A Worst Case, Constant Time Priority Queue

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Presentation transcript:

A Worst Case, Constant Time Priority Queue Ian Munro University of Waterloo Joint work with Andrej Brodnik (Ljubljana & Luleå) Svante Carlsson (Blekinge Inst. Tech.) Johan Karlsson (Luleå)

The Problem van Emde Boas (SWAT i.e. FOCS 1975) Universe integers [1,..m] {n of which are present} Operations: insert / delete find least value  x (or greatest ) Bound: O(lg lg m) time Space: Improved to O(m) bits Model: Standard RAM, with bit twiddling

Some Subsequent Work Kurt Mehlhorn, Stephan Näher and Helmut Alt (SiComp ‘88): vEB is optimal - Lower bound (lg lg m) on pointer machine Peter Miltersen (STOC ‘94): Lower bound ( lg lg m) on a RAM Paul Beame and Faith Fich (STOC ‘99): return to parameterization by number of values actually present - matching upper & lower bounds- (lg n/ lg lg n)

Another Model: Rambo Random Access Machine with Byte Overlap Mike Fredman and Dan Willard Several words can share bits:

Can we do better under this model? Consider elements at bottom an internal node is flagged if it has a descendant lg n bit RAMBO word at leaf Can find lowest ancestor in O(1) time

The vEB Stratified Tree Internal node keeps track of “outside” bottom elements 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A Problem Any individual leaf may be referred to by many ancestors So one insertion/deletion can require up to 2lg n reference changes Modify the approach

Each leaf can be referred to by at most two ancestors Split Tagged Tree Keep track of left / right inside leaf … if different from parent’s Each leaf can be referred to by at most two ancestors

So what do we have? Constant time “extended priority queue” … two memory accesses for search, three for update on our model How much space do we need? 2m + O(lg m) bits of ordinary RAM m bits of RAMBO memory in a particular configuration we call Yggdrasil and it has been implemented in hardware