# A capacity scaling algorithm for M-convex submodular flow by Satoru Iwata, Satoko Moriguchi, Kazuo Murota

By Satoru Iwata, Satoko Moriguchi, Kazuo Murota

This paper provides a swifter set of rules for the M-convex submodular How challenge, that's a generalization of the minimum-cost How challenge with an M-convex rate functionality for the How-boundary, the place an M-convex functionality is a nonlinear nonseparable cliserete convex functionality on integer issues. The set of rules extends the ability sealing strategy lor the submodular How challenge by means of Fleischer. Iwata and MeCormiek (2002) by using a unique means of altering the aptitude through fixing greatest submodular How difficulties.

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**Extra info for A capacity scaling algorithm for M-convex submodular flow**

**Example text**

3 The message complexity of wake-up under R in a k-dimensional hypercube with a dimensional labeling is ⌰(n log n). Complete Graphs Let us focus on wake-up in a complete graph. The use of the generic protocolWFlood will require O(n2 ) messages. We can obviously use the simplified broadcast protocol KBcast we developed for complete graphs. The number of messages transmitted will be k (n − 1), where k denotes the number of initiators. Even in the worst case (when every entity is independently awake and they all simultaneously start the protocol) O(n2 ) messages will be transmitted.

Consider the following simple strategy. 34 BASIC PROBLEMS AND PROTOCOLS Strategy HyperFlood: 1. The initiator sends the message to all its neighbors. 2. A node receiving a message from the link labeled l will send the messages only to those neighbors with label l < l. NOTE. The only difference between HyperFlood and the normal Flooding is in step 2: Instead of sending the message to all neighbors except the sender, the entity will forward it only to some of them, which will depend on the label of the port from where the message is received.

3. If an action contains a change of status, this operation will be the last one before exiting the action. 4. The set of status values of the protocol, and the set of restrictions under which the protocol operates will be explicit. 5. Precedence The external events are as follows: spontaneous impulse (Spontaneously), reception of a message (Receiving), and alarm clock ring (When). Different types of external events can occur simultaneously; for example, the alarm clock might ring at the same time a message arrives.