of moving points of the potential locations for …loffl001/publications/slides/... · 2020. 2....
TRANSCRIPT
David KirkpatrickFrank Staals
Maarten Loffler
COMPETITIVE QUERY STRATEGIESFOR MINIMISING THE PLY
OF THE POTENTIAL LOCATIONSOF MOVING POINTS
Will Evans
A BRIEF ELUCIDATIONOF THE TITLE
“MOVING POINTS”
“MOVING POINTS”• Location-aware mobile devices
“MOVING POINTS”• Location-aware mobile devices• GPS systems
“MOVING POINTS”• Location-aware mobile devices• GPS systems
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• New geometric data sets• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Often very large• New geometric data sets• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Often very large• New geometric data sets
• Imprecise in nature
• Flock tracking• Wireless sensor networks
“MOVING POINTS”
• Smart phones
• Location-aware mobile devices• GPS systems
• Often very large• New geometric data sets
• Imprecise in nature
• Flock tracking
• Dynamic / unpredictable
• Wireless sensor networks
“POTENTIAL LOCATIONS”
“POTENTIAL LOCATIONS”• Motion causes imprecision
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
p
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
• We know an upper bound on its speed
p
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
• We know an upper bound on its speed
• In 1 time step p can move at most d
p
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
• We know an upper bound on its speed
• In 1 time step p can move at most d
?
p
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
?
• We know an upper bound on its speed
• In 1 time step p can move at most d
?
p
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
?
?
• We know an upper bound on its speed
• In 1 time step p can move at most d
?
p
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
?
?
• We know an upper bound on its speed
• In 1 time step p can move at most d
?
• We model the potential locations of p inthe next time step by a disk of radius d
p
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
?
?
• We know an upper bound on its speed
• In 1 time step p can move at most d
? d
• We model the potential locations of p inthe next time step by a disk of radius d
p
“POTENTIAL LOCATIONS”
• Suppose we have a moving point p• Motion causes imprecision
?
?
• We know an upper bound on its speed
• In 1 time step p can move at most d
? d
• We model the potential locations of p inthe next time step by a disk of radius d
p
OBSERVATION:If we knew where pwas t time ago, we know it is nowsomewhere in a disk of radius td
“QUERY”
“QUERY”• Consider n moving points
“QUERY”• Consider n moving points
“QUERY”• Consider n moving points• Suppose that for each point we know a
location pi and an associated time stamp ti
“QUERY”• Consider n moving points• Suppose that for each point we know a
location pi and an associated time stamp ti• Then the current potential location region
of each point is a disk centered at pi
“QUERY”• Consider n moving points• Suppose that for each point we know a
location pi and an associated time stamp ti• Then the current potential location region
of each point is a disk centered at pi
“QUERY”
• Each unit of time, all regions grow by d
• Consider n moving points• Suppose that for each point we know a
location pi and an associated time stamp ti• Then the current potential location region
of each point is a disk centered at pi
“QUERY”
• Each unit of time, all regions grow by d
• Consider n moving points• Suppose that for each point we know a
location pi and an associated time stamp ti• Then the current potential location region
of each point is a disk centered at pi
“QUERY”
• Each unit of time, all regions grow by d
• Consider n moving points
• We can query a point to update it
• Suppose that for each point we know alocation pi and an associated time stamp ti
• Then the current potential location regionof each point is a disk centered at pi
“QUERY”
• Each unit of time, all regions grow by d
• Consider n moving points
• We can query a point to update it
• Suppose that for each point we know alocation pi and an associated time stamp ti
• Then the current potential location regionof each point is a disk centered at pi
“QUERY”
• Each unit of time, all regions grow by d
• Consider n moving points
• We can query a point to update it
• Suppose that for each point we know alocation pi and an associated time stamp ti
• Then the current potential location regionof each point is a disk centered at pi
• Takes 1 unit of time to execute
“QUERY”
