# Why do we use priority queues when implementing Dijkstra's Algorithm? [closed]

Title says it all. Why is the choice of data structure for Dijkstra's algorithm a priority queue, rather than a simple sorted list?

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## closed as off topic by David Eppstein, Harry Gindi, Noah Snyder, Reid Barton, S. Carnahan♦Mar 10 '10 at 2:44

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This is about undergraduate level algorithms rather than research level mathematics. And it reads like a homework question. –  David Eppstein Mar 10 '10 at 2:02
You're right about undergraduate level algorithms, but it's not homework. –  munch Mar 10 '10 at 2:07
Did you try StackOverflow? –  Harry Gindi Mar 10 '10 at 2:15
@fpqc it's not really programming. i'd call this math.. –  munch Mar 10 '10 at 2:17

Consider the running time for adding a new element to a sorted list, keeping the list sorted. If the list is an array, you can find the insertion point in $O(\log n)$ steps, where $n$ is the current size of the list. But then you have to make room for the new element by shifting all the elements behind it one step back, and that takes $n/2$ steps on the average. Or you could use a linked list, but then binary search is not available, and it takes $n/2$ steps (on the average) to find the insertion point (and $O(1)$ to do the actual insertion). For a properly implemented priority queue, insertion is $O(\log n)$, and so is fixing up the queue after removing the smallest member.
But in Dijkstra's algorithm, you don't just want the minimum member once. Instead, you have a collection of edges, and repeatedly remove one (the minimum) and add more, looking for the minimum all over again, and so forth. Estimating the running time of all these operations seems a bit more involved than just waving your arms and saying $O(n\log n)$. –  Harald Hanche-Olsen Mar 10 '10 at 2:07