Journal of the ACM

We propose techniques to improve the usage of wireless spectrum in the context of wireless local area networks (WLANs) using new channel assignment methods among interfering Access Points (APs). We identify new ways of channel re-use that are based on realistic interference scenarios in WLAN environments. We formulate a weighted variant of the graph coloring problem that takes into account realistic channel interference observed in wireless environments, as well as the impact of such interference on wireless users. We prove that the weighted graph coloring problem is NP-hard and propose scalable distributed algorithms that achieve significantly better performance than existing techniques for channel assignment. We evaluate our algorithms through extensive simulations and experiments over an in-building wireless testbed.

/pdf/weighted-coloring-based-channel-assignment-for-wlans-mafmzqxez4.pdf

Weighted coloring based channel assignment for WLANs

The state of the union

We propose an efficient client-based approach for channel management (channel assignment and load balancing) in 802.11-based WLANs that lead to better usage of the wireless spectrum. This approach is based on a “conflict set coloring” formulation that jointly performs load balancing along with channel assignment. Such a formulation has a number of advantages. First, it explicitly captures interference effects at clients. Next, it intrinsically exposes opportunities for better channel re-use. Finally, algorithms based on this formulation do not depend on specific physical RF models and hence can be applied efficiently to a wide-range of in-building as well as outdoor scenarios. We have performed extensive packet-level simulations and measurements on a deployed wireless testbed of 70 APs to validate the performance of our proposed algorithms. We show that in addition to single network scenarios, the conflict set coloring formulation is well suited for channel assignment where multiple wireless networks share and contend for spectrum in the same physical space. Our results over a wide range of both simulated topologies and in-building testbed experiments indicate that our approach improves application level performance at the clients by upto three times (and atleast 50%) in comparison to current best-known techniques.

/pdf/a-client-driven-approach-for-channel-management-in-wireless-40q8jndgzk.pdf

A Client-Driven Approach for Channel Management in Wireless LANs

Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call minimum conflict-free coloring (min-CF-coloring). In its general form, the input of the min-CF-coloring problem is a set system $(X,{\cal S})$, where each $S \in {\cal S}$ is a subset of X. The output is a coloring $\chi$ of the sets in ${\cal S}$ that satisfies the following constraint: for every $x \in X$ there exists a color $i$ and a unique set $S \in {\cal S}$ such that $x \in S$ and $\chi(S) = i$. The goal is to minimize the number of colors used by the coloring $\chi$. Min-CF-coloring of general set systems is not easier than the classic graph coloring problem. However, in view of our motivation, we consider set systems induced by simple geometric regions in the plane. In particular, we study disks (both congruent and noncongruent), axis-parallel rectangles (with a constant ratio between the smallest and largest rectangle), regular hexagons (with a constant ratio between the smallest and largest hexagon), and general congruent centrally symmetric convex regions in the plane. In all cases we have coloring algorithms that use O(log n) colors (where n is the number of regions). Tightness is demonstrated by showing that even in the case of unit disks, $\Theta(\log n)$ colors may be necessary. For rectangles and hexagons we also obtain a constant-ratio approximation algorithm when the ratio between the largest and smallest rectangle (hexagon) is a constant. We also consider a dual problem of CF-coloring points with respect to sets. Given a set system $(X,{\cal S})$, the goal in the dual problem is to color the elements in X with a minimum number of colors so that every set $S \in {\cal S}$ contains a point whose color appears only once in S. We show that O(log |X|) colors suffice for set systems in which X is a set of points in the plane and the sets are intersections of X with scaled translations of a convex region. This result is used in proving that O(log n) colors suffice in the primal version.

Conflict-Free Colorings of Simple Geometric Regions with Applications to Frequency Assignment in Cellular Networks

We study conflict-free colorings, where the underlying set systems
 arise in geometry. Our main result is a general framework for
 conflict-free coloring of regions with low union complexity. A
 coloring of regions is conflict-free if for any covered point in
 the plane, there exists a region that covers it with a unique
 color (i.e., no other region covering this point has the same
 color). For example, we show that we can conflict-free color any
 family of n pseudo-discs with O(log n) colors.

