R – A min-heap with better than O(logn) increase key

You can do following: store a hashmap inside your heap that maps your heap values to heap indexes. Then you should extend your usual heap-logic just a bit:

on Swap(i, j): 
    map[value[i]] = j; 
    map[value[j]] = i;

on Insert(key, value): 
    map.Add(value, heapSize) in the beginning;

on ExtractMin: 
    map.Remove(extractedValue) in the end;

on UpdateKey(value, newKey): 
    index = map[value]; 
    keys[index] = newKey; 

BubbleUp(index) in case of DecreaseKey, and BubbleDown/Heapify(index) in case of IncreaseKey, to restore min-heap-property.

Here's my C# implementation: http://pastebin.com/kkZn123m

Insert and ExtractMin call Swap log(N) times, when restoring heap property. And you're adding O(1) overhead to Swap, so both operations remain O(log(n)). UpdateKey is also log(N): first you lookup index in a hashmap for O(1), then you're restoring heap property for O(log(N)) as you do in Insert/ExtractMin.

Important note: using values for index lookup will require that they are UNIQUE. If you're not ok with this condition, you will have to add some uniqueue id to your key-value pairs and maintain mapping between this uniqueue id and heap index instead of value-index mapping. But for Dijkstra it's not needed, as your values will be graph nodes and you don't want duplicate nodes in your priority queue.

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