A Bounding Box Overlay for Competitive Routing in Hybrid Communication Networks1footnote 11footnote 1This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center ’On-The-Fly Computing’ (SFB 901).

A Bounding Box Overlay for Competitive Routing in Hybrid Communication Networks111This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center ’On-The-Fly Computing’ (SFB 901).

Christina Kolb Computer Science Department, Paderborn University, Paderborn, Germany
ckolb@mail.upb.de
Christian Scheideler Computer Science Department, Paderborn University, Paderborn, Germany
scheideler@upb.de
Jannik Sundermeier Heinz Nixdorf Institute & Computer Science Department, Paderborn University, Paderborn, Germany
janniksu@mail.upb.de
Abstract

In this work, we present a new approach for competitive geometric routing in wireless ad hoc networks. In general, it is well-known that any online routing strategy performs very poor in the worst case. The main difficulty are uncovered regions within the wireless ad hoc network, which we denote as radio holes. Complex shapes of radio holes, for example zig-zag-shapes, make local geometric routing even more difficult, i.e., forwarded messages in direction to the destination might get stuck in a dead end or are routed along very long detours, when there is no knowledge about the ad hoc network. To obtain knowledge about the position and shape of radio holes, we make use of a hybrid network approach. This approach assumes that we can not just make use of the ad hoc network but also of some cellular infrastructure, which is used to gather knowledge about the underlying ad hoc network. Communication via the cellular infrastructure incurs costs as cell phone providers are involved. Therefore, we use the cellular infrastructure only to compute routing paths in the ad hoc network. The actual data transmission takes place in the ad hoc network. In order to find good routing paths we aim at computing an abstraction of the ad hoc network in which radio holes are abstracted by bounding boxes. The advantage of bounding boxes as hole abstraction is that we only have to consider a constant number of nodes per hole. We prove that bounding boxes are a suitable hole abstraction that allows us to find -competitive paths in the ad hoc network in case of non-intersecting bounding boxes. In case of intersecting bounding boxes, we show via simulations that our routing strategy significantly outperforms the so far best online routing strategies for wireless ad hoc networks. Finally, we also present a routing strategy that is -competitive in case of pairwise intersecting bounding boxes.

wireless ad hoc networks, -competitive routing, bounding boxes, visibility graphs
\EventEditors

John Q. Open and Joan R. Acces \EventNoEds2 \EventLongTitle42nd Conference on Very Important Topics (CVIT 2016) \EventShortTitleCVIT 2016 \EventAcronymCVIT \EventYear2016 \EventDateDecember 24–27, 2016 \EventLocationLittle Whinging, United Kingdom \EventLogo \SeriesVolume42 \ArticleNo

1 Introduction

Imagine yourself walking through the city center with your smartphone. Because there are crowds of people with smartphones walking around as well, the density of smartphones is very high. In practice, whenever there are smartphones close by, i.e., one smartphone is in the WiFi range of another phone and vice versa, they can be connected via freely available direct wireless connections (e.g., WiFi Direct or Bluetooth). Thus, one can set up a wireless ad hoc network between smartphones, where the direct wireless communication mode enables the phones to send large amounts of data to each other. We assume routing in the ad hoc network to be for free as messages are transmitted directly and no third party is involved.

In general, it would be much easier to communicate only via a cellular network since every node would be able to directly communicate with every other node (given that the cell phone infrastructure covers all nodes). This, however, is only possible up to a limited amount of data. Usually, smartphone owners have a contract with cellphone providers which offers a limited data volume. Once the data volume has been exceeded, messages can only be exchanged at very low speed in the cellular network. To maximize the lifetime of the data contracts while also being able to exchange almost unlimited data, it is evident to exchange all data via the ad hoc network whereas the cellular infrastructure is only used to find nearly optimal routing paths. Finding nearly optimal routing paths in the ad hoc network is a non-trivial task, since sparse regions of the ad hoc network can lead to radio holes. In general, natural and human made obstacles like buildings cause radio holes in the ad hoc network of smartphones. Complex shapes of radio holes, e.g., zig-zag shapes, make competitive local roting extremely difficult [15]. Messages that are simply forwarded into the direction of the destination might get stuck in a dead end or are routed on very long detours, when there is no knowledge about the ad hoc network. Unfortunately, collecting global knowledge about the entire ad hoc network, i.e., knowledge about the exact location and shape of radio holes, would be too expensive when only using cellular communication since potentially many people are located on the boundaries of holes. Therefore, we address the following question: Can cellular communication be used effectively to find near-shortest paths in the ad hoc network?

