Three weeks after the slaying of Howard Beach jogger Karina Vetrano, investigators say it may take a long time to solve the case as detectives expand their search beyond the dead woman’s close-knit Queens neighborhood to parts of Brooklyn.

“It’s a tough one,” one law enforcement official familiar with the case said. “It isn’t going to be a question of good detective work, it is going to be a question of luck.”

The shift of part of the homicide investigation to areas of East New York and further west was signaled last week by NYPD Chief of Detectives Robert Boyce. The change in focus came about because Boyce himself recognized that a bicycle path at the north end of Spring Creek Park — where Vetrano, 30, was strangled — goes west into Brooklyn and could have afforded the killer a quick exit from the park. The path also goes a shorter distance east into South Ozone Park.

“I walked it several times, you walk to the Belt Parkway, to Fountain Avenue and Erskine Street,” Boyce said. “You would have to walk, you couldn’t drive a car — you could ride a bicycle.”

The bike path actually extends five miles into Canarsie and beyond, creating more square miles through which the killer could have fled. Immediately after Vetrano’s battered and sexually abused body was found the night of Aug. 2, police scoured Howard Beach in search of video footage from surveillance cameras in the hopes of catching the suspect in the neighborhood. Boyce said nothing of interest was found.

But with the shift to the west, the search for Vetrano’s killer is dealing with a larger and more desolate area along the Belt Parkway without a great deal of surveillance camera coverage. The three nearest cameras to the park appear to be a security post to a city landfill at the foot of Fountain Avenue, a city-run facility for the developmentally disabled and mentally ill also on Fountain Avenue and a shopping mall on Erskine Street.

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Police said they have all the physical evidence they will find related to the crime, including three good samples of DNA from a possible suspect. However, the DNA profile has not matched any of the more than 600,000 DNA samples in the New York State database, which means the material didn’t come from someone with a felony or recent misdemeanor record in the state. There also has been no hit with a database maintained by the FBI, police noted.

Vetrano’s family has been pleading with the public for information and last week her father Philip said investigators had some persons of interest and were within days of making an arrest. But NYPD Commissioner William Bratton said that was not the case.

“We have no suspects. We are not close to an arrest,” Bratton said. “We need the public’s assistance finding a potential suspect that might lead to an arrest. I certainly feel for the family of this young woman.”

In an effort to shake loose useful tips, Vetrano’s family started a GoFundMe campaign which amassed nearly $260,000, which police said helped generate 70 tips. But none of the tips have proved useful so far, the law enforcement official said.

Detectives also have gone through Vetrano’s social and work life — she worked at Vetro Restaurant & Lounge and was involved in filmmaking — but have found nothing of note, the official said. Some investigators said Vetrano’s last run was a random event which any killer wouldn’t have known about in advance.

“The wrong time, wrong place,” the official said.

Detectives also have been questioning people with felony and misdemeanor sex crimes records. Police also have questioned motorists whose cars were seen on videos in the neighborhood the night of Aug. 2, one resident said.