Naming and Routing Using Global Topology Discovery

Transcription

1 Naming and Routing Using Global Topology Discovery Qing Fang Department of Electrical Engineering Stanford University Stanford, CA Jie Gao, Leonidas J. Guibas Department of Computer Science Stanford University Stanford, CA Li Zhang Information Dynamics Laboratory, HP Labs, Palo Alto, CA Abstract We present a novel approach to providing a naming structure and associated routing algorithms for a network of nodes distributed over a 2-d field. We assume that these nodes communicate wirelessly with some of the other nodes in their proximity but do not know their geographic location; such scenarios are common in sensor networks. In a preprocessing phase we partition the nodes into routable tiles regions where the node placement is sufficiently dense and regular that local greedy methods can work well. The adjacency relations between these tiles capture the global topology of the field. This includes not just connectivity but also the presence of holes, etc. Each node is named using the name of the tile containing it and a set of local coordinates derived from connectivity graph path distances within its tile and those that form its topological neighbors. We show that efficient and loadbalanced global routing can be implemented by using the tile adjacency graph for global route planning and then these local coordinates for realizing actual inter- and intratile routes. Keywords: System design, Graph theory, Combinatorics I. INTRODUCTION The need to impose a network infrastructure on large collections of simple nodes communicating with their neighbors arises in many contexts. For example, in sensor networks nodes commonly communicate via wireless links and, in any given deployment, a node can reach some (though typically not all) neighbors within a certain range. Across all of networking, naming schemes and routing algorithms are closely related appropriate naming schemes enable efficient routing. In more structured settings, such as the Internet, IP addresses exhibit a natural hierarchical structure which is heavily exploited in the data structures (routing tables) and algorithms used by routers. In cases where deployment is more ad hoc and the link state more fragile, as in sensor networks, local greedy forwarding protocols have been proposed based on treating the geographic location of the node as its network address. Such protocols can operate directly using local information and do not need more global state that can be expensive to update and maintain. Although geographical location gives the nodes natural names and enables efficient routing in many cases, it is not always easy or inexpensive to obtain. GPS receivers can be costly and lead to cumbersome form factors; furthermore, they do not work indoors, or in conditions under heavy foliage, etc. Since in most settings it is not feasible to have each node be equipped with a GPS receiver, various localization algorithms have been developed [14], [15]. In these methods the geographic location of certain anchor nodes is assumed to be known (either manually, or through GPS). Then other nodes determine their location by estimating their distances to three anchors and become anchors themselves, and so on. Still, such techniques are both expensive and inaccurate. What s more, geographic information may not correlate that well with communication and connectivity. The assumption that the wireless communication range be modeled by a unit disk is too simple and unrealistic [6], [10]. Interference between wireless channels also plays an important role in determining the available connectivity and bandwidth. In in-door situations or for nonuniformly distributed sensors, nodes that are geographically close may actually be distant in the communication graph. That is why simple greedy forwarding schemes may encounter local minima where packets get stuck. When the geographical location of the sensors does not map well to the communication graph, location based routing might be misleading and inefficient. In contrast with geographical location, link information and connectivity is more reliable in designing routing algorithms. In fact, localization algorithms exist that take link information and predict the geographical location of the sensor nodes [11], [12], which are then used by geographical routing schemes to select routing

2 paths. Obtaining the geographical information is only an intermediate step and an unnecessary detour. Recently there is work on geographical routing without location information [13], [16]. However, this approach still tries to embed the sensors in two dimensional Euclidean plane and then use geographic forwarding as the routing scheme. The key observation we start from is that since nodes only make local routing decisions, global coordinates are not necessary for geographic routing. We aim to further decouple the dependency of geographic routing on Euclidean coordinates by inventing new local coordinates based entirely on the link information of the node connectivity graph with no attempt to produce a 2-d embedding, locally or globally. To do so, we first need to discover the global topology of the node field. We do this