The authors in [10] were the first to provide an approach that combines the ad hoc and the cellular communication mode in a Hybrid Communication Network. The cellular infrastructure is used to establish an overlay network that computes the convex hulls of each radio hole to find near shortest routing paths that only consist of ad hoc links (i.e., the WiFi-interfaces of nodes). They assume that convex hulls of holes do not intersect. In this work, we improve their results concerning two aspects. We propose an overlay network based on bounding boxes which only requires a constant number of nodes per hole for finding routing paths. Moreover, we consider intersections of bounding boxes, where the difficulty is that a path leading through an area of intersecting bounding boxes can be arbitrarily complex (see Figure 1). We prove both theoretically and with simulations that our approach outperforms classical online routing strategies for geometric ad hoc routing significantly.

Figure 1: The shortest path between and (red arrows) leads through an area of intersecting bounding boxes. This path can be arbitrarily complex since the shapes of the radio holes can interleave each other as depicted in the figure.

1.1 Motivation

Classical research about ad hoc networks always assumes that no global infrastructure exists and nodes are only able to communicate via their ad hoc interface. In this work, we consider a real-life scenario of people using their smartphones in urban areas. A smartphone can be interpreted as ad hoc device since it is able to communicate for instance via WiFi Direct or Bluetooth. Smartphones, however, are different as they combine multiple communication modes in one device and hence are also able to use a cellular infrastructure to communicate. In metropolitan areas, the density of smartphones is sufficiently high such that the ad hoc network of smartphones is connected and, in principle, the entire data transmission between smartphones could be solely carried out via ad hoc links. The main challenge of data transmission in an ad hoc network of smartphones in a metropolitan area is routing. Usually, human made obstacles like buildings interfere wireless communication leading to radio holes in the ad hoc network. It is well known that routing in an ad hoc network without knowledge about shapes and locations of radio holes potentially leads to very long detours [13]. To overcome this drawback, the locations and shapes of radio holes can be determined efficiently via the cellular infrastructure which is available in metropolitan areas. For a fast computation of routing paths, it makes sense to not consider the exact shape of a hole but a more coarse-grained abstraction like a bounding box. In metropolitan areas, buildings are usually convex and rectangular shaped such that the bounding boxes of radio holes only rarely intersect, rendering the bounding box a practical hole abstraction in the described scenario.

1.2 Model

The model and definitions of this work are close to those of [10]. We model the participants of the network as a set of nodes in the Euclidean plane, where . Each node is associated with a unique ID (e.g., its phone number). For any given pair of nodes , we denote the Euclidean distance between and by . Nodes have different communication modes. For short distances, they can communicate via their WiFi-interface. Additionally, a node can communicate with every other node whose ID is known via the cellular infrastructure. More formally, we model our network as a hybrid directed graph where the node set represents the set of cell phones, an edge is in whenever knows the phone number (or simply ID) of , and an edge is also in the ad hoc edge set whenever can send a message to using its Wifi interface. For all edges , can only use a long-range link to directly send a message to . The concrete edges contained in and are defined in later sections. Since WiFi-communication can only be used over short distances, can only contain edges which are part of the Unit Disk Graph of (). , is a bi-directed graph that contains all edges with . Assume to be connected so that a message can be sent from every node to every other node in by just using ad hoc edges.

While the potential ad hoc edges are fixed, the nodes can change over time: If a node knows the IDs of nodes and , then it can send the ID of to , which adds to . This procedure is called ID-introduction. Alternatively, if deletes the address of some node with , then is removed from . There are no other means of changing , i.e., a node cannot learn about an ID of a node unless is in ’s UDG-neighborhood or the ID of is sent to by some other node.
Moreover, we consider synchronous message passing in which time is divided into rounds. We assume that every message initiated in round is delivered at the beginning of round .

1.3 Objective

Our objective is to design a competitive routing algorithm for ad hoc networks, where the source of a message knows the ID of the destination , or in other words, . We call a routing strategy -competitive, if the length of a path obtained by the strategy has length at most times the length of an optimal path for a constant . The authors in [13] have shown that any online routing algorithm that only has local knowledge about the network cannot be -competitive. Based on these results, the authors in [10] proposed a strategy that makes use of a Hybrid Communication Network to obtain information about location and shapes of holes. They have proven that in case the convex hulls of radio holes do not intersect, their approach finds -competitive paths in the ad hoc network.

In this paper, we aim for a reduction of the number of cellular infrastructure nodes that have to be considered for the computation of -competitive paths. To do so, we replace the computation of convex hulls of holes by the computation of bounding boxes. In addition to [10], we also propose a strategy for intersecting bounding boxes.

1.4 Our Contributions

We consider any hybrid graph where the Unit Disk Graph of is connected. Let be the set of radio holes in and denotes the length of the perimeter of a radio hole . For every radio hole, the nodes with maximal/minimal - and -coordinates are called extreme points. Our main contribution is:

{theorem}

For any distribution of the nodes in that ensures that UDG() is connected and of bounded degree, where the bounding boxes of the radio holes do not overlap, our algorithm computes an abstraction of UDG() in communication rounds using only polylogarithmic communication work at each node so that -competitive paths between all source-destination pairs outside of bounding boxes can be found in an online fashion.

