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--- courses:cs211:winter2018:journals:ahmadh:ch1 [2018/01/16 21:34] – created ahmadh
+++ courses:cs211:winter2018:journals:ahmadh:ch1 [2018/01/17 00:21] (current) – ahmadh
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 ====== Chapter 1 ======
-===== Section 1.1 =====
+===== Section 1.1: The Stable Matching Problem =====
-This is a test.
+The problem stems from the question that could one design a college admissions process, or a job recruiting process, that was self-enforcing, i.e. self-interest would prevent offers from being retracted and redirected, and all the offers would be stable?
+This problem can be reduced to a simple problem, the solution to which can be extended to solve the Stable Matching Problem in general: given a set of n men and n women and an order of preference for each man and woman, find a stable matching such that everyone ends up married to somebody and nobody is married to more than one person. This problem is solved--efficiently--by the Gale-Shapley algorithm.
+==== Section 1.1.1: Pseudo-code for the Gale-Shapley Algorithm ====
+A sketch of the Gale-Shapley algorithm (from Page 6 of the textbook) to this problem is given below:
+	Initially all m and w are free
+	While there is a man m who is free and hasn’t proposed to every woman
+	   Choose such a man m
+	   Let w be the highest-ranked woman in m’s preference list to whom m has not yet proposed
+	   If w is free then
+	      (m, w) become engaged
+	   Else w is currently engaged to m’
+	      If w prefers m’ to m then
+	         m remains free
+	      Else w prefers m to m’
+	         (m,w) become engaged
+	         m’ becomes free
+	      Endif
+	   Endif
+	Endwhile
+	Return the set S of engaged pairs
+==== Section 1.1.2: Analysis the Gale-Shapley Algorithm ====
+The algorithm terminates after n^2 steps at most, since every man can at most propose to each of n women. Therefore, it is O(n^2). The proof for the algorithm is given on Page 7 of the textbook.