President Obama and Mitt Romney hunted for last-minute support on Sunday in a frenetic sprint across battleground states, even as their parties faced off in the first of what could be a growing number of legal disputes over presidential ballots and how they are counted.

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In Florida, the state's Democratic Party filed a lawsuit on Sunday morning that would force the Republican-led government to extend early voting in South Florida after complaints that extremely long lines on Saturday had prevented some people from casting their ballots. The Republican-controlled Legislature cut back early voting, which ended Saturday, from 14 days to eight.

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The lawsuit was followed by a chaotic day in the Democratic stronghold of Miami-Dade County, which opened one of its election offices for two hours to accept completed absentee ballots and then shut down only to reopen again on Sunday afternoon. Three counties said they would open again on Monday, but Democratic lawyers will continue to argue in court that in-person early voting should continue through Tuesday in Broward County.

In Ohio, Republican election officials will go to court on Monday to defend an 11th-hour directive to local election officials that critics say could invalidate thousands of provisional ballots by forcing voters to attest to the type of identification they provide.

Together, were a potential preview of the clashes that could emerge in as many as a half-dozen swing states over Tuesday's voting. The closeness of the races in those states has intensified the stakes of voter turnout, smooth operations at polling places, ballot problems and recounts.

In the battles, Republicans are mobilizing to defend against what they say is the potential for voter fraud, and Democrats are preparing to protect against what they say are efforts to suppress voting rights.

"The larger issue, in my view, is the scale of the effort that is required to have Election Day run smoothly," said Robert Bauer, the chief counsel for Mr. Obama's campaign. "Any number of things can go wrong, not by anybody's fault or intention, but we are fully prepared and so, we believe, are election officials around the country."

On the campaign trail, Mr. Obama and Mr. Romney presented themselves as agents of change while painting the other as an obstacle — final arguments in a nip-and-tuck race that may hinge on the success of the campaigns' elaborate turnout operations.

"The question of this election comes down to this," Mr. Romney told a crowd of about 4,500 on Sunday morning in Des Moines. "Do you want four more years like the last four years, or do you want real change? President Obama promised change, but he couldn't deliver it."

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Speaking to 23,000 people on a high school football field in Hollywood, Fla., Mr. Obama scoffed at Mr. Romney's bid to claim the mantle of change, deriding him as a political quick-change artist who is repackaging the failed policies of previous Republican administrations.

"When you make this choice, part of what you're choosing is who do you trust," Mr. Obama said. "After four years as president, you know me by now. You know I mean what I say and I say what I mean."

Racing the sun as well as the clock, both campaigns brimmed with confidence about their chances on Tuesday, though polls showed Mr. Obama holding a slender lead in several of the battleground states he needs to win the Electoral College. That advantage was evident in the itineraries of the two men on Sunday.

While each covered familiar ground from New Hampshire to Ohio, Mr. Romney sought to open a new front with a rally in Pennsylvania. Tightening polls have given him hope that he can take the state from Mr. Obama, who won there by a double-digit margin in 2008. But the president's advisers dismissed the foray as a desperate move by a challenger running out of other paths to victory.