Does Too many ids hurt performance

If I have in the page unnecessary ids on elements, like the HTML Helper does in ASP.Net-MVC.
Does it reduce the performace of my id selectors? (I have a page with enormous amout of elements)

Example:

// First DOM   
<HTML>
...
    <input type="text" value="first" id="useless" />
    <input type="text" value="second" id="useful" />
</HTML>

// Second  DOM  
<HTML>
...
    <input type="text" value="first"/>
    <input type="text" value="second" id="useful" />
</HTML>

Script:

<script>
    alert($('#useful').val());
    // never select the first element (with the useless id)
</script>