Problems on One Way Road Networks

Let $OWRN = \left\langle W_x,W_y \right\rangle$ be a One Way Road Network where $W_x$ and $W_y$ are the sets of directed horizontal and vertical roads respectively. $OWRN$ can be considered as a variation of directed grid graph. The intersection of the horizontal and vertical roads are the vertices of $OWRN$ and any two consecutive vertices on a road is connected by an edge. In this work, we analyze the problem of collision free Traffic configuration in an $OWRN$. A traffic configuration is a two-tuple $TC=\left\langle OWRN, C\right\rangle$, where $C$ is a set of cars travelling on a pre-defined path. We prove that finding a maximum cardinality subset $C_{sub}\subseteq C$ such that $TC=\left\langle OWRN, C_{sub}\right\rangle$ is collision-free, is NP-hard. Lastly we investigate the properties of connectedness, shortest paths in an $OWRN$.


Introduction
The rapid development in the existing motor vehicle technology has led to the increase in demand of automated vehicles, which are in themselves capable of various decision activities such as motion-controlling , path planning etc .This has motivated many to address a large number of algorithmic and optimisation problems. The 1939 paper by Robbins [2], which gives the idea of orientable graphs and the paper by Masayoshi et.al [5], are to a certain extent an inspiration to formulating our graph network. The work by Dasler and Mount [1], which basically considers motion coordination of a set of vehicles at a traffic-crossing (intersection), has been a huge motivation and a much closer approach to that of ours. But unlike their work, we consider a much simpler version of a grid graph and mainly concentrate on analysing essential properties , and deriving suitable algorithms and structures to have a collision-free movement of traffic in the given graph network. Further, our work also extends in mentioning the only possible shortest path configuartions in our defined graph. We now discuss some definitions and notations that are referred to in the rest of the paper.

Definitions and Notations
A road is a directed line, which is either parallel to X-axis (X i ) or Y-axis (Y i ) and it is uniquely defined by its direction * Tezpur University, ak.jammi@gmail.com † Tezpur University, adas33745@gmail.com ‡ Tezpur University, saikia.navaneeta@gmail.com § Tezpur University, karmarind@gmail.com and distance from the corresponding parallel axis. Here direction is the constraint which restricts the movement on the road. Formally, a road is defined as a 2-tuple, where i, j are the indices with respect to their parallel axis, d k is the direction of the road i.e., d k ∈ {0, 1} (where 0 represents −ve direction and 1 represents +ve direction of the respective axis) , and x i is the distance of the road X i from X-axis, similarly for y j .
We define a One Way Road Network (OWRN) as a network with a set of n horizontal and m vertical Roads. Formally a OWRN is a 2-tuple, A junction or vertex v ij is defined as the intersection of X i and Y j . Formally, v ij ∈ (W x × W y ).
An edge is a connection between two adjacent vertices of a road and all the edges on a road are in the same direction as that of the road.
The boundary roads of a OWRN are the furthest and nearest roads from the X-axis and Y-axis, i.e., X 1 , X n , Y 1 and Y m . In this paper we term each vertex on the boundary roads as boundary vertex, i.e., all the vertices with degrees 2 and 3.
A vehicle c is defined as a 3-tuple t, s, P , where t is the starting time of the vehicle, s is the speed of the vehicle (throughout the journey), and P is the path to be travelled by the vehicle. Formally, c = t, s, P .
A path P r of a vehicle c r is defined as the ordered set of vertices through which it traverses the OWRN, Formally, A traffic configuration is defined as a collection of vehicles over a OWRN. Formally a T C is a 2-tuple, T C = OW RN, C , where C is the set of vehicles {c 1 , c 2 , c 3 . . . , c k }. Now, we define a collision as two vehicles c i and c j (i = j) reaching the same vertex orthogonally at the same time. So a collision-free traffic configuration is a TC without any collisions.

Results
Before considering the traffic configuration problem, we define the connectivity of a OWRN.

Connectivity of a One Way Road Network
In this section, we consider a general OWRN of n×m roads, and show the conditions for it to be strongly-connected.
The reachability to (and from) the non-boundary vertices is evident from the following lemmas.