• Each unit of time, all regions grow by d
• Consider n moving points
• We can query a point to update it
• Suppose that for each point we know alocation pi and an associated time stamp ti
• Then the current potential location regionof each point is a disk centered at pi
• Takes 1 unit of time to execute
“QUERY”
• Each unit of time, all regions grow by d
• Consider n moving points
• After a query, one region collapses to apoint, but all other regions grow
• We can query a point to update it
• Suppose that for each point we know alocation pi and an associated time stamp ti
• Then the current potential location regionof each point is a disk centered at pi
• Takes 1 unit of time to execute
“QUERY”
• Each unit of time, all regions grow by d
• Consider n moving points
• After a query, one region collapses to apoint, but all other regions grow
• We can query a point to update it
• Suppose that for each point we know alocation pi and an associated time stamp ti
• Then the current potential location regionof each point is a disk centered at pi
• Takes 1 unit of time to execute
“QUERY”
• Each unit of time, all regions grow by d
• Consider n moving points
• After a query, one region collapses to apoint, but all other regions grow
• We can query a point to update it
• Suppose that for each point we know alocation pi and an associated time stamp ti
• Then the current potential location regionof each point is a disk centered at pi
• Takes 1 unit of time to execute
“PLY”
“PLY”• Ply of a point
“PLY”
• Given a set of regions• Ply of a point
“PLY”
• Given a set of regions• Ply of a point
“PLY”
• Given a set of regions• δ(p) = |R : p ∈ R|
• Ply of a point
“PLY”
• Given a set of regions• δ(p) = |R : p ∈ R|
• Ply of a point
δ = 1
δ = 3
“PLY”
• Ply of the set of regions
• Given a set of regions• δ(p) = |R : p ∈ R|
• Ply of a point
δ = 1
δ = 3
“PLY”
• Ply of the set of regions
• Given a set of regions• δ(p) = |R : p ∈ R|
• Ply of a point
• Maximum ply of all points
δ = 1
δ = 3
“PLY”
• Ply of the set of regions
• Given a set of regions• δ(p) = |R : p ∈ R|
• Ply of a point
• Maximum ply of all points• ∆ = maxp δ(p)
δ = 1
δ = 3
“PLY”
• Ply of the set of regions
• Given a set of regions• δ(p) = |R : p ∈ R|
• Ply of a point
• Maximum ply of all points• ∆ = maxp δ(p)
δ = 1
δ = 3
∆ = 4
“PLY”
• Ply of the set of regions
• Given a set of regions• δ(p) = |R : p ∈ R|
• Ply of a point
• Maximum ply of all points• ∆ = maxp δ(p)
δ = 1
δ = 3
∆ = 4
POSTULATION:Low ply is good!
“COMPETITIVE”
“COMPETITIVE”• Classical, “offline” algorithm
“COMPETITIVE”
• Receives the whole input at once• Classical, “offline” algorithm
“COMPETITIVE”
• Receives the whole input at once• Classical, “offline” algorithm
• Online algorithm
“COMPETITIVE”
• Receives the whole input at once• Classical, “offline” algorithm
• Receives a sequence of input events• Online algorithm
“COMPETITIVE”
• Receives the whole input at once• Classical, “offline” algorithm
• Receives a sequence of input events• Online algorithm
• Needs to react to each event beforeknowing what the next event will be
“COMPETITIVE”
• Receives the whole input at once• Classical, “offline” algorithm
• Receives a sequence of input events• Online algorithm
• Needs to react to each event beforeknowing what the next event will be
• Competitive ratio
“COMPETITIVE”
• Receives the whole input at once• Classical, “offline” algorithm
• Receives a sequence of input events• Online algorithm
• Needs to react to each event beforeknowing what the next event will be
• Competitive ratio• Performance measure for online algorithms
“COMPETITIVE”
• Receives the whole input at once• Classical, “offline” algorithm
• Receives a sequence of input events• Online algorithm
• Needs to react to each event beforeknowing what the next event will be
• Competitive ratio• Performance measure for online algorithms• Cost of online algorithm divided by cost of
best possible offline algorithm
David KirkpatrickFrank Staals
Maarten Loffler
COMPETITIVE QUERY STRATEGIESFOR MINIMISING THE PLY
OF THE POTENTIAL LOCATIONSOF MOVING POINTS
Will Evans
David KirkpatrickFrank Staals
Maarten Loffler
COMPETITIVE QUERY STRATEGIESFOR MINIMISING THE PLY
OF THE POTENTIAL LOCATIONSOF MOVING POINTS
Will Evans
PROBLEM STATEMENT
PROBLEM STATEMENT• Input
PROBLEM STATEMENT• Input• Set of n moving points with initial regions
PROBLEM STATEMENT• Input• Set of n moving points with initial regions• Target time t∗
PROBLEM STATEMENT• Input• Set of n moving points with initial regions• Target time t∗
• Objective