by partitioning the node field into tiles, and then extracting the adjacency relations among these tiles. Our local coordinates are defined within a tile, while the global tile connectivity structure provides a high-level map of the field. This global understanding allows our naming and routing scheme to avoid some of the common pitfalls and limitations of current global coordinate-based schemes, such as the need to construct a planar graph so as to avoid local minima [9], [1], or the explicit discovery of holes [4], or the limitation to two dimensions. In our work we use the local link connectivity relationships among the nodes to discover the global topology of the sensor field. It is a bit unfortunate that the current usage of the term topology discovery in the networking community refers to just discovering these local link relationships among the nodes and only loworder topological invariants such as path connectivity. We use the term topology discovery here to mean understanding the global topology of the field in the sense of algebraic topology, so we also want to obtain higher order topological features, such as holes (in 2-d, and more in higher dimensions). Historically, algebraic topology has attempted to characterize the connectivity of continuous spaces by focusing on discrete invariants, like homology groups. Since continuous spaces are hard to compute with, an essential step in deriving these invariants is discretization: the replacement of the continuous structure by a discrete structure that provably has the same topology. The most commonly used discrete structure is that of a simplicial complex [8], a collection of simplices (points, segments, triangles, tetrahedra, etc.) of the appropriate dimension that are properly glued together as in the triangle meshes used in graphics, for example. Because of recent interest in shapes described by point cloud data (scattered collections of point samples), the same paradigm has been extended to capturing the topology of discrete point collections. When we talk of the topology of a point cloud, we really mean the topology of the underlying space from which those points were sampled. That can be captured by fattening the points into small balls and then defining a combinatorial complex according to the intersection patterns of these balls. In this approach the issue of what ball size to use is crucial. This is typically addressed by considering several sizes at once and looking at the persistence of topological features as the size changes [2]. These same ideas can be applied to a field of communication nodes, to learn its global topology. However, the lack of positional information, the need to further reduce computational costs, and the desire to perform the topology estimation in a distributed way all make the network case more challenging. II. OVERVIEW We construct a routing complex based on the communication graph of the nodes. A routing complex consists of routable tiles, whose topological adjacencies define the global tile connectivity graph. Each tile itself has trivial topology, so that simple greedy routing schemes work well within the tile. Our particular routing complex is based on using Voronoi cells of landmark nodes, and is denoted as the landmark Voronoi complex (LVC). The corresponding tile connectivity graph is denoted as the combinatorial Delaunay triangulation. The idea is that a small subset of the nodes are declared as landmarks and these are used to partition the remaining nodes according to the nearest landmark. Within the routing complex, we give each node a topology-aware name, which has a combinatorial part and a numerical part. The combinatorial part of the name is based on the global topology of the communication graph. Each node is given the name of the landmark to whose tile this node belongs. The numerical part of the naming scheme is a set of local coordinates of a node within its tile, which is defined by graph distances to certain nearby landmarks. Specifically, the coordinates are the squares of the distances to the landmarks in the containing tile and its neighbors in the tile connectivity graph. Such a tiling with a local landmark distance coordinate system within each cell is analogous to an atlas with local chart structures in differential topology [7]. With the global topology discovered and captured by the landmark Voronoi complex, we design topology-

3 time: 0 time: 0 (i) (ii) Fig. 1. Landmarks are shown by triangles. The nodes are divided into tiles. The gray nodes are the boundaries of the tiles. (i) The landmark Voronoi complex; (ii) The combinatorial Delaunay triangulation. aware routing protocols. Inside a tile all the nodes share a common local coordinate system. We design a simple local greedy algorithm by using only local landmark distance coordinates to deliver a packet inside a tile. Our virtual distance between two nodes in the same tile is defined as the sum of squared differences between corresponding coordinates. The greedy algorithm simply selects the neighbor closer to the packet s destination under this distance measure. Since a tile usually has smoothly distributed nodes and trivial topology, such a greedy approach rarely stuck. By