Linear Path Optimization with Two Dependent Variables Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?Minimising sum of consecutive points distances Manhattan metricEvolutionary algorithm for the Physical Travelling Salesman ProblemHow to order objects to minimize non-adjacency costFinding the best combinations between items of 2 arrays in a sequential mannerAlgorithm to collect items before they expireGetting maximum number of pairs in a setMinimizing cost of bus travelAlgorithm for finding the set of minimum coordinate pairsMaximize pairings subject to distance constraintFind minimum time path between two nodesSingle pair shortest path algorithm with time a constraint
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Linear Path Optimization with Two Dependent Variables
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?Minimising sum of consecutive points distances Manhattan metricEvolutionary algorithm for the Physical Travelling Salesman ProblemHow to order objects to minimize non-adjacency costFinding the best combinations between items of 2 arrays in a sequential mannerAlgorithm to collect items before they expireGetting maximum number of pairs in a setMinimizing cost of bus travelAlgorithm for finding the set of minimum coordinate pairsMaximize pairings subject to distance constraintFind minimum time path between two nodesSingle pair shortest path algorithm with time a constraint
$begingroup$
Alright, so this is a fairly interesting problem I have but also slightly difficult to explain so I will try my best.
There are two runners on a line that goes from $x=0$ to $x=100$. The two runners start at $x=50$. The runners are then given an array of coordinate pairs that they must visit. The catch is, the coordinate pair contains the $x$ value for locations runner 1 and runner 2 must be at the same time. So for example, if they are given a coordinate pair $(40, 70)$, to successfully "complete" that coordinate, runner 1 must go to $x=40$, and runner 2 must go to $x=70$. They can't move on to the next coordinate pair until both have reached their destination.
So given a large array of coordinate pairs, the runners have to visit each coordinate pair in any order they chose. The runners can move at the same time and have the same speed. The trick is how to optimize the order in which they visit the coordinates. For example, if runner 1 is at $x=10$, and runner 2 is at $x=90$, it would be inefficient to chose a coordinate pair like $(80,80)$, because runner 2 would only travel $10$ units, and spend a long time waiting while runner 1 is moving $70$ units. This is sort of like the travelling salesman problem, except there are two people involved dependent on each other, and they can visit any point from any other given point in any order.
Does anyone have any ideas how to create an algorithm that would return the best (or at least good) optimized order in which they would visit these coordinate pairs?
algorithms optimization traveling-salesman
edited Apr 5 at 12:01
xskxzr
4,36721033
asked Apr 5 at 11:08
user102516user102516
241
$endgroup$
add a comment |
$begingroup$
Alright, so this is a fairly interesting problem I have but also slightly difficult to explain so I will try my best.
There are two runners on a line that goes from $x=0$ to $x=100$. The two runners start at $x=50$. The runners are then given an array of coordinate pairs that they must visit. The catch is, the coordinate pair contains the $x$ value for locations runner 1 and runner 2 must be at the same time. So for example, if they are given a coordinate pair $(40, 70)$, to successfully "complete" that coordinate, runner 1 must go to $x=40$, and runner 2 must go to $x=70$. They can't move on to the next coordinate pair until both have reached their destination.
So given a large array of coordinate pairs, the runners have to visit each coordinate pair in any order they chose. The runners can move at the same time and have the same speed. The trick is how to optimize the order in which they visit the coordinates. For example, if runner 1 is at $x=10$, and runner 2 is at $x=90$, it would be inefficient to chose a coordinate pair like $(80,80)$, because runner 2 would only travel $10$ units, and spend a long time waiting while runner 1 is moving $70$ units. This is sort of like the travelling salesman problem, except there are two people involved dependent on each other, and they can visit any point from any other given point in any order.
Does anyone have any ideas how to create an algorithm that would return the best (or at least good) optimized order in which they would visit these coordinate pairs?
algorithms optimization traveling-salesman
edited Apr 5 at 12:01
xskxzr
4,36721033
asked Apr 5 at 11:08
user102516user102516
241
$endgroup$
add a comment |
$begingroup$
Alright, so this is a fairly interesting problem I have but also slightly difficult to explain so I will try my best.
There are two runners on a line that goes from $x=0$ to $x=100$. The two runners start at $x=50$. The runners are then given an array of coordinate pairs that they must visit. The catch is, the coordinate pair contains the $x$ value for locations runner 1 and runner 2 must be at the same time. So for example, if they are given a coordinate pair $(40, 70)$, to successfully "complete" that coordinate, runner 1 must go to $x=40$, and runner 2 must go to $x=70$. They can't move on to the next coordinate pair until both have reached their destination.
So given a large array of coordinate pairs, the runners have to visit each coordinate pair in any order they chose. The runners can move at the same time and have the same speed. The trick is how to optimize the order in which they visit the coordinates. For example, if runner 1 is at $x=10$, and runner 2 is at $x=90$, it would be inefficient to chose a coordinate pair like $(80,80)$, because runner 2 would only travel $10$ units, and spend a long time waiting while runner 1 is moving $70$ units. This is sort of like the travelling salesman problem, except there are two people involved dependent on each other, and they can visit any point from any other given point in any order.