PROBLEM STATEMENT• Input• Set of n moving points with initial regions• Target time t∗
• Objective• Select one region to query each time step
PROBLEM STATEMENT• Input• Set of n moving points with initial regions• Target time t∗
• Objective• Select one region to query each time step• Minimize the ply of the regions at time t∗
PROBLEM STATEMENT• Input• Set of n moving points with initial regions• Target time t∗
• Objective• Select one region to query each time step• Minimize the ply of the regions at time t∗
• Competitive analysis
PROBLEM STATEMENT• Input• Set of n moving points with initial regions• Target time t∗
• Objective• Select one region to query each time step• Minimize the ply of the regions at time t∗
• Competitive analysis• Compare against adversary that knows the
full trajectories of the points in advance
TECHNICAL STUFF
PROJECTED REGIONS
PROJECTED REGIONS• View points in time-space
PROJECTED REGIONS• View points in time-space
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory• Because of speed limit, they stay in cones
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory• Because of speed limit, they stay in cones
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory• Because of speed limit, they stay in cones
• Now consider t∗
• Cones intersect the plane z = t∗ in disks
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory• Because of speed limit, they stay in cones
• Now consider t∗
• Cones intersect the plane z = t∗ in disks
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory• Because of speed limit, they stay in cones
z = t∗
• Now consider t∗
• These are the projected uncertainty regions• Cones intersect the plane z = t∗ in disks
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory• Because of speed limit, they stay in cones
z = t∗
• Now consider t∗
• These are the projected uncertainty regions• Cones intersect the plane z = t∗ in disks
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory• Because of speed limit, they stay in cones
z = t∗
• Now consider t∗
• These are the projected uncertainty regions• Cones intersect the plane z = t∗ in disks
PROJECTED REGIONS• View points in time-space• Points follow some unknown z-monotone
trajectory• Because of speed limit, they stay in cones
z = t∗
• Now consider t∗
• Projected regions never grow, and theyshrink to radius t∗ − t when we query them
STRATEGY ATTEMPT 1
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri
3
12
22 3
3
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
3
12
22 3
3
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
3
12
22 3
3
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi• Of course, Ri is not unique
3
12
22 3
3
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi• Of course, Ri is not unique
3
12
22 3
3
• Query any of them
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi• Of course, Ri is not unique
3
12
22 3
3
• Query any of them
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi• Of course, Ri is not unique• Query any of them
1
12
22 2
2
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
CLAIM:Resulting ply ∆ can be afactor n larger than optimal!
• Of course, Ri is not unique• Query any of them
1
12
22 2
2
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
CLAIM:Resulting ply ∆ can be afactor n larger than optimal!
• Of course, Ri is not unique• Query any of them
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
CLAIM:Resulting ply ∆ can be afactor n larger than optimal!
• Of course, Ri is not unique• Query any of them
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
CLAIM:Resulting ply ∆ can be afactor n larger than optimal!
• Of course, Ri is not unique
∆ = 9
• Query any of them
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
CLAIM:Resulting ply ∆ can be afactor n larger than optimal!
• Of course, Ri is not unique
∆ = 9
• Query any of them
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
CLAIM:Resulting ply ∆ can be afactor n larger than optimal!
• Of course, Ri is not unique
∆ = 9
• Query any of them
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
CLAIM:Resulting ply ∆ can be afactor n larger than optimal!
• Of course, Ri is not unique
∆ = 9
• Query any of them
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
CLAIM:Resulting ply ∆ can be afactor n larger than optimal!
• Of course, Ri is not unique
∆ = 9
• Query any of them
STRATEGY ATTEMPT 1• Let’s just always query the region
with the largest ply.• Let δi be the max ply in region Ri• Find region Ri with largest δi
CLAIM:Resulting ply ∆ can be afactor n larger than optimal!