switching landmarks properly, we modify this greedy approach for routing across tiles and produce natural and near optimal global routing paths. Our naming and routing scheme are designed to perform well for dense node distributions over a field with a few large holes [5]. That is, nodes are densely distributed except in a few number of large areas where node density is substantially smaller. Such kind of node distributions appear naturally in real settings where obstacles or occlusions can create large areas without enough node coverage. Because we assume that the global topology is relatively simple, the number of landmark nodes needed to capture it is also relatively small. In summary, our naming and routing protocol has two phases. The preprocessing phase discovers the global topology by constructing the landmark Voronoi complex. Every node is given a name, and everyone knows the global field topology a compact, lightweight structure, because the number of landmark nodes is small. When a node has a routing request, it first uses the combinatorial structure, the landmark Voronoi complex, to determine a short sequence of tiles that the routing path is going to cross. Then the Voronoi landmark-based routing protocol is deployed to actually guide a packet across tiles and towards the destination. III. LANDMARK VORONOI COMPLEX (LVC) For a set of nodes S and a communication graph G, the landmark Voronoi complex is a combinatorial structure that captures the global topology of the network by using only local link connectivity. We denote by τ(u, v) the topological length (hop count) of the shortest path between u, v in the communication graph. A. Topology-based tiling A tiling is a subdivision of the nodes into a small number of tiles, each of them with trivial topology. The adjacency of the tiles reflects the topology of the communication graph. We make use of the combinatorial

4 Delaunay triangulation [3] to obtain such a tiling 1. A combinatorial Delaunay triangulation is defined on an abstract graph G. A few vertices of graph G are marked as landmarks L. A combinatorial Delaunay triangulation is an abstract graph on the landmarks. Two landmarks u, v are connected by an edge in the combinatorial Delaunay triangulation if there exists a vertex w, called a witness, such that u, v are the closest two landmarks of w. More precisely, we denote by N 1 (w) the set of closest landmarks, i.e., no landmark has a hop count strictly smaller than the landmarks in N 1 (w). We denote by N 2 (w) the set of second closest landmarks such that no landmark other than nodes in N 1 (w) has a hop count w strictly smaller than those in N 2 (w). Then w is a witness of edge u, v if u, v N 1 (w) or u, v belong to N 1 (w) and N 2 (w) respectively. An edge forms a 1- simplex, and higher order abstract simplices are defined analogously. In view of the landmark Voronoi complex, a tile consists of all nodes nearest to a particular landmark. Adjacency between tiles is defined via the combinatorial Delaunay triangulation: two tiles are neighbors if and only if their landmarks are joined by and an edge in the combinatorial Delaunay. Figure 1 shows an example of a combinatorial Delaunay triangulation and a corresponding tiling. For dense distribution of nodes with a small number of large holes, the combinatorial Delaunay triangulation on a few number of landmarks fully captures the global topology of the communication network. In the following theorem we show that the combinatorial Delaunay triangulation maintains connectivity. Theorem 3.1. If graph G is connected, then the combinatorial Delaunay triangulation on an arbitrary and nonempty subset of landmarks is also connected. Proof: We prove by contradiction. Assume that the combinatorial Delaunay triangulation is not connected, i.e., there exist two landmarks in different connected components although in graph G they are connected by a path. We take the closest disconnected landmarks u, v such that the shortest path P to connect them in G is the shortest among all such pairs. P doesn t contain any other landmarks. There must exist a node w such that the hop count between w, u doesn t differ from the hop count between w, v by more than 1 (w might 1 Our definition of the combinatorial Delaunay triangulation is different from the definition in [3]. We made the modification with the intention of maintaining the same connectivity with the original graph. be u or v themselves). See Figure 2. We assume that u w x Fig. 2. The combinatorial Delaunay triangulation on G keeps the connectivity of graph G. τ(w, u) τ(w, v) τ(w, u) + 1. Since w does not witness the edge uv, w is closer to another landmark, say x. Therefore the hop count from u, v (say u) to w must be strictly greater than the hop count from x to w. Therefore the shortest paths between w, v and w, u are shorter than the shortest path between u, v. By the hypothesis, w and u, as well as w and v, must be in the same connected component. So u, v must be connected. This contradicts with the assumption. Although the combinatorial Delaunay triangulation on any set of landmarks keeps the same connectivity of the original graph, we do need enough landmarks so that the Voronoi cells of