How do I get my one year old to quit clinging to family members??

My little girl is one year and one month old. She is attached to Grandpa's side 24-7 and throws tantrums whenever he tries to leave the house or even put her down. I know he likes holding her all the time and loves playing with her, but she doesn't realize that people cannot be with her constantly. Whenever grandpa isn't around, she finds somebody else to cling to. Usually me or grandma. I try to get her to be more independent when I am home with her, by giving her toys and encouraging her to stay in the same room with me and play quietly. I even try enclosing her in her crib with toys, which only works sometimes and usually for a few minutes. She hates being enclosed and being alone. She seems to have a very short attention span. She will play with one toy for one minute and either move to the next toy or start exploring and crawling into another room to see what she can get into. TV only holds her attention for a minute or two which is no complaint to me, I'd rather her play outside or with toys. She's just a little ball of energy who wants to get into everything. I struggle to do things around the house or do homework when I'm home alone with her, because I'm always chasing her around or trying to get her to play by herself. She only seems to want to play with her toys if I play with her, which I don't mind for a while, but I have other things to get done! We live with her daddy (my fiancee) and his parents temporarily and if it wasn't for them when they are home, i would never accomplish cleaning or homework. I'm worried when we move back out this summer that my daughter is not only going to struggle emotionally by not seeing grandpa and grandma everyday, but that she will transfer her 24-7 clingy habits to me and will never develop the independence we both need!What should I do?