A. Panconesi and A. Srinivasan, The local nature of \Delta-coloring and its algorithmic applications, Combinatorica 15(2) (1995), 255--280.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... does not help [6, 21] Generalizing from rings to bounded degree graphs one sees that several classical graph structures of both theoretical and practical interest, including MIS s, maximal matchings, Delta 1) and even Delta vertex colorings, can be computed in poly logarithmic time [1, 2, 7, 25]. In fact, many of these algorithms are very satisfactory because they are both quite simple and really of low complexity, i.e. with small exponents and no hidden large constants. Further generalizing from bounded degree graphs to general topologies has proven elusive, in spite of several efforts ....

....In fact, many of these algorithms are very satisfactory because they are both quite simple and really of low complexity, i.e. with small exponents and no hidden large constants. Further generalizing from bounded degree graphs to general topologies has proven elusive, in spite of several efforts [1, 2, 17, 22, 25, 26]. The situation here is, more or less, as follows. For a reasonably large class of graph structures, the asymptotically best deterministic algorithm known to date uses O(n ffl(n) rounds, where ffl(n) is a function which (very slowly) goes to 0 as n, the size of the network, grows. These ....

A. Panconesi and A. Srinivasan, The Local Nature of \Delta-coloring and Its Algorithmic Applications, Combinatorica 15 (2) 1995, 255-280.

....property of Delta colourings, yielding the following: There is no o(n) randomized, synchronous protocol to Delta colour paths, cycles or cliques. For all other graphs, there is a randomized protocol which, with high probability, computes a Delta colouring in polylogarithmically many rounds [21]. The property in question, holding for graphs which are neither cliques, paths nor cycles, is this: If G is Delta coloured except for one last vertex, it is possible to complete the colouring by a simple recolouring operation along an augmenting path of length O(log Delta n) starting from ....

.... holding for graphs which are neither cliques, paths nor cycles, is this: If G is Delta coloured except for one last vertex, it is possible to complete the colouring by a simple recolouring operation along an augmenting path of length O(log Delta n) starting from the uncoloured vertex [21]. It is an open problem whether randomization is necessary in all of the above algorithmic results; the asymptotically best deterministic protocols known to date need O(n ffl(n) rounds, where ffl(n) tends (very slowly) to zero as the number of vertices grows [1, 22] In a 1968 paper Vizing ....

A. Panconesi and A. Srinivasan, The local nature of \Delta-coloring and its algorithmic applications, Combinatorica 15(2) (1995), 255--280.

.... does not help [20] Generalizing from rings to bounded degree graphs one sees that several classical graph structures of both theoretical and practical interest, including MIS s, maximal matchings, Delta 1) and even Delta vertex colorings, can be computed in polylogarithmic time [1, 2, 7, 23]. In fact, many of these algorithms are very satisfactory because they are both quite simple and really of low complexity, i.e. with small exponents and no hidden large constants. Further generalizing from bounded degree graphs to general topologies has proven elusive, in spite of several efforts ....

....In fact, many of these algorithms are very satisfactory because they are both quite simple and really of low complexity, i.e. with small exponents and no hidden large constants. Further generalizing from bounded degree graphs to general topologies has proven elusive, in spite of several efforts [1, 2, 16, 21, 23, 24]. The situation here is, more or less, as follows. For a reasonably large class of graph structures, the asymptotically best deterministic algorithm known to date uses O(n ffl(n) rounds, where ffl(n) is a function which (very slowly) goes to 0 as n, the size of the network, grows. These ....

A. Panconesi and A. Srinivasan, The Local Nature of \Delta-coloring and Its Algorithmic Applications, Combinatorica 15 (2) 1995, 255-280.

....an Alexander von Humboldt research fellowship. following: There is no o(n) randomized, synchronous protocol to Delta colour paths, cycles or cliques. For all other graphs, there is a randomized protocol which, with high probability, computes a Delta colouring in polylogarithmically many rounds [17]. The property in question, holding for graphs which are neither cliques, paths nor cycles, is this: If G is Delta coloured except for one last vertex, it is possible to complete the colouring by a simple recolouring operation along an augmenting path of length O(log Delta n) starting from ....

.... holding for graphs which are neither cliques, paths nor cycles, is this: If G is Delta coloured except for one last vertex, it is possible to complete the colouring by a simple recolouring operation along an augmenting path of length O(log Delta n) starting from the uncoloured vertex [17]. It is an open problem whether randomization is necessary in all of the above algorithmic results; the asymptotically best deterministic protocols known to date need O(n ffl(n) rounds, where ffl(n) tends (very slowly) to zero as the number of vertices grows [1, 18] In a 1968 paper Vizing ....