The storage needed by the four extreme points of each radio hole is . For every other node, the space requirement is constant.

Note that we do not consider source or destination nodes inside of bounding boxes in this work. Nevertheless, these can be efficiently handled by a straightforward extension of the protocols presented in this paper by the routing algorithm of [10] for the case that at the source, the target or both nodes lie inside of a hole abstraction. We also consider intersecting bounding boxes. For pairwise intersecting bounding boxes, we prove the following {theorem} For any distribution of the nodes in that ensures that UDG() is connected and of bounded degree, where the bounding boxes of the radio holes do pairwise overlap and the convex hulls of holes do not overlap, our algorithm computes an abstraction of UDG() in communication rounds using only polylogarithmic communication work at each node so that -competitive paths between all source-destination pairs outside of bounding boxes can be found in an online fashion.

The storage needed by the four extreme points of each radio hole is . For every other node, the space requirement is constant. For multiple bounding box intersections, we prove: We prove that in case we can find a -competitive path between outer intersection points of bounding boxes, we can also find a -competitive path between all source-destination outside of bounding boxes. Since the computation of -competitive paths between outer intersection points is a hard problem, we provide a heuristic solution in this paper. We show via simulations that our approach outperforms classical online routing strategies for ad hoc network with holes significantly, both for intersecting and non-intersecting bounding boxes.

1.5 Related Work

In the context of geometric routing in ad hoc networks, several routing techniques have been investigated. One of the early approaches is GPSR [11], in which greedy routing is used whenever possible. In case a packet reaches a dead end, the packet is routed along the perimeter of the hole via the right-hand rule. As soon as greedy routing is applicable again, the routing mode is changed to greedy routing. A similar approach is Compass Routing [12]. The algorithm considers the direct line segment connecting the source node and the target node . At every step the edge with smallest slope to the direct line segment is chosen. This, however, does not lead to a delivery guarantee in all kinds of graphs. An example for a graph with delivery guarantee is the Delaunay Graph which in addition is a -spanner of the Euclidean metric [19]. The value for is . MixedChordArc is the latest -competitive routing strategy for Delaunay Graphs which has been recently published by Bonichon et al. [3]. The authors in [12] introduce a strategy that combines compass routing with face routing to obtain a routing strategy with delivery guarantee for all kinds of connected geometric graphs. Several extensions of these original ideas have been investigated. Some of these extensions are FACE-I, FACE-II, AFR, OAFR, GOAFR and GOAFR+ [5, 14, 15, 13]. In [15, 13] it is proven that the strategies GOAFR and GOAFR+ are asymptotically optimal. GOAFR and GOAFR+ achieve path length which have a quadratic competitiveness compared to the shortest path. INF [7] is an approach that combines greedy forwarding with randomness. In case a packet gets stuck via greedy routing, a random intermediate location is chosen. This, however, requires some global knowledge to choose a random node which is not too far away from the location. Additionally, INF does not ensure delivery guarantee.

In [20], also greedy routing with a modification of face routing is used. To overcome potential bottlenecks which avoid a guaranteed delivery, a restricted flooding procedure is used. In a slightly different setting, namely nodes on the grid, a packet is routed along multiple paths and is hence comparable to a restricted flooding procedure [18]. In their model, alive node and crashed nodes exist on the grid. The crashed nodes behave like obstacles on the grid which have to be avoided by routing paths.

In addition to the just mentioned local routing, there are also routing strategies that use a portion of global knowledge about the network. BoundHole [8], for instance, uses a preprocessing phase at each node which is located at the boundary of a hole. These hole nodes send out a packet which is routed using the right hand rule around the perimeter of the hole until it reaches the source of the message. On the way, the packet collects information about the boundary of the hole. With knowledge about the boundary, the authors are able to find better paths than strategies which only use local information. For a survey on all mentioned strategies, we refer the reader to [1].

To combine local and global routing strategies, where the goal is to use only few global knowledge, Hybrid Communication Networks have been introduced [10]. Hybrid Communication Networks have also been proposed in different contexts. In practical applications, the term Hybrid Communication Network usually combines wired with wireless networks like in [17, 6]. Closer to our application is the scenario presented in [9]. The authors assume an external network which is not under control of the network participants. The participants can, however, control an internal network. The authors show that the combination of both networks allows to evaluate monitoring problems of the external network much faster than in classical approaches which only use the links of the external network.

The approach we extend in this work makes use of global information as well [10]. The global information is gathered via a Hybrid Communication Network. In a Hybrid Communication Network, nodes can communicate with other nodes in their ad hoc range for free. In addition, they can use long-range links to communicate with any other node of the network. These long-range links, the Cellular Infrastructure, are costly. The solution they propose is to compute an Overlay Network in which holes are represented by their convex hulls. It is assumed that the convex hulls of the holes do not intersect. The storage requirements for some nodes are asymptotically in the size of the sum of all holes. In this work, we aim to reduce the storage requirements for these nodes and investigate also the challenging question of -competitive routing through intersections of hole abstractions.

