Algorithm Design - Kleinberg and Tardos

Chapter 3 Problem 12

Page 112 Youre helping a group of ethnographers analyze some oral history data

theyve collected by interviewing members of a village to learn about the

lives of people whove lived there over the past two hundred years.

From these interviews, theyve learned about a set of n people (all

of them now deceased), whom well denote P1, P2,..., Pn. Theyve also

collected facts about when these people lived relative to one another.

Each fact has one of the following two forms:

. For some i and j, person Pi died before person Pj was born; or

. for some i and j, the life spans of Pi and Pj overlapped at least partially.

Naturally, theyre not sure that all these facts are correct; memories

are not so good, and a lot of this was passed down by word of mouth. So

what theyd like you to determine is whether the data theyve collected is

at least internally consistent, in the sense that there could have existed a

set of people for which all the facts theyve learned simultaneously hold.

Give an efficient algorithm to do this: either it should produce pro- posed dates of birth and death for each of the n people so that all the facts

hold true, or it should report (correctly) that no such dates can existthat

is, the facts collected by the ethnographers are not internally consistent.