|Consider a more general form of the hospital, medical student problem from Lecture 1. There are m hospitals, each with a certain number of available positions. There are n medical students who are graduating and interested in joining one of the hospitals. Each hospital has a ranking of the students and each student has a ranking of the hospitals. Assume that there are more students graduating that there are slots available in the hospitals. The goal is to find a way of assigning each student to at most one hospital in such a way that all available positions in all hospitals are filled. |(i) What are the two types of instability for this problem? (ii) Extend the Gale-Shapley algorithm to find a stable assignment of students to hospitals. Prove that it always outputs a stable matching. |Consider a more general form of the hospital, medical student problem from Lecture 1. There are m hospitals, each with a certain number of available positions. There are n medical students who are graduating and interested in joining one of the hospitals. Each hospital has a ranking of the students and each student has a ranking of the hospitals. Assume that there are more students graduating that there are slots available in the hospitals. The goal is to find a way of assigning each student to at most one hospital in such a way that all available positions in all hospitals are filled. |(i) What are the two types of instability for this problem? (ii) Extend the Gale-Shapley algorithm to find a stable assignment of students to hospitals. Prove that it always outputs a stable matching