Location Functions on Trees: A Survey and Some Recent Results

Transcription

1 Location Functions on Trees: A Survey and Some Recent Results F.R. McMorris Department of Applied Mathematics Illinois Institute of Technology Chicago, Illinois & Department of Mathematics University of Louisville Louisville, Kentucky

2 Anonymous OR Researcher Working with trees

3 Anonymous OR Researcher Working with trees is just monkey business

4 Non-anonymous OR Researchers During the past 15 years, this area of network location has experienced a fairly rapid growth in terms of both potential applications and theoretical development. D.R. Shier & P.M Dearing, Operations Research 1983

5 Outline of the talk Introduction

6 Outline of the talk Introduction Three standard location functions on finite metric spaces

7 Outline of the talk Introduction Three standard location functions on finite metric spaces Properties ( axioms ) used to characterize these functions

8 Outline of the talk Introduction Three standard location functions on finite metric spaces Properties ( axioms ) used to characterize these functions Some old and very new results

9 The general setting Let (X, d) be a finite metric space and X = k>1 elements of X are called profiles and are denoted π = (x 1,..., x k ), π = (y 1,..., y m ), etc. X k. The

10 The general setting Let (X, d) be a finite metric space and X = k>1 X k. The elements of X are called profiles and are denoted π = (x 1,..., x k ), π = (y 1,..., y m ), etc. For the profile π, think k users (voters, customers, clients, agents, etc.) with each user having a preferred location point in X.

11 The general setting Let (X, d) be a finite metric space and X = k>1 X k. The elements of X are called profiles and are denoted π = (x 1,..., x k ), π = (y 1,..., y m ), etc. For the profile π, think k users (voters, customers, clients, agents, etc.) with each user having a preferred location point in X. A location function on X is be a function of the form L : X 2 X \{ }.

12 The general setting Let (X, d) be a finite metric space and X = k>1 X k. The elements of X are called profiles and are denoted π = (x 1,..., x k ), π = (y 1,..., y m ), etc. For the profile π, think k users (voters, customers, clients, agents, etc.) with each user having a preferred location point in X. A location function on X is be a function of the form L : X 2 X \{ }. Our interest is in location functions that return, for any profile π, a set of points that minimize an objective criterion of remoteness from π.

13 Location functions on metric space (X, d) Locating a fire station versus locating a mall or distribution center.

14 Location functions on metric space (X, d) Locating a fire station versus locating a mall or distribution center. Median Function: Med(π) = {x : k d(x, x i ) is minimum}, i=1 for any profile π = (x 1,..., x k ). (mall)

15 Location functions on metric space (X, d) Locating a fire station versus locating a mall or distribution center. Median Function: Med(π) = {x : k d(x, x i ) is minimum}, i=1 for any profile π = (x 1,..., x k ). (mall) Center Function: Cen(π) = {x X : e(x, π) is minimum}, where e(x, π) = max{d(x, x 1 ), d(x, x 2 ),..., d(x, x k )} for any profile π. (fire station)

16 Location functions on metric space (X, d) Locating a fire station versus locating a mall or distribution center. Median Function: Med(π) = {x : k d(x, x i ) is minimum}, i=1 for any profile π = (x 1,..., x k ). (mall) Center Function: Cen(π) = {x X : e(x, π) is minimum}, where e(x, π) = max{d(x, x 1 ), d(x, x 2 ),..., d(x, x k )} for any profile π. (fire station) Mean Function: Mean(π) = {x X : k i=1 d 2 (x, x i ) is minimum}, for any profile π.

17 l p -function The mean and median functions are special instances of the following location function that is inspired by the l p -norm. p with π p = p k i=1 d p (x, x i ). l p -function: l p (π) = {x X : k i=1 d p (x, x i ) is minimum}, for any profile π.

18 Properties of Med It is not hard to show that Med always satisfies the following properties on any metric space.

19 Properties of Med It is not hard to show that Med always satisfies the following properties on any metric space. Anonymity (A): For every profile π = (x 1,... x k ) X and permutation σ of {1,... k}, Med(π) = Med(π σ ), where π σ = (x σ(1),..., x σ(k) ).

20 Properties of Med It is not hard to show that Med always satisfies the following properties on any metric space. Anonymity (A): For every profile π = (x 1,... x k ) X and permutation σ of {1,... k}, Med(π) = Med(π σ ), where π σ = (x σ(1),..., x σ(k) ). Betweenness (B): Med((x, y)) = {z : d(x, y) = d(x, z) + d(z, y)}.