2 Preliminaries

Initially, we define our ad hoc network topology and provide some general results about routing in the ad hoc network.

2.1 Properties of the ad hoc network

In this paper, we assume that the nodes of the ad hoc network are in general position, i.e., there are no three nodes on a line and no four nodes on a cycle. Moreover, we assume that the coordinates of each node are unique and thus there are no two nodes on the same position. We consider a 2-localized Delaunay Graph as topology for the ad hoc network which is related to the Delaunay Graph. Let be the unique circle through the nodes and and be the triangle formed by the nodes and . For any , the Delaunay Graph of contains all triangles for which does not contain any further node besides and . The -localized Delaunay Graph is a structure that only allows edges which do not exceed the transmission range of a node. In -localized Delaunay Graphs, a triangle for nodes of satisfies that all edges of have length at most and the interior of the disk does not contain any node which can be reached within hops from or in UDG(). The -localized Delaunay Graph is defined to consist of all edges of -localized triangles and all edges for which the circle with diameter does not contain any further node . For , we obtain the -localized Delaunay Graph which is also a planar graph [16]. 2-localized Delaunay Graphs can be constructed in a constant number of communication rounds [10]. Since -localized Delaunay Graphs do not contain all edges of a corresponding Delaunay Graph, one cannot simply use routing strategies for Delaunay Graphs in our scenario. We denote faces of the -localized Delaunay Graph which are not triangles as holes. For the formal definition of holes, we distinguish between inner and outer holes. The definition of inner holes is similar to the definition used in [8]. {definition} [Hole] Let . An inner hole is a face of with at least nodes. Furthermore, let be the set of all edges of the convex hull of . Define to be the graph that contains all edges of the 2-localized Delaunay Graph and . An outer hole is a face in with at least nodes, that contains an edge with . Nodes lying on the perimeter of a hole are called hole nodes. Note that the hole nodes of the same hole form a ring, i.e., each hole node is adjacent to exactly two other hole nodes for each hole it is part of. The choice of the -localized Delaunay Graph as network topology is motivated by its spanner-property. The Delaunay Graph contains paths between every pair of nodes and of which are not longer than times their Euclidean distance. Delaunay Graphs are proven to be geometric -spanners [19]. Xia argues that the bound of also relates to -localized Delaunay Graphs [19]. However, these graphs are not spanners of the Euclidean metric but of the Unit Disk Graph. For the ease of notation, whenever we say that there is a -competitive path in the 2-localized Delaunay Graph we mean that the path has length at most times the length of the shortest possible path in the Unit Disk Graph of the same node set.

2.2 Competitive Routing in 2-localized Delaunay Graphs

In general, we cannot apply routing strategies for the Delaunay Graph in 2-localized Delaunay Graphs since 2-localized Delaunay Graphs contain holes. In this section, however, we prove that 2-localized Delaunay Graphs and Delaunay Graphs do not differ in dense regions and hence we can apply routing strategies for the Delaunay Graph between visible nodes, i.e., pairs of nodes which direct line segment does not intersect any hole.
{theorem} Let be a 2-localized Delaunay Graph and such that the line segment does not intersect any hole of . Then, there exists a path between and in such that

To prove Section 2.2, we make use of a definition which was introduced by Bose et al. [4]. Let and be nodes of a Delaunay Graph. Bose et al. considered the chain of triangles intersected by the line segment . Each of these triangles contains an edge which either lies completely above or below . If we consider only these edges, we can see that these edges form a polygon. Walking along all edges lying above describes a path between and . This path is called upper chain of and () and the corresponding path for all edges which lie below is called lower chain of and (). Xia has proven that between any pair of nodes and in a Delaunay Graph a path with length at most exists [19]. The path construction of Xia uses only edges which connect nodes of and . We use this knowledge and show that in Delaunay Graphs a polygon described by an upper and a lower chain of nodes and never contains any edge with a length larger than , provided and are visible from each other in the corresponding 2-localized Delaunay Graph. Afterward, we conclude that between any pair of visible nodes and in a 2-localized Delaunay Graph a path with length at most exists.

Lemma 1.

Given a 2-localized Delaunay Graph and two nodes and such that the line segment does not intersect any hole of . Let be the Delaunay Graph to the same point set . The polygon described by and in does not contain any edge with .

Proof.

There are three types of edges which are part of the polygon with boundaries and in . Edges that cross the line segment , edges which lie completely above and edges that lie completely below . We prove for every type of edges that these cannot be larger than one in case and are visible from each other in .

Case 1: Edges crossing :

Without loss of generality, let be a triangle which is intersected by and an edge that crosses . Assume . This immediately implies that crosses a hole since intersects a face with at least nodes which is a contradiction to our assumption.