/pdf/conflict-free-coloring-of-points-and-simple-regions-in-the-1ru9rkovpn.pdf

Conflict-Free Coloring of Points and Simple Regions in the Plane

Let H = (V, E) be a hypergraph. A conflict-free coloring of H is an assignment of colors to V such that, in each hyperedge e ∈ E, there is at least one uniquely-colored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to cellular antennae, in battery consumption aspects of sensor networks, in RFID protocols, and several other fields. Conflict-free coloring has been the focus of many recent research papers. In this paper, we survey this notion and its combinatorial and algorithmic aspects.

https://www.cs.bgu.ac.il/~shakhar/my_papers/reprint.pdf

Conflict-Free Coloring and its Applications

Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem called minimum conflict-free coloring (min-CF-coloring). In its general form, the input of the min-CF-coloring problem is a set system (X, S), where each S /spl isin/ S is a subset of X. The output is a coloring X of the sets in S that satisfies the following constraint: for every x /spl isin/ X there exists a color i and a unique set S /spl isin/ S, such that x /spl isin/ S and /spl chi/(S) = i. The goal is to minimize the number of colors used by the coloring X. Min-CF-coloring of general set systems is not easier than the classic graph coloring problem. However, in view of our motivation, we consider set systems induced by simple geometric regions in the plane. In particular, we study disks (both congruent and non-congruent), axis-parallel rectangles (with a constant ratio between the smallest and largest rectangle) regular hexagons (with a constant ratio between the smallest and largest hexagon), and general congruent centrally-symmetric convex regions in the plane. In all cases we have coloring algorithms that use O(log n) colors (where n is the number of regions). For rectangles and hexagons we obtain a constant-ratio approximation algorithm when the ratio between the largest and smallest rectangle (hexagon) is a constant. We also show that, even in the case of unit disks, /spl Theta/(log n) colors may be necessary.

/pdf/conflict-free-colorings-of-simple-geometric-regions-with-2qr271jrff.pdf

Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks

In this thesis we study a variety of problems in combinatorial and computational geometry, which deal with various aspects of arrangements of geometric objects, in the plane and in higher dimensions Some of these problems have algorithmic applications, while others provide combinatorial bounds for various structures in such arrangements The thesis involves two main themes: (i) Counting Crossing Configurations in Geometric Settings and its Applications: Suppose we “draw” a simple undirected graph G = (V, E) in the plane using points to represent vertices, and Jordan arcs connecting them to represent edges Assume that G has n vertices and m edges and that m ≥ 4n Then, using a planarity argument, there must exist two crossing arcs in this drawing This fact can be exploited to show that the number of such crossings is Ω(m/n), no matter how the graph is drawn The proof of this “Crossing Lemma” is due to Leighton [Lei83] and to Ajtai et al [ACNS82] A probabilistic proof of this fact was entitled “A proof from the book” [AZ98] Adapting and extending the proof technique of the Crossing Lemma, we provide improved asymptotic bounds on well-studied geometric combinatorial problems, such as the “k-set” problem (Chapter 4), the complexity of polytopes spanned by sets of points in the plane and in space (Chapter 3), etc In Chapter 2 we provide some sharp asymptotic Ramsey type theorems for intersection patterns of “nice” objects that are spanned by finite point sets: For example, we prove that for any dimension d, there exists a constant c = c(d) such that for any set P of n points in IR and any set S of m > cn distinct balls, each bounded by a sphere passing through a distinct pair of points of P , there exists a subset S ′ of S of size at least Ω(m/n) with nonempty intersection This is asymptotically tight and improves the previously best known bound (see [CEG94]) We extend this result to other families of objects, including pseudo-disks in the plane and axis-parallel boxes in any dimension The proofs rely on the same probabilistic proof technique of the Crossing Lemma, and can be regarded as extensions of that lemma The results of this chapter are joint work with Micha Sharir and appear in [SS03b] In Chapter 3 we prove that the maximum total complexity of k non-overlapping convex polygons in a set of n points in the plane is Θ(n √ k) This bound was already proved in the dual plane by Halperin and Sharir [HS92] However, our proof is much simpler and uses the Crossing Lemma applied to the collection of edges of the given polygons Similar results are obtained for more restricted collections of polygons We then generalize these results to bound the total complexity of k distinct non-overlapping

http://cs.tau.ac.il/thesis/thesis/smorodinsky-shakhar-phd.pdf

Shakhar Smorodinsky

Papers

Conflict-Free Colorings of Simple Geometric Regions with Applications to Frequency Assignment in Cellular Networks

Conflict-Free Coloring of Points and Simple Regions in the Plane

Conflict-Free Coloring and its Applications

Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks

Combinatorial Problems in Computational Geometry