Does anyone have any ideas how to create an algorithm that would return the best (or at least good) optimized order in which they would visit these coordinate pairs?
algorithms optimization traveling-salesman
edited Apr 5 at 12:01
xskxzr
4,36721033
asked Apr 5 at 11:08
user102516user102516
241
$endgroup$
Alright, so this is a fairly interesting problem I have but also slightly difficult to explain so I will try my best.
There are two runners on a line that goes from $x=0$ to $x=100$. The two runners start at $x=50$. The runners are then given an array of coordinate pairs that they must visit. The catch is, the coordinate pair contains the $x$ value for locations runner 1 and runner 2 must be at the same time. So for example, if they are given a coordinate pair $(40, 70)$, to successfully "complete" that coordinate, runner 1 must go to $x=40$, and runner 2 must go to $x=70$. They can't move on to the next coordinate pair until both have reached their destination.
So given a large array of coordinate pairs, the runners have to visit each coordinate pair in any order they chose. The runners can move at the same time and have the same speed. The trick is how to optimize the order in which they visit the coordinates. For example, if runner 1 is at $x=10$, and runner 2 is at $x=90$, it would be inefficient to chose a coordinate pair like $(80,80)$, because runner 2 would only travel $10$ units, and spend a long time waiting while runner 1 is moving $70$ units. This is sort of like the travelling salesman problem, except there are two people involved dependent on each other, and they can visit any point from any other given point in any order.
Does anyone have any ideas how to create an algorithm that would return the best (or at least good) optimized order in which they would visit these coordinate pairs?
algorithms optimization traveling-salesman
algorithms optimization traveling-salesman
edited Apr 5 at 12:01
xskxzr
4,36721033
asked Apr 5 at 11:08
user102516user102516
241
edited Apr 5 at 12:01
xskxzr
4,36721033
asked Apr 5 at 11:08
user102516user102516
241
edited Apr 5 at 12:01
xskxzr
4,36721033
edited Apr 5 at 12:01
xskxzr
4,36721033
edited Apr 5 at 12:01
xskxzr
4,36721033
4,36721033
asked Apr 5 at 11:08
user102516user102516
241
asked Apr 5 at 11:08
user102516user102516
241
asked Apr 5 at 11:08
user102516user102516
241
241
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
You can consider the 1D-position of the 2 runners as one 2D-position.
X-coordinate and Y-coordinate for respectively runners 1 and 2. So in your instance, the starting point is (0, 100).
Then all the goal points coordiantes can have a 2D-position in the same way, for instance (40, 70). Now the Travelling salesman problem has to be solved using the Tchebychev distance (infinite norm). I am pretty sure this is NP-complete.
A simple heuristic approach may be to always run to the next closest point (greedy nearest neighboor). Or you can either look for a more sophisticated one...
answered Apr 5 at 12:20
VinceVince
74128
$endgroup$
add a comment |
$begingroup$
As Vince observes, your problem is TSPP (traveling salesman path problem) on the plane with respect to the $L_infty$ metric. On the plane, the $L_infty$ and $L_1$ metrics are equivalent (the unit balls differ by a rotation of $45^circ$), so your problem is equivalent to TSPP on the plane with respect to the $L_1$ metric. This problem has been addressed on this question.
answered Apr 5 at 12:27
Yuval FilmusYuval Filmus
197k15186350
$endgroup$
1
$begingroup$
For those readers who have less of a background with metrics and unit circles/spheres: Wikipedia article about unit spheres where you can see the $L_infty$ and $L_1$ circles.
$endgroup$
– einpoklum
Apr 5 at 15:31
add a comment |
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2 Answers
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oldest
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2 Answers
2
active
oldest
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votes
$begingroup$
You can consider the 1D-position of the 2 runners as one 2D-position.
X-coordinate and Y-coordinate for respectively runners 1 and 2. So in your instance, the starting point is (0, 100).
Then all the goal points coordiantes can have a 2D-position in the same way, for instance (40, 70). Now the Travelling salesman problem has to be solved using the Tchebychev distance (infinite norm). I am pretty sure this is NP-complete.
A simple heuristic approach may be to always run to the next closest point (greedy nearest neighboor). Or you can either look for a more sophisticated one...
answered Apr 5 at 12:20
VinceVince
74128
$endgroup$
add a comment |
$begingroup$
You can consider the 1D-position of the 2 runners as one 2D-position.
X-coordinate and Y-coordinate for respectively runners 1 and 2. So in your instance, the starting point is (0, 100).
Then all the goal points coordiantes can have a 2D-position in the same way, for instance (40, 70). Now the Travelling salesman problem has to be solved using the Tchebychev distance (infinite norm). I am pretty sure this is NP-complete.
A simple heuristic approach may be to always run to the next closest point (greedy nearest neighboor). Or you can either look for a more sophisticated one...
answered Apr 5 at 12:20
VinceVince
74128
$endgroup$
add a comment |
$begingroup$
You can consider the 1D-position of the 2 runners as one 2D-position.