Case 2: Edges above :

Without loss of generality, let be a triangle which is intersected by and an edge that lies above with . With this knowledge we can conclude that and potentially lie on the perimeter of a hole if and are not hole edges themselves. Again we can easily see that would cross a hole in this case which is a contradiction to our assumption.

Case 3: Edges below :

We can apply the same argumentation as for Case here.

Since every possible type of edges cannot have a length larger than , we have proven Lemma 1.


Lemma 1 implies that we can apply routing strategies for Delaunay Graphs also between visible nodes in 2-localized Delaunay Graphs. This leads to the relation between our routing strategy and Visibility Graphs. In the Visibility Graph of a set of polygons, represents the set of corners of the polygons, and there is an edge in if and only if a line can be drawn from to without crossing any polygon, i.e., is visible from . De Berg et al. showed that it is enough to consider nodes of obstacle polygons for finding shortest paths in polygonal domains [2]. Hence, if we consider the Visibility Graph of holes of the 2-localized Delaunay Graph, we can translate a path in the Visibility Graph to a path in 2-localized Delaunay Graph by applying a routing strategy for Delaunay Graphs along every edge on the path in the Visibility Graph. As we do not want to store large routing tables, we are interested in online routing strategies for the Delaunay Graph. In this work, we make use of the online strategy MixedChordArc [3] which finds -competitive paths between every source and target node in the Delaunay Graph. To sum it up, knowledge about the Visibility Graph of holes enables us to find -competitive paths in the 2-localized Delaunay Graph between any pair of nodes by applying the MixedChordArc-strategy along every edge of the shortest path between and in the Visibility Graph.

3 Geometric properties of bounding box paths

We have seen that knowledge about the Visibility Graph of holes enables us to find -competitive paths in the 2-localized Delaunay Graph. In general, however, the node set of a Visibility Graph can be very large since potentially many nodes could lie on the boundary of holes. To reduce the space constraints and to speed up the computation of -competitive paths, we aim for a reduction of the number of nodes in the Visibility Graph while still being able to find -competitive paths. To do so, we reduce the Visibility Graph by only considering the bounding boxes of holes. The following definition defines the axis-parallel bounding box of a hole.
{definition}[Bounding Box]
Let be a polygon. Let and and be defined analogously. These points are called extreme points of . The (axis-parallel) Bounding Box of is a polygon with the following nodes:

  1. (top-left)

  2. (top-right)

  3. (bottom-left)

  4. (bottom-right)

The nodes are connected via the direct line segments and . In the following, we see how we can embed bounding boxes of holes in the 2-localized Delaunay Graph and that considering only bounding boxes of holes allows us to find -competitive paths between every source and target node that lies outside of any bounding box.

3.1 Embedding of Bounding Boxes

In general, nodes of bounding boxes of holes do not match with any nodes of the ad hoc network (see Figures 3 and 3). In this section, we propose an embedding of bounding boxes in the 2-localized Delaunay Graph and prove later on that we can find -competitive paths with help of the embedding. Since we consider a given 2-localized Delaunay Graph with node set and edge set , we have to find nodes in that represent nodes of bounding boxes. Nodes of bounding boxes of holes are called real bounding box nodes whereas nodes of that represent real bounding box nodes are denoted as representatives of a real bounding box.

Figure 2: The nodes of the red bounding box are not part of the 2-localized Delaunay Graph.
Figure 3: A bounding box described by its representatives of the ad hoc network.

The solution is to choose those nodes of as representatives of real bounding box nodes which have the shortest distance to a real bounding box node. More formally, let be a 2-localized Delaunay Graph with corresponding Voronoi Diagram . A real bounding box node is represented by the node of the Voronoi Cell with . The resulting bounding box (see Figure 3) does not necessarily enclose the entire hole anymore but we prove that it has similar properties as a real bounding box. Since we have proven that -competitive paths between visible nodes in 2-localized Delaunay Graph exist, our idea is to use the direct line between real bounding box nodes for routing decisions. We call this direct line virtual Axis.
{definition}[Virtual Axis]
Consider a 2-localized Delaunay Graph with nodes . Let and be the cells of the corresponding Voronoi Diagram with and . Additionally let with and but and . We call the line segment a virtual Axis between and in . For the ease of notation, we simply write .
In our scenario, we use a virtual Axis between visible real bounding box nodes.

After clarifying the definition of virtual Axes, we can introduce the main theorem of this section. We prove that there exists a path with length at most times the Euclidean distance between the real bounding box nodes between two nodes and representing two adjacent real bounding box nodes.

{theorem}

Let be a 2-localized Delaunay Graph with . For any with endpoints and that does not intersect any hole of , there exists a path between and in with length at most:


We split the proof of Theorem 3.1 into two lemmas. Lemma 2 bounds the length of the line segment and Lemma 3 proves that is a candidate for the shortest polyline (see creftypecap 2) between and . The combination of both yields Section 3.1.

Lemma 2.