the landmarks and their adjacencies correctly capture the global field topology. Especially we want a few landmarks that are close to topological features, such as hole boundaries. Hand picked landmarks are one option, since in may case the presence of holes is known a priori to those deploying the network. We can also discover the boundaries of holes automatically [5], or at least some nodes on the boundary [13]. Then we can make the probability of a node being selected as a landmark high if it is near the boundary. Since the node distribution we are mostly interested in has a small number of large holes, the number of landmarks necessary is proportional to the number of holes. The landmark Voronoi complex, as a combinatorial summarization of the global sensor field, has small size and can be made available to each node. IV. NAMING AND ROUTING STRUCTURES In this section we introduce a naming and routing structure by using the landmark Voronoi complex. How to construct the LVC and the routing structures will be described in the next section. A. Naming of nodes A node is named by two parts. Firstly a node is associated with the tile that it lies in. This tile is denoted as the resident tile. The landmark of the tile is denoted as the home landmark. A node also calculates and stores the hop count to the landmark of its resident v

5 tile and the landmarks adjacent to its resident tile. Such landmarks are called reference landmarks. Specifically, the second part of the name of node x is actually a k- tuple: (d 1, d 2,, d k ), where d i = τ(u i, x) 2, τ(u i, x) is the minimum number of hops from x to a reference landmark u i. This k-tuple forms the virtual local coordinates of x, called the local landmark distance coordinates. For nodes within the same resident tile, their coordinates are with respect to the same set of landmarks. We note that in continuous domain, if we have at least three reference landmarks, the node defined by the distances to its reference landmarks is uniquely determined. However, such naming may have ambiguity in the discrete domain. Different nodes might have the same names. If the graph connectivity maps well with the underlying Euclidean space, we can expect that nodes with the same names are also close in space. B. Routing within a tile greedy routing via distances to landmarks By subdividing the node regions into tiles, the nodes within a tile are generally nicely distributed. Therefore we can expect greedy routing schemes to work well since the sensor field is rather smooth. The greedy method we adopt here is different from the previously proposed greedy schemes, which are all based on Euclidean coordinates. Our greedy method uses the coordinate system based on distances to landmarks, such that the distances (hop counts) between nodes and their reference landmarks are sufficient. 1) Continuous version: Our greedy routing algorithm is motivated by the following result that shows the continuous version will never get stuck under certain circumstances. Precisely, consider a destination and k landmarks {u i } (not necessarily in convex position) in the Euclidean plane. A point p in the Euclidean plane has virtual coordinates C(p) = ( pu 1 2, pu 2 2,, pu k 2 ), where pu i is the Euclidean distance between p and the i-th landmark u i. The virtual distance between p, q under this coordinate system is defined as d(p, q) = k i=1 ( pu i 2 qu i 2 ) 2. Given a destination q, the greedy routing algorithm follows the gradient such that the virtual distance to q is minimized. Lemma 4.1. In continuous domain, the greedy routing algorithm will never stuck, if 3 k 9, and the destination q is inside the convex hull of the reference landmarks. Proof: All we need to prove is that for any point p in the Euclidean plane, there is always a direction such that the virtual distance to q is decreasing. In fact, we ll show that if we move along the straight line connecting p, q, the virtual distance to the destination q is monotonically decreasing. We can transform the coordinate system so that the destination q stays at the origin and p = (x, 0). Suppose the reference landmark u i has Euclidean coordinate (x i, y i ). The virtual distance between p and q is: d(p, q) = k i=1 ( pu i 2 qu i 2 ) 2 = k i=1 (x2 2xx i ) 2. Then its partial derivative with respect to x is: d(p, q)/ x = 4 k i=1 x(x 2x i)(x x i ) = 4x(kx 2 3( k i=1 x i)x + 2 k i=1 x2 i ). Next we show that for any x > 0, d(p, q)/ x is not zero. That is, (3 k i=1 x i) 2 4k 2 k i=1 x2 i < 0. Since q = (0, 0) is inside the convex hull of the landmarks, x i s cannot be all positive or all negative. Therefore, by Cauchy-Schwartz inequality: (3 k i=1 x i) 2 < 9(k 1) k i=1 x2 i. As long as 9(k 1) 8k, i.e. k 9, the virtual distance to q is always decreasing on the straight line from p to q. Therefore the greedy algorithm will never get stuck. 