Given two sets of points $A$ and $B$ ($|A| = m$, $|B| = n$), we seek to find a minimum-weight many-to-many matching which seeks to match each point in $A$ to at least one point in $B$ and vice versa. Each matched pair (an edge) has a weight. The goal is to find the matching that minimizes the total weight. We study two kinds of problems depending on the edge weight used. The first edge weight is the Euclidean distance, $d_1$. The second is edge weight is the square of the Euclidean distance, $d_2$. There already exists an $O(k\log k)$ algorithm for $d_1$, where $k=m+n$. We provide an $O(mn)$ algorithm for the $d_2$ problem. We also solve the problem of finding the minimum-weight matching when the sets $A$ and $B$ are allowed to be translated on the real line. We present an $O(mnk \log k)$ algorithm for the $d_1$ problem and an $O(3^{mn})$ algorithm for the $d_2$. Furthermore, we also deal with the special case where $A$ and $B$ lie on a circle of a specific circumference. We present an $O(k^2 \log k)$ algorithm and $O(kmn)$ algorithm for solving the minimum-weight matching for the $d_1$, and $d_2$ weights respectively. Much like the problem on the real line, we extend this problem to allow the sets $A$ and $B$ to be rotated on the circle. We try to find the minimum-weight many-to-many matching when rotations are allowed. For $d_1$ we present an $O(k^2mn \log k)$ algorithm and a $O(3^{mn})$ algorithm for $d_2$. / Thesis (Master, Computing) -- Queen's University, 2010-11-08 20:48:18.968
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OKQ.1974/6202 |
| Date | 17 November 2010 |
| Creators | MOHAMAD, Mustafa Amid |
| Contributors | Queen's University (Kingston, Ont.). Theses (Queen's University (Kingston, Ont.)) |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Rights | This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner. |
| Relation | Canadian theses |
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