SHOW AND EXPLAIN WORK FOR THE ANSWER, PLEASE!
11. You're helping some security analysts monitor a collection of networked computers, tracking the spread of an online virus. There are n computers in the system, labeled C1,C2,..., Cn, and as input you're given a collection of trace data indicating the times at which pairs of computers commu nicated. Thus the data is a sequence of ordered triples (C,, Cj, tk); such a triple indicates that C and C, exchanged bits at time tk. There are m triples total We'll assume that the triples are presented to you in sorted order of time. For purposes of simplicity, we'll assume that each pair of computers communicates at most once during the interval you're observing The security analysts you're working with would like to be able to answer questions of the following form: If the virus was inserted into computer Ca at time x, could it possibly have infected computer C, by time y? The mechanics of infection are simple: if an infected computer Ci communicates with an uninfected computer C at time tk (in other words, if one of the triples (G, G, tk) or (G.?.tk ) appears in the trace data), then computer C, becomes infected as well, starting at time tk Infection can thus spread from one machine to another across a sequence of communications, provided that no step in this sequence involves a move backward in time. Thus, for example, if C is infected by time tk and the trace data contains triples (C,, C,tk) and C, Cqt), where tk S tr. then C will become infected via C. (Note that it is okay for t to be equal to t,; this would mean that C, had open connections to both C and C? at the same time, and so a virus could move from C to Cq) For example, suppose n 4, the trace data consists of the triples and the virus was inserted into computer C at time 2. Then C3 would be infected at time 8 by a sequence of three steps: first C2 becomes infected at time 4, then C4 gets the virus from C2 at time 8, and then C3 gets the virus from C4 at time 8. On the other hand, if the trace data were (C2, C3, 8), (C, C4,12), (C1, C2, 14), and again the virus was inserted into computer C a time 2, then C3 would not become infected during the period of observation: although C2 becomes infected at time 14, we see that C3 only communicates with C2 before C2 was infected. There is no sequence of communications moving forward in time by which the virus could get from C to C in this second example Design an algorithm that answers questions of this type: given a collection of trace data, the algorithm should decide whether a virus introduced at computer Ca at time x could have infected computer Cb by time y. The algorithm should run in time O(m+n)
