CS6170 - Algorithms Design To Elect Leader Among The Agents And Probability In Time That Is Logarithmic - IT/Computer Science Assignment Help

Assignment Task -

 Randomized Algorithms ;

1. Consider the (?+1)-colouring algorithm discussed above without the sleeping step, i.e., all  vertices are awake in every step and pick a tentative colour uniformly at random from its list  of colours. Show that this algorithm also finds a (?+1)-coloring in ??(log ??) rounds with high  probability (whp). (Exercise 6.7 from Pandurangan.) 

2. A matching of a graph G = (V,E) is a subset of edges M ⊆ E such that no two edges in M share  a common vertex. A matching M is maximal if no more edges can be added to M while  keeping M as a matching. Give an O(logn)-round (whp) distributed algorithm for finding a  maximal matching. (Exercise 6.11 from Pandurangan.) 

3. In the gossip model of communication, we have  nodes with unique IDs that operate in the  following synchronous manner. In each round, the nodes are randomly paired up to form a  matching with pairs and each pair of matched nodes can exchange messages of size logarithmic in Note that there is no underlying network graph, so any node can be paired  with any other node. Design an algorithm for leader election and prove that it terminates  with high probability in time that is logarithmic in . 

4. Consider a set of mobile agents that are present in a ring of vertices. You may assume  that and are sufficiently large. The vertices in the ring can be viewed as just rooms with a  door pointing clockwise and another door pointing counterclockwise. The mobile agents  have unique IDs and operate in synchronous rounds. In each round, each agent is

(i) aware  of all other agents in the vertex that  is currently in,

(ii) can send an (log + log )-bits  message to each co-occupant (i.e., other agents in the same room as ),

(iii) receive  messages sent by its co-occupants, and finally

(iv) move one step either clockwise or  counterclockwise. Each time an agent moves from one vertex to a neighbouring vertex, it  must spend one unit of energy.  

Design algorithms to elect a leader among the agents using the least total energy under the  following two assumptions:

(i) Only is known to the agents and

(ii) only is known to the  agents. What are the best guarantees that you can achieve in terms of

(a) correctness  probability and

(b) total energy expenditure? Note that the question is left a bit open-ended  on purpose.

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