We consider (n+1) runners with given constant unique integer speeds running along the circumference of a circle whose circumferential length is one, and all runners starting from the same point. We define and give lower bounds to a first problem PMAX of finding, for every runner r, the maximum number of runners that can be simultaneously separated from runner r by a distance of atleast d. For d=1/(2^(floor(lg(n)))), a lower bound for PMAX is ( n - ((n-1)/floor(lg(n))) ), which makes the fraction of simultaneously separated runners tend to 1 as n tends to infinity. Next, we define and give upper bounds to a second problem ISOLATE of finding, for every runner r, the minimum number of steps needed to isolate r, assuming that the runners that can be simultaneously separated from r by atleast d, are removed at each step. For d=1/(2^(floor(lg(n)))), an upper bound for ISOLATE is ( lg(n - 1)/lg(floor(lg(n))) ).
Given (n+1) points (called runners) with unique constant integer speeds moving along the circumference of a circle of unit length of circumference, starting from the same point at time t=0, a runner is said to be lonely at a time t if it is simultaneously separated from all other n runners by a distance of atleast 1/(n+1) at t. The well known Lonely Runner Conjecture (LRC) is that for every runner r, there exists a time t r at which r becomes lonely [1][2] [3].
Though inspired from the LRC, this paper defines and focuses on two different problems.
The first problem PMAX is to find, for every runner r, the lower bound for the maximum number of runners that can be simultaneously separated from r by a distance > d, where d is a given constant. Clearly, the answer to PMAX =n for d=(1/(n+1)), if and only if, the LRC is true.
The second problem ISOLATE is to find, for given distance d, for every runner r, the upper bound for the minimum number of steps needed to isolate runner r, assuming that the runners that can be simultaneously separated from runner r by a distance > d, are removed at each step.
We denote: 1. the total number of runners, including runner r, as (n+1) 2. runner r as simply runner 0.
- G as the set of runners other than runner 0. 4. each runner initially in G, as a unique integer in [1, n]. 5. the speed of runner 0 as 0 (i.e. s 0 = 0), and the integer speeds of the other runners (i.e. s i for each runner i in [1,n]) as the absolute value of their corresponding relative speeds with respect to s 0 . This approach is inspired from papers on the LRC [1][2] [3]. 6. runner 0’s position as permanently fixed at zero, conveniently denoted as the top of the circle. 7. the distance between any two runners i and j, as the shortest distance along the circumference of the circle between runners i and j. For example, if at time t, runner i is located at the left most point of the circle, and since runner 0 is fixed at the top of the circle, we say that the distance between runner i and runner 0 is 0.25 at time t, and not 0.75. 8. d as a given constant that is a negative integer power of two, where 0 < d < 0.5. 9. for each runner i in [1, n], E i as the positive integer representing the position (also called index) of the least significant “1” of s i from the right most 0 bit of s i , when s i is written in binary format. For example, if n=3, s 1 =2, s 2 =7, s 3 =16 and s 0 =0, then in binary format, s 1 =0010, s 2 =0111, s 3 =1000, so E 1 = 2, E 2 = 1 and E 3 = 4. 10. p as max(E i , over all runners i in [1, n]). For the previous example, p = 4. 11. any time t as a sum of some non-negative integer plus a fraction that is the sum of negative integer powers of two, i.e. t = NNI + b 1 2 -1 + b 2 2 -2 + b 3 2 -3 + … + b p-1 2 -p+1 + b p 2 -p , where b j ∈ {0, 1} for each integer j in [1, p], and where NNI denotes some non-negative integer. This is the notation we shall follow when we try to prove the existence of time t in this paper. Note that when t is a positive integer, all n runners are at the same position on the circle as the position they are at t=0, as they all have integer speeds. The existence of a vector <b p , b p-1 , b p-2 , … , b 2 , b 1 > that satisfies the conditions of any Theorem in our paper obviously implies the existence of some real time t satisfying that Theorem, though we know that the converse is not necessarily true. 12. using the above definition of t, the position of any runner i at time t, as the fractional part of (t s i ). 13. the status of a runner i as SAVED if it is separated from runner 0 by a distance > d, given the current values of the binary variable vector <b p , b p-1 , b p-2 , … , b 2 , b 1 >. 14. the status of a runner i as UNSAVED, if it is not SAVED, given the current values of the binary variable vector <b p , b p-1 , b p-2 , … , b 2 , b 1 >. 15. floor(x) as the greatest integer smaller than x. 16. ceiling(x) as the smallest integer greater than x. 17. lg(x) as the logarithm of x to the base 2. 18. a^b as a b , and we shall use both notations where convenient. 19. LCM as Least Common Multiple.
We first state and prove a trivial Theorem 1.
Theorem 1 : For each runner i of G, there exists a time t i at which runner i is separated from runner 0 by d = 0.5. Proof : The t i defined by setting the b j = 1 for j = E i , and setting the b j = 0 for all integers j in [1, p] and j ≠ E i , puts the position of runner i at 0.5, while runner 0 continues to remain at 0. At this t i , the runners k, where k ≠ i and k ≠ 0, can be anywhere on the circumference of the circle. Hence Proved Theorem 1 .
Next, we state and prove Theorems 2 and 3, which respectively give some lower bounds for PMAX, for d = 0.25 and d = 1/( 2 floor(lg(n)) ) .
Theorem 2 : There exists a time t at which atleast ceiling(n/2) runners of G are simultaneously separated from runner 0 by d > 0.25. Proof : The time t can be defined by the following algorithm:
- Initialize b j = 0 for all integers
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