evacuaiton problem: group search on the line leszek gąsieniec, thomas gorry, russell martin, marek...
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Evacuaiton Problem: Group Search on the LineLeszek Gąsieniec, Thomas Gorry, Russell Martin, Marek Chrobak
[email protected] HTTP://WWW.CSC.LIV.AC.UK/~TGORRY/
The General Evacuation Problem
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• There are K Mobile Entities located at some point of origin.
• They are tasked with locating an Evacuation Point.
• Once located all K Mobile Entities must occupy the Evacuation Point simultaneously.
Possible Models• Environment settings
• Fixed or mobile target
• Randomised or Deterministic strategies
• Variance of communication
• Number of Mobile Entities
• Variance on speed of Mobile Entities
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Group Search on the Line• K Mobile Entities all initially located at a point of origin on a line.
• Evacuation Point is fixed at an unknown direction and distance d from the origin point.
• Communication is limited to when two or more Mobile Entities occupy a location simultaneously.
AIM: All Mobile Entities must simultaneously occupy the Evacuation Point.
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OriginsThe cow-path problem was introduced by Baeza-Yates, et al. in 1988 [1] .
1. The cow does not know the value of d2. Does not know which of the w paths leads to the goal
3. The cow’s eyesight is not very good, so it will not know it has found the goal until it is standing on it.
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“A single cow stands at a crossroads (defined as the origin) with w paths leading off into unknown territory. Traveling with unit speed, the goal of the cow is to locate a
destination that is at distance d from the origin in as little time as possible.”
[1] R.A. Baeza-Yates , J.C. Culberson , and G.J.E. Rawlins, Searching with uncertainty, Proc. SWAT 88: 1st Scandinavian workshop on algorithm theory, no. 318 pp. 176–189, 1988.
OriginsBaeza-Yates, et al. [1, 2] studied the cow-path problem, and proposed a deterministic algorithm as a solution.
In the case that w = 2 (two paths), their algorithm will find the goal in time at most 9d and that this is optimal up to lower order terms.
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In the same work, the authors considered the case of w > 2 paths, showing they could find the destination with an optimal (up to lower order terms) result of
[2] R.A. Baeza-Yates , J.C. Culberson , and G.J.E. Rawlins, Searching in the plane, Information and Computation, vol. 106, no. 2, pp. 234–252, 1993.
One Mobile Entity with Uniform Speed
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This deterministic search strategy for a single Mobile Entity yields the search time of 9d, which is optimal up to lower order terms [2, Theorem 2.1].
Multiple Mobile Entities with Uniform Speeds
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Strategy 1: Ignore everyone else!Strategy 2: Teamwork
Multiple Mobile Entities with Uniform Speeds
d1 = The distance from the origin to the destination (= d).
t1 = Time to discover the destination by one Mobile Entity.
t2 = Additional time for this Mobile Entity to inform the other Mobile Entity.
d1 + d2 = The distance from the origin where the two Mobile Entities will meet in this scenario.
α = = = Speed used during initial exploration, and by the second Mobile Entity until it is informed of the location of the destination.
Total Evacuation Time is t1 + t2 + 2d1 + d29 /24 EVACUATION PROBLEM: GROUP SEARCH ON THE LINE - LESZEK GĄSIENIEC, THOMAS GORRY, RUSSELL MARTIN,, MAREK CHROBAK
HTTP://WWW.CSC.LIV.AC.UK/~TGORRY/
Multiple Mobile Entities with Uniform Speeds
d1 = The distance from the origin to the destination (= d).
t1 = Time to discover the destination by one Mobile Entity.
t2 = Additional time for this Mobile Entity to inform the other Mobile Entity.
d1 + d2 = The distance from the origin where the two Mobile Entities will meet in this scenario.
α = = = Speed used during initial exploration, and by the second Mobile Entity until it is informed of the location of the destination.
Total Evacuation Time is t1 + t2 + 2d1 + d210 /24
Theorem 1: Algorithm 2, with α = gives an evacuation procedure with time bound 9d, where d is the distance from the origin to the destination.
EVACUATION PROBLEM: GROUP SEARCH ON THE LINE - LESZEK GĄSIENIEC, THOMAS GORRY, RUSSELL MARTIN,, MAREK CHROBAK HTTP://WWW.CSC.LIV.AC.UK/~TGORRY/
Multiple Mobile Entities with Uniform Speeds
d1 = The distance from the origin to the destination (= d). t1 = Time to discover the destination by one Mobile Entity. t2 = Additional time for this Mobile Entity to inform the
other Mobile Entity. d1 + d2 = The distance from the origin where the two
Mobile Entities will meet in this scenario.
α = = = Speed used during initial exploration, and by the second Mobile Entity until it is informed of the location of the destination.
Only satisfied when α = So using an “exploration speed” of α = gives an evacuation
procedure for 2 Mobile Entities that works in 9d.
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Proof 1:
t1 + t2 + 2d1 + d2 ≤ 9d1
EVACUATION PROBLEM: GROUP SEARCH ON THE LINE - LESZEK GĄSIENIEC, THOMAS GORRY, RUSSELL MARTIN,, MAREK CHROBAK HTTP://WWW.CSC.LIV.AC.UK/~TGORRY/
Two Mobile Entities with Different Speeds• K = 2• Different maximum speeds. (maximum speed of the faster Mobile Entity is 1, and the speed of the second Mobile Entity is some value 0 < s < 1).