A. Panconesi and A. Srinivasan, The local nature of \Delta-coloring and its algorithmic applications, Combinatorica 15(2) (1995), 255--280.

....n) algorithm that runs on the CREW PRAM, has been devised by Han [13] We improve these time complexities by presenting an ( n m)n ffi ; log 2 n) algorithm where ffi 0 is any constant. This gives faster algorithms for other problems such as the much harder Delta vertex coloring problem [25]. We use similar ideas to derive a faster algorithm for approximating the maximum acyclic subgraph problem [9] incurring a small processor penalty. Thus, our first method yields the fastest known NC algorithms for a large class of problems, without a big processor penalty. Though decreasing the ....

.... for any fixed ffi 0, there exists an ( n m)2 ffiq n ffi ; t log n q) algorithm to find such a labeling, if P rofit and Cost are bounded by n O(1) As instantiations of this general framework we can derive the following as corollaries, via Theorem 6 and some results from [20] and [25]. Recall that any graph with maximum degree Delta can be colored using Delta 1 colors. The first linear processor NC algorithm for ( Delta 1) coloring was presented in [20] with a running time of O(log 3 n log log n) It is shown in [20] that: a) the General Profit Cost problem can be ....

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A. Panconesi and A. Srinivasan, The local nature of \Delta--colorings and its algorithmic applications, Combinatorica, 15:255--280, 1995.

.... 1:58 Delta log n colours and runs in O(log n) time [24] For the interesting special case of bipartite graphs Lev, Pippinger, and Valiant show that Delta colourings can be computed in NC, but this is provably impossible in the distributed model of computation even if randomness is allowed (see [17, 25]) The result of Panconesi and Srinivasan was greatly improved by Alon [1] and Dubhashi and Panconesi [7] who showed how to compute nearly optimal colourings in O(log n) rounds with high probability. These solutions are based on a strategy known as the Rodl Nibble, a powerful probabilistic ....

.... best determinstic algorithms known so far have time complexity O(n ffl(n) where n is the number of vertices of the network and ffl(n) is a function which goes to zero as n grows [4, 23] On the other hand, if randomization is allowed all of these functions can be computed in O(polylog n) time [18, 15, 20, 25]. This paper shows that the improvement brought by randomization can be as big as doubly exponential. The question remains whether this is inherently the case. We believe that our result is also interesting from a technical point of view. The difficulty of the analysis is to show that the graph ....

A. Panconesi and A. Srinivasan, The local nature of \Delta--coloring and its algorithmic applications, Combinatorica 15(2) (1995), 255--280.

.... show that Delta colourings can be computed in polylogarithmic time in the PRAM model, whereas this is provably impossible in the distributed model of computation even if randomness is allowed, since in this case the number of rounds is at least on the order of the diameter of the input graph (see [22]) In this paper, we improve on the previous state of the art by giving a distributed randomized algorithm that computes a near optimal edge colouring in time O(log n) provided the maximum degree is large enough . More precisely, let G be a Delta regular graph with n nodes. We prove the ....

A. Panconesi and A. Srinivasan, The local nature of \Delta--colorings and its algorithmic applications, Combinatorica 15 (2) 1995, pp. 255-280.

.... 1:58 Delta log n colours and runs in O(logn) time [25] For the interesting special case of bipartite graphs Lev, Pippinger, and Valiant show that Delta colourings can be computed in NC, but this is provably impossible in the distributed model of computation even if randomness is allowed (see [18, 26]) The result of Panconesi and Srinivasan was greatly improved by Alon [1] and Dubhashi and Panconesi [7] who showed how to compute nearly optimal colourings in O(logn) rounds with high probability. These solutions are based on a strategy known as the Rodl Nibble, a powerful probabilistic ....

.... determinstic algorithms known so far have time complexity O(n ffl(n) where n is the number of vertices of the network and ffl(n) is a function which goes to zero 3 as n grows [4, 24] On the other hand, if randomization is allowed all of these functions can be computed in O(polylog n) time [19, 16, 21, 26]. This paper shows that the improvement brought by randomization can be as big as doubly exponential. The question remains whether this is inherently the case. We believe that our result is also interesting from a technical point of view. The difficulty of the analysis is to show that the graph ....

A. Panconesi and A. Srinivasan, The local nature of \Delta--coloring and its algorithmic applications, Combinatorica 15(2) (1995), 255--280.

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