1. update Persistent Data Structures (Version Control) v0v0 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 Ephemeral query v0v0 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 Partial persistence query only update & query v0v0 v1v1 v2v2 v3v3 v4v4 v6v6 Full persistence updates at leaves any version can be copied query all versions v5v5 v0v0 v1v1 v2v2 v3v3 Confluently persistence update/merge/query all versions v5v5 Purely functional car cdr never modify only create new pairs only DAGs version DAG version tree version list 1 Retroactive v0v0 v1v1 v2v2 v3v3 v4v4 update & query all versions updates in the past propragate

2. Retroactive Data Structures 2 v0v0 v1v1 v2v2 v3v3 v4v4 mTotal number of updates/versions rDistance from current time nMaximal data structure size at any time Partial retroactiveUpdate all versions & query current Full retroactiveUpdate & query all versions r m [E.D. Demaine, J. Iacono, S. Langerman, Retroactive Data Structures, Proc. 15 th Annual ACM- SIAM Symposium on Discrete Algorithms, 274-283, 2004]

3. Rollback  Full Retroactivity 3 v0v0 v1v1 v2v2 v3v3 v4v4 Update u 1 u2u2 u3u3 u4u4 Change  1  2 2  3 3  4 4 current + Generic, can always be applied, space efficient - Slow retroactive operations

4. Lower bounds for Retroactivity 4 a 0 + a 1 x + a 2 x 2 + ∙∙∙ + a n x n ( requires Ω(n) multiplications given x by Motzkin’s theorem ) x0x0 x1x1 x2x2 ∙∙∙xixi xnxn * [M. Patrascu, E.D. Demaine, Logarithmic Lower Bounds in the Cell-Probe Model, SIAM J. of Computing 35(4): 932-963, 2006] ( prefix sum queries require Ω(log n) ) *

5. Partial  Full Retroactivity 5 v5v5 v7v7 v8v8 v1v1 v2v2 v4v4  (  m) u1u1 u2u2 u3u3 partial retroactive structures (remember using full persistence) v0v0 v9v9 current u7u7 u8u8 u9u9 ½ v3v3 v6v6 u4u4 u5u5 u6u6 v5v5 v4v4 temporary during queries to V 5 (rollback)

6. Partial Retroactive Commutative Data Structures 6

7. Decomposable Search Problems 7 n T(2n)≥(1+  )T(n) A B C ADAD A BB C CDCD DD time delete(D)insert(D)

8. Specific Retroactive Data Structures 8 * [D.D. Sleator, R.E. Tarjan, A Data Structure for Dynamic Trees, Proc. 13 th Annual ACM Symposium on Theory of Computing, 114-122, 1981] * ?