21 Properties of Med It is not hard to show that Med always satisfies the following properties on any metric space. Anonymity (A): For every profile π = (x 1,... x k ) X and permutation σ of {1,... k}, Med(π) = Med(π σ ), where π σ = (x σ(1),..., x σ(k) ). Betweenness (B): Med((x, y)) = {z : d(x, y) = d(x, z) + d(z, y)}. Consistency (C): If Med(π 1 ) Med(π 2 ) for profiles π 1 and π 2, then Med(π 1 π 2 ) = Med(π 1 ) Med(π 2 ), where π 1 π 2 is the concatenation of π 1 and π 2, i.e., if π 1 = (x 1,..., x k ) and π 2 = (y 1,..., y m ) then π 1 π 2 = (x 1,..., x k, y 1,..., y m ).

22 Proof of (C) Let D(x, π) = k d(x, x i ). i=1 Assume Med(π 1 ) Med(π 2 ). Let x Med(π 1 ) Med(π 2 ), and z Med(π 1 π 2 ). Then D(x, π 1 π 2 ) = D(x, π 1 ) + D(x, π 2 ) D(z, π 1 ) + D(z, π 2 ) = D(z, π 1 π 2 ) D(x, π 1 π 2 ), so x Med(π 1 π 2 ). From above, D(x, π 1 ) + D(x, π 2 ) = D(z, π 1 ) + D(z, π 2 ) so D(x, π 1 ) D(z, π 1 ) and D(x, π 2 ) D(z, π 2 ) imply D(x, π 1 ) = D(z, π 1 ) and D(x, π 2 ) = D(z, π 2 ). i.e., z Med(π 1 ) Med(π 2 ).

23 Axioms for location function Let L be a location function on X.

24 Axioms for location function Let L be a location function on X. Anonymity (A): For every profile π = (x 1,... x k ) X and permutation σ of {1,... k}, L(π) = L(π σ ), where π σ = (x σ(1),..., x σ(k) ).

25 Axioms for location function Let L be a location function on X. Anonymity (A): For every profile π = (x 1,... x k ) X and permutation σ of {1,... k}, L(π) = L(π σ ), where π σ = (x σ(1),..., x σ(k) ). Betweenness (B): L((x, y)) = {z : d(x, y) = d(x, z) + d(z, y)}.

26 Axioms for location function Let L be a location function on X. Anonymity (A): For every profile π = (x 1,... x k ) X and permutation σ of {1,... k}, L(π) = L(π σ ), where π σ = (x σ(1),..., x σ(k) ). Betweenness (B): L((x, y)) = {z : d(x, y) = d(x, z) + d(z, y)}. Consistency (C): If L(π 1 ) L(π 2 ) for profiles π 1 and π 2, then L(π 1 π 2 ) = L(π 1 ) L(π 2 ).

27 ABC Theorem Let L be a location function on X satisfying (A), (B) and (C). Is this enough to guarantee that L = Med?

28 ABC Theorem Let L be a location function on X satisfying (A), (B) and (C). Is this enough to guarantee that L = Med? Answer:

29 ABC Theorem Let L be a location function on X satisfying (A), (B) and (C). Is this enough to guarantee that L = Med? Answer: Sometimes yes!

30 ABC Theorem Consider a finite connected graph G = V (, E) with d the usual graph distance. So (V, d) is a metric space.

31 ABC Theorem Consider a finite connected graph G = V (, E) with d the usual graph distance. So (V, d) is a metric space. Note that the points in this space are V and that all location points must be on the vertices G. (There is a large literature concerned with location on networks where edges have lengths and location points can be placed on the edges.)

32 ABC Theorem Consider a finite connected graph G = V (, E) with d the usual graph distance. So (V, d) is a metric space. Note that the points in this space are V and that all location points must be on the vertices G. (There is a large literature concerned with location on networks where edges have lengths and location points can be placed on the edges.) Further assume the graph is a tree T.

33 ABC Theorem Consider a finite connected graph G = V (, E) with d the usual graph distance. So (V, d) is a metric space. Note that the points in this space are V and that all location points must be on the vertices G. (There is a large literature concerned with location on networks where edges have lengths and location points can be placed on the edges.) Further assume the graph is a tree T. A more general result in (McM, H.M. Mulder, F.S. Roberts, DAM 1998) gives the Collorary: Let L be a location function on the tree T. Then L = Med if and only if L satisfies (A), (B) and (C).