• Of course, Ri is not unique
∆ = 9
• Query any of them
STRATEGY ATTEMPT 2
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
STRATEGY ATTEMPT 2• Why not just always query the
largest region?• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Final regions at t∗ will have sizes 1 to n
• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Final regions at t∗ will have sizes 1 to n
• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Final regions at t∗ will have sizes 1 to n• Configuration depends on order
• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Final regions at t∗ will have sizes 1 to n• Configuration depends on order
CLAIM:Resulting ply ∆ can be afactor
√n larger than optimal!
• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Final regions at t∗ will have sizes 1 to n• Configuration depends on order
CLAIM:Resulting ply ∆ can be afactor
√n larger than optimal!
• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Final regions at t∗ will have sizes 1 to n• Configuration depends on order
CLAIM:Resulting ply ∆ can be afactor
√n larger than optimal!
• Never query the same region twice
STRATEGY ATTEMPT 2• Why not just always query the
largest region?
• Effectively, query cyclicy• Final regions at t∗ will have sizes 1 to n• Configuration depends on order
CLAIM:Resulting ply ∆ can be afactor
√n larger than optimal!
• Never query the same region twice
STRATEGY ATTEMPT 3
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order• Throw away the others; set n = 1
2n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order• Throw away the others; set n = 1
2n
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order• Throw away the others; set n = 1
2n• Repeat
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order• Throw away the others; set n = 1
2n• Repeat
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order• Throw away the others; set n = 1
2n• Repeat
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order• Throw away the others; set n = 1
2n• Repeat
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order• Throw away the others; set n = 1
2n• Repeat
STRATEGY ATTEMPT 3• Keep querying cyclicly until τ = n• Now regions have sizes n+ 1 to 2n
• While there are still regions left• Find the 1
2n regions with highest ply• Query all of them once, in any order• Throw away the others; set n = 1
2n
CLAIM:Resulting ply ∆ is within aconstant factor of optimal!
• Repeat
. . .WHY WOULD THAT WORK?
. . .WHY WOULD THAT WORK?• We may need a bit of notation
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n
R
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n
R
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n
RS
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h
RS
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h
h = 3
RS
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h• ∆∗: optimal ply (the ply of S)
h = 3
RS
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h• ∆∗: optimal ply (the ply of S)
h = 3
∆∗ = 2 RS
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h• ∆∗: optimal ply (the ply of S)• R′: half of R with highest ply (at least h)
h = 3
∆∗ = 2 RS
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h• ∆∗: optimal ply (the ply of S)• R′: half of R with highest ply (at least h)
h = 3
∆∗ = 2 SR′
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h• ∆∗: optimal ply (the ply of S)• R′: half of R with highest ply (at least h)
h = 3
∆∗ = 2
• S ′: optimal regions corresponding to R′
SR′
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h• ∆∗: optimal ply (the ply of S)• R′: half of R with highest ply (at least h)
h = 3
∆∗ = 2
• S ′: optimal regions corresponding to R′
R′
S ′
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h• ∆∗: optimal ply (the ply of S)• R′: half of R with highest ply (at least h)
h = 3
∆∗ = 2
• S ′: optimal regions corresponding to R′• Ξ: a grid with cells of size Θ(n)
R′
S ′
. . .WHY WOULD THAT WORK?• We may need a bit of notation• R: n regions of sizes n+ 1 . . . 2n• S: optimal final regions of sizes 1 . . . n• h: half of the R has ply at least h• ∆∗: optimal ply (the ply of S)• R′: half of R with highest ply (at least h)
