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m C(n m 1, n 1) m n m m = C(m, n) m n = Thus, the algorithms updates, on the average, less than m/n + 1<2 elements and therefore the average delay is constant for any m and n(m n) 14 LISTING PERMUTATIONS A sequence p1 , p2 , , pn of mutually distinct elements is a permutation of S = {s1 , s2 , , sn } if and only if {p1 , p2 , , pn } = {s1 , s2 , , sn } = S In other words, an n-permutation is an ordering, or arrangement, of n given elements For example, there are six permutations of the set {A, B, C} These are ABC, ACB, BAC, BCA, CAB, and CBA Many algorithms have been published for generating permutations.



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We introduce the algorithm of Karapetian but omit the proof details due to the limited space Readers may refer to the work by Karapetian [31] for details The key components of the algorithm are the clockwise sweep and counter-clockwise sweep In each run of the clockwise sweep, a set of paths that can share a same color is found To do so, a path p is rst included into a set A Then another path that can share a color with the paths in A is searched in the clockwise direction If there are multiple candidates then the path q with the smallest end node q is included into A The process is repeated until no path can be included in A Similarly, in each run of the counter-clockwise sweep, a set of paths that can share a same color is found.

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To do so, a path p is rst included into a set B Then another path that can share a color with the paths in B is searched in the counter-clockwise direction If there are multiple candidates then the path q with the largest end node q is included into B The process is repeated until no path can be included in B The algorithm calls the clockwise sweep at most (GP )/2 times and calls the counter-clockwise sweep at most (GP ) times The algorithm of Karapetian [31] is given in Figure 185 The algorithm of Karapetian [31] is a 15-approximation algorithm The upper bound 3 (GP )/2 is tight in the sense that there are instances of the WA problem on Cn that require at least 3 (GP )/2 colors.

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With MaxNet, only the most severely bottlenecked link on the end-to-end path generates the congestion signal that controls the source rate. This approach is unlike the previously described protocols, for which all of the bottlenecked links on the endto-end path add to the congestion signal (by independent random packet marking or dropping at each link), which is termed SumNet. To achieve this result, the packet format must include bits to communicate the complete congestion price (Figure 8.12). This information may be carried in a 32-bit field in a new IPv4 option, an IPv4 TCP option or in the IPv6 per-hop options field, or even in an out-of-band control packet. Each link replaces the current congestion price in packet j Mj , with the link s congestion price Pl t , if it is greater than the one in the packet. In this way, the maximum congestion price on the path is communicated to the destination, which relays the information back to the source in acknowledgment packets. The link price is determined by an AQM algorithm: P1 t + 1 = P1 t + Y 1 t C1 t

An example of such instances can be constructed as follows: Let n = 5k (k 1) and P = P0 P1 P2 P3 P4 , where Pi (0 i 4) is a set of L paths between node i k and node (i + 1) k + 1 (arithmetic operations are modulo n) Then the clique number (GP ) is 2L and any valid coloring for P requires at least 3L colors 18432 RWA Problem For the RWA problem on the ring network, a well-used approach for paths selection is the edge avoidance routing in which every routing path.

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