Since the discovery of the "Philadelphia chromosome" in chronic myelogenous leukemia in 1960, there is an ongoing intensive research of chromosomal aberrations in cancer. These aberrations, which result in abnormally structured genomes, became a hallmark of cancer. Many studies give evidence to the connection between chromosomal alterations and aberrant genes involved in the carcinogenesis process. An important problem in the analysis of cancer genomes, is inferring the history of events leading to the observed aberrations. Cancer genomes are usually described in form of karyotypes, which present the global changes in the genomes' structure. In this study, we propose a mathematical framework for analyzing chromosomal aberrations in cancer karyotypes. We introduce the problem of sorting karyotypes by elementary operations, which seeks for a shortest sequence of elementary chromosomal events transforming a normal karyotype into a given (abnormal) cancerous karyotype. Under certain assumptions, we prove a lower bound for the elementary distance, and present a polynomial-time 3-approximation algorithm. We applied our algorithm to karyotypes from the Mitelman database, which records cancer karyotypes reported in the scientific literature. Approximately 94% of the karyotypes in the database, totalling 57,252 karyotypes, supported our assumptions, and each of them was subjected to our algorithm. Remarkably, even though the algorithm is only guaranteed to generate a 3-approximation, it produced a sequence whose length matches the lower bound (and hence optimal) in 99.9% of the tested karyotypes.
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