Lemma 1. For every non-boundary vertex
Proof. we prove this lemma by considering the two roads which intersect to form the vertex v ij .

For the road
2. For the road Y j , if d j = 0 then by definition we can reach v 1j from v ij , i.e., f = 1. Otherwise we can reach v mj from v ij , i.e., f = m.
From the above conditions we can clearly see that for any non-boundary vertex v ij , ∃e, f such that v ie , v ej are reachable from v ij .

Lemma 2. For every non-boundary vertex
Proof. We prove this lemma by considering the two roads which intersect to form the vertex v ij .
From the above conditions we can clearly see that for every non-boundary vertex v ij , ∃e, f such that v ij is reachable from v ie and v f j .

Theorem 3. A One Way Road Network is strongly-connected iff the boundary roads form a cycle.
Proof. The proof of this theorem follows from Lemma 4 and Lemma 5

Lemma 4. If all the boundary vertices of a OWRN form a cycle, then it is strongly-connected.
Proof. Consider two vertices v ij , v kl in a OWRN, to reach from v ij to v kl , we have four different possibilities 1. Both boundary vertices: Any boundary vertex is reachable from any other boundary vertex, since they all form a cycle. Therefore a path exists.
2. v ij non-boundary vertex, v kl boundary vertex: From Lemma 1 we know that, from any non-boundary vertex v ij we can always reach a boundary vertex, and from that vertex we can reach v kl as shown in 1. Therefore a path exists.
3. v ij boundary vertex, v kl non-boundary vertex: From Lemma 2 we know that any non-boundary vertex v kl is always reachable from a boundary vertex, and which in turn is reachable from v ij as shown in 1. Therefore a path exists.
4. Both non-boundary vertices: From 1, 2 and 3 it is implied that there exists a path in this case too.

Lemma 5. If a given One Way Road Network is stronglyconnected, then all the boundary vertices form a cycle.
Proof. Let us assume on the contrary that the boundary vertices do not form a cycle in the OWRN. Then there will exist a boundary vertex of degree 2 such that either both the boundary roads are incoming or outgoing.
1. Both incoming roads: In this case, we will not be able to reach any other vertex from that vertex.
2. Both outgoing roads: In this case, we will not be able to reach that vertex from any other vertex.
So there will exist atleast one vertex which is not stronglyconnected. Therefore, the OWRN is not strongly-connected. Hence, by contradiction, we can claim that the boundary vertices of a strongly-connected OWRN will always form a cycle.

Traffic Configuration
We now define the traffic configuration problem in a connected OWRN. Problem 1. Given a traffic configuration OW RN, C , our objective is to find a maximum cardinality subset C sub , C sub ⊆ C , such that the new traffic configuration OW RN, C sub is collision-free.
In the following sections we discuss the hardness of the above problem, and also mention some of the restricted versions of the same.

Hardness of Collision-Free Traffic Configuration
In this section we show that finding a solution to the traffic configuration problem is NP-Hard. For this , we have the following theorem. Theorem 6. Given an undirected graph G = V, E , there exists a traffic configuration OW RN, C , computable in polynomial-time, such that the cardinality of Maximum Independent Set is k iff the maximum cardinality of C sub is k.
To prove this theorem, we reduce Maximum Independent Set problem to the Traffic Configuration problem, which is achieved with the help of the following lemmas and algorithms.