Let be a 2-localized Delaunay Graph with a pair of nodes and let be a virtual Axis between and that does not intersect any hole of . The endpoints of are denoted as and . Then,


Proof.

By the triangle inequality . Next, we bound the length of the line segments and . We argue that the length of each line segment is at most . is the representative of since is the node with smallest distance to of all nodes in . lies either in or on the boundary of a triangle that has as a node. Each edge of this triangle has length at most due to the properties of 2-localized Delaunay Graphs. When considering triangles, it is easy to see that the endpoints of a triangle – in our case – are those nodes which have the largest distances to each other in the triangle. Consequently, the worst possible case is that falls exactly on the half of an edge with length . In this case the closest point is at distance which is an upper bound for . We can use the same arguments for the line segment .

Thus, we can bound the length of as follows:

Further, , due to the definition of holes. Thus we obtain a final bound on :

After being able to express the length of in terms of , we start with proving that a -competitive path between and exists. The proof is inspired by the path construction for Delaunay Graphs introduced by Xia. We need two definitions which have been introduced by Xia [19].

Definition 1 (Chain of Disks).

A finite sequence of disks is called chain of disks if it has the following two properties:

Property 1:
Every pair of consecutive disks and intersects but neither disk contains the other.

Denote by and the arcs on the boundary of that are in and respectively. These arcs are denoted as connecting arcs of .

Property 2:
The connecting arcs of do not overlap for , however they can share an endpoint.

Two points and are called terminals of if lies on the boundary of and is not in the interior of and lies on the boundary of and is not in the interior of .


Definition 2 (Shortest polyline between and ).

Given a chain of disks with terminals and . Let be the centers of . The polyline is called the centered polyline between and . For , let and be the intersections of the boundaries of and . Without loss of generality, all ’s are assumed to be on one side of the centered polyline and all ’s are on the other side. For notational convenience, define and . Let be the shortest polyline from to that consists of line segments where for .

Figure 4: The Chain of Disks from to along .

.

With these definitions, we can state Lemma 3.

Lemma 3.

Let be a 2-localized Delaunay Graph with a pair of nodes and let be a virtual Axis that does not intersect any hole of with endpoints and . Further let be the chain of disks with terminals and obtained by the circumcircles of all triangles intersected by . Then, we can bound the shortest polyline as follows:

Proof.

In [19], the author argues that any sequence of disks obtained by the circumcircles of triangles along a line segment in a Delaunay Graph is a chain of disks. Due to Lemma 1 we know that does not intersect any hole. Consider the chain of disks with terminals and obtained by the circumcircles of all triangles intersected by . See Figure 4 for a visualization. The main observation for our proof is that the polyline fulfills the requirements of creftypecap 2 and is a candidate for the shortest polyline between and . Thus, is an upper bound for . Hence:

The combination of Lemma 2 and Lemma 3 helps us to prove Section 3.1. Xia states that the shortest connection between two terminal nodes and along a chain of disks is at most [19]. Thus, we obtain that there exists a path between and with length at most:


So far, we concentrated on proving the existence of such a path. Nevertheless, we are also able to find a -competitive path via the MixedChordArc-algorithm. To do so, we slightly modify the algorithm such that we do not use the direct line segment between two representatives as referencing segment but the virtual axis connecting the real bounding box vertices. The analysis of MixedChordArc [3] proves that the path found along the virtual axis has length at most times the length of the virtual axis. The entire path has length at most times the length of the virtual axis since the connection between and the first node along the path and and the last node on the path has length at most times the length of the virtual axis. This leads to the following corollary.
{corollary} Let be a 2-localized Delaunay Graph with . For any with endpoints and that does not intersect any hole of , there exists an online routing strategy that finds a path between and in with length at most:


The results of this section enable us to reduce the problem of finding -competitive paths in the 2-localized Delaunay Graph to finding -competitive paths in Visibility Graphs as it is done in the next section.

3.2 Competitive Paths via non-intersecting Bounding Boxes

Based on our results of Section 3.1, we reduce the problem of finding -competitive paths via bounding boxes to finding -competitive paths in Visibility Graphs. Therefore, we introduce a special class of Visibility Graphs, namely Bounding Box Visibility Graphs. In Bounding Box Visibility Graphs, each obstacle (holes in our case) is represented by its axis-parallel bounding box. Consequently, consists of the nodes of the axis-parallel bounding box of each obstacle. Moreover, consists of the edges of each bounding box as well as of edges between visible nodes of different bounding boxes. In this setting, we call two nodes to be visible from each other in case their direct line segment does not intersect any bounding box.
Let be a set of polygonal obstacles, and a source- and a target-location. Further let and be the nodes of an axis-parallel bounding box representing a polygon . A Bounding Box Visibility Graph is defined as follows: {definition}[Bounding Box Visibility Graph] A geometric graph is called Bounding Box Visibility Graph if and . Additionally, and . For two nodes of different bounding boxes , the edge if does not intersect the bounding box of any obstacle .