h = 3
∆∗ = 2
• S ′: optimal regions corresponding to R′• Ξ: a grid with cells of size Θ(n)
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .• S ′ covers an area of at least Ω(n3/∆∗)
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .• S ′ covers an area of at least Ω(n3/∆∗)• S ′ intersects at least Ω(n/∆∗) cells of Ξ
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .• S ′ covers an area of at least Ω(n3/∆∗)• S ′ intersects at least Ω(n/∆∗) cells of Ξ
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .• S ′ covers an area of at least Ω(n3/∆∗)• S ′ intersects at least Ω(n/∆∗) cells of Ξ• There is an independent set I of cells of Ξ
that are intersected by S ′ of size Ω(n/∆∗)
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .• S ′ covers an area of at least Ω(n3/∆∗)• S ′ intersects at least Ω(n/∆∗) cells of Ξ• There is an independent set I of cells of Ξ
that are intersected by S ′ of size Ω(n/∆∗)
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .• S ′ covers an area of at least Ω(n3/∆∗)• S ′ intersects at least Ω(n/∆∗) cells of Ξ• There is an independent set I of cells of Ξ
that are intersected by S ′ of size Ω(n/∆∗)• Each cell of I is intersected by at least h
unique regions of R′
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .• S ′ covers an area of at least Ω(n3/∆∗)• S ′ intersects at least Ω(n/∆∗) cells of Ξ• There is an independent set I of cells of Ξ
that are intersected by S ′ of size Ω(n/∆∗)• Each cell of I is intersected by at least h
unique regions of R′• There are Ω(h · n/∆∗) regions in R′
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .• S ′ covers an area of at least Ω(n3/∆∗)• S ′ intersects at least Ω(n/∆∗) cells of Ξ• There is an independent set I of cells of Ξ
that are intersected by S ′ of size Ω(n/∆∗)• Each cell of I is intersected by at least h
unique regions of R′• There are Ω(h · n/∆∗) regions in R′• So. . .h ∈ O(∆∗)
R′
S ′
Ξ
. . .WHY WOULD THAT WORK?
h = 3
∆∗ = 2
• Now for the argument. . .• S ′ covers an area of at least Ω(n3/∆∗)• S ′ intersects at least Ω(n/∆∗) cells of Ξ• There is an independent set I of cells of Ξ
that are intersected by S ′ of size Ω(n/∆∗)• Each cell of I is intersected by at least h
unique regions of R′• There are Ω(h · n/∆∗) regions in R′• So. . .h ∈ O(∆∗)• !!!
R′
S ′
Ξ
FURTHER RESULTS
FURTHER RESULTS• Limited time τ
FURTHER RESULTS
• Depends average time ` since last query• Limited time τ
FURTHER RESULTS
• Depends average time ` since last query• Limited time τ
• Competitive ratio: O(√`/τ + 1)
FURTHER RESULTS
• Depends average time ` since last query• Limited time τ
• Competitive ratio: O(√`/τ + 1)
• Dimension d
FURTHER RESULTS
• Depends average time ` since last query• Limited time τ
• Competitive ratio: O(√`/τ + 1)
• Dimension d• Analysis goes through mostly unchanged
FURTHER RESULTS
• Depends average time ` since last query• Limited time τ
• Competitive ratio: O(√`/τ + 1)
• Dimension d• Analysis goes through mostly unchanged• Competitive ratio: O((`/τ + 1)d−
dd+1 )
FURTHER RESULTS
• Depends average time ` since last query• Limited time τ
• Competitive ratio: O(√`/τ + 1)
• Dimension d• Analysis goes through mostly unchanged• Competitive ratio: O((`/τ + 1)d−
dd+1 )
• Lower bounds
FURTHER RESULTS
• Depends average time ` since last query• Limited time τ
• Competitive ratio: O(√`/τ + 1)
• Dimension d• Analysis goes through mostly unchanged• Competitive ratio: O((`/τ + 1)d−
dd+1 )
• Lower bounds• Ratios are tight up to constant factor
FURTHER RESULTS
• Depends average time ` since last query• Limited time τ
• Competitive ratio: O(√`/τ + 1)
• Dimension d• Analysis goes through mostly unchanged• Competitive ratio: O((`/τ + 1)d−
dd+1 )
• Lower bounds• Ratios are tight up to constant factor
• NP-hardness
FURTHER RESULTS
• Depends average time ` since last query• Limited time τ
• Competitive ratio: O(√`/τ + 1)
• Dimension d• Analysis goes through mostly unchanged• Competitive ratio: O((`/τ + 1)d−
dd+1 )
• Lower bounds• Ratios are tight up to constant factor
• NP-hardness• Even knowing the full trajectories
THANK YOU!