In certain situations, an evacuation time of 9d is still achievable.
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Two Mobile Entities with Different Speeds
• FME = red
• SME = blue
• Yellow lines = turning points in the SME’s trajectory are the turning points of the FME from its previously completed stage in its own trajectory.
• During the kth stage of the FME, it is exploring up to a distance of 2k (on one side of the origin).
• FME and SME will meet at distance 2k−2 from the origin,
• After which the FME is exploring virgin territory up to a distance of 2k from the origin.
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Two Mobile Entities with Different Speeds
Theorem 2: The strategy outlined for the SME and FME gives a 9d bound for the evacuation problem.
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Two Mobile Entities with Different Speeds
• Figure 3 shows in detail the paths taken by the two Mobile Entities once the evacuation point has been found.
• Evacuation point = orange line.
• The path the FME took = red
• The path that the SME took = blue
• Originally intended paths shown as a dashed lines.
• The purple line = where the two Mobile Entities walked together at s = to the evacuation point.
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Two Mobile Entities with Different Speeds
Proof 2:
Think of the evacuation procedure as a three-step process:
1. The FME locates the evacuation point
2. The FME informs the SME of that location
3. The two entities proceed (back) to the evacuation point.
Assume that d ≥ 2. (The 9d bound for small d is easy to verify.)k = The integer such that 2k−2 < d ≤ 2k. In particular, we can write d = 2k−2 + ε for some0 < ε ≤ 3·2k−2.
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Two Mobile Entities with Different Speeds
Proof 2:
Discovery phase time:
Inform phase time: = 2 ε Evacuation phase time: 2 ε = 6 ε
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Two Mobile Entities with Different Speeds
Proof 2:
Therefore, the entire evacuation procedure (in the worst-case, with a speed for the SME) will take
time at most 9d.
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Three Mobile Entities with Different Speeds• K = 3• Different maximum speeds. (maximum speed of two Mobile Entities is 1, and the speed of the third Mobile Entity is some value 0 < s < 1.
In certain situations, an evacuation time of 9d is still achievable.
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Three Mobile Entities with Different Speeds
• FME_1 = red
• FME_2 = green
• SME = blue
• Evacuation Point = Yellow
• FME_1 & FME_2 = orange
1. The two FMEs explore the line with speed = while the SME remains stationary at the origin.
2. Once a FME finds the Evacuation Point they sprint with speed = 1 to tell the other Mobile Entities.
3. The SME learns of the location of the Evacuation Point and proceeds there with its maximum speed.
4. As before once FME_2 is notified by FME_1 they sprint with their maximum speed of 1 to the Evacuation Point.
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Three Mobile Entities with Different Speeds
Theorem 3: The strategy outlined for the SME, FME_1 and FME_2 gives a 9d bound for the evacuation problem if the SME travels with a speed .
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Three Mobile Entities with Different Speeds
Proof 3:
Again think of the evacuation procedure as a three step process:
1. FME_1 locates the evacuation point
2. FME_1 informs the SME of that location as it passes on its way to inform FME_2.
3. FME_1 and FME_2 then proceed (back) to the evacuation point.
Also from Proof 1 we know that this strategy for FME_1 and FME_2 gives 9d.
Therefore, the SME must simply move 1d at least as fast as the FME_1 can get the FME_2 and return to the Evacuation Point.
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Three Mobile Entities with Different Speeds
Proof 3:
Time for FME_1 to find d:
Time to inform SME:
Time for FME_1 to inform FME_2 and return to the Evacuation Point:
Time for SME to get from the origin to the Evacuation Point:
Therefore the speed of the SME must satisfy the following: 5dThis means that must be at least .23 / 24EVACUATION PROBLEM: GROUP SEARCH ON THE LINE - LESZEK GĄSIENIEC, THOMAS GORRY, RUSSELL MARTIN,, MAREK CHROBAK
HTTP://WWW.CSC.LIV.AC.UK/~TGORRY/
Thank You
Summary• Using an exploration speed of α = gives an evacuation procedure for 2
Mobile Entities that share the same maximum speed of 1 that works in 9d.
• For two Mobile Entities with different speeds the entire evacuation procedure (in the worst-case, with a speed for the SME) will take time at
most 9d.
• Therefore, the entire evacuation procedure (in the worst-case, with a speed for the SME) will take time at most 9d when there are three Mobile
Entities, two with the maximum speed of 1 and one with a slower speed.
Other work on the Evacuation ProblemJ. Czyzowicz, L. Gąsieniec, T. Gorry, E. Kranakis, R. Martin, D. Pająk, Evacuating
Robots from an Unknown Exit in a Disk.
EVACUATION PROBLEM: GROUP SEARCH ON THE LINE - LESZEK GĄSIENIEC, THOMAS GORRY, RUSSELL MARTIN,, MAREK CHROBAK HTTP://WWW.CSC.LIV.AC.UK/~TGORRY/ 24 / 24