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Evacuaiton Problem: Group Search on the Line

Evacuaiton Problem: Group Search on the Line. Leszek Gąsieniec, Thomas Gorry , Russell Martin, Marek Chrobak. The General Evacuation Problem. There are K Mobile Entities located at some point of origin. They are tasked with locating an Evacuation Point.

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Evacuaiton Problem: Group Search on the Line

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  1. Evacuaiton Problem: Group Search on the Line Leszek Gąsieniec, Thomas Gorry, Russell Martin, Marek Chrobak t.gorry@Liverpool.ac.uk http://www.csc.liv.ac.uk/~tgorry/

  2. The General Evacuation Problem • 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. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  3. 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 Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  4. 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 dfrom 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. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  5. Origins The cow-path problem was introduced by Baeza-Yates, et al. in 1988 [1] . • The cow does not know the value of d • Does not know which of the w paths leads to the goal • The cow’s eyesight is not very good, so it will not know it has found the goal until it is standing on it. “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. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  6. Origins Baeza-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. 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. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  7. One Mobile Entity with Uniform Speed 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]. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  8. Multiple Mobile Entities with Uniform Speeds Strategy 1: Ignore everyone else! Strategy 2: Teamwork Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  9. 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 Entityuntil it is informed of the location of the destination. • Total Evacuation Time is t1 + t2 + 2d1 + d2 Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  10. 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 Entityuntil it is informed of the location of the destination. • Total Evacuation Time is t1 + t2 + 2d1 + d2 • 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/

  11. 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 Entityuntil 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. • 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/

  12. 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. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  13. Two Mobile Entities with Different Speeds • FME= red • SME = blue • Yellow lines = turning points in theSME’strajectory 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). • FMEand SME will meet at distance 2k−2 from the origin, • After which the FMEis exploring virgin territory up to a distance of 2k from the origin. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  14. Two Mobile Entities with Different Speeds Theorem 2: The strategy outlined for the SME and FME gives a 9d bound for the evacuation problem. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  15. 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. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  16. Two Mobile Entities with Different Speeds Proof 2: Think of the evacuation procedure as a three-step process: • The FME locates the evacuation point • The FMEinforms the SME of that location • 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 some 0 < ε ≤ 3·2k−2. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  17. Two Mobile Entities with Different Speeds Proof 2: Discovery phase time: Inform phase time: = 2 ε Evacuation phase time: 2 ε = 6 ε Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  18. 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. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  19. 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. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  20. Three Mobile Entities with Different Speeds • FME_1= red • FME_2= green • SME = blue • Evacuation Point = Yellow • FME_1& FME_2 = orange • The two FMEs explore the line with speed = while the SME remains stationary at the origin. • Once a FME finds the Evacuation Point they sprint with speed = 1 to tell the other Mobile Entities. • The SME learns of the location of the Evacuation Point and proceeds there with its maximum speed. • As before once FME_2 is notified by FME_1 they sprint with their maximum speed of 1 to the Evacuation Point. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  21. Three Mobile Entities with Different Speeds Theorem 3: The strategy outlined for the SME, FME_1andFME_2gives a 9d bound for the evacuation problem if the SMEtravels with a speed . Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  22. Three Mobile Entities with Different Speeds Proof 3: Again think of the evacuation procedure as a three step process: • FME_1locates the evacuation point • FME_1informs the SME of that location as it passes on its way to inform FME_2. • 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. Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  23. 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: 5d This means that must be at least . Evacuation Problem: Group Search on the Line - Leszek Gąsieniec, Thomas Gorry, Russell Martin,, Marek Chrobak http://www.csc.liv.ac.uk/~tgorry/

  24. Thank You Summary Using an exploration speed of α = gives an evacuation procedure for 2 Mobile Entities that share the same maximum speed of 1that 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 Problem J. 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/

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