2) Discrete version: In a graph setting, the discrete version of the greedy routing algorithm uses the hop counts to the landmarks as a replacement to the Euclidean distances. In situations where nodes are densely distributed, the minimum number of hops to the landmark approximates the Euclidean distance well. Given two nodes p, q with virtual coordinates (τ(p, u 1 ) 2, τ(p, u 2 ) 2,, τ(p, u k ) 2 ) and (τ(q, u 1 ) 2, τ(q, u 2 ) 2,, τ(q, u k ) 2 ) respectively, we define their (discrete) virtual distance to be d(p, q) = k i=1 (τ(p, u i) 2 τ(q, u i ) 2 ) 2. Given a destination t, our greedy routing algorithm always chooses the neighbor q of p such that d(q, t) is minimized. In other words, we move the packets by minimizing the Euclidean distance under the virtual coordinate system greedily. This greedy algorithm is local and efficient such that only the virtual coordinates of the neighbor nodes are needed. The discrete version of the greedy routing algorithm no longer guarantees that a packet will be delivered without getting stuck. So a packet may hit a local minimum such that all the neighbor nodes have virtual distances further away from the destination. A backup method at the local minimum is to flood within the tile.

6 C. Routing across tiles After the initial preprocessing stage, the landmark Voronoi complex is constructed and stored at each node. Upon a routing request, the landmark Voronoi complex determines the sequence of tiles that a routing path should cross. If the destination and the source are inside the same tile, greedy routing via distances to landmarks as described previously is used. If the destination is in a different tile other than the source, we propose a way to construct natural routing paths across tiles. u 1 p u 2 u 3 Fig. 3. Routing across tiles. Assume that the sequence of tiles, {T i }, that a packet will cross is generated by using LVC. If the packet is currently residing at node p, which is in a tile T i different than the resident tile of the destination, we use the landmark u i+1 in the next tile T i+1 as a temporary destination and send the packet along the shortest path from p to u i+1. Once the packet is inside the tile T i+1, we switch the temporary destination to the landmark in T i+2 until the packet is in the same tile as the destination, where the greedy routing scheme is used to deliver the packet to the destination. We note that when a packet travels through the intermediate tile, it will never get stuck. V. PROTOCOLS Our topology based routing include two phases: topology discovery and routing. Corresponding to the two phases, we introduce the Naming Protocol and the Voronoi Landmark-based Routing Protocol (VLRP). A. Naming protocol After landmarks are selected, the network starts topology discovery phase. The naming protocol is designed to carry out the following tasks: 1. Construct the Landmark Voronoi Complex (LVC) in a distributed fashion; 2. Compute routing table on the dual graph of the landmark voronoi complex, the combinatorial delaunay triangulation (CDT); q 3. Assign each node its local landmark distance coordinate from its reference landmarks; The number of landmarks is small compared to the total number of nodes in the network. In the simulation results shown in this paper, out of 2000 nodes, 23 landmarks are elected. In the topology discovery phase, each landmark floods the whole network. After the flooding, each node learns its minimal path distance to the landmarks. Such information enables a node to determine if it is on the boundary of a tile, i.e. a node is on the boundary of a tile if and only if it is a witness. If a node identifies itself as a witness of, for example, landmark u 1 and u 2, it sends a packet to u 1 and u 2. When u 1 receives the packet, it marks u 2 as its neighbor on the CDT, and vice versa. This concludes the distributed construction of the CDT and its dual, the LVC. In the process of landmark flooding, each landmark learns its hop distance to any other landmark in the network. After the CDT is constructed, each landmark deletes its distance entries to the other landmarks that are not its neighbors in the CDT. We designated one node to poll the landmark nodes and collect their CDT neighbor information. After this node gathers such information from all the landmarks, it floods these information to all the nodes in the network. Therefore every node in the network has the combinatorial Delaunay Triangulation, which captures the global topology. Each node then computes the global routing table on landmarks using Dijkstra s algorithm. Alternatively, computation of landmark routing table can also be carried out at the polling node. The polling node floods the computation result back to all the other nodes in the network. For each node, the home landmark is the landmark that is closest to it. Therefore, each node knows its landmark ID. After CDT is constructed, each landmark floods its own tile to distribute the IDs of all the reference landmarks of this tile. Using these information, each node deletes its distance entries that record hop counts to landmarks that are not its reference landmarks. B. Voronoi landmark-based routing protocol At the preprocessing stage, the network builds necessary infrastructure to support routing. After successful completion of the Naming Protocol, the global topology of the network is captured by the CDT, and the landmark distance coordinates of each node with respect to its local reference landmarks are stored at each node. VLRP runs on top of the infrastructure built by the Naming Protocol.