Lemma 7. Any complete graph K n can be converted to an equivalent traffic configuration T C.
Proof. We prove this lemma using proof by construction. The following steps show how to construct a T C from K n .
1. We construct a OWRN of 2n × n roads, with 2n horizontal roads and n vertical roads in which and y j = 0 j = 1 y j−1 + δ 1 < j ≤ n where δ is a numeric constant.
2. The set of vehicles C is defined as {c 1 , c 2 , c 3 . . . , c n } and for each vehicle c i ∈ C we assume (a) The start time to be 0 and the velocity to be ω.
(c) c i = 0, ω, P i 3. Now we can observe that two vehicles {c i , c j } ∈ C collide at vertex v (n−i+1)(j) , where i < j.
4. We assume that each node l in K n corresponds to a vehicle c l , and each edge between two nodes α and γ in K n corresponds to the collision of the respective vehicles c α ,c γ .
∴ We obtain the corresponding T C = OW RN, C of K n . Now to reduce any simple graph G, we first compute the corresponding TC for the complete graph K n (G). We then introduce 4 equi-spaced roads with directions (d k 's) {0, 1, 0, 1} between every two adjacent roads X i ,X i+1 and Y j , Y j+1 , respectively, in the above formed OWRN, the path of each vehicle is to be modified accordingly.
We define method DELAY(α, β, P i , ∆),where α and β are the vertices in the path of c i , and ∆ is the total number of delays. Which modify the path P i to introduce a ∆ number of small time delays in between the vertices α,β, this delay will also be propogated to all the sucessive vertices of β in P i .
The reduction algorithm is constructed using the following properties: Property 1: The number of delays introduced in the path of a vehicle c i is equal to i.

Property 2:
If there is an edge between two nodes i, j, j > i in G, then c i and c j will have collision in the TC. The number of delays introduced in the path P j before the collision of c i ,c j is i.

Property 3:
If there is no edge between two nodes i, j, j > i in G, then c i and c j will not have a collision in the TC. The number of delays introduced in the path P j before the collision of c i ,c j is i − 1.
The method HASEDGE(i, j) will return value true if there is an edge between i and j in the graph G, else false.
Lemma 8. The maximum number of delays introduced between the two collision vertices α and β as defined in the reduction algorithm, will be two.
Proof. The proof of this lemma follows from the above stated properties. The number of delays introduced in the path P j , before collision of vehicles c i and c j is either i,i − 1. The number of delays introduced in the path P j , before collision of vehicles c i+1 and c j is either i + 1,i.
∴ the maximum number of delays that can be introduced between α and β is two.
From the above Lemma and the reduction algorithm, we have the following Lemma Lemma 9. The above Reduction algorithm can be solved using Dynamic Programming approach in polynomial-time O(n 2 ), and the space complexity of both TC and OWRN created is O(n 2 ).

Lemma 10.
If C sub be any subset of C in TC such that T C new = OW RN, C sub is collision-free, then C sub corresponds to Independent Set of G.
Proof. Since T C new is collision-free, so no two nodes in the graph G, which corresponds to respective cars in Csub, consists of an edge. Thus, we can claim that C sub corresponds to an independent set in G.
From Lemma 10 we can say that maximum C sub corresponds to Maximum Independent Set in G. Now, using Lemma 9 and Lemma 10 we can prove that the traffic configuration problem is NP-Hard.

Restricted Version
If we constrain our vehicles to move in a straight line motion, then the corresponding graph to T C will be a Bipartite Graph . And, Maximum Independent Set of a Bipartite Graph can be computed using Konig's Theorem and Network-Flow Algorithm in polynomial-time. Hence, the restricted version of the problem is solvable in polynomial-time.

Shortest Path Properties
Suppose in a city of only One Way Road Network ,a person wants to move from one point to another in minimum time. Now, the objective would be to compute the shortest path to the destination in least possible time. Designing efficient algorithms to compute the shortest path in a One Way Road Network would be useful in many applications in the areas of facility location, digital micro-fluidic bio-chips,etc. The length of the shortest path between two vertices in a OWRN may not be the Manhattan distance. There may be a pair of neighbouring vertices which are the farthest pair of vertices in the OWRN metric. A turn in a path is defined when two consecutive pair of edges are from different roads. We have the following properties for shortest path between any two vertices in a OWRN: 1. Between any pair of vertices u, v , there exists a shortest path of atmost four turns.
2. The upperbound on the length of the shortest path between any pair of vertices u, v is the perimeter of the boundary of the OWRN.
We observe that any shortest path between every pair of vertices in the OWRN will be a rotationaly symmetric to one of the paths shown below:

Remarks
We have shown all the possible configurations of the path, that connects two vertices in a OWRN. In the future we will extend this work to compute various kind of facility location problems on a OWRN. It will be interesting to investigate the time complexity of one-centre or k-centre problems with respect to OWRN metric. Other interesting problems may be to design an efficient data structure for dynamic maintenance of shortest path in directed grid graphs.