New York City’s budget for the upcoming fiscal year is $82.1 billion, about 4.59 percent higher than the one adopted last year worth $78.5 billion, under a deal announced Wednesday.

It was the third budget for Bill de Blasio’s mayoralty and Speaker Melissa Mark-Viverito’s speakership, and both claimed it was free of the so-called budget dance, in which a mayor purposefully proposes dramatic cuts to cherished public entities, such as firehouses, so the council would have something to fight over.

“We’ve definitely come a long way since talking about those firehouses,” she said.

Both de Blasio and Mark-Viverito are Democrats.

When promised savings is factored in, the budget is about $100 million less than the executive budget de Blasio proposed earlier in the year of $82.2 billion. The budget calls for no layoffs, staff reductions or new taxes.

This year’s big-ticket items include $39.5 million a year to fund summer jobs for youths 14 to 24, which would grow the program to 60,000 slots from 50,000; $21 million more for six-day library service, and $1.7 million more to extend beach and pool season for a week past Labor Day.

The deal was cemented late Wednesday afternoon with the traditional handshake beneath the City Hall’s rotunda between the City Council speaker and the mayor.

It came after weeks of backroom negotiations. Except for a double-sided press release distributed by the mayor’s office, the actual budget documents were not made available, administration officials said, because they remain subject to final negotiation before the budget is formally passed by the council and signed into law by the mayor.

June 8 is the earliest budget deal since 2001, when Rudy Giuliani was mayor, de Blasio said. By law, the budget must be finalized before July 1, the beginning of the fiscal year.

Soon after the handshake, de Blasio left City Hall: he had a plane to catch for his daughter’s college graduation in California.