After the Columbine High School massacre in 1999 and the Mumbai attack in 2008, police departments across the United States adopted a new mindset on how to deal with what they call "active shooter" incidents in which people are trapped in restaurants, theaters or other soft targets.

Long gone are the days of establishing a perimeter and waiting for SWAT teams and hostage negotiators to save the day. The new formula for preserving lives calls for ordinary police officers to go on offense and take the attackers' lives, with an emphasis on speed and force.

It's an approach reinforced by the recent multipronged attacks in Paris and the hotel siege in Mali. After those attacks, authorities have been hammering home the dangers of responding to mass homicides in progress, even suggesting that hostages should fight for themselves as a last resort.

With the horrors of Paris still fresh, Washington, D.C., Police Chief Cathy Lanier said in a recent television interview that she's worried about a creeping "numbness to what potentially is a reality" on U.S. soil, adding, "Just ignoring it and not preparing yourself — that's not an option anymore." Said New York Police Commissioner William Bratton: "It's a new world we're living in; it's a very troubling world," he said. He added that it's the police's job to worry about the threat but for "the population to be aware."

Here's a look at some of the hard realities:

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NO HESITATION, NO NEGOTIATION

The hostage-taking in Paris by heavily armed attackers linked to the Islamic State group at the Bataclan concert hall had nothing to do with using victims as bargaining chips. It was, in the words of New York Police Department officials, "a media event" intended to buy time for more killing. "It's not 'Dog Day Afternoon,'" said security expert Scott Stewart, of the Stratfor strategic intelligence firm, of Austin, Texas, referring to the Al Pacino film about a prolonged hostage standoff. With that in mind, police are making the calculation that, "if we move swiftly, there still be casualties, but it will be fewer casualties," said Maki Haberfeld, a professor at New York's John Jay College of Criminal Justice.

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COMMANDO COPS

Under the new normal, police departments nationwide have increasingly armed ordinary patrol officers with high-powered rifles to match extremists' firepower in a domestic brand of urban warfare. The NYPD has taken the extra step of forming a quick-strike force of 500 officers specially trained to combat terrorism and has run tactical drills to improve response times to multiple locations, especially soft targets such as theaters and restaurants. The approach puts ordinary cops at greater risk, "something that police officers accept on a daily basis. ... They live with this possibility of losing their lives on any given day, whether a benign traffic stop or a terrorist attack," Haberfeld said.

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SHOOTERS AND BOMBERS

When it comes to neutralizing terrorists armed with suicide vests, there are no clear answers. Experts say police are trained to keep their distance, yet shoot to kill. The good news in Paris was the suicide bombers — as opposed to the shooters — proved the lesser evil. It was further proof that the new brand of grass-roots jihadists "have struggled to make vests and bombs" that could cause even worse casualties, Stewart said. Two prime examples: the failed attempt by Najibullah Zazi to assemble suicide vests and deploy them in New York City subways in 2009 and the botched Times Square car bombing by Faisal Shahzad in 2010.

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GOING DOWN WITH A FIGHT

Authorities have long said that running and hiding could help people survive a mass-shooting siege. Those who die often are shot at close range in the first few minutes after they freeze in fear. The shooters typically "are not good marksmen, so the more distance you can put between yourself and them, the better," Stewart said. But post-Paris, law enforcement officials have talked more openly about a third way to react: fighting back. Lanier, the D.C. chief, caused a stir by telling CNN that hostages trapped in the line of fire might need to team up and "take the gunman out." It's a new concept for American civilians, but "not necessarily a shift in counties that experience terrorism for decades. ... The best way to save your life is to try to do yourself and not wait until law enforcement arrives," Haberfeld said.

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Associated Press writers Jake Pearson and Colleen Long contributed to this report.