1

Qn :

• Pos. 1
• Pos. 2

ALG3: Phase Pn: Qn = {unscheduled jobs Ji with ai ≤ (n − 1)k} For t = (n − 1)k, . . . , nk − 1, schedule m jobs of highest per-unit value Preempt jobs at the end of Pn Implementation: Units of a job are placed on same ad position

Algorithm for m ad positions

(n − 1)k nk − 1

Qn :

• Pos. 1
• Pos. 2

ALG3: Phase Pn: Qn = {unscheduled jobs Ji with ai ≤ (n − 1)k} For t = (n − 1)k, . . . , nk − 1, schedule m jobs of highest per-unit value Preempt jobs at the end of Pn Thm: c = 1/βk−1 · (1 + 1/(1 − βk)) c = (1 + 1/(1 − β(2 − √ 2)))/(2 − √ 2) for k = ⌈logβ(2 − √ 2)⌉ 3.414 ≈ 2/(2 − √ 2) ≤ c ≤ 1/(3 − 2 √ 2) ≈ 5.828

Simple algorithm

Time:

(n − 1)k nk − 1

Qn : Pn

ALG1: Phase Pn: Qn = {unscheduled jobs Ji with ai ≤ (n − 1)k} Schedule jobs of Qn in order of non-increasing per-unit value Jobs of same value sorted in order of increasing arrival time Preempt job at the end of Pn

Analysis simple algorithm

I = J1, . . . , JN Ji = (ai, li, vi) k-quantized input Ik = J′

1, . . . , J′ N

J′

i = (a′ i, li, vi)

a′

i = k⌈ai/k⌉

(n − 1)k nk − 1

Pn

Analysis simple algorithm

Lemma: OPT(Ik) ≥ βk−1 · OPT(I) Proof: Shift optimal schedule for I by k − 1 time units to the right. Observation: ALG1(Ik) = ALG1(I)

Relaxed offline algorithm

CHOP: Optimal algorithm that may resume preempted jobs Always schedule job with highest per-unit value. Jobs with same per-unit value are scheduled in the same order as in ALG1. Observation: CHOP(Ik) ≥ OPT(Ik)

Timing property

S = ALG1’s schedule for Ik tS(i) = start time of Ji in S S∗ = CHOP’s schedule for Ik tS∗(i) = start time of Ji in S∗ Lemma: tS(i) ≤ tS⋆(i)

tS(i) ≤ tS⋆(i)

Let Ji⋆ be first job in S⋆ with tS(i⋆) > tS⋆(i⋆) = t⋆

Ji

CHOP ALG1:

t⋆ Ji⋆

• vi ≥ vi⋆
• At time t⋆ CHOP has finished Ji

vi > vi⋆ : √ vi = vi⋆ : CHOP schedules jobs of value vi in the same order as ALG1

• tS⋆(i) ≤ t⋆ − li

tS(i) ≥ t⋆ − li + 1

Phase analysis

Time:

(n − 1)k nk − 1

Pn

Jn = {jobs scheduled by ALG1 in Pn} ALG1(Pn) = value achieved for Jn ALG1(Ik) =

• n

ALG1(Pn) CHOP(Pn) = value achieved for Jn Lemma: CHOP(Ik) =

• n

CHOP(Pn) Proof: For any Ji scheduled by CHOP, tS(i) ≤ tS⋆(i) {jobs scheduled by CHOP} ⊆

• n

Jn

Phase analysis

Time:

(n − 1)k nk − 1

Pn

Lemma: CHOP(Pn) ≤ ALG1(Pn)/(1 − βk) Proof: J = last job of Pn with value v CHOP(Pn) ≤ ALG1(Pn) + extra value in scheduling preempted portion of J ALG1(Pn) =

nk−1

• t=(n−1)k

βtv = (β(n−1)k − βnk)/(1 − β) · v extra value =

• t≥nk

βtv = βnk/(1 − β) · v CHOP(Pn)/ALG1(Pn) ≤ 1 + βnk/(β(n−1)k − βnk) = 1/(1 − βk)

Wrapping up

CHOP(Ik) ≤ ALG1(Ik)/(1 − βk) βk−1 · OPT(I) ≤ CHOP(Ik) OPT(I) ≤ ALG1(I)/(βk−1(1 − βk))

Analysis refined algorithm

• Take care of delays if last job of previous phase is continued.
• Loss of preempted jobs have to amortized over several phases;

segments of up to three phases.

Analysis m ad positions

• ∃ S⋆ value is at least as high as that of optimal schedule
• S⋆ schedules up to 2m jobs at any time
• ∀t: each Ji scheduled in S⋆ but not in S can be mapped

(a) to a unit vi′ ≥ vi scheduled in S or (b) to a job preempted by S at time t′ < t

Open problems

• Tight bounds for deterministic algorithms
• Design randomized algorithms