34 ABC Theorem Consider a finite connected graph G = V (, E) with d the usual graph distance. So (V, d) is a metric space. Note that the points in this space are V and that all location points must be on the vertices G. (There is a large literature concerned with location on networks where edges have lengths and location points can be placed on the edges.) Further assume the graph is a tree T. A more general result in (McM, H.M. Mulder, F.S. Roberts, DAM 1998) gives the Collorary: Let L be a location function on the tree T. Then L = Med if and only if L satisfies (A), (B) and (C). The ABC saga over more than a decade resulted in the best ABC theorem to date (H.M. Mulder & B. Novick, DAM 2012?): Theorem Let L be a location function on a median graph G. Then L = Med on G if and only if L satisfies (A), (B) and (C).

35 The ABC Theorem Bob Powers found an example on the covering graph of the 5 element non-distributive lattice M 3 (the diamond lattice ) where the ABC Theorem does not hold. i.e., there exists a location function that is not the median function, but does satisfy the three axioms.

36 The ABC Theorem Bob Powers found an example on the covering graph of the 5 element non-distributive lattice M 3 (the diamond lattice ) where the ABC Theorem does not hold. i.e., there exists a location function that is not the median function, but does satisfy the three axioms. Open Problem: Characterize those graphs where the ABC theorem is true.

37 The Mean Function It would appear that moving from p = 1 to p = 2 in the l p -function world would yield similar, easy to interpret, results on trees.

38 The Mean Function It would appear that moving from p = 1 to p = 2 in the l p -function world would yield similar, easy to interpret, results on trees. Recall that the mean function is defined by Mean(π) = {x X : k i=1 d 2 (x, x i ) is minimum}.

39 The Mean Function It would appear that moving from p = 1 to p = 2 in the l p -function world would yield similar, easy to interpret, results on trees. Recall that the mean function is defined by Mean(π) = {x X : k i=1 d 2 (x, x i ) is minimum}. Obviously Mean satisfies (A), and can be easily shown to satisfy (C).

40 The Mean Function It would appear that moving from p = 1 to p = 2 in the l p -function world would yield similar, easy to interpret, results on trees. Recall that the mean function is defined by Mean(π) = {x X : k i=1 d 2 (x, x i ) is minimum}. Obviously Mean satisfies (A), and can be easily shown to satisfy (C). Mean does not satisfy (B), but does satisfy the analogous axiom (J. Biagi, UofL MA Thesis, 2000): Middleness (Mid): Let x, y V. If d(x, y) is even, then L((x, y)) = K 1 where d(x, K 1 ) = d(y, K 1 ) = d(x, y)/2. If d(x, y) is odd, then L((x, y)) = K 2 where d(x, K 2 ) = d(y, K 2 ).

41 Reasonable conjecture Conjecture: Let L be a location function on a tree T. Then L = Mean if and only if L satisfies (A), (Mid) and (C).

42 Reasonable conjecture Conjecture: Let L be a location function on a tree T. Then L = Mean if and only if L satisfies (A), (Mid) and (C). Unfortunately ALL the l p -functions for p > 1 satisfy these three axioms.

43 Reasonable conjecture Conjecture: Let L be a location function on a tree T. Then L = Mean if and only if L satisfies (A), (Mid) and (C). Unfortunately ALL the l p -functions for p > 1 satisfy these three axioms. We needed to add a strong Property Z in (McM, H.M. Mulder, O. Ortega, DMAA 2010) to get Theorem: Let L be a location function on a tree T. Then L = Mean if and only if L satisfies (A), (Mid), (C) and (Z).

44 Property Z R π (a, b) = d(b, x) d(b, x), x π ba x π ab D π (a, b, c) = 2 d(b, x). x π abc Property (Z) : Let π = (x 1, x 2,..., x n ) be a profile and L be a location function on T. If L(π) = {a}, then R π (b, a) + R π (c, b) > D π (a, b, c) whenever ab, bc E.

45 In another direction Perhaps looking for an analog to ABC was the wrong way to go.

46 In another direction Perhaps looking for an analog to ABC was the wrong way to go. For a profile π = (x 1,..., x k ) and vertex z, let π z = (x 1,..., x k, z). Invariance (In): Let L be a location function on a tree T. L satisfies (In) if: a L(π) if and only if L(π a) = {a} for any profile π.