Figure 5: A Bounding Box Visibility Graph for three obstacles.

Figure 5 provides a visualization of a Bounding Box Visibility Graph. To provide an intuition about the proof ideas, we initially assume that the considered Bounding Box Visibility Graph contains only a single bounding box. Additionally, we assume that the starting-location and the target-location do not lie inside of the bounding box. The node-set of the corresponding Bounding Box Visibility Graph contains , and the nodes of the bounding box. More formally: . The rest of the section deals with proving Theorem 3.2 which states that there always exists a path of length at most between and in the described setting.

{theorem}

Let be a Bounding Box Visibility Graph containing a single bounding box with a starting-location and a target-location . Then, there exists a path between and in with length at most .
For the proof of Theorem 3.2 we need a useful property of right triangles. Whenever we consider a right triangle with legs and and hypotenuse , we can prove that walking along and is not longer than a constant times walking along . Lemma 4 deals with this property.

Lemma 4.

Let be the legs of a right triangle with hypotenuse . Then, .

Proof.

The Pythagorean Theorem states . We compute the ratio of and and bound it afterward.

The equivalence holds as , and are greater than zero. After applying the Binomial Theorem, we obtain:

We analyze the properties of the latter addend and prove:

Since quadratic numbers are always positive, our claim holds and we can finally plug all results together and finish the proof.

With the knowledge of Lemma 4 we are able to prove Theorem 3.2.

Proof of Theorem 3.2.

Without loss of generality, we assume . We distinguish two cases concerning the size of the bounding box. In Case , we consider bounding boxes that fit completely into the bounding box around and . Case deals with bounding boxes that exceed the bounding box around and . We consider the surrounding bounding box with nodes and . See Figure 6 for a visualization.

Figure 6: The surrounding bounding box around and .

Case 1: The bounding box of the obstacle is completely cointained in the bounding box with nodes and .

The worst case is that the bounding box of the hole and the bounding box with nodes and coincide. In that case, we see that the shortest connection from to is the combination of the line segments and . The combination of and has the same length. By applying Lemma 4, we obtain:


The last inequality holds because the Euclidean distance between and is the shortest possible length of a path between and . The distance of and in the Unit Disk Graph has to be larger or equal. If the two mentioned bounding boxes do not coincide, the shortest path among nodes of bounding boxes is still smaller than a path along the comprising bounding box. Thus, we obtain as an upper bound for the length of the shortest path among nodes of a bounding box.

Case 2: The bounding box of the obstacle exceeds the bounding box with nodes and .

In this case, it is enough to consider the cases in which and or and are intersected by . Otherwise, we can apply the same argumentation as in Case since there is a path which is completely contained in the box with nodes and . Without loss of generality, we assume that both and are intersected. We can use the same proof for the other case by turning the view about 90 degrees. Let be the highest point of the hole polygon and the lowest point respectively. The shortest geometric connection between and has to pass either or . Without loss of generality, we assume that the shortest geometric connection passes . The shortest possible (not necessarily realistic) connection would be and . We can upper bound the length of this connection by giving the legs of two right triangles as maximal path length. Consider the points and . The longest possible path over bounding box points would be , and . Since this path uses the legs of right triangles with hypotenuses and , we can apply Lemma 4 and obtain that the maximal length is at most . See Figure 7 for a visualization of both right triangles. The last inequality holds since the distance between and in the Unit Disk Graph cannot be smaller than the shortest possible geometric connection between and .

Figure 7: A visualization of Case .

So far, we have analyzed the maximal length of -competitive paths we can achieve for a single bounding box in Bounding Box Visibility Graphs. It remains to combine these insights with our knowledge about virtual Axes to bound the length of bounding box-paths in 2-localized Delaunay Graphs.

{corollary}

Consider a 2-localized Delaunay Graph which contains a single hole with bounding box . Between any pair of nodes and with but , there exists a path from to that contains representatives of bounding boxes such that:

Additionally, there exists an online routing strategy that finds a path from to such that:

Proof.

If we reduce the 2-localized Delaunay Graph with to a Bounding Box Visibility Graph only containing and all nodes of , we know that it contains a path between and with length at most . If we use the nodes in that are closest to the nodes of as representatives for , we can apply virtual Axis routing and obtain that between two adjacent representatives of a path of length at most times their Euclidean distance exists (Section 3.1). Finally, we can combine both insights and conclude that in 2-localized Delaunay Graphs containing a single hole with bounding box there is a path between any pair of nodes of length at most which proves the Corollary. For the online routing strategy, we can conclude based on Section 3.1 that we can find paths of length .


