Some security consultants working in the financial domain are currently advising a client who is investigating a potential money-laundering scheme. The investigation thus far has indicated that n suspicious transactions took place in recent days, each involving money transferred into a single account. Unfortunately, the sketchy nature of the evidence to date means that they don’t know the identity of the account, the amounts of the transactions, or the exact times at which the transactions took place. What they do have is an approximate time-stamp for each transaction; the evidence indicates that transaction i took place at time ti ± ei, for some "margin of error" ei. (In other words, it took place sometime between tt – et and tt + ei.) Note that different transactions may have different margins of error.
In the last day or so, they’ve come across a bank account that (for other reasons we don’t need to go into here) they suspect might be the one involved in the crime. There are n recent events involving the account, which took place at times x1, x2, …, xn To see whether it’s plausible that this really is the account they’re looking for, they’re wondering whether it’s possible to associate each of the account’s n events with a distinct one of the n suspicious transactions in such a way that, if the account event at time xt is associated with the suspicious transaction that occurred approximately at time tj, then |tj – xi|≤ e,. (In other words, they want to know if the activity on the account lines up with the suspicious transactions to within the margin of error; the tricky part here is that they don’t know which account event to associate with which suspicious transaction.)
Give an efficient algorithm that takes the given data and decides whether such an association exists. If possible, you should make the running time be at most O(n2).
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