Mayor Bill de Blasio came under attack on multiple fronts Tuesday, including by former Mayor Rudy Giuliani over the shooting of the governor's aide, and former Police Commissioner Ray Kelly over the city's current NYPD policies.

The Giuliani quarrel was the latest between the 107th and 109th mayors over the city's direction under progressive Democrat de Blasio.

"There's been this tremendous increase in shootings," Giuliani, a Republican, said Tuesday. "And now we have this terrible shooting at the West Indian parade."

Giuliani was referring to the shooting of Carey Gabay -- a lawyer in Gov. Andrew M. Cuomo's administration -- who was caught in a crossfire during a pre-parade party Monday.

Giuliani said de Blasio "should do a better job of policing the West Indian parade," including cracking down on gun possession in the days before.

Police Commissioner William Bratton said the annual parade poses "probably our most problematic event in the city" and was a challenge even when Bratton was Giuliani's top cop two decades ago. Both spoke outside a House terrorism hearing where they testified.

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Giuliani also kept up his criticism of de Blasio on the city's rising homelessness. He said de Blasio made "a very nasty comment about me" -- de Blasio had said Giuliani's defense of his record on homelessness is "delusional," given a 40 percent increase during his mayoralty, according to de Blasio spokeswoman Karen Hinton.

"When he reduces crime 65 percent, then he can criticize me," Giuliani said.

Earlier Tuesday, de Blasio stuck to his position blaming the homelessness problem on the 2008 recession and skyrocketing housing costs. Appearing on MSNBC's "Morning Joe," de Blasio said, "The fact is, the Great Recession led to something we hadn't seen before."

Host Joe Scarborough, who had delivered a blistering critique of the mayor last month on the issue, noted that now, "the economy's not worse; it's better."

De Blasio responded that unlike in previous recoveries, fewer people bounced back and that housing costs in New York City continue to climb.

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About 57,000 people live in the city's shelters, lower than the record 60,000 who were there in December but higher than the 52,000 when de Blasio took office 21 months ago.

Kelly, on a book tour for his memoir, "Vigilance: My Life Serving America and Protecting Its Empire City," faulted de Blasio's shifts in NYPD strategy to stress better community relations.

"This quest to have the police loved, it's never going to be achieved, and what I say is, the goal is to have mutual respect," Kelly said in a New York Observer interview.

In The New York Times, Kelly linked the de-emphasis of stop-and-frisk tactics to the slight increase in killings. De Blasio has pointed to an overall decline in crime.

Kelly also tacitly criticized Bratton for suggesting that a Times Square pedestrian plaza be removed to shoo away topless women who hustle for tips.

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"I don't think you make a $40 million change ripping up . . . [the plaza] because of perceived aggressive panhandling," Kelly said. "You enforce that with undercover officers."