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There is currently an urgent need for effective solutions against distributed denial-of-service (DDoS) attacks directed at many well-known Web sites. Because of increased sophistication and severity of these attacks, the system administrator of a victim site needs to quickly and accurately identify the probable attackers and eliminate the attack traffic. Our work is based on a probabilistic marking algorithm in which an attack graph can be constructed by a victim site. We extend the basic concept such that one can quickly and efficiently deduce the intensity of the "local traffic" generated at each router in the attack graph based on the volume of received marked packets at the victim site. Given the intensities of these local traffic rates, we can rank the local traffic and identify the network domains generating most of the attack traffic. We present our trace back and attacker identification algorithms. We also provide a theoretical framework to determine the minimum stable time tmin, which is the minimum time needed to accurately determine the locations of attackers and local traffic rates of participating routers in the attack graph. Extensive experiments are carried out to illustrate that one can accurately determine the minimum stable time tmin and, at the same time, determine the location of attackers under various threshold parameters, network diameters, attack traffic distributions, on/off patterns, and network traffic conditions.