47 In another direction Perhaps looking for an analog to ABC was the wrong way to go. For a profile π = (x 1,..., x k ) and vertex z, let π z = (x 1,..., x k, z). Invariance (In): Let L be a location function on a tree T. L satisfies (In) if: a L(π) if and only if L(π a) = {a} for any profile π. It can be shown with some work that Mean satisfies (In); and also the following: Unanimity(U): For any vertex a T, L((a, a,..., a)) = {a}.

48 New Theorems In (McM, H.M. Mulder, O. Ortega, Networks 2012) we showed: Theorem: If L is a location function on the tree T and p > 1, then L = l p if and only if L satisfies (In), (U), and p-projection.

49 New Theorems In (McM, H.M. Mulder, O. Ortega, Networks 2012) we showed: Theorem: If L is a location function on the tree T and p > 1, then L = l p if and only if L satisfies (In), (U), and p-projection. (p-projection is another technical axiom that does the heavy lifting)

50 New Theorems In (McM, H.M. Mulder, O. Ortega, Networks 2012) we showed: Theorem: If L is a location function on the tree T and p > 1, then L = l p if and only if L satisfies (In), (U), and p-projection. (p-projection is another technical axiom that does the heavy lifting) Theorem: If L is a location function on the tree T, then L = Mean if and only if L satisfies (In), (U), and 2-Projection.

51 p-projection mess p-projectiveness (p-p): Let π = (x 1, x 2,..., x k ) be a profile of odd length, let L be a location function on T, and let p be an integer with p > 1. Let β = (z 1, z 2,..., z k ) be an out-projected profile of π, and assume L(π) = {a}. If 0 < ( ) p 1 lp 1 S β (a) + + ( ) p p 1 l1 S β (a) 2 ( p 1) lp 1 S (a) βba 2 ( ) p 3 lp 3 S (a) 2( ) p βba p 1 l1 S (a) + π βba ab + π ba ( 1) p for any vertex b adjacent to a, then L(β) = {a}.

52 p-projection mess p-projectiveness (p-p): Let π = (x 1, x 2,..., x k ) be a profile of odd length, let L be a location function on T, and let p be an integer with p > 1. Let β = (z 1, z 2,..., z k ) be an out-projected profile of π, and assume L(π) = {a}. If 0 < ( ) p 1 lp 1 S β (a) + + ( ) p p 1 l1 S β (a) 2 ( p 1) lp 1 S (a) βba 2 ( ) p 3 lp 3 S (a) 2( ) p βba p 1 l1 S (a) + π βba ab + π ba ( 1) p for any vertex b adjacent to a, then L(β) = {a}. Open Problems: Find properties much easier to grasp than (Z) and (p-p) that allow for a characterization of Mean on trees.

53 p-projection mess p-projectiveness (p-p): Let π = (x 1, x 2,..., x k ) be a profile of odd length, let L be a location function on T, and let p be an integer with p > 1. Let β = (z 1, z 2,..., z k ) be an out-projected profile of π, and assume L(π) = {a}. If 0 < ( ) p 1 lp 1 S β (a) + + ( ) p p 1 l1 S β (a) 2 ( p 1) lp 1 S (a) βba 2 ( ) p 3 lp 3 S (a) 2( ) p βba p 1 l1 S (a) + π βba ab + π ba ( 1) p for any vertex b adjacent to a, then L(β) = {a}. Open Problems: Find properties much easier to grasp than (Z) and (p-p) that allow for a characterization of Mean on trees. Extending beyond trees is open.

54 Center Function on Trees Recall the definition: Cen(π) = {x X : e(x, π) is minimum}, where e(x, π) = max{d(x, x 1 ), d(x, x 2 ),..., d(x, x k )}.

55 Center Function on Trees Recall the definition: Cen(π) = {x X : e(x, π) is minimum}, where e(x, π) = max{d(x, x 1 ), d(x, x 2 ),..., d(x, x k )}. On trees, Cen clearly satisfies (Mid), and hence not (B) and the following example shows that it does not satisfy (C):

56 Center Function on Trees Recall the definition: Cen(π) = {x X : e(x, π) is minimum}, where e(x, π) = max{d(x, x 1 ), d(x, x 2 ),..., d(x, x k )}. On trees, Cen clearly satisfies (Mid), and hence not (B) and the following example shows that it does not satisfy (C): Let T be the path x 1 x 2 x 3 x 4, π 1 = (x 1, x 4 ) and π 2 = (x 1, x 3 ). Then Cen(π 1 ) = {x 2, x 3 } and Cen(π 2 ) = {x 2 } which gives Cen(π 1 ) Cen(π 2 ) = {x 2 }. But Cen((x 1, x 4, x 1, x 3 )) = {x 2, x 3 }.

