Introduction to algorithms
Text algorithms
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In a computer network that consists of M subnetworks, the L-bit address of a machine consists of two parts: A prefix si that contains the address of the subnetwork to which the machine belongs, and a suffix (of length L$-$ |si|) containing the address of that particular machine within its subnetwork. In fixed-length subnetwork addressing, |si| is independent of i, whereas, in variable-length subnetwork addressing, |si| varies from one subnetwork to another. To avoid ambiguity when decoding addresses, there is a requirement that no si be a prefix of another sj. The practical problem is how to find a suitable set of sis in order to maximize the total number of addressable machines, when the ith subnetwork contains ni machines. Not all of the ni machines of a subnetwork i need be addressable in a solution: If $n_i 2^{L-|s_i|},$ then only $2^{L-|s_i|}$ machines of that subnetwork are addressable (none is addressable if the solution assigns no address si to that subnetwork). The abstract problem implied by this formulation is: Given an integer L, and given M (not necessarily distinct) positive integers $n_1, \cdots, n_M,$ find M binary strings $s_1, \cdots, s_M$ (some of which may be empty) such that 1) no nonempty string si is prefix of another string sj, 2) no si is more than L bits long, and 3) the quantity $\sum \nolimits _{|s_k|\ne0} \min \left\{ n_k, 2^{L-|s_k|} \right\}$ is maximized. We generalize the algorithm to the case where each ni also has a priority pi associated with it and there is an additional constraint involving priorities: Some subnetworks are then more important than others and are treated preferentially when assigning addresses. The algorithms can be used to solve the case when L itself is a variable; that is, when the input no longer specifies L but, rather, gives a target integer 驴 for the number of addressable machines, and the goal is to find the smallest L whose corresponding optimal solution results in at least 驴 addressable machines.