X-coordinate and Y-coordinate for respectively runners 1 and 2. So in your instance, the starting point is (0, 100).
Then all the goal points coordiantes can have a 2D-position in the same way, for instance (40, 70). Now the Travelling salesman problem has to be solved using the Tchebychev distance (infinite norm). I am pretty sure this is NP-complete.
A simple heuristic approach may be to always run to the next closest point (greedy nearest neighboor). Or you can either look for a more sophisticated one...
answered Apr 5 at 12:20
VinceVince
74128
$endgroup$
You can consider the 1D-position of the 2 runners as one 2D-position.
X-coordinate and Y-coordinate for respectively runners 1 and 2. So in your instance, the starting point is (0, 100).
Then all the goal points coordiantes can have a 2D-position in the same way, for instance (40, 70). Now the Travelling salesman problem has to be solved using the Tchebychev distance (infinite norm). I am pretty sure this is NP-complete.
A simple heuristic approach may be to always run to the next closest point (greedy nearest neighboor). Or you can either look for a more sophisticated one...
answered Apr 5 at 12:20
VinceVince
74128
answered Apr 5 at 12:20
VinceVince
74128
answered Apr 5 at 12:20
VinceVince
74128
answered Apr 5 at 12:20
VinceVince
74128
74128
add a comment |
add a comment |
$begingroup$
As Vince observes, your problem is TSPP (traveling salesman path problem) on the plane with respect to the $L_infty$ metric. On the plane, the $L_infty$ and $L_1$ metrics are equivalent (the unit balls differ by a rotation of $45^circ$), so your problem is equivalent to TSPP on the plane with respect to the $L_1$ metric. This problem has been addressed on this question.
answered Apr 5 at 12:27
Yuval FilmusYuval Filmus
197k15186350
$endgroup$
1
$begingroup$
For those readers who have less of a background with metrics and unit circles/spheres: Wikipedia article about unit spheres where you can see the $L_infty$ and $L_1$ circles.
$endgroup$
– einpoklum
Apr 5 at 15:31
add a comment |
$begingroup$
As Vince observes, your problem is TSPP (traveling salesman path problem) on the plane with respect to the $L_infty$ metric. On the plane, the $L_infty$ and $L_1$ metrics are equivalent (the unit balls differ by a rotation of $45^circ$), so your problem is equivalent to TSPP on the plane with respect to the $L_1$ metric. This problem has been addressed on this question.
answered Apr 5 at 12:27
Yuval FilmusYuval Filmus
197k15186350
$endgroup$
1
$begingroup$
For those readers who have less of a background with metrics and unit circles/spheres: Wikipedia article about unit spheres where you can see the $L_infty$ and $L_1$ circles.
$endgroup$
– einpoklum
Apr 5 at 15:31
add a comment |
$begingroup$
As Vince observes, your problem is TSPP (traveling salesman path problem) on the plane with respect to the $L_infty$ metric. On the plane, the $L_infty$ and $L_1$ metrics are equivalent (the unit balls differ by a rotation of $45^circ$), so your problem is equivalent to TSPP on the plane with respect to the $L_1$ metric. This problem has been addressed on this question.
answered Apr 5 at 12:27
Yuval FilmusYuval Filmus
197k15186350
$endgroup$
As Vince observes, your problem is TSPP (traveling salesman path problem) on the plane with respect to the $L_infty$ metric. On the plane, the $L_infty$ and $L_1$ metrics are equivalent (the unit balls differ by a rotation of $45^circ$), so your problem is equivalent to TSPP on the plane with respect to the $L_1$ metric. This problem has been addressed on this question.
answered Apr 5 at 12:27
Yuval FilmusYuval Filmus
197k15186350
answered Apr 5 at 12:27
Yuval FilmusYuval Filmus
197k15186350
answered Apr 5 at 12:27
Yuval FilmusYuval Filmus
197k15186350
answered Apr 5 at 12:27
Yuval FilmusYuval Filmus
197k15186350
197k15186350
1
$begingroup$
For those readers who have less of a background with metrics and unit circles/spheres: Wikipedia article about unit spheres where you can see the $L_infty$ and $L_1$ circles.
$endgroup$
– einpoklum
Apr 5 at 15:31
add a comment |
1
$begingroup$
For those readers who have less of a background with metrics and unit circles/spheres: Wikipedia article about unit spheres where you can see the $L_infty$ and $L_1$ circles.
$endgroup$
– einpoklum
Apr 5 at 15:31
1
1
$begingroup$
For those readers who have less of a background with metrics and unit circles/spheres: Wikipedia article about unit spheres where you can see the $L_infty$ and $L_1$ circles.
$endgroup$
– einpoklum
Apr 5 at 15:31
$begingroup$
For those readers who have less of a background with metrics and unit circles/spheres: Wikipedia article about unit spheres where you can see the $L_infty$ and $L_1$ circles.
$endgroup$
– einpoklum
Apr 5 at 15:31
add a comment |
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