We continue with considering multiple non-intersecting bounding boxes. {theorem} Let be a Bounding Box Visibility Graph that contains multiple non-intersecting bounding boxes and a source- and a target-location and . There exists a path between and in with:


To prove Section 3.2, we define a special class of paths in geometric graphs which helps us to construct paths in Bounding Box Visibility Graphs which are -competitive to the shortest path in usual Visibility Graphs. Therefore, we compare the covered distance in vertical direction as well as the covered distance in horizontal direction of both paths.
{definition}[Monotone Paths]
A path in a geometric graph is called increasing x-monotone if for all . Analogously, such a path is called increasing y-monotone if for all . Similarly, paths are called decreasing -/-monotone if for all . A path is called -monotone if it is either increasing or decreasing -monotone. Analogously, a path is called -monotone if it is either increasing or decreasing -monotone. For the proof of Theorem 3.2, we compare the shortest path between a pair of nodes and in a Visibility Graph to a path between and in the corresponding Bounding Box Visibility Graph . Observe that walks along a sequence of polygons from to . Whenever walks from a polygon to a polygon , is - and -monotone for that part, as follows a direct line segment between and . The key idea for our proof is to construct a path in that has the same monotonicity properties as for every pair of consecutive visited polygons and of . Therefore, we introduce a greedy routing strategy for that constructs paths having the same monotonicity properties as . Our greedy strategy is called Greedy Visibility Routing (GreViRo) and is defined as follows:
Let be a shortest path between two points and in a Visibility Graph that contains polygons with non-intersecting bounding boxes. The sequence of polygons visited by is denoted as and the direct line segment walked by from polygon to is denoted as . In addition, the intersection points of with and are defined as and respectively. Further, let be the corresponding Bounding Box Visibility Graph. Consider a line segment of . GreViRo connects two nodes and of provided and as well as and are visible from each other and the path has the same monotonicity properties as .

GreViRo always chooses the node of a bounding box intersecting the face with nodes and that does not violate the monotonicity properties of and minimizes the distance to until is visible.

Figure 8: A path construction of GreViRo.

Figure 8 depicts a path construction of GreViRo. Observe that GreViRo is defined to fulfill the same monotonicity properties as . We are left with proving the correctness of GreViRo.

Lemma 5.

Let and be defined as described above. GreViRo constructs a path in between and .

Proof.

Without loss of generality, assume is increasing - and increasing -monotone. Observe that the same proof can be used for all other cases by turning the view and degrees around. Turning the view degrees around yields an increasing - and decreasing -monotone path, degrees yields a decreasing - and decreasing -monotone path and degrees yields a decreasing - and increasing -monotone path.

We prove by contradiction that GreViRo constructs a path between and . Assume GreViRo has reached a node and cannot proceed further because is not visible and all other visible nodes violate increasing - and -monotonicity. Since is not visible from , there has to be a bounding box which is intersected by the line segment . Consider a visible node of . Due to our assumption, and . Further consider the node which has been visited before proceeding to . Due to our assumption, GreViRo gets stuck at node hence, and . The crucial observation is that has already been visible from . Observe that there cannot be a third bounding box intersecting the line segment since nodes of this bounding box would have been preferred by GreViRo to as these are closer to . Hence, GreViRo would have chosen the node instead of which is a contradiction to our assumption. We refer to Figure 9 for a visualization of the contradiction.


Figure 9: The contradiction described in the proof of Lemma 5. The contradicting path is marked in blue and thick. The reader can see that has already been visible from .

GreViRo is an analysis tool that allows us to construct a path in that fulfills the same monotonicity properties as the original path in . For structuring the proof of Theorem 3.2, we split it into two lemmas. Initially, we assume that is - and -monotone (Lemma 6). Using this assumption makes the proof easier and thus helps to understand the overall proof ideas. Afterward, we drop this assumption in Lemma 7 and assume that is either - or -monotone but not both. Finally we drop any monotonicity assumption and can prove Theorem 3.2 with our knowledge of Lemmas 6 and 7.

Lemma 6.

If is - and -monotone, then there exists a path between and in with length at most .

Proof.

The proof idea is to construct a path in which is also - and -monotone such that we can conclude that does neither walk a longer distance in horizontal nor in vertical direction than . Without loss of generality, we assume is both increasing - and -monotone. The proofs for the other cases can be obtained by turning the view , and degrees around. Consider the sequence of hole polygons () which is visited by the shortest path . Observe that when walking from to , intersects either the lower edge of or the left edge of . Note that the edges of are not part of but are used here to understand the path construction in . Due to our monotonicity assumption, has to intersect either the upper edge of or the right edge of when proceeding to .

The concrete path construction works as follows: Assume our path has brought us to and is heading to polygon next. There are two cases to consider. Either the left or the lower edge of is intersected by . Whenever the left edge is intersected, we know due to our monotonicity assumption, that the upper edge has to be intersected afterward. Hence, we can use the top left node of as next point of our path-construction. The same argumentation can be used if intersects the lower edge of . Due to the monotonicity assumption, we know that the right edge of has to be intersected when proceeds to . Hence, we can walk to the lower right node of . More formally, the next node of our path construction is:

  • Case 1: if intersects