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    <p>Have you considered using a directed acyclic graph instead of a tree? In other words, each node has multiple "parents" that it can send messages to -- the acyclic requirement ensures that all messages eventually arrive. I ask because it sounds like you have a wireless network and because there's an efficient approach to computing the optimum solution.</p> <p>The approach is linear programming. Let r be the index of the root node. For nodes i, j, let cij be the energy cost of sending a message from i to j. Let xij be a variable that will represent the number of messages sent by node i to node j in each time step. Let z be a variable that will represent the maximum rate of energy consumption at each node.</p> <p>The linear program is</p> <pre><code>minimize z subject to # the right hand side is the rate of energy consumption by i (for all i) z &gt;= sum over all j of cij * xij # every node other than the root sends one more message than it receives (for all i != r) sum over all j of xij == 1 + sum over all j of xji # every link has nonnegative utilization (for all i, j) xij &gt;= 0 </code></pre> <p>You can write a code that generates this LP in something very much like this format, whereupon it can be solved by an off-the-shelf LP solver (e.g., the free GLPK).</p> <p>There are a couple of features of the LP worth mentioning. First, it may seem odd that we haven't "forced" messages to go to the root. It turns out that as long as the cij constants are positive, it just wastes energy to send messages in cycles, so there's no point. This also ensures that the directed graph we've constructed is acyclic.</p> <p>Second, the xij variables are in general not integers. How do we send half a message? One possible solution is randomization: if M is the total rate of messages sent by node i, then node i sends each message to node j with probability xij/M. This ensures that the averages work out over time. Another alternative is to use some sort of weighted round-robin scheme.</p>
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    1. POAlgorithm to calculate the most energy efficient ad-hoc network
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    1. USuser287792
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    1. COFairly certain a max flow (a special case of LP) would do it, even if the problem asker abandoned the problem!
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      1. USLarry
    2. COThe constraints are flow constraints, but the objective is different. I agree that the primal-dual method would likely be in play.
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