Wireless sensor networks are normally characterized by resource challenged nodes. Since communication costs the most in terms of energy in these networks, minimizing this overhead is important. We consider minimum length node scheduling in regular multi-hop wireless sensor networks. We present collision-free decentralized scheduling algorithms based on TDMA with spatial reuse that do not use message passing, this saving communication overhead. We develop the algorithms using graph-based k-hop interference model and show that the schedule complexity in regular networks is independent of the number of nodes and varies quadratically with k which is typically a very small number. We follow it by characterizing feasibility regions in the SINR parameter space where the constant complexity continues to hold while simultaneously satisfying the SINR criteria. Using simulation, we evaluate the efficiency of our solution on random network deployments.
McDiarmid and Reed showed that the chromatic number of general hexagonal graphs is bounded by 4 3 times the clique number @cite_13 . This result suggests that our hexagonal scheduling algorithm, the complexity of which is also bounded by 4 3 times the clique number, might indeed be very close to optimal. A 4 3-approximate distributed channel assignment algorithm that uses several rounds of message passing is presented in @cite_18 .
@cite_18 @cite_13
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We deterministically compute a @D+1 coloring and a maximal independent set(MIS) in time O(@D^1^ ^2^+^@Q^(^1^ ^h^)+log^*n) for @D"1"+"[email protected][email protected]^1^+^i^ ^h, where @D"j is defined as the maximal number of nodes within distance j for a node and @D:[email protected]"1. Our greedy coloring and MIS algorithms improve the state-of-the-art algorithms running in O(@D+log^*n) for a large class of graphs, i.e., graphs of (moderate) neighborhood growth with h>=36. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. Our algorithm runs in time O([email protected]+log^*n) for @[email protected][email protected](log^1^+^1^ ^l^o^g^^^*^nn) and @[email protected]?O(@D log^1^+^1^ ^l^o^g^^^*^nn). For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest @D+1 coloring algorithm running in time O([email protected]+logn). The algorithm works without knowledge of @g and uses less than @D colors, i.e., (1-1 O(@g))@D with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number @g into account. We also improve on the state of the art deterministic computation of (2,c)-ruling sets. We show an algorithm that finds cliques of size (log n log log n)2 whenever a graph has a clique of size at least n (log n)b for an arbitrary constant b. This leads to an algorithm that approximates max clique within a factor of O(n(log log n)2 (log n)3), which matches the best approximation ratio known for the chromatic number. The previously best approximation ratio known for max clique was O(n (log n)2). We present and analyse a very simple randomised distributed vertex colouring algorithm for arbitrary graphs of size n that halts in time O(logn) with probability 1-o(n^-^1). Each message containing 1 bit, its bit complexity per channel is O(logn). From this algorithm, we deduce and analyse a randomised distributed vertex colouring algorithm for arbitrary graphs of maximum degree @D and size n that uses at most @D+1 colours and halts in time O(logn) with probability 1-o(n^-^1). We also obtain a partition algorithm for arbitrary graphs of size n that builds a spanning forest in time O(logn) with probability 1-o(n^-^1). We study some parameters such as the number, the size and the radius of trees of the spanning forest. We study the computation power of the congested clique, a model of distributed computation where n players communicate with each other over a complete network in order to compute some function of their inputs. The number of bits that can be sent on any edge in a round is bounded by a parameter b We consider two versions of the model: in the first, the players communicate by unicast, allowing them to send a different message on each of their links in one round; in the second, the players communicate by broadcast, sending one message to all their neighbors. It is known that the unicast version of the model is quite powerful; to date, no lower bounds for this model are known. In this paper we provide a partial explanation by showing that the unicast congested clique can simulate powerful classes of bounded-depth circuits, implying that even slightly super-constant lower bounds for the congested clique would give new lower bounds in circuit complexity. Moreover, under a widely-believed conjecture on matrix multiplication, the triangle detection problem, studied in [8], can be solved in O(ne) time for any e > 0. The broadcast version of the congested clique is the well-known multi-party shared-blackboard model of communication complexity (with number-in-hand input). This version is more amenable to lower bounds, and in this paper we show that the subgraph detection problem studied in [8] requires polynomially many rounds for several classes of subgraphs. We also give upper bounds for the subgraph detection problem, and relate the hardness of triangle detection in the broadcast congested clique to the communication complexity of set disjointness in the 3-party number-on-forehead model.