57 Center Function on Trees Recall the definition: Cen(π) = {x X : e(x, π) is minimum}, where e(x, π) = max{d(x, x 1 ), d(x, x 2 ),..., d(x, x k )}. On trees, Cen clearly satisfies (Mid), and hence not (B) and the following example shows that it does not satisfy (C): Let T be the path x 1 x 2 x 3 x 4, π 1 = (x 1, x 4 ) and π 2 = (x 1, x 3 ). Then Cen(π 1 ) = {x 2, x 3 } and Cen(π 2 ) = {x 2 } which gives Cen(π 1 ) Cen(π 2 ) = {x 2 }. But Cen((x 1, x 4, x 1, x 3 )) = {x 2, x 3 }. But Cen does satisfy (on any finite metric space): Quasi-Consistency (QC): If L(π) = L(π ) for profiles π and π, then L(ππ ) = L(π).

58 Center Function on Trees Recall the definition: Cen(π) = {x X : e(x, π) is minimum}, where e(x, π) = max{d(x, x 1 ), d(x, x 2 ),..., d(x, x k )}. On trees, Cen clearly satisfies (Mid), and hence not (B) and the following example shows that it does not satisfy (C): Let T be the path x 1 x 2 x 3 x 4, π 1 = (x 1, x 4 ) and π 2 = (x 1, x 3 ). Then Cen(π 1 ) = {x 2, x 3 } and Cen(π 2 ) = {x 2 } which gives Cen(π 1 ) Cen(π 2 ) = {x 2 }. But Cen((x 1, x 4, x 1, x 3 )) = {x 2, x 3 }. But Cen does satisfy (on any finite metric space): Quasi-Consistency (QC): If L(π) = L(π ) for profiles π and π, then L(ππ ) = L(π). Clearly if a location function L satisfies (C) then it satisfies (QC).

59 Characterization of Cen For a profile π, let {π} be the set of elements making up π. Then Cen clearly satisfies the next property (on any finite metric space) for a location function L.

60 Characterization of Cen For a profile π, let {π} be the set of elements making up π. Then Cen clearly satisfies the next property (on any finite metric space) for a location function L. Population Invariant (PI): If {π} = {π }, then L(π) = L(π ).

61 Characterization of Cen For a profile π, let {π} be the set of elements making up π. Then Cen clearly satisfies the next property (on any finite metric space) for a location function L. Population Invariant (PI): If {π} = {π }, then L(π) = L(π ). Final axiom: Redundancy (R): Let L be a location function on a tree T. If x T ({π} x), then L(π x) = L(π).

62 Characterization of Cen For a profile π, let {π} be the set of elements making up π. Then Cen clearly satisfies the next property (on any finite metric space) for a location function L. Population Invariant (PI): If {π} = {π }, then L(π) = L(π ). Final axiom: Redundancy (R): Let L be a location function on a tree T. If x T ({π} x), then L(π x) = L(π). The following is in (McM, F.S. Roberts, C. Wang, Networks, 2001) with a shorter proof in (H.M. Mulder, M.J. Pelsmajer, K.B. Reid, Aust. J. Comb. 2008)

63 Characterization of Cen For a profile π, let {π} be the set of elements making up π. Then Cen clearly satisfies the next property (on any finite metric space) for a location function L. Population Invariant (PI): If {π} = {π }, then L(π) = L(π ). Final axiom: Redundancy (R): Let L be a location function on a tree T. If x T ({π} x), then L(π x) = L(π). The following is in (McM, F.S. Roberts, C. Wang, Networks, 2001) with a shorter proof in (H.M. Mulder, M.J. Pelsmajer, K.B. Reid, Aust. J. Comb. 2008) Theorem: Let L be a location function on a tree T. Then L is the center function Cen on T if and only if L satisfies properties (Mid), (PI ), (QC) and (R).

64 Work to do Characterize Cen on a more general class of graphs than trees.

65 Work to do Characterize Cen on a more general class of graphs than trees. Results needed for 2-median, 2-mean, 2-center, etc.