7 In VLRP, when the destination of a packet is given, by comparing the destination landmark with the home landmark of the current tile, a node can decide if the destination is within the same tile. If the destination is not within the same tile, the current node can decide the virtual landmark route, i.e. a sequence of tiles, to reach the resident tile of the destination by checking its landmark routing table. This is expressed as R =u 1,..., u i, where R is the route given by the landmark routing table. u s are the landmark IDs on the route. VLRP is designed to handle the following routing scenarios: Routing within a tile. When the destination is within the current tile, our greedy routing algorithm is used to route to the destination node. Routing across tiles. If the destination is in a tile that is not the current tile, the method shown in section IV-C is used to help routing. For example, R = u 1, u 2, u 3 is obtained from the landmark routing table. This means that the destination is in the tile of landmark u 3. In addition, the routing path to reach that tile is to transit through tile u 1 and u 2. Before the packet reaches tile u 1, it uses u 1 as its temporary destination. The greedy routing now is carried out by choosing one of the current node s neighbors which has the shortest path distance to u 1. This is always possible because the two tiles are adjacent and u 1 is always a CDT neighbor of the node s home landmark. We showed in Section IV-B, that the distance function of our greedy routing algorithm has no local minimal when the number of landmarks is greater or equal to 3 and less or equal to 9 in continuous domain. Therefore, local greedy routing always makes progress. However, in discrete case, it is possible for a packet to get stuck simply due to the lack of neighbors to chose from in routing. When this happens, local flooding within a tile is used to help deliver the packet to the destination. C. Data structures Landmark are used as reference in determining nodes local coordinates. However, from programming point of view, they are just ordinary nodes. No extra processing power or memory are required. Each node stores locally the information as shown in Figure 4. VI. SIMULATIONS We implemented the Naming protocol and VLRP using C++. Although our simulations do not take into con- Node{ the global routing table on landmarks; hop counts to its reference landmarks; a bit to record if the node is on the boundary of a tile; the ID of its 1-hop neighbors; } Fig. 4. Information stored at a node. sideration detailed networking behaviors, such as packet loss, packet delay and timing, etc., our simulations verify the correctness of the algorithm and feasibility of the protocols. As our routing scheme is designed to be practical for real world deployment, we intend to do network level simulations using ns-2 in the future. We simulated a network with 2000 nodes, of which 23 are landmarks. The landmarks are chosen randomly with 5 boundary nodes added after the random selection. The communication graph is the unit disk graph on the nodes. Two nodes can communicate directly if their Euclidean distance is within 1. After the communication graph is generated, the Euclidean coordinates are discarded since our protocols only require a communication graph. Although this work is intended for dense sensor field with large holes, we actually find out that our algorithms are pretty tolerant to sparse nodes with lots of small holes. In our simulation, on average each node has only hop neighbors. We also created tougher and more interesting scenarios to show the power of our routing algorithms. In all the figures, we show landmarks by big triangles. Gray landmarks are nodes on the boundaries of Voronoi cells, i.e., witnesses of the edges in combinatorial Delaunay triangulation. By simulation we find out that our routing algorithm generates natural, near optimal and load balanced routing paths. Figure 5 shows the routing algorithm for cases where the destination is within the same tile and where the destination is inside a different tile. Figure 6 shows two parallel routing paths. The landmark Voronoi complex indeed captures the true topology of the network. Therefore our routing algorithm is very powerful in discovering routes in very hard situations. In the scenario as shown in Figure 7, two big rooms are connected by a narrow and long corridor. Landmarks are selected randomly. The routing path that goes from one room to the other through the corridor is correctly discovered by using our routing algorithms. We further notice that we do not actually need landmark nodes be placed in the corridor (For randomly selected landmarks, the chance that a landmark is placed in the corridor is small.). As long as the original

8 time: 0 time: 0 (i) (ii) Fig. 5. (i) Routing within a tile; (ii) Routing across tiles. time: 0 time: 0 (i) (ii) Fig. 6. Two parallel paths.

9 network is connected, its connectivity is inherited by the combinatorial Delaunay triangulation. Fig. 7. A narrow corridor connects two rooms. Our routing algorithm discovers a route that goes through the corridor. A. Quality of routes VII. DISCUSSIONS The routing scheme proposed in this paper produces natural and load balanced routes that get around routing holes. Previous geographic routing schemes either use planar graphs [9], [1] or explicitly discovered hole boundaries [4] to get around holes. When a packet gets stuck, it is then routed along the boundary of a hole until greedy forwarding can be done again. Such methods tend to overload the nodes on the boundaries of holes and increase the size of holes. Our topology based routing algorithm keeps in mind the global topological features and derives natural and load balanced routes to get around holes. (i) q q p p Fig. 8. (i) Paths generated between two pairs p, q and p, q by the Voronoi landmark-based routing protocol; (ii) Paths generated between two pairs p, q and p, q by geographical routing algorithms. Nodes along the boundaries of holes tend to be overloaded. (ii) q q p p B. Multi-resolution LVC hierarchies In this paper we focus on dense node distributions with a small number of large holes. However, the techniques can be generalized to multi-resolution hierarchies to capture holes with different sizes. The landmark Voronoi complex is just a 2-level hierarchy. If the tile of the landmark Voronoi complex has non-trivial topology such that local greedy routing gets stuck frequently, we can pick a set of 2-nd level landmarks and construct 2nd level landmark Voronoi complex. In other words, smaller holes inside a tile is captured by the 2nd level landmark Voronoi complex. In summary, a LVC hierarchy can be constructed to capture multi-resolution holes. VIII. SUMMARY AND FUTURE WORK In this paper we propose a topology-based naming and routing structure by using only the link connectivity of the network. We partition the network into tiles by using the landmark Voronoi complex such that within each tile local greedy routing expects to work well. We completely discard the Euclidean coordinates and invent more robust local landmark coordinate system within each tile. Our Voronoi landmark-based routing protocol generates natural, near optimal and load balanced routing paths. The algorithms and protocols we propose in this paper work for sensor nodes in three dimensions as well. In fact, what underlying space the sensors lie in doesn t really matter. We also note that this paper is very preliminary work. Network dynamics (node addition and failure) and a thorough network-level simulation will be presented in the near future. Furthermore, it opens a door to a big space where we should and will explore. Within this topology-based naming and routing framework, a few questions need to answered. For example, what s the best way to select landmarks? Are there other local coordinate systems with good properties within a tile? With so little information such as only the connectivity graph of the network, what s the limitation that we can do? All these questions will be studied and addressed in the future. REFERENCES [1] P. Bose, P. Morin, I. Stojmenovic, and J. Urrutia. Routing with guaranteed delivery in ad hoc wireless networks. In 3rd Int. Workshop on Discrete Algorithms and methods for mobile computing and communications (DialM 99), pages 48 55, [2] G. Carlsson, A. Collins, L. Guibas, and A. Zomorodian. Persistence barcodes for shapes. In Symposium on Geometry Processing, to appear. [3] G. Carlsson and V. de Silva. Topological approximation by small simplicial complexes, preprint.

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