A dynamic algorithm for topologically sorting directed acyclic graphs David J. Pearce and Paul H.J. Kelly Imperial College, London, UK

S Introduction  Topologically sorting a directed acyclic graph G=(V,E)  Sort nodes so X before Y, if X  Y  E  Well-known algorithms taking  (v + e) time > E.g. using depth-first search Y U X T W S Z TVWZYUX V

S Introduction  Topologically sorting a directed acyclic graph G=(V,E)  Sort nodes so X before Y, if X  Y  E  Well-known algorithms taking  (v + e) time > E.g. using depth-first search Y U X T W S Z TVWZYUX V

S  How to update topological sort after edge insertion? Problem Definition Y U X T W S Z TVWZYUX V  Invalidating or non-invalidating? >Adding Y  V does not invalidate sort, but X  Y does  How to deal with invalidating edge insertions? >Re-sorting entire graph takes  (v+e) time again

S  How to update topological sort after edge insertion? Problem Definition Y U X T W S Z TVWZYUX V  Invalidating or non-invalidating? >Adding Y  V does not invalidate sort, but X  Y does  How to deal with invalidating edge insertions? >Re-sorting entire graph takes  (v+e) time again

 How to update topological sort after edge insertion? Problem Definition Y U X T W S Z TVWZYUXS V  Invalidating or non-invalidating? >Adding Y  V does not invalidate sort, but X  Y does  How to deal with invalidating edge insertions? >Re-sorting entire graph takes  (v+e) time again

Performing less work  How to avoid re-sorting entire graph after edge insertion? Affected region Y U X T W S Z TVWZ V USYX  Affected region is all nodes between Y and X >We denote this set as AR XY  Only AR XY needs reordering to obtain valid sort >Can eliminate many nodes from consideration >Proof by Marchetti-Spaccamela et al. [MNR96]

SU  How to re-sort affected region? Performing less work (continued)  Could just move Y to right of X! > But, V now incorrectly prioritised with respect to Y > Problem, cannot move Y past nodes it reaches in affected region Y U X T W S Z TVWZYX V SUTVWZYX Before After

 Algorithm MNR due to Marchetti-Spaccamela et al. [MNR96] Algorithm MNR SU Y U X T W S Z TVWZYX V SUTVWZYX Before After  Depth-First Search from Y identifies reachable set > Only visits those in affected region  Shift other nodes to the left > Every node in affected region moved – so at least O(AR XY ) time

 Algorithm PK - main contribution of this work Algorithm PK SU Y U X T W S Z TVWZYX V SUTVZY Before After  Key insight: can avoid resorting entire affected region > {Z,T} remain untouched, saving work compared with MNR > Only set  xy  AR XY needs re-sorting (i.e. nodes in grey) WX

 Define  XY = R F  R B Algorithm PK – what is  XY ? SU Y U X T W S Z TVWZYX V  Observation : nodes in R B must come before those in R F > R B = all nodes reaching X (including X) > R F = all nodes reachable from Y (including Y) VWYX RBRB RFRF

 Begins with two Depth-First Searches to identify R F and R B Algorithm PK – How does it work? SU Y U X T W S Z TVWZYX V Before  Use forward and backward Depth-First Search  Forward search to determine R F (same as MNR)  Backward search to determine R B Forward DFS Backward DFS

 Next, re-sort members of  XY Algorithm PK – how does it work? Y U X T W S Z V  Place R B and R F into slots previously held by R B  R F > R B goes into leftmost slots, R F into rightmost slots VWYX RBRB RFRF SUT?Z???

 PK needs at most ~O(  XY + E(  XY )) per edge insertion > In contrast, MNR needs O(AR XY + E(  XY )) time > Thus, PK should win when  XY much smaller than AR XY  Algorithm AHRSZ due to Alpern et al. [AHRSZ90] > Worst-case time complexity ~O(K min + E(K min )) > Where parameter K min   XY is the minimal cover > So, tighter bound than algorithm PK – but is it practical? - Employs Dietz and Sleator ordered list structure [DS87] - Permits new priorities values to be created in O(1) time - But, this complex to implement and has relatively high overheads Algorithm PK – Complexity ? Definition : E(K) = { X  Y  E | X  K  Y  K }

Experimental Study – Part I  Experiment >Measure cost per insertion over 5000 insertions into random DAG >Invalidating and non-invalidating included to reflect actual performance

Experimental Study – Part II  Experiment >Measure cost per insertion over 5000 insertions into random DAG >Invalidating and non-invalidating included to reflect actual performance

Conclusion  Algorithm PK >Theoretical complexity marginally inferior to AHRSZ algorithm >But, simplicity of PK yields more efficient algorithm in practice  Algorithm MNR >Worst theoretical complexity overall >However, outperforms the others on dense graphs  Other Work … >Extended algorithms to incremental strongly connected components problem – see [PKH03a,Pea04] >Developed batch variant of MNR – see [Pea04] -Obtains O(b + v + e) bound on time to insert batch of b edges -In contrast, MNR/PK/AHRSZ need O(b(v+e)) worst-case time

References  [MNR96] – A. Marchetti-Spaccamela, U. Nanni and H. Rohnert, “Maintaining a topological order under edge insertions”. Information Processing Letters,  [AHRSZ90] - B. Alpern, R. Hoover, B.K. Rosen, P.F. Sweeny and F.K. Zadec, “Incremental evaluation of computational circuits”, In Proc. ACM Symposium on Discrete Algorithms,  [PKH03a] – D. J. Pearce, P. H.J. Kelly and Chris Hankin, “Online Cycle Detection and Difference Propagation for Pointer Analysis”, In Proc. IEEE Workshop on Source Code Analysis and Manipulation,  [Pea04] – D. J. Pearce, “Some directed graph algorithms and their application to pointer analysis”, PhD Thesis, Imperial College.

Q) Is  XY minimal ? Y U X W  The answer is no. For example: XWUY YXUW  Key point : U not repositioned > But, under PK it would be since U   XY (because it reaches X) > Scope for an even better algorithm ?