WASHINGTON, November 4 (RIA Novosti) - The Democratic and Republican party candidates for U.S. president and their running mates voted on Tuesday morning before returning home to await the results.

The Democratic front-runner, Illinois Sen. Barack Obama, voted in his hometown of Chicago, while his Republican rival, Arizona Sen. John McCain, cast his ballot in Phoenix with his wife, Cindy.

Democratic vice presidential candidate Sen. Joe Biden voted in Delaware before flying to join Obama in Chicago, while Alaska Gov. Sarah Palin went to the polls in her hometown of Wasilla, where she was once mayor, before heading off to meet McCain in Arizona.

Opinion polls show Obama solidly ahead nationwide, and up or level in all the swing states that will decide the election. The McCain campaign says its internal polling shows the race is much closer than media surveys suggest.

The U.S. ambassador to Russia said on Tuesday that whoever wins he would bring new ideas to the world, although the country's national interests would not change.

Ambassador John Beyrle also told an audience at his official Spaso House residence that the new president would strengthen U.S.-Russian relations.

The Russian parliament's leading authority on foreign relations said last week that an Obama victory would give him hope for better Russian-U.S. relations, while in a recent Russian opinion poll, of the 50% of respondents who expressed a preference, Obama was favored over McCain by almost two to one.

Konstantin Kosachyov, the head of the State Duma international affairs committee, told journalists that he thought Obama would be less set in his ideas and more likely to reach agreement with Russia than McCain.

"I, as a Russian politician, certainly link Barack Obama with hopes for an improvement in Russian-American relations," he said.

Polls have now opened in Hawaii, the last state to join the voting. Palin's home state of Alaska is the last to finish voting, at 8:00 p.m. local time (06:00 GMT Wednesday), although the result is likely to be known several hours before then.

A good indication of the race will arrive by 8:00 p.m. East Coast time (01:00 GMT Wednesday), by which time voting will have ended in states accounting for almost half the electoral college votes that actually elect the president, including the key swing states of Ohio and Pennsylvania